Chen, S J; Perng, S Y; Kuan, C K; Tseng, T C; Wang, D J
2001-01-01
An active polynomial grating has been designed for use in synchrotron radiation soft-X-ray monochromators and spectrometers. The grating can be dynamically adjusted to obtain the third-order-polynomial surface needed to eliminate the defocus and coma aberrations at any photon energy. Ray-tracing results confirm that a monochromator or spectrometer based on this active grating has nearly no aberration limit to the overall spectral resolution in the entire soft-X-ray region. The grating substrate is made of a precisely milled 17-4 PH stainless steel parallel plate, which is joined to a flexure-hinge bender shaped by wire electrical discharge machining. The substrate is grounded into a concave cylindrical shape with a nominal radius and then polished to achieve a roughness of 0.45 nm and a slope error of 1.2 mu rad rms. The long trace profiler measurements show that the active grating can reach the desired third-order polynomial with a high degree of figure accuracy.
Apostol, Tom M. (Editor)
1991-01-01
In this 'Project Mathematics! series, sponsored by California Institute for Technology (CalTech), the mathematical concept of polynomials in rectangular coordinate (x, y) systems are explored. sing film footage of real life applications and computer animation sequences, the history of, the application of, and the different linear coordinate systems for quadratic, cubic, intersecting, and higher degree of polynomials are discussed.
Constructive feedforward neural networks using hermite polynomial activation functions.
Ma, Liying; Khorasani, K
2005-07-01
In this paper, a constructive one-hidden-layer network is introduced where each hidden unit employs a polynomial function for its activation function that is different from other units. Specifically, both a structure level as well as a function level adaptation methodologies are utilized in constructing the network. The functional level adaptation scheme ensures that the "growing" or constructive network has different activation functions for each neuron such that the network may be able to capture the underlying input-output map more effectively. The activation functions considered consist of orthonormal Hermite polynomials. It is shown through extensive simulations that the proposed network yields improved performance when compared to networks having identical sigmoidal activation functions.
Gratings in passive and active optical waveguides
DEFF Research Database (Denmark)
Berendt, Martin Ole
1999-01-01
attenuated. In either case the cladding mode coupling gives loss on the short wavelength side of the reflection band. The cladding mode coupling loss is a major problem for the utilization of fiber Bragg gratings in wavelength division multiplexed (WDM) system. In this project, a numerical model for cladding...... investigated. Pump induced temperature gratdient in the DFB structure has been found to degrade the output power. Selective cooling to compensate the temperature gradient more than doubled the output power. The experiment indicates that ouput power in excess of 10mW, with 80 mW of 980 nm pumping is feasible...
Rivera, Ferdinand; Becker, Joanne, Rossi
2004-01-01
In this report, we give a sociocultural account of the mediating functions handheld graphing calculators and social interaction play in students' mathematical understanding. We discuss the evolution of students' abilities to symbolize, model, and develop collective mathematical practices about polynomial inequalities in instrumented activity. In…
Measuring water activity of aviation fuel using a polymer optical fiber Bragg grating
Zhang, Wei; Webb, David J.; Carpenter, Mark; Williams, Colleen
2014-05-01
Poly(methyl methacrylate) (PMMA) based polymer optical fiber Bragg gratings have been used for measuring water activity of aviation fuel. Jet A-1 samples with water content ranging from 100% ERH (wet fuel) to 10 ppm (dried fuel), have been conditioned and calibrated for measurement. The PMMA based optical fiber grating exhibits consistent response and a good sensitivity of 59±3pm/ppm (water content in mass). This water activity measurement allows PMMA based optical fiber gratings to detect very tiny amounts of water in fuels that have a low water saturation point, potentially giving early warning of unsafe operation of a fuel system.
Active temperature compensation design of sensor with fiber gratings
Institute of Scientific and Technical Information of China (English)
Xingfa Dong(董兴法); Yonglin Huang(黄勇林); Li Jiang(姜莉); Guiyun Kai(开桂云); Xiaoyi Dong(董孝义)
2004-01-01
A technique for compensation of temperature effects in fiber grating sensors is reported. For strain sensors and other sensors related to strain such as electromagnetic sensors, a novel structure is designed, which uses two fiber Bragg gratings (FBGs) as strain differential sensor and has temperature effects cancelled. Using this technique, the stress sensitivity has been amplified and gets up to 0.226 nm/N, the total variation in wavelength difference within the range of 3-45 ℃ is 0.03 nm, 1/14 of the uncompensated FBG.The structure can be used in the temperature-insensitive static strain measurement and minor-vibration measurement.
Alessi, David A; Rosso, Paul A; Nguyen, Hoang T; Aasen, Michael D; Britten, Jerald A; Haefner, Constantin
2016-12-26
Laser energy absorption and subsequent heat removal from diffraction gratings in chirped pulse compressors poses a significant challenge in high repetition rate, high peak power laser development. In order to understand the average power limitations, we have modeled the time-resolved thermo-mechanical properties of current and advanced diffraction gratings. We have also developed and demonstrated a technique of actively cooling Petawatt scale, gold compressor gratings to operate at 600W of average power - a 15x increase over the highest average power petawatt laser currently in operation. Combining this technique with low absorption multilayer dielectric gratings developed in our group would enable pulse compressors for petawatt peak power lasers operating at average powers well above 40kW.
Freud, Géza
1971-01-01
Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of these polynomials forms an orthogonal system with respect to a non-negative m-distribution defined on the real numerical axis. Comprised of five chapters, the book begins with the fundamental properties of orthogonal polynomials. After discussing the momentum problem, it then explains the quadrature procedure, the convergence theory, and G. Szegő's theory. This book is useful for those who intend to use it as referenc
Mason, JC
2002-01-01
Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Yet no book dedicated to Chebyshev polynomials has been published since 1990, and even that work focused primarily on the theoretical aspects. A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject''s state of the art, Chebyshev Polynomials is just such a treatment. It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations. Problems in each chapter, ranging in difficulty from elementary to quite advanced, reinforce the concepts and methods presented.Far from being an esoteric subject, Chebysh...
Two-dimensional gain cross-grating based on spatial modulation of active Raman gain
Wang, Li; Zhou, Feng-Xue; Guo, Hong-Ju; Niu, Yue-Ping; Gong, Shang-Qing
2016-11-01
Based on the spatial modulation of active Raman gain, a two-dimensional gain cross-grating is theoretically proposed. As the probe field propagates along the z direction and passes through the intersectant region of the two orthogonal standing-wave fields in the x-y plane, it can be effectively diffracted into the high-order directions, and the zero-order diffraction intensity is amplified at the same time. In comparison with the two-dimensional electromagnetically induced cross-grating based on electromagnetically induced transparency, the two-dimensional gain cross-grating has much higher diffraction intensities in the first-order and the high-order directions. Hence, it is more suitable to be utilized as all-optical switching and routing in optical networking and communication. Project supported by the National Natural Science Foundation of China (Grant Nos. 11274112 and 11347133).
Energy Technology Data Exchange (ETDEWEB)
Blackett, S.A. [Univ. of Auckland (New Zealand). Dept of Engineering Science
1996-02-01
Numerical analysis is an important part of Engineering. Frequently relationships are not adequately understood, or too complicated to be represented by theoretical formulae. Instead, empirical approximations based on observed relationships can be used for simple fast and accurate evaluations. Historically, storage of data has been a large constraint on approximately methods. So the challenge is to find a sufficiently accurate representation of data which is valid over as large a range as possible while requiring the storage of only a few numerical values. Polynomials, popular as approximation functions because of their simplicity, can be used to represent simple data. Equation 1.1 shows a simple 3rd order polynomial approximation. However, just increasing the order and number of terms included in a polynomial approximation does not improve the overall result. Although the function may fit exactly to observed data, between these points it is likely that the approximation is increasingly less smooth and probably inadequate. An alternative to adding further terms to the approximation is to make the approximation rational. Equation 1.2 shows a rational polynomial, 3rd order in the numerator and denominator. A rational polynomial approximation allows poles and this can greatly enhance an approximation. In Sections 2 and 3 two different methods for fitting rational polynomials to a given data set are detailed. In Section 4, consideration is given to different rational polynomials used on adjacent regions. Section 5 shows the performance of the rational polynomial algorithms. Conclusions are presented in Section 6.
Dobbs, David E.
2010-01-01
This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of…
[Biologically active substances in grated cocoa and cocoa butter].
Kosman, V M; Stankevich, N M; Makarov, V G; Tikhonov, V P
2007-01-01
In the article results of comparative analysis of grated cocoa and cocoa butter samples are presented. The investigation was done by modern instrumental methods such as HPLC, GC, UV- VIS-spectroscopy, and also with application of titrimetric and grarimetric methods. In the analyzed samples contents of total phenolics changes in an interval 1,0-3,2%, including monomeric proantocyanidins 0,6-1,35%; pyrroloquinoline quinine (PQQ) 0,34-0,76 microg/g; phenyl ethylamine from 2,79 to 14,97 microg/g, tyramine from 9,56 to 71,68 microg/g, dopamine from 5,3 to 25,85 microg/g; theobromine from 3,3 to 8%, caffeine from 0,49 to 0,70%; among the amino acids at the greatest quantities were presented glutaminic and asparaginic acids, arginin and leucin; three main fatty acids were determined - palmitinic (31+/-2% rel.), oleinic (35+/-2% rel.) and stearinic (35+/-2% rel.); the main phytosterins were sytosterin (up to 192 mg%) and obtusifoliol (up to 198,5 mg%).
Directory of Open Access Journals (Sweden)
Sven Svebak
2016-08-01
Full Text Available Results from two studies of biological consequences of laughter are reported. A proposed inhibitory brain mechanism was tested in Study 1. It aims to protect against trunk compression that can cause health hazards during vigorous laughter. Compression may be maximal during moderate durations and, for protective reasons, moderate in enduring vigorous laughs. Twenty-five university students volunteered to see a candid camera film. Laughter responses (LR and the superimposed ha-responses were operationally assessed by mercury-filled strain gauges strapped around the trunk. On average, the thorax compression amplitudes exceeded those of the abdomen, and greater amplitudes were seen in the males than in the females after correction for resting trunk circumference. Regression analyses supported polynomial relations because medium LR durations were associated with particularly high thorax amplitudes. In Study 2, power changes were computed in the beta and alpha EEG frequency bands of the parietal cortex from before to after exposure to the comedy “Dinner for one” in 56 university students. Highly significant linear relations were calculated between the number of laughs and post-exposure cortical activation (increase of beta, decrease of alpha due to high activation after frequent laughter. The results from Study 1 supported the hypothesis of a protective brain mechanism that is activated during long LRs to reduce the risk of harm to vital organs in the trunk cavity. The results in Study 2 supported a linear cortical activation and, thus, provided evidence for a biological correlate to the subjective experience of mental refreshment after laughter.
Institute of Scientific and Technical Information of China (English)
LI Chunxiang; ZHOU Dai
2004-01-01
The polynomial matrix using the block coefficient matrix representation auto-regressive moving average (referred to as the PM-ARMA) model is constructed in this paper for actively controlled multi-degree-of-freedom (MDOF) structures with time-delay through equivalently transforming the preliminary state space realization into the new state space realization. The PM-ARMA model is a more general formulation with respect to the polynomial using the coefficient representation auto-regressive moving average (ARMA) model due to its capability to cope with actively controlled structures with any given structural degrees of freedom and any chosen number of sensors and actuators. (The sensors and actuators are required to maintain the identical number.) under any dimensional stationary stochastic excitation.
Institute of Scientific and Technical Information of China (English)
王雷
2008-01-01
<正>Polynomial functions are among the sim- plest expressions in algebra.They are easy to evaluate:only addition and repeated multipli- cation are required.Because of this,they are often used to approximate other more compli-
Koornwinder, T.H.
2012-01-01
Askey-Wilson polynomial refers to a four-parameter family of q-hypergeometric orthogonal polynomials which contains all families of classical orthogonal polynomials (in the wide sense) as special or limit cases.
Reddy, A Satyanarayana
2011-01-01
A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the graphs which are polynomials in the pattern polynomial graph have been studied. We also identify known graph classes which are pattern polynomial graphs.
New classes of test polynomials of polynomial algebras
Institute of Scientific and Technical Information of China (English)
冯克勤; 余解台
1999-01-01
A polynomial p in a polynomial algebra over a field is called a test polynomial if any endomorphism of the polynomial algebra that fixes p is an automorphism. some classes of new test polynomials recognizing nonlinear automorphisms of polynomial algebras are given. In the odd prime characteristic case, test polynomials recognizing non-semisimple automorphisms are also constructed.
Fiber Bragg grating strain sensors to monitor and study active volcanoes
Sorrentino, Fiodor; Beverini, Nicolò; Carbone, Daniele; Carelli, Giorgio; Francesconi, Francesco; Gambino, Salvo; Giacomelli, Umberto; Grassi, Renzo; Maccioni, Enrico; Morganti, Mauro
2016-04-01
Stress and strain changes are among the best indicators of impending volcanic activity. In volcano geodesy, borehole volumetric strain-meters are mostly utilized. However, they are not easy to install and involve high implementation costs. Advancements in opto-electronics have allowed the development of low-cost sensors, reliable, rugged and compact, thus particularly suitable for field application. In the framework of the EC FP7 MED-SUV project, we have developed strain sensors based on the fiber Bragg grating (FBG) technology. In comparison with previous implementation of the FBG technology to study rock deformations, we have designed a system that is expected to offer a significantly higher resolution and accuracy in static measurements and a smooth dynamic response up to 100 Hz, implying the possibility to observe seismic waves. The system performances are tailored to suit the requirements of volcano monitoring, with special attention to power consumption and to the trade-off between performance and cost. Preliminary field campaigns were carried out on Mt. Etna (Italy) using a prototypal single-axis FBG strain sensor, to check the system performances in out-of-the-lab conditions and in the harsh volcanic environment (lack of mains electricity for power, strong diurnal temperature changes, strong wind, erosive ash, snow and ice during the winter time). We also designed and built a FBG strain sensor featuring a multi-axial configuration which was tested and calibrated in the laboratory. This instrument is suitable for borehole installation and will be tested on Etna soon.
Improved polynomial remainder sequences for Ore polynomials.
Jaroschek, Maximilian
2013-11-01
Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.
Factoring Polynomials and Fibonacci.
Schwartzman, Steven
1986-01-01
Discusses the factoring of polynomials and Fibonacci numbers, offering several challenges teachers can give students. For example, they can give students a polynomial containing large numbers and challenge them to factor it. (JN)
Palindromic random trigonometric polynomials
Conrey, J. Brian; Farmer, David W.; Imamoglu, Özlem
2008-01-01
We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at l...
Polynomial Datapaths Optimization
Parta, Hojat
2014-01-01
The research presented focuses on optimization of polynomials using algebraic manipulations at the high level and digital arithmetic techniques at the implementation level. Previous methods lacked any algebraic understanding of the polynomials or only exposed limited potential. We have treated the polynomial optimization problem in abstract algebra allowing us algebraic freedom to transform polynomials. Unlike previous attempts where only a set of limited benchmarks have been used, we have fo...
Multiplication of a Schubert polynomial by a Stanley symmetric polynomial
Assaf, Sami
2017-01-01
We prove, combinatorially, that the product of a Schubert polynomial by a Stanley symmetric polynomial is a truncated Schubert polynomial. Using Monk's rule, we derive a nonnegative combinatorial formula for the Schubert polynomial expansion of a truncated Schubert polynomial. Combining these results, we give a nonnegative combinatorial rule for the product of a Schubert and a Schur polynomial in the Schubert basis.
Rivera, Ferdinand D.
2007-01-01
This paper provides an instrumental account of precalculus students' graphical process for solving polynomial inequalities. It is carried out in terms of the students' instrumental schemes as mediated by handheld graphing calculators and in cooperation with their classmates in a classroom setting. The ethnographic narrative relays an instrumental…
Blankertz, Raoul
2011-01-01
This diploma thesis is concerned with functional decomposition $f = g \\circ h$ of polynomials. First an algorithm is described which computes decompositions in polynomial time. This algorithm was originally proposed by Zippel (1991). A bound for the number of minimal collisions is derived. Finally a proof of a conjecture in von zur Gathen, Giesbrecht & Ziegler (2010) is given, which states a classification for a special class of decomposable polynomials.
Polynomial Fibonacci-Hessenberg matrices
Energy Technology Data Exchange (ETDEWEB)
Esmaeili, Morteza [Dept. of Mathematical Sciences, Isfahan University of Technology, 84156-83111 Isfahan (Iran, Islamic Republic of)], E-mail: emorteza@cc.iut.ac.ir; Esmaeili, Mostafa [Dept. of Electrical and Computer Engineering, Isfahan University of Technology, 84156-83111 Isfahan (Iran, Islamic Republic of)
2009-09-15
A Fibonacci-Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci-Hessenberg matrix. Several classes of polynomial Fibonacci-Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci-Hessenberg matrices satisfying this property are given.
Laser-induced gratings in the gas phase excited via Raman-active transitions
Energy Technology Data Exchange (ETDEWEB)
Kozlov, D.N. [General Physics Inst., Russian Academy of Sciences, Moscow (Russian Federation); Bombach, R.; Hemmerling, B.; Hubschmid, W. [Paul Scherrer Inst. (PSI), Villigen (Switzerland)
1997-06-01
We report on a new time resolved coherent Raman technique that is based on the generation of thermal gratings following a population change among molecular levels induced by stimulated Raman pumping. This is achieved by spatially and temporally overlapping intensity interference patterns generated independently by two lasers. When this technique is used in carbon dioxide, employing transitions which belong to the Q-branches of the {nu}{sub 1}/2{nu}{sub 2} Fermi dyad, it is possible to investigate molecular energy transfer processes. (author) 2 figs., 10 refs.
Polynomial Graphs and Symmetry
Goehle, Geoff; Kobayashi, Mitsuo
2013-01-01
Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…
Nonnegativity of uncertain polynomials
Directory of Open Access Journals (Sweden)
iljak Dragoslav D.
1998-01-01
Full Text Available The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.
Yu, Jiun-Hung
2012-01-01
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work. We explicitly allow the code symbols to be polynomials of different degrees, which leads to two different notions of weight and distance. Algebraic decoding is studied in detail. If the moduli are not irreducible, the notion of an error locator polynomial is replaced by an error factor polynomial. We then obtain a collection of gcd-based decoding algorithms, some of which are not quite standard even when specialized to Reed-Solomon codes.
Li, Weijie; Ho, Siu Chun Michael; Song, Gangbing
2016-04-01
Steel reinforcement corrosion is one of the dominant causes for structural deterioration for reinforced concrete structures. This paper presents a novel corrosion detection technique using an active thermal probe. The technique takes advantage of the fact that corrosion products have poor thermal conductivity, which will impede heat propagation generated from the active thermal probe. At the same time, the active thermal probe records the temperature response. The presence of corrosion products can thus be detected by analyzing the temperature response after the injection of heat at the reinforcement-concrete interface. The feasibility of the proposed technique was firstly analyzed through analytical modeling and finite element simulation. The active thermal probe consisted of carbon fiber strands to generate heat and a fiber optic Bragg grating (FBG) temperature sensor. Carbon fiber strands are used due to their corrosion resistance. Wet-dry cycle accelerated corrosion experiments were performed to study the effect of corrosion products on the temperature response of the reinforced concrete sample. Results suggest a high correlation between corrosion severity and magnitude of the temperature response. The technique has the merits of high accuracy, high efficiency in measurement and excellent embeddability.
Additive and polynomial representations
Krantz, David H; Suppes, Patrick
1971-01-01
Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utiliz
STABILITY OF SWITCHED POLYNOMIAL SYSTEMS
Institute of Scientific and Technical Information of China (English)
Zhiqiang LI; Yupeng QIAO; Hongsheng QI; Daizhan CHENG
2008-01-01
This paper investigates the stability of (switched) polynomial systems. Using semi-tensor product of matrices, the paper develops two tools for testing the stability of a (switched) polynomial system. One is to convert a product of multi-variable polynomials into a canonical form, and the other is an easily verifiable sufficient condition to justify whether a multi-variable polynomial is positive definite. Using these two tools, the authors construct a polynomial function as a candidate Lyapunov function and via testing its derivative the authors provide some sufficient conditions for the global stability of polynomial systems.
Arkwright, J. W.; Blenman, N. G.; Underhill, I. D.; Maunder, S. A.; Spencer, N. J.; Costa, M.; Brooks, S. J.; Szczesniak, M. M.; Dinning, P. G.
2010-09-01
Diagnostic catheters based on fibre Bragg gratings (FBG's) are proving to be highly effective for measurement of the muscular activity associated with peristalsis in the human gut. The primary muscular contractions that generate peristalsis are circumferential in nature; however, it has long been known that there is also a component of longitudinal contractility present, acting in harmony with the circumferential component to improve the overall efficiency of material movement. To date, there have been relatively few reports on the measurement or inference of longitudinal contractions in humans and all have been limited to detection at a single location only. This is due to the lack of a viable recording technique suitable for real-time in-vivo measurement of this type of activity over extended lengths of the gut. We report the detection of longitudinal motion in lengths of excised mammalian colon using an FBG technique that should be viable for similar detection in humans. The longitudinal sensors have been combined with our previously reported FBG pressure sensing elements to form a composite catheter that allows the relative phase between the two components to be detected. The catheter output has been validated using digital video mapping in an ex-vivo animal preparation using lengths of rabbit ileum.
On Generalized Bell Polynomials
Directory of Open Access Journals (Sweden)
Roberto B. Corcino
2011-01-01
Full Text Available It is shown that the sequence of the generalized Bell polynomials Sn(x is convex under some restrictions of the parameters involved. A kind of recurrence relation for Sn(x is established, and some numbers related to the generalized Bell numbers and their properties are investigated.
Tricubic polynomial interpolation.
Birkhoff, G
1971-06-01
A new triangular "finite element" is described; it involves the 12-parameter family of all quartic polynomial functions that are "tricubic" in that their variation is cubic along any parallel to any side of the triangle. An interpolation scheme is described that approximates quite accurately any smooth function on any triangulated domain by a continuously differentiable function, tricubic on each triangular element.
Calculators and Polynomial Evaluation.
Weaver, J. F.
The intent of this paper is to suggest and illustrate how electronic hand-held calculators, especially non-programmable ones with limited data-storage capacity, can be used to advantage by students in one particular aspect of work with polynomial functions. The basic mathematical background upon which calculator application is built is summarized.…
Hetyei, Gábor
2010-01-01
We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley's toric polynomials may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric $h$-vector in terms of the $cd$-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric $h$-vector of a dual simplicial Eulerian poset in terms of its $f$-vector. This formula implies Gessel's formula for the toric $h$-vector of a cube, and may be used to prove that the nonnegativity of the toric $h$-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for ...
Computing the Alexander Polynomial Numerically
DEFF Research Database (Denmark)
Hansen, Mikael Sonne
2006-01-01
Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically.......Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically....
ON PROPERTIES OF DIFFERENCE POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
Chen Zongxuan; Huang Zhibo; Zheng Xiumin
2011-01-01
We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f(z)f(z+c).
Dziuda, Łukasz; Skibniewski, Franciszek W; Krej, Mariusz; Baran, Paulina M
2013-05-01
We present a fiber-optic sensor for monitoring respiration and heart activity designed to operate in the magnetic resonance imaging (MRI) environment. The sensor employs a Plexiglas springboard, which converts movements of the patient's body lying on the board (i.e., lung- and heart-induced vibrations) to strain, where a fiber Bragg grating attached to the board is used to measure this strain. Experimental studies are carried out during thoracic spine MRI examinations. The presence of the metal-free sensor construction in the MRI environment does not pose a threat to the patient and has no influence over the quality of imaging, and the signal is identical to that obtained without any electromagnetic interference. The results show that the sensor is able to accurately reflect the ballistocardiographic signal, enabling determinations of the respiration rate (RR) and heart rate (HR). The data delivered by the sensor are normally distributed on the Bland-Altman plot for the characteristic point determination and exhibit clear dependence on the RR and HR values for the RR and HR determinations, respectively. Measurement accuracies are better than 7% of the average values, and thus, with further development, the sensor will be implemented in routine MRI examinations.
Chromatic polynomials for simplicial complexes
DEFF Research Database (Denmark)
Møller, Jesper Michael; Nord, Gesche
2016-01-01
In this note we consider s s -chromatic polynomials for finite simplicial complexes. When s=1 s=1 , the 1 1 -chromatic polynomial is just the usual graph chromatic polynomial of the 1 1 -skeleton. In general, the s s -chromatic polynomial depends on the s s -skeleton and its value at r r is the n...
R.J. Stroeker (Roel)
2002-01-01
textabstractA Q-derived polynomial is a univariate polynomial, defined over the rationals, with the property that its zeros, and those of all its derivatives are rational numbers. There is a conjecture that says that Q-derived polynomials of degree 4 with distinct roots for themselves and all their
R.J. Stroeker (Roel)
2006-01-01
textabstractA Q-derived polynomial is a univariate polynomial, defined over the rationals, with the property that its zeros, and those of all its derivatives are rational numbers. There is a conjecture that says that Q-derived polynomials of degree 4 with distinct roots for themselves and all their
Determinants and Polynomial Root Structure
De Pillis, L. G.
2005-01-01
A little known property of determinants is developed in a manner accessible to beginning undergraduates in linear algebra. Using the language of matrix theory, a classical result by Sylvester that describes when two polynomials have a common root is recaptured. Among results concerning the structure of polynomial roots, polynomials with pairs of…
Kuipers, J.
2012-06-01
New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.
Interpolation and Polynomial Curve Fitting
Yang, Yajun; Gordon, Sheldon P.
2014-01-01
Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…
Schemes for Deterministic Polynomial Factoring
Ivanyos, Gábor; Saxena, Nitin
2008-01-01
In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n-1) is a smooth number.
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...... of parameter spaces into structurally stable domains, and a description of the bifurcations. For this reason, the talk will focus on these questions for complex polynomial vector fields....
A Characterization of Polynomials
DEFF Research Database (Denmark)
Andersen, Kurt Munk
1996-01-01
Given the problem:which functions f(x) are characterized by a relation of the form:f[x1,x2,...,xn]=h(x1+x2+...+xn), where n>1 and h(x) is a given function? Here f[x1,x2,...,xn] denotes the divided difference on n points x1,x2,...,xn of the function f(x).The answer is: f(x) is a polynomial of degree...
Some discrete multiple orthogonal polynomials
Arvesú, J.; Coussement, J.; van Assche, W.
2003-04-01
In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317-347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2.
Spreading lengths of Hermite polynomials
Sánchez-Moreno, P; Manzano, D; Yáñez, R; 10.1016/j.cam.2009.09.043
2009-01-01
The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (w...
Oblivious Polynomial Evaluation
Institute of Scientific and Technical Information of China (English)
Hong-Da Li; Dong-Yao Ji; Deng-Guo Feng; Bao Li
2004-01-01
The problem of two-party oblivious polynomial evaluation(OPE)is studied,where one party(Alice)has a polynomial P(x)and the other party(Bob)with an input x wants to learn P(x)in such an oblivious way that Bob obtains P(x)without learning any additional information about P except what is implied by P(x)and Alice does not know Bob's input x.The former OPE protocols are based on an intractability assumption except for OT protocols.In fact,evaluating P(x)is equivalent to computing the product of the coefficient vectors(a0,...,an)and(1,...,xn).Using this idea,an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol.Compared with the existing OPE protocol,another characteristic of the proposed protocols is the degree of the polynomial is private.Another OPE protocol works in case of existence of untrusted third party.
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...
Symmetric functions and Hall polynomials
MacDonald, Ian Grant
1998-01-01
This reissued classic text is the acclaimed second edition of Professor Ian Macdonald's groundbreaking monograph on symmetric functions and Hall polynomials. The first edition was published in 1979, before being significantly expanded into the present edition in 1995. This text is widely regarded as the best source of information on Hall polynomials and what have come to be known as Macdonald polynomials, central to a number of key developments in mathematics and mathematical physics in the 21st century Macdonald polynomials gave rise to the subject of double affine Hecke algebras (or Cherednik algebras) important in representation theory. String theorists use Macdonald polynomials to attack the so-called AGT conjectures. Macdonald polynomials have been recently used to construct knot invariants. They are also a central tool for a theory of integrable stochastic models that have found a number of applications in probability, such as random matrices, directed polymers in random media, driven lattice gases, and...
Polynomial Regression on Riemannian Manifolds
Hinkle, Jacob; Fletcher, P Thomas; Joshi, Sarang
2012-01-01
In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein as well as the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer's study.
Witt Rings and Permutation Polynomials
Institute of Scientific and Technical Information of China (English)
Qifan Zhang
2005-01-01
Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over Z/p2Z and 3-tuples of polynomial functions over Z/pZ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod pl. The main new result is the characterization of strong orthogonal systems over Z/p1Z.
Superoscillations with arbitrary polynomial shape
Chremmos, Ioannis; Fikioris, George
2015-07-01
We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functions are obtained as the product of the polynomial with a sufficiently flat, bandlimited envelope function whose Fourier transform has at least N-1 continuous derivatives and an Nth derivative of bounded variation, N being the order of the polynomial. Polynomials of arbitrarily high order can be approximated if the Fourier transform of the envelope is smooth, i.e. a bump function.
Reflectivity-modulated grating-mirror
DEFF Research Database (Denmark)
2012-01-01
The invention relates to vertical cavity lasers (VCL) incorporating a reflectivity-modulated grating mirror (1) for modulating the laser output. A cavity is formed by a bottom mirror (4), an active region (3), and an outcoupling top grating mirror (1) formed by a periodic refractive index grating...... to the oscillation axis. A modulated voltage (91) is applied in reverse bias between the n- and p-doped layers to modulate the refractive index of the electrooptic material layer (12) and thereby the reflectivity spectrum of the grating mirror (1). The reflectivity of the grating mirror (1) can be modulated between...... a reflectivity with little or no out coupling and a reflectivity with normal out coupling, wherein lasing in the VCL is supported at both the first and the second reflectivity. As the out coupling mirror modulates the output, the lasing does not need to be modulated, and the invention provides the advantage...
Derivations and identities for Kravchuk polynomials
Bedratyuk, Leonid
2012-01-01
We introduce the notion of Kravchuk derivations of the polynomial algebra. We prove that any element of the kernel of the derivation gives a polynomial identity satisfied by the Kravchuk polynomials. Also, we prove that any kernel element of the basic Weitzenb\\"ok derivations yields a polynomial identity satisfied by the Kravchuk polynomials. We describe the corresponding intertwining maps.
Some New Formulae for Genocchi Numbers and Polynomials Involving Bernoulli and Euler Polynomials
Directory of Open Access Journals (Sweden)
Serkan Araci
2014-01-01
Full Text Available We give some new formulae for product of two Genocchi polynomials including Euler polynomials and Bernoulli polynomials. Moreover, we derive some applications for Genocchi polynomials to study a matrix formulation.
Parallel Construction of Irreducible Polynomials
DEFF Research Database (Denmark)
Frandsen, Gudmund Skovbjerg
Let arithmetic pseudo-NC^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) ...
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...
The number of polynomial solutions of polynomial Riccati equations
Gasull, Armengol; Torregrosa, Joan; Zhang, Xiang
2016-11-01
Consider real or complex polynomial Riccati differential equations a (x) y ˙ =b0 (x) +b1 (x) y +b2 (x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η ≥ 2 (resp. η = 1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
Prime power polynomial maps over finite fields
Berson, Joost
2012-01-01
We consider polynomial maps described by so-called prime power polynomials. These polynomials are defined using a fixed power of a prime number, say q. Considering invertible polynomial maps of this type over a characteristic zero field, we will only obtain (up to permutation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F_q automorphisms of this type have (in general) an entirely different structure. Namely, we will show that the prime power polynomial maps over F_q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F_q. Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible prime power polynomial maps correspond to invertible matrices. This alternate description of the prime power polynomial automorphism subgroup leads to the solution of many famous conjectures for this kind of polynomials and polynomial maps.
Iwata, Fujio
2001-06-01
Toppan Printing Co., Ltd. originated the name of 'grating image'. It means an image that consists of diffraction grating dots that look similar to the halftone dots of conventional printing. We proposed this new display method using simple gratings in order to enhance the visual effects when illumination is made by a fluorescent lamp. We considered the use of simple gratings as elemental dots, and used a number of elemental dots to display a 2D image. This method produces an effect something like the halftone dots of printing. The grating image technology grows from its starting to become able to produce 3D images and a 3D-video system using an electron beam grating-writing system.
Iwata, Fujio
1995-07-01
The word 'grating image' was first named by Toppan Printing Company, Ltd. It means that an image consists of grating dots. In 1988, we presented this new technology at the Optical Security Systems Symposium, in Switzerland. Then it was improved and applied in display application. Recently, it was further applied in 3D video systems. In this report, the development history and the recent situations of grating image technology are described.
Hadamard Factorization of Stable Polynomials
Loredo-Villalobos, Carlos Arturo; Aguirre-Hernández, Baltazar
2011-11-01
The stable (Hurwitz) polynomials are important in the study of differential equations systems and control theory (see [7] and [19]). A property of these polynomials is related to Hadamard product. Consider two polynomials p,q ∈ R[x]:p(x) = anxn+an-1xn-1+...+a1x+a0q(x) = bmx m+bm-1xm-1+...+b1x+b0the Hadamard product (p × q) is defined as (p×q)(x) = akbkxk+ak-1bk-1xk-1+...+a1b1x+a0b0where k = min(m,n). Some results (see [16]) shows that if p,q ∈R[x] are stable polynomials then (p×q) is stable, also, i.e. the Hadamard product is closed; however, the reciprocal is not always true, that is, not all stable polynomial has a factorization into two stable polynomials the same degree n, if n> 4 (see [15]).In this work we will give some conditions to Hadamard factorization existence for stable polynomials.
Befriending Askey-Wilson polynomials
Szabłowski, Paweł J
2011-01-01
Although our main interest is with the Askey-Wilson (AW) polynomials we recall and review four other families of the so-called Askey-Wilson scheme of polynomials. We do this for completeness as well as for better exposition of AW properties. Our main results concentrate on the complex parameters case, revealing new fascinating symmetries between the variables and some of the parameters. In particular we express Askey-Wilson polynomials as linear combinations of Al-Salam--Chihara (ASC) polynomials which together with the obtained earlier expansion of the Askey-Wilson density forms complete generalization of the situation met in the case of Al-Salam--Chihara and q-Hermite polynomials and the Poisson-Mehler expansion formula. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we arrive to AW polynomials and AW densities that have clear probabilistic interpretation. We recall some known and present some believed to be unknown identities an...
DEFF Research Database (Denmark)
Zhang, C.; Webb, D.J.; Kalli, K.
We report for the first time fibre Bragg grating inscription in microstructured optical fibre fabricated from Topas® cyclic olefin copolymer. The temperature sensitivity of the grating was studied revealing a positive Bragg wavelength shift of approximately 0.8 nmK-1,the largest sensitivity yet...
An elastomeric grating coupler
Kocabas, A.; Ay, F.; Dana, A.; Aydinli, A.
2006-01-01
We report on a novel nondestructive and reversible method for coupling free space light to planar optical waveguides. In this method, an elastomeric grating is used to produce an effective refractive index modulation on the surface of the optical waveguide. The external elastomeric grating binds to
Polynomial Regressions and Nonsense Inference
Directory of Open Access Journals (Sweden)
Daniel Ventosa-Santaulària
2013-11-01
Full Text Available Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis and psychology, just to mention a few examples. In many cases, the data employed to estimate such specifications are time series that may exhibit stochastic nonstationary behavior. We extend Phillips’ results (Phillips, P. Understanding spurious regressions in econometrics. J. Econom. 1986, 33, 311–340. by proving that an inference drawn from polynomial specifications, under stochastic nonstationarity, is misleading unless the variables cointegrate. We use a generalized polynomial specification as a vehicle to study its asymptotic and finite-sample properties. Our results, therefore, lead to a call to be cautious whenever practitioners estimate polynomial regressions.
Locally tame plane polynomial automorphisms
Berson, Joost; Furter, Jean-Philippe; Maubach, Stefan
2010-01-01
For automorphisms of a polynomial ring in two variables over a domain R, we show that local tameness implies global tameness provided that every 2-generated invertible R-module is free. We give many examples illustrating this property.
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.
Stochastic Estimation via Polynomial Chaos
2015-10-01
TΨ is a vector with P+1 elements. With these dimensions, (29) is solvable by standard numerical linear algebra techniques. The specific matrix...initial conditions for partial differential equations. Here, the elementary theory of the polynomial chaos is presented followed by the details of a...the elementary theory of the polynomial chaos is presented followed by the details of a number of example calculations where the statistical mean and
Properties of Leach-Flessas-Gorringe polynomials
Pursey, D. L.
1990-09-01
A generating function is obtained for the polynomials recently introduced by Leach, Flessas, and Gorringe [J. Math. Phys. 30, 406 (1989)], and is then used to relate the Leach-Flessas-Gorringe (or LFG) polynomials to Hermite polynomials. The generating function is also used to express a number of integrals involving the LFG polynomials as finite sums of parabolic cylinder functions.
Birth-death processes and associated polynomials
Doorn, van Erik A.
2003-01-01
We consider birth-death processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birth-death polynomials. The sequence of associated polynomials linked with a sequence of birth-death polynomials and its orthogonalizing measure can be used in the analysis
On the Hermite-Apostol-Genocchi Polynomials
Kurt, Veli; Kurt, Burak
2011-09-01
In this study, we introduce and investigate the Hermite-Apostol-Genocchi polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials. Also, we prove two theorems between 2-dimensional Hermite polynomials and Hermite-Apostol-Genocchi polynomials.
Uniqueness and Zeros of -Shift Difference Polynomials
Indian Academy of Sciences (India)
Kai Liu; Xin-Ling Liu; Ting-Bin Cao
2011-08-01
In this paper, we consider the zero distributions of -shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to -shift difference polynomials. We also investigate the uniqueness problem of -shift difference polynomials that share a common value.
Multi-particle dynamical systems and polynomials
Demina, Maria V.; Kudryashov, Nikolai A.
2016-05-01
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi-particle dynamical system by finding polynomial solutions of partial differential equations is introduced. The method enables one to integrate a wide class of polynomial multi-particle dynamical systems. The general solutions of certain dynamical systems related to linear second-order partial differential equations are found. As a by-product of our results, new families of orthogonal polynomials are derived.
Using Tutte polynomials to analyze the structure of the benzodiazepines
Cadavid Muñoz, Juan José
2014-05-01
Graph theory in general and Tutte polynomials in particular, are implemented for analyzing the chemical structure of the benzodiazepines. Similarity analysis are used with the Tutte polynomials for finding other molecules that are similar to the benzodiazepines and therefore that might show similar psycho-active actions for medical purpose, in order to evade the drawbacks associated to the benzodiazepines based medicine. For each type of benzodiazepines, Tutte polynomials are computed and some numeric characteristics are obtained, such as the number of spanning trees and the number of spanning forests. Computations are done using the computer algebra Maple's GraphTheory package. The obtained analytical results are of great importance in pharmaceutical engineering. As a future research line, the usage of the chemistry computational program named Spartan, will be used to extent and compare it with the obtained results from the Tutte polynomials of benzodiazepines.
Plain Polynomial Arithmetic on GPU
Anisul Haque, Sardar; Moreno Maza, Marc
2012-10-01
As for serial code on CPUs, parallel code on GPUs for dense polynomial arithmetic relies on a combination of asymptotically fast and plain algorithms. Those are employed for data of large and small size, respectively. Parallelizing both types of algorithms is required in order to achieve peak performances. In this paper, we show that the plain dense polynomial multiplication can be efficiently parallelized on GPUs. Remarkably, it outperforms (highly optimized) FFT-based multiplication up to degree 212 while on CPU the same threshold is usually at 26. We also report on a GPU implementation of the Euclidean Algorithm which is both work-efficient and runs in linear time for input polynomials up to degree 218 thus showing the performance of the GCD algorithm based on systolic arrays.
Gratings in polymeric waveguides
Mishakov, G.; Sokolov, V.; Kocabas, A.; Aydinli, A.
2007-04-01
Laser-induced formation of polymer Bragg grating filters for Dense Wavelength Division Multiplexing (DWDM) applications is discussed. Acrylate monomers halogenated with both fluorine and chlorine, which possess absorption losses less than 0.25 dB/cm and wide choice of refractive indices (from 1.3 to 1.5) in the 1.5 μm telecom wavelength region were used. The monomers are highly intermixable thus permitting to adjust the refractive index of the composition within +/-0.0001. Moreover they are photocurable under UV exposure and exhibit high contrast in polymerization. These properties make halogenated acrylates very promising for fabricating polymeric waveguides and photonic circuits. Single-mode polymer waveguides were fabricated on silicon wafers using resistless contact lithography. Submicron index gratings have been written in polymer waveguides using holographic exposure with He-Cd laser beam (325 nm) through a phase mask. Both uniform and apodized gratings have been fabricated. The gratings are stable and are not erased by uniform UV exposure. The waveguide gratings possess narrowband reflection spectra in the 1.5 μm wavelength region of 0.4 nm width, nearly rectangular shape of the stopband and reflectivity R > 99%. The fabricated Bragg grating filters can be used for multiplexing/demultiplexing optical signals in high-speed DWDM optical fiber networks.
Derivations and identities for Fibonacci and Lucas polynomials
Bedratyuk, Leonid
2012-01-01
We introduce the notion of Fibonacci and Lucas derivations of the polynomial algebras and prove that any element of kernel of the derivations defines a polynomial identity for the Fibonacci and Lucas polynomials. Also, we prove that any polynomial identity for Appel polynomial yields a polynomial identity for the Fibonacci and Lucas polynomials and describe the corresponding intertwining maps.
Tree modules and counting polynomials
Kinser, Ryan
2011-01-01
We give a formula for counting tree modules for the quiver S_g with g loops and one vertex in terms of tree modules on its universal cover. This formula, along with work of Helleloid and Rodriguez-Villegas, is used to show that the number of d-dimensional tree modules for S_g is polynomial in g with the same degree and leading coefficient as the counting polynomial A_{S_g}(d, q) for absolutely indecomposables over F_q, evaluated at q=1.
Orthogonal Polynomials and their Applications
Dehesa, Jesús; Marcellan, Francisco; Francia, José; Vinuesa, Jaime
1988-01-01
The Segovia meeting set out to stimulate an intensive exchange of ideas between experts in the area of orthogonal polynomials and its applications, to present recent research results and to reinforce the scientific and human relations among the increasingly international community working in orthogonal polynomials. This volume contains original research papers as well as survey papers about fundamental questions in the field (Nevai, Rakhmanov & López) and its relationship with other fields such as group theory (Koornwinder), Padé approximation (Brezinski), differential equations (Krall, Littlejohn) and numerical methods (Rivlin).
Symbolic computation of Appell polynomials using Maple
Directory of Open Access Journals (Sweden)
H. Alkahby
2001-07-01
Full Text Available This work focuses on the symbolic computation of Appell polynomials using the computer algebra system Maple. After describing the traditional approach of constructing Appell polynomials, the paper examines the operator method of constructing the same Appell polynomials. The operator approach enables us to express the Appell polynomial as Bessel function whose coefficients are Euler and Bernuolli numbers. We have also constructed algorithms using Maple to compute Appell polynomials based on the methods we have described. The achievement is the construction of Appell polynomials for any function of bounded variation.
Modeling Component-based Bragg gratings Application: tunable lasers
Directory of Open Access Journals (Sweden)
Hedara Rachida
2011-09-01
Full Text Available The principal function of a grating Bragg is filtering, which can be used in optical fibers based component and active or passive semi conductors based component, as well as telecommunication systems. Their ideal use is with lasers with fiber, amplifiers with fiber or Laser diodes. In this work, we are going to show the principal results obtained during the analysis of various types of grating Bragg by the method of the coupled modes. We then present the operation of DBR are tunable. The use of Bragg gratings in a laser provides single-mode sources, agile wavelength. The use of sampled grating increases the tuning range.
Polynomial Regressions and Nonsense Inference
DEFF Research Database (Denmark)
Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir
Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis, and psychology, just to mention a few examples. In many cases, the data employed to estimate such estimations are time series that may exhibit stochastic nonstationary ...
Uniform approximation by (quantum) polynomials
Drucker, A.; de Wolf, R.
2011-01-01
We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson's Theorem, which gives a nearly-optimal quantitative version of Weierstrass's Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectivel
On Modular Counting with Polynomials
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt
2006-01-01
For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z_m of degree O(\\sqrt n) that is a generalized representation (commonly also called weak representation) of the MODl...
Two polynomial division inequalities in
Directory of Open Access Journals (Sweden)
Goetgheluck P
1998-01-01
Full Text Available This paper is a first attempt to give numerical values for constants and , in classical estimates and where is an algebraic polynomial of degree at most and denotes the -metric on . The basic tools are Markov and Bernstein inequalities.
Global Polynomial Kernel Hazard Estimation
DEFF Research Database (Denmark)
Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch
2015-01-01
This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically redu...
Polynomial J-spectral factorization
Kwakernaak, Huibert; Sebek, Michael
1994-01-01
Several algorithms are presented for the J-spectral factorization of a para-Hermitian polynomial matrix. The four algorithms that are discussed are based on diagonalization, successive factor extraction, interpolation, and the solution of an algebraic Riccati equation, respectively. The paper includ
Global Polynomial Kernel Hazard Estimation
DEFF Research Database (Denmark)
Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch;
2015-01-01
This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically...
A New Generalisation of Macdonald Polynomials
Garbali, Alexandr; de Gier, Jan; Wheeler, Michael
2017-01-01
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters (q, t) and polynomial in a further two parameters (u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.
A Summation Formula for Macdonald Polynomials
de Gier, Jan; Wheeler, Michael
2016-03-01
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases {t = 1} and {q = 0}, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q-Whittaker polynomials.
An analysis on the inversion of polynomials
M. F. González-Cardel; R. Díaz-Uribe
2006-01-01
In this work the application and the intervals of validity of an inverse polynomial, according to the method proposed by Arfken [1] for the inversion of series, is analyzed. It is shown that, for the inverse polynomial there exists a restricted domain whose longitude depends on the magnitude of the acceptable error when the inverse polynomial is used to approximate the inverse function of the original polynomial. A method for calculating the error of the approximation and its use in determini...
Application of Chebyshev Polynomial to simulated modeling
Institute of Scientific and Technical Information of China (English)
CHI Hai-hong; LI Dian-pu
2006-01-01
Chebyshev polynomial is widely used in many fields, and used usually as function approximation in numerical calculation. In this paper, Chebyshev polynomial expression of the propeller properties across four quadrants is given at first, then the expression of Chebyshev polynomial is transformed to ordinary polynomial for the need of simulation of propeller dynamics. On the basis of it,the dynamical models of propeller across four quadrants are given. The simulation results show the efficiency of mathematical model.
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2007-01-01
Presents the results on positive trigonometric polynomials within a unitary framework; the theoretical results obtained partly from the general theory of real polynomials, partly from self-sustained developments. This book provides information on the theory of sum-of-squares trigonometric polynomials in two parts: theory and applications.
Lattice Platonic Solids and their Ehrhart polynomial
Ionascu, Eugen J
2011-01-01
First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and those for regular lattice octahedrons. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
General Eulerian Numbers and Eulerian Polynomials
Directory of Open Access Journals (Sweden)
Tingyao Xiong
2013-01-01
Full Text Available We will generalize the definitions of Eulerian numbers and Eulerian polynomials to general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian numbers and polynomials to the general Eulerian polynomials and numbers.
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Universite de Montreal, PO Box 6128, Station Centre-ville, Montreal QC H3C 3J7 (Canada); Zhedanov, Alexei [Donetsk Institute for Physics and Technology, Donetsk 83114 (Ukraine)
2009-10-30
We construct new families of elliptic solutions of the restricted Toda chain. The main tool is a special (so-called Stieltjes) ansatz for the moments of corresponding orthogonal polynomials. We show that the moments thus obtained are related to three types of Lame polynomials. The corresponding orthogonal polynomials can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials.
Frobenious-Euler Type Polynomials Related to Hermite-Bernoulli Polynomials
Kurt, Burak; Simsek, Yilmaz
2011-09-01
The aim of this paper is to define and investigate a new generating functions of the Frobenious-Euler polynomials and numbers. We establish some fundamental properties of these numbers and polynomials. We also derive relationship between these polynomials and Hermite-Apostol-Bernoulli polynomials and numbers. We also give some remarks and applications.
Gao, Nan; Zhang, Yuchao; Xie, Changqing
2011-11-01
We introduce circular Fibonacci gratings (CFGs) that combine the concept of circular gratings and Fibonacci structures. Theoretical analysis shows that the diffraction pattern of CFGs is composed of fractal distributions of impulse rings. Numerical simulations are performed with two-dimensional fast Fourier transform to reveal the fractal behavior of the diffraction rings. Experimental results are also presented and agree well with the numerical results. The fractal nature of the diffraction field should be of great theoretical interest, and shows potential to be further developed into practical applications, such as in laser measurement with wideband illumination.
Maystre, Daniel
2014-01-01
The chapter contains a detailed presentation of the surface integral theory for modelling light diffraction by surface-relief diffraction gratings having a one-dimensional periodicity. Several different approaches are presented, leading either to a single integral equation, or to a system of coupled integral equations. Special attention is paid to the singularities of the kernels, and to different techniques to accelerate the convergence of the numerical computations. The theory is applied to gratings having different profiles with or without edges, to real metal and dielectrics, and to perfectly conducting substrates.
An elastomeric grating coupler
Kocabas, Askin; Ay, Feridun; Dâna, Aykutlu; Aydinli, Atilla
2006-01-01
We report on a novel nondestructive and reversible method for coupling free space light to planar optical waveguides. In this method, an elastomeric grating is used to produce an effective refractive index modulation on the surface of the optical waveguide. The external elastomeric grating binds to the surface of the waveguide with van der Waals forces and makes conformal contact without any applied pressure. As a demonstration of the feasibility of the approach, we use it to measure the refractive index of a silicon oxynitride film. This technique is nondestructive, reversible, low cost and can easily be applied to the characterization of optical materials for integrated optics.
Ultra-High Temperature Gratings
Institute of Scientific and Technical Information of China (English)
John Canning; Somnath Bandyopadhyay; Michael Stevenson; Kevin Cook
2008-01-01
Regenerated gratings seeded by type-Ⅰ gratings are shown to withstand temperatures beyond 1000 ℃. The method of regeneration offers a new approach to increasing temperature resistance of stable fibre Bragg and other gratings. These ultra-high temperature (UHT) gratings extend the applicability of silicate based components to high temperature applications such as monitoring of smelters and vehicle and aircraft engines to high power fibre lasers.
Normal BGG solutions and polynomials
Cap, A; Hammerl, M
2012-01-01
First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries, and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of local frames for any natural vector bundle associated to a parabolic geometry. We then prove that the coefficient functions of any normal solution of a first BGG operator with respect to such a frame are polynomials in the normal coordinates of the parabolic geometry. A bound on the degree of these polynomials in terms of representation theory data is derived. For geometries locally isomorphic to the homogeneous model of ...
BSDEs with polynomial growth generators
Directory of Open Access Journals (Sweden)
Philippe Briand
2000-01-01
Full Text Available In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.
Polynomial-Chaos-based Kriging
Schöbi, R; Sudret, B.; Wiart, J.
2015-01-01
International audience; Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. Optimization and uncertainty quantification problems typically require a large number of runs of the computational model at hand, which may not be feasible with high-fidelity models directly. Thus surrogate models (a.k.a metamodels) have been increasingly investigated in the last decade. Polynomial Chaos Expansion...
Leont'ev, V. K.
2015-11-01
A pseudo-Boolean function is an arbitrary mapping of the set of binary n-tuples to the real line. Such functions are a natural generalization of classical Boolean functions and find numerous applications in various applied studies. Specifically, the Fourier transform of a Boolean function is a pseudo-Boolean function. A number of facts associated with pseudo-Boolean polynomials are presented, and their applications to well-known discrete optimization problems are described.
Weak lensing tomography with orthogonal polynomials
Schaefer, Bjoern Malte
2011-01-01
The topic of this article is weak cosmic shear tomography where the line of sight-weighting is carried out with a set of specifically constructed orthogonal polynomials, dubbed TaRDiS (Tomography with orthogonAl Radial Distance polynomIal Systems). We investigate the properties of these polynomials and employ weak convergence spectra, which have been obtained by weighting with these polynomials, for the estimation of cosmological parameters. We quantify their power in constraining parameters in a Fisher-matrix technique and demonstrate how each polynomial projects out statistically independent information, and how the combination of multiple polynomials lifts degeneracies. The assumption of a reference cosmology is needed for the construction of the polynomials, and as a last point we investigate how errors in the construction with a wrong cosmological model propagate to misestimates in cosmological parameters. TaRDiS performs on a similar level as traditional tomographic methods and some key features of tomo...
Weak lensing tomography with orthogonal polynomials
Schäfer, Björn Malte; Heisenberg, Lavinia
2012-07-01
The topic of this paper is weak cosmic shear tomography where the line-of-sight weighting is carried out with a set of specifically constructed orthogonal polynomials, dubbed Tomography with Orthogonal Radial Distance Polynomial Systems (TaRDiS). We investigate the properties of these polynomials and employ weak convergence spectra, which have been obtained by weighting with these polynomials, for the estimation of cosmological parameters. We quantify their power in constraining parameters in a Fisher matrix technique and demonstrate how each polynomial projects out statistically independent information, and how the combination of multiple polynomials lifts degeneracies. The assumption of a reference cosmology is needed for the construction of the polynomials, and as a last point we investigate how errors in the construction with a wrong cosmological model propagate to misestimates in cosmological parameters. TaRDiS performs on a similar level as traditional tomographic methods and some key features of tomography are made easier to understand.
On Ternary Inclusion-Exclusion Polynomials
Bachman, Gennady
2010-01-01
Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials $\\Phi_{pqr}$, where $p
Stable piecewise polynomial vector fields
Directory of Open Access Journals (Sweden)
Claudio Pessoa
2012-09-01
Full Text Available Let $N={y>0}$ and $S={y<0}$ be the semi-planes of $mathbb{R}^2$ having as common boundary the line $D={y=0}$. Let $X$ and $Y$ be polynomial vector fields defined in $N$ and $S$, respectively, leading to a discontinuous piecewise polynomial vector field $Z=(X,Y$. This work pursues the stability and the transition analysis of solutions of $Z$ between $N$ and $S$, started by Filippov (1988 and Kozlova (1984 and reformulated by Sotomayor-Teixeira (1995 in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields $Z_{epsilon}$, defined by averaging $X$ and $Y$. This family approaches $Z$ when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002 providing conditions on $(X,Y$ for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on $mathbb{R}^2$. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here.
DEFF Research Database (Denmark)
Kaiser, W.; Bach, L.; Reithmaier, J. P.;
2003-01-01
37 GHz direct-modulation bandwidth could be obtained by a multi-section design with an integrated weakly coupled DBR grating. The laser shows side mode suppression ratios of 45 dB and output powers exceeding 20 mW.......37 GHz direct-modulation bandwidth could be obtained by a multi-section design with an integrated weakly coupled DBR grating. The laser shows side mode suppression ratios of 45 dB and output powers exceeding 20 mW....
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, we mainly study the relation of two cyclically reduced words w and w' on the condition they have the same trace polynomial (i.e., tr w= tr w' ). By defining an equivalence relation through such operators on words as inverse, cyclically left shift, and mirror, it is straightforward to get that w ～ w' implies tr w = tr w'. We show by a counter example that tr w = tr w' does not imply w ～ w'. And in two special cases, we prove that tr w = tr w' if and only if w ～ w'.
The stable computation of formal orthogonal polynomials
Beckermann, Bernhard
1996-12-01
For many applications - such as the look-ahead variants of the Lanczos algorithm - a sequence of formal (block-)orthogonal polynomials is required. Usually, one generates such a sequence by taking suitable polynomial combinations of a pair of basis polynomials. These basis polynomials are determined by a look-ahead generalization of the classical three term recurrence, where the polynomial coefficients are obtained by solving a small system of linear equations. In finite precision arithmetic, the numerical orthogonality of the polynomials depends on a good choice of the size of the small systems; this size is usually controlled by a heuristic argument such as the condition number of the small matrix of coefficients. However, quite often it happens that orthogonality gets lost.
Optimizing polynomials for floating-point implementation
De Dinechin, Florent
2008-01-01
The floating-point implementation of a function on an interval often reduces to polynomial approximation, the polynomial being typically provided by Remez algorithm. However, the floating-point evaluation of a Remez polynomial sometimes leads to catastrophic cancellations. This happens when some of the polynomial coefficients are very small in magnitude with respects to others. In this case, it is better to force these coefficients to zero, which also reduces the operation count. This technique, classically used for odd or even functions, may be generalized to a much larger class of functions. An algorithm is presented that forces to zero the smaller coefficients of the initial polynomial thanks to a modified Remez algorithm targeting an incomplete monomial basis. One advantage of this technique is that it is purely numerical, the function being used as a numerical black box. This algorithm is implemented within a larger polynomial implementation tool that is demonstrated on a range of examples, resulting in ...
Haglund's conjecture on 3-column Macdonald polynomials
Blasiak, Jonah
2014-01-01
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Mac...
A new Arnoldi approach for polynomial eigenproblems
Energy Technology Data Exchange (ETDEWEB)
Raeven, F.A.
1996-12-31
In this paper we introduce a new generalization of the method of Arnoldi for matrix polynomials. The new approach is compared with the approach of rewriting the polynomial problem into a linear eigenproblem and applying the standard method of Arnoldi to the linearised problem. The algorithm that can be applied directly to the polynomial eigenproblem turns out to be more efficient, both in storage and in computation.
Exceptional polynomials and SUSY quantum mechanics
Indian Academy of Sciences (India)
K V S Shiv Chaitanya; S Sree Ranjani; Prasanta K Panigrahi; R Radhakrishnan; V Srinivasan
2015-07-01
We show that for the quantum mechanical problem which admit classical Laguerre/Jacobi polynomials as solutions for the Schrödinger equations (SE), will also admit exceptional Laguerre/Jacobi polynomials as solutions having the same eigenvalues but with the ground state missing after a modification of the potential. Then, we claim that the existence of these exceptional polynomials leads to the presence of non-trivial supersymmetry.
On the verification of polynomial system solvers
Institute of Scientific and Technical Information of China (English)
Changbo CHEN; Marc MORENO MAZA; Wei PAN; Yuzhen XI
2008-01-01
We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of well-known polynomial systems. Our experimental results illustrate the high effi-ciency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve.
Control to Facet for Polynomial Systems
DEFF Research Database (Denmark)
Sloth, Christoffer; Wisniewski, Rafael
2014-01-01
for the controller design are solved by searching for polynomials in Bernstein form. This allows the controller design problem to be formulated as a linear programming problem. Examples are provided that demonstrate the efficiency of the method for designing controls for polynomial systems.......This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up...
Multi-indexed (q)-Racah Polynomials
Odake, Satoru
2012-01-01
As the second stage of the project $multi-indexed orthogonal polynomials$, we present, in the framework of `discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q)-Racah polynomials. They are obtained from the (q)-Racah polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state' vectors of type I and II, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the `solutions' of the matrix Schr\\"odinger equation with negative `eigenvalues', except for one of the two boundary points.
Polynomial chaotic inflation in supergravity revisited
Directory of Open Access Journals (Sweden)
Kazunori Nakayama
2014-10-01
Full Text Available We revisit a polynomial chaotic inflation model in supergravity which we proposed soon after the Planck first data release. Recently some issues have been raised in Ref. [12], concerning the validity of our polynomial chaotic inflation model. We study the inflaton dynamics in detail, and confirm that the inflaton potential is very well approximated by a polynomial potential for the parameters of our interest in any practical sense, and in particular, the spectral index and the tensor-to-scalar ratio can be estimated by single-field approximation. This justifies our analysis of the polynomial chaotic inflation in supergravity.
Asymptotics for a generalization of Hermite polynomials
Alfaro, M; Peña, A; Rezola, M L
2009-01-01
We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.
Relative risk regression models with inverse polynomials.
Ning, Yang; Woodward, Mark
2013-08-30
The proportional hazards model assumes that the log hazard ratio is a linear function of parameters. In the current paper, we model the log relative risk as an inverse polynomial, which is particularly suitable for modeling bounded and asymmetric functions. The parameters estimated by maximizing the partial likelihood are consistent and asymptotically normal. The advantages of the inverse polynomial model over the ordinary polynomial model and the fractional polynomial model for fitting various asymmetric log relative risk functions are shown by simulation. The utility of the method is further supported by analyzing two real data sets, addressing the specific question of the location of the minimum risk threshold.
Directory of Open Access Journals (Sweden)
Ryoo CS
2010-01-01
Full Text Available The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.
Algorithms in Solving Polynomial Inequalities
Directory of Open Access Journals (Sweden)
Christopher M. Cordero
2015-11-01
Full Text Available A new method to solve the solution set of polynomial inequalities was conducted. When −1 −2 >0 ℎ 1,2∈ ℝ 10 if n is even. Then, the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+∞ ∪ ,+1 : }. Thus, when −1−2…−≥0, the solution is ∈ ℝ ∈−∞, 1∪ ,+∞∪, +1: }. If is odd, then the solution set is ∈ ℝ ∈ ,+∞ ∪ ,+1 : }. Thus, when −1 −2…−≥0, the solution set is ∈ ℝ ∈ ,+∞∪, +1: }. Let −1−2…−<0 if n is even. Then, the solution set is ∈ ℝ ∈ ,+1 ∶ }. Thus, when −1 −2…−≤0, then the solution set is ∈ ℝ ∈, +1: }. If is an odd, then the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+1 : }. Thus, when −1 −2 … − ≤0, the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+1 : }. This research provides a novel method in solving the solution set of polynomial inequalities, in addition to other existing methods.
Classification based polynomial image interpolation
Lenke, Sebastian; Schröder, Hartmut
2008-02-01
Due to the fast migration of high resolution displays for home and office environments there is a strong demand for high quality picture scaling. This is caused on the one hand by large picture sizes and on the other hand due to an enhanced visibility of picture artifacts on these displays [1]. There are many proposals for an enhanced spatial interpolation adaptively matched to picture contents like e.g. edges. The drawback of these approaches is the normally integer and often limited interpolation factor. In order to achieve rational factors there exist combinations of adaptive and non adaptive linear filters, but due to the non adaptive step the overall quality is notably limited. We present in this paper a content adaptive polyphase interpolation method which uses "offline" trained filter coefficients and an "online" linear filtering depending on a simple classification of the input situation. Furthermore we present a new approach to a content adaptive interpolation polynomial, which allows arbitrary polyphase interpolation factors at runtime and further improves the overall interpolation quality. The main goal of our new approach is to optimize interpolation quality by adapting higher order polynomials directly to the image content. In addition we derive filter constraints for enhanced picture quality. Furthermore we extend the classification based filtering to the temporal dimension in order to use it for an intermediate image interpolation.
Low crosstalk Arrayed Waveguide Grating with Cascaded Waveguide Grating Filter
Energy Technology Data Exchange (ETDEWEB)
Deng Yang; Liu Yuan; Gao Dingshan, E-mail: dsgao@mail.hust.edu.cn [Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074 (China)
2011-02-01
We propose a highly compact and low crosstalk arrayed waveguide grating (AWG) with cascaded waveguide grating (CWGF). The side lobes of the silicon nanowire AWG, which are normally introduced by fabrication errors, can be effectively suppressed by the CWGF. And the crosstalk can be improved about 15dB.
ON FIRST INTEGRALS OF POLYNOMIAL AUTONOMOUS SYSTEMS
Institute of Scientific and Technical Information of China (English)
WANG Yuzhen; CHENG Daizhan; LI Chunwen
2002-01-01
Using Carleman linearization procedure, this paper investigates the problemof first integrals of polynomial autonomous systems and proposes a procedure to find thefirst integrals of polynomial family for the systems. A generalized eigenequation is obtainedand then the problem is reduced to the solvability of the eigenequation. The result is ageneralization of some known results.
Reliability polynomials crossing more than twice
Brown, J.I.; Koç, Y.; Kooij, R.E.
2011-01-01
In this paper we study all-terminal reliability polynomials of networks having the same number of nodes and the same number of links. First we show that the smallest possible size for a pair of networks that allows for two crossings of their reliability polynomials have seven nodes and fifteen edges
Sums of Powers of Fibonacci Polynomials
Indian Academy of Sciences (India)
Helmut Prodinger
2009-11-01
Using the explicit (Binet) formula for the Fibonacci polynomials, a summation formula for powers of Fibonacci polynomials is derived straightforwardly, which generalizes a recent result for squares that appeared in Proc. Ind. Acad. Sci. (Math. Sci.) 118 (2008) 27--41.
Notes on Schubert, Grothendieck and Key Polynomials
Kirillov, Anatol N.
2016-03-01
We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.
BOUNDS FOR THE ZEROS OF POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
W. M. Shah; A.Liman
2004-01-01
Let P(z) =n∑j=0 ajzj be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrom-Kakeya theorem.
Several explicit formulae for Bernoulli polynomials
Komatsu, Takao; Pita Ruiz V., Claudio de J.
2016-01-01
We prove several explicit formulae for the $n$-th Bernoulli polynomial $B_{n}(x)$, in which $B_{n}(x)$ is equal to an affine combination of the polynomials $(x-1)^{n}$, $(x-2)^{n}$, $ldots$, $(x-k-1)^{n}$, where $k$ is any fixed positive integer greater or equal than $n$.
Elementary combinatorics of the HOMFLYPT polynomial
Chmutov, Sergei
2009-01-01
We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.
A Note on Solvable Polynomial Algebras
Directory of Open Access Journals (Sweden)
Huishi Li
2014-03-01
Full Text Available In terms of their defining relations, solvable polynomial algebras introduced by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990] are characterized by employing Gr\\"obner bases of ideals in free algebras, thereby solvable polynomial algebras are completely determinable and constructible in a computational way.
Connections between the matching and chromatic polynomials
Directory of Open Access Journals (Sweden)
E. J. Farrell
1992-01-01
Full Text Available The main results established are (i a connection between the matching and chromatic polynomials and (ii a formula for the matching polynomial of a general complement of a subgraph of a graph. Some deductions on matching and chromatic equivalence and uniqueness are made.
On Polynomial Functions over Finite Commutative Rings
Institute of Scientific and Technical Information of China (English)
Jian Jun JIANG; Guo Hua PENG; Qi SUN; Qi Fan ZHANG
2006-01-01
Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.
A polynomial approach to nonlinear system controllability
Zheng, YF; Willems, JC; Zhang, CH
2001-01-01
This note uses a polynomial approach to present a necessary and sufficient condition for local controllability of single-input-single-output (SISO) nonlinear systems. The condition is presented in terms of common factors of a noncommutative polynomial expression. This result exposes controllability
The topology of Julia sets for polynomials
Institute of Scientific and Technical Information of China (English)
尹永成
2002-01-01
We prove that wandering components of the Julia set of a polynomial are singletons provided each critical point in a wandering Julia component is non-recurrent. This means a conjecture of Branner-Hubbard is true for this kind of polynomials.
Fractal Trigonometric Polynomials for Restricted Range Approximation
Chand, A. K. B.; Navascués, M. A.; Viswanathan, P.; Katiyar, S. K.
2016-05-01
One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.
New pole placement algorithm - Polynomial matrix approach
Shafai, B.; Keel, L. H.
1990-01-01
A simple and direct pole-placement algorithm is introduced for dynamical systems having a block companion matrix A. The algorithm utilizes well-established properties of matrix polynomials. Pole placement is achieved by appropriately assigning coefficient matrices of the corresponding matrix polynomial. This involves only matrix additions and multiplications without requiring matrix inversion. A numerical example is given for the purpose of illustration.
Distortion control of conjugacies between quadratic polynomials
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
We use a new type of distortion control of univalent functions to give an alternative proof of Douady-Hubbard’s ray-landing theorem for quadratic Misiurewicz polynomials. The univalent maps arise from Thurston’s iterated algorithm on perturbation of such polynomials.
Uniqueness of meromorphic functions concerning differential polynomials
Institute of Scientific and Technical Information of China (English)
QIAO Lei
2007-01-01
Based on a unicity theorem for entire funcitions concerning differential polynomials proposed by M. L. Fang and W. Hong, we studied the uniqueness problem of two meromorphic functions whose differential polynomials share the same 1-point by proving two theorems and their related lemmas. The results extend and improve given by Fang and Hong's theorem.
Fostering Connections between Classes of Polynomial Functions.
Buck, Judy Curran
The typical path of instruction in high school algebra courses for the study of polynomial functions has been from linear functions, to quadratic functions, to polynomial functions of degree greater than two. This paper reports results of clinical interviews with an Algebra II student. The interviews were used to probe into the student's…
Colored HOMFLY polynomials can distinguish mutant knots
Nawata, Satoshi; Singh, Vivek Kumar
2015-01-01
We illustrate from the viewpoint of braiding operations on WZNW conformal blocks how colored HOMFLY polynomials with multiplicity structure can detect mutations. As an example, we explicitly evaluate the (2,1)-colored HOMFLY polynomials that distinguish a famous mutant pair, Kinoshita-Terasaka and Conway knot.
Indian Academy of Sciences (India)
V K Jain
2009-02-01
For a polynomial of degree , we have obtained an upper bound involving coefficients of the polynomial, for moduli of its zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).
Differential Krull dimension in differential polynomial extensions
Smirnov, Ilya
2011-01-01
We investigate the differential Krull dimension of differential polynomials over a differential ring. We prove a differential analogue of Jaffard's Special Chain Theorem and show that differential polynomial extensions of certain classes of differential rings have no anomaly of differential Krull dimension.
Polynomial weights and code constructions
DEFF Research Database (Denmark)
Massey, J; Costello, D; Justesen, Jørn
1973-01-01
polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any primepgreater than 2, a new extensive class ofp-ary "Reed-Muller codes," 2) a new class of "repeated-root" cyclic codes...... that are subcodes of the binary Reed-Muller codes and can be very simply instrumented, 3) a new class of constacyclic codes that are subcodes of thep-ary "Reed-Muller codes," 4) two new classes of binary convolutional codes with large "free distance" derived from known binary cyclic codes, 5) two new classes...... of long constraint length binary convolutional codes derived from2^r-ary Reed-Solomon codes, and 6) a new class ofq-ary "repeated-root" constacyclic codes with an algebraic decoding algorithm....
Fuzzy Morphological Polynomial Image Representation
Directory of Open Access Journals (Sweden)
Chin-Pan Huang
2010-01-01
Full Text Available A novel signal representation using fuzzy mathematical morphology is developed. We take advantage of the optimum fuzzy fitting and the efficient implementation of morphological operators to extract geometric information from signals. The new representation provides results analogous to those given by the polynomial transform. Geometrical decomposition of a signal is achieved by windowing and applying sequentially fuzzy morphological opening with structuring functions. The resulting representation is made to resemble an orthogonal expansion by constraining the results of opening to equate adapted structuring functions. Properties of the geometric decomposition are considered and used to calculate the adaptation parameters. Our procedure provides an efficient and flexible representation which can be efficiently implemented in parallel. The application of the representation is illustrated in data compression and fractal dimension estimation temporal signals and images.
DEFF Research Database (Denmark)
Marckmann, Carl Johan
2003-01-01
The subject of this ph.d. thesis was the development of an electrically switchable Bragg grating made in an optical waveguide using thermal poling to be applied within optical telecommunication systems. The planar waveguides used in this thesis were fabricated at the Micro- and Nanotechnology....... This result is very useful in the production of telecommunication devices since polarization independence of the second-order nonlinearity is wanted. In order to increase the second-order nonlinearity, it was found that the introduction of a high refractive index trapping layer was favorable. During...
Sobolev orthogonal polynomials on a simplex
Aktas, Rabia
2011-01-01
The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_\\bg(x) = x_1^{\\g_1} ... x_d^{\\g_d} (1- |x|)^{\\g_{d+1}}$ when all $\\g_i > -1$ and they are eigenfunctions of a second order partial differential operator $L_\\bg$. The singular cases that some, or all, $\\g_1,...,\\g_{d+1}$ are -1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of $L_\\bg$ in each singular case is found. Secondly, these polynomials are shown to be orthogonal with respect to an inner product which is explicitly determined. This inner product involves derivatives of the functions, hence the name Sobolev orthogonal polynomials.
On Chebyshev polynomials and torus knots
Gavrilik, A M
2009-01-01
In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s,2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, with the help of q,p-numbers, the generalized two-variable Alexander polynomials, and prove their direct connection with the HOMFLY polynomials and the skein relation of the latter.
Baxter operator formalism for Macdonald polynomials
Gerasimov, Anton; Oblezin, Sergey
2012-01-01
We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials. Precisely we construct a dual pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter operators is closely related to the dual pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-facto...
Orthogonal Polynomials from Hermitian Matrices II
Odake, Satoru
2016-01-01
This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended $\\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schr\\"odinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out...
Polynomials with Palindromic and Unimodal Coeﬃ cients
Institute of Scientific and Technical Information of China (English)
Hua SUN; Yi WANG; Hai Xia ZHANG
2015-01-01
Let f(q) = arqr +· · ·+asqs, with ar = 0 and as = 0, be a real polynomial. It is a palindromic polynomial of darga n if r+s = n and ar+i = as−i for all i. Polynomials of darga n form a linear subspace Pn(q) of R(q)n+1 of dimension ? n2 ?+1. We give transition matrices between two bases ?qj(1+q+· · ·+qn−2j)? , ?qj(1+q)n−2j? and the standard basis ?qj(1+qn−2j)? of Pn(q). We present some characterizations and sufficcient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.
Jacob's ladders and new orthogonal systems generated by Jacobi polynomials
Moser, Jan
2010-01-01
Is is shown in this paper that there is a connection between the Riemann zeta-function $\\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
HIGHER ORDER MULTIVARIABLE NORLUND EULER-BERNOULLI POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
刘国栋
2002-01-01
The definitions of higher order multivariable Norlund Euler polynomials and Norlund Bernoulli polynomials are presented and some of their important properties are expounded. Some identities involving recurrence sequences and higher order multivariable Norlund Euler-Bernoulli polynomials are established.
Stretchable diffraction gratings for spectrometry
Simonov, A.N.; Grabarnik, S.; Vdovine, G.V
2007-01-01
We have investigated the possibility of using transparent stretchable diffraction gratings for spectrometric applications. The gratings were fabricated by replication of a triangular-groove master into a transparent viscoelastic. The sample length, and hence the spatial period, can be reversibly cha
Picosecond Holographic-Grating Spectroscopy
Duppen, K.
1987-01-01
Interfering light waves produce an optical interference pattern in any medium that interacts with light. This modulation of some physical parameter of the system acts as a classical holographic grating for optical radiation. When such a grating is produced through interaction of pulsed light waves w
Slow plasmons in grating cavities
Aydinli, Atilla; Karademir, Ertugrul; Balci, Sinan; Kocabas, Coskun
2016-03-01
Recent research on surface plasmon polaritons and their applications have brought forward a wealth of information and continues to be of interest to many. In this report, we concentrate on propagating surface plasmon polaritons (SPPs) and their interaction with matter. Using grating based metallic structures, it is possible to control the electrodynamics of propagating SPPs. Biharmonic gratings loaded with periodic Si stripes allow excitation of SPPs that are localized inside the band gap with grating coupling. The cavity state is formed due to periodic effective index modulation obtained by one harmonic of the grating and loaded Si stripes. More complicated grating structures such as metallic Moiré surfaces have also been shown to form a localized state inside the band gap when excited with Kretschmann configuration.
Polynomial Interpolation in the Elliptic Curve Cryptosystem
Directory of Open Access Journals (Sweden)
Liew K. Jie
2011-01-01
Full Text Available Problem statement: In this research, we incorporate the polynomial interpolation method in the discrete logarithm problem based cryptosystem which is the elliptic curve cryptosystem. Approach: In this study, the polynomial interpolation method to be focused is the Lagrange polynomial interpolation which is the simplest polynomial interpolation method. This method will be incorporated in the encryption algorithm of the elliptic curve ElGamal cryptosystem. Results: The scheme modifies the elliptic curve ElGamal cryptosystem by adding few steps in the encryption algorithm. Two polynomials are constructed based on the encrypted points using Lagrange polynomial interpolation and encrypted for the second time using the proposed encryption method. We believe it is safe from the theoretical side as it still relies on the discrete logarithm problem of the elliptic curve. Conclusion/Recommendations: The modified scheme is expected to be more secure than the existing scheme as it offers double encryption techniques. On top of the existing encryption algorithm, we managed to encrypt one more time using the polynomial interpolation method. We also have provided detail examples based on the described algorithm.
On Chebyshev polynomials and torus knots
Gavrilik, A. M.; Pavlyuk, A. M.
2009-01-01
In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s,2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, wit...
Cycles are determined by their domination polynomials
Akbari, Saieed
2009-01-01
Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ is the polynomial $D(G,x)=\\sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. In this paper we show that cycles are determined by their domination polynomials.
Limits of zeros of polynomial sequences
Zhu, Xinyun; Grossman, George
2007-01-01
In the present paper we consider $F_k(x)=x^{k}-\\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \\to \\infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also ...
A bivariate chromatic polynomial for signed graphs
Beck, Matthias
2012-01-01
We study Dohmen--P\\"onitz--Tittmann's bivariate chromatic polynomial $c_\\Gamma(k,l)$ which counts all $(k+l)$-colorings of a graph $\\Gamma$ such that adjacent vertices get different colors if they are $\\le k$. Our first contribution is an extension of $c_\\Gamma(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_\\Gamma(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynomials to acyclic orientations.
A Polynomial Preconditioner for the CMRH Algorithm
Directory of Open Access Journals (Sweden)
Jiangzhou Lai
2011-01-01
Full Text Available Many large and sparse linear systems can be solved efficiently by restarted GMRES and CMRH methods Sadok 1999. The CMRH(m method is less expensive and requires slightly less storage than GMRES(m. But like GMRES, the restarted CMRH method may not converge. In order to remedy this defect, this paper presents a polynomial preconditioner for CMRH-based algorithm. Numerical experiments are given to show that the polynomial preconditioner is quite simple and easily constructed and the preconditioned CMRH(m with the polynomial preconditioner has better performance than CMRH(m.
On Calculation of Adomian Polynomials by MATLAB
Directory of Open Access Journals (Sweden)
Hossein ABOLGHASEMI
2011-01-01
Full Text Available Adomian Decomposition Method (ADM is an elegant technique to handle an extensive class of linear or nonlinear differential and integral equations. However, in case of nonlinear equations, ADM demands a special representation of each nonlinear term, namely, Adomian polynomials. The present paper introduces a novel MATLAB code which computes Adomian polynomials associated with several types of nonlinearities. The code exploits symbolic programming incorporated with a recently proposed alternative scheme to be straightforward and fast. For the sake of exemplification, Adomian polynomials of famous nonlinear operators, computed by the code, are given.
The Translated Dowling Polynomials and Numbers.
Mangontarum, Mahid M; Macodi-Ringia, Amila P; Abdulcarim, Normalah S
2014-01-01
More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.
Polynomial threshold functions and Boolean threshold circuits
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.
2013-01-01
We study the complexity of computing Boolean functions on general Boolean domains by polynomial threshold functions (PTFs). A typical example of a general Boolean domain is 12n . We are mainly interested in the length (the number of monomials) of PTFs, with their degree and weight being...... of secondary interest. We show that PTFs on general Boolean domains are tightly connected to depth two threshold circuits. Our main results in regard to this connection are: PTFs of polynomial length and polynomial degree compute exactly the functions computed by THRMAJ circuits. An exponential length lower...
Laurent polynomial moment problem: a case study
Pakovich, F; Zvonkin, A
2009-01-01
In recent years, the so-called polynomial moment problem, motivated by the classical Poincare center-focus problem, was thoroughly studied, and the answers to the main questions have been found. The study of a similar problem for rational functions is still at its very beginning. In this paper, we make certain progress in this direction; namely, we construct an example of a Laurent polynomial for which the solutions of the corresponding moment problem behave in a significantly more complicated way than it would be possible for a polynomial.
Exponential Polynomial Approximation with Unrestricted Upper Density
Institute of Scientific and Technical Information of China (English)
Xiang Dong YANG
2011-01-01
We take a new approach to obtaining necessary and sufficient conditions for the incompleteness of exponential polynomials in Lp/α, where Lp/α is the weighted Banach space of complex continuous functions f defined on the real axis (R)satisfying (∫+∞/-∞|f(t)|pe-α(t)dt)1/p, 1 < p < ∞, and α(t) is a nonnegative continuous function defined on the real axis (R). In this paper, the upper density of the sequence which forms the exponential polynomials is not required to be finite. In the study of weighted polynomial approximation, consideration of the case is new.
More on rotations as spin matrix polynomials
Energy Technology Data Exchange (ETDEWEB)
Curtright, Thomas L. [Department of Physics, University of Miami, Coral Gables, Florida 33124-8046 (United States)
2015-09-15
Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined.
Modular polynomials via isogeny volcanoes
Broker, Reinier; Sutherland, Andrew V
2010-01-01
We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also consider several modular functions g for which Phi_n^g is smaller than Phi_n, allowing us to handle n over 60000.
M-Polynomials and Topological Indices of Titania Nanotubes
Directory of Open Access Journals (Sweden)
Mobeen Munir
2016-10-01
Full Text Available Titania is one of the most comprehensively studied nanostructures due to their widespread applications in the production of catalytic, gas sensing, and corrosion-resistant materials. M-polynomial of nanotubes has been vastly investigated, as it produces many degree-based topological indices, which are numerical parameters capturing structural and chemical properties. These indices are used in the development of quantitative structure-activity relationships (QSARs in which the biological activity and other properties of molecules, such as boiling point, stability, strain energy, etc., are correlated with their structure. In this report, we provide M-polynomials of single-walled titania (SW TiO2 nanotubes and recover important topological degree-based indices to theoretically judge these nanotubes. We also plot surfaces associated to single-walled titania (SW TiO2 nanotubes.
Local Polynomial Estimation of Distribution Functions
Institute of Scientific and Technical Information of China (English)
LI Yong-hong; ZENG Xia
2007-01-01
Under the condition that the total distribution function is continuous and bounded on (-∞,∞), we constructed estimations for distribution and hazard functions with local polynomial method, and obtained the rate of strong convergence of the estimations.
Tutte Polynomial of Scale-Free Networks
Chen, Hanlin; Deng, Hanyuan
2016-05-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics. The computation of this invariant for a graph is NP-hard in general. In this paper, we focus on two iteratively growing scale-free networks, which are ubiquitous in real-life systems. Based on their self-similar structures, we mainly obtain recursive formulas for the Tutte polynomials of two scale-free networks (lattices), one is fractal and "large world", while the other is non-fractal but possess the small-world property. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at ( x, y)-plane, such as, the number of spanning trees, the number of acyclic orientations, etc.
Generation of multivariate Hermite interpolating polynomials
Tavares, Santiago Alves
2005-01-01
Generation of Multivariate Hermite Interpolating Polynomials advances the study of approximate solutions to partial differential equations by presenting a novel approach that employs Hermite interpolating polynomials and bysupplying algorithms useful in applying this approach.Organized into three sections, the book begins with a thorough examination of constrained numbers, which form the basis for constructing interpolating polynomials. The author develops their geometric representation in coordinate systems in several dimensions and presents generating algorithms for each level number. He then discusses their applications in computing the derivative of the product of functions of several variables and in the construction of expression for n-dimensional natural numbers. Section II focuses on the construction of Hermite interpolating polynomials, from their characterizing properties and generating algorithms to a graphical analysis of their behavior. The final section of the book is dedicated to the applicatio...
Superconformal minimal models and admissible Jack polynomials
Blondeau-Fournier, Olivier; Ridout, David; Wood, Simon
2016-01-01
We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.
Hermite polynomials and quasi-classical asymptotics
Energy Technology Data Exchange (ETDEWEB)
Ali, S. Twareque, E-mail: twareque.ali@concordia.ca [Department of Mathematics and Statistics, Concordia University, Montréal, Québec H3G 1M8 (Canada); Engliš, Miroslav, E-mail: englis@math.cas.cz [Mathematics Institute, Silesian University in Opava, Na Rybníčku 1, 74601 Opava, Czech Republic and Mathematics Institute, Žitná 25, 11567 Prague 1 (Czech Republic)
2014-04-15
We study an unorthodox variant of the Berezin-Toeplitz type of quantization scheme, on a reproducing kernel Hilbert space generated by the real Hermite polynomials and work out the associated quasi-classical asymptotics.
Concentration for noncommutative polynomials in random matrices
2011-01-01
We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries and unitary or orthogonal matrices.
Thermodynamic characterization of networks using graph polynomials
Ye, Cheng; Peron, Thomas K DM; Silva, Filipi N; Rodrigues, Francisco A; Costa, Luciano da F; Torsello, Andrea; Hancock, Edwin R
2015-01-01
In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the...
Characteristic Polynomials of Complex Random Matrix Models
Akemann, G
2003-01-01
We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite-N, where N is the size of the matrix. We give some explicit examples at finite-N for specific weight functions. The characteristic polynomials in the large-N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large-N limit.
Twisted Polynomials and Forgery Attacks on GCM
DEFF Research Database (Denmark)
Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey;
2015-01-01
nonce misuse resistance, such as POET. The algebraic structure of polynomial hashing has given rise to security concerns: At CRYPTO 2008, Handschuh and Preneel describe key recovery attacks, and at FSE 2013, Procter and Cid provide a comprehensive framework for forgery attacks. Both approaches rely...... heavily on the ability to construct forgery polynomials having disjoint sets of roots, with many roots (“weak keys”) each. Constructing such polynomials beyond naïve approaches is crucial for these attacks, but still an open problem. In this paper, we comprehensively address this issue. We propose to use...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...
Limits of zeros of polynomial sequences
Zhu, Xinyun
2007-01-01
In the present paper we consider $F_k(x)=x^{k}-\\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \\to \\infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also shown, in both cases, that the convergence is monotone for sufficiently large degree. We give an upper bound for the modulus of the complex zeros of the polynomials for each sequence. This gives a general solution related to problems considered by Dubeau 1989, 1993, Miles 1960, Flores 1967, Miller 1971 and later by the second author in the present paper, and Narayan 1997.
Voltage-controlled compression for period tuning of optical surface relief gratings.
Kollosche, Matthias; Döring, Sebastian; Stumpe, Joachim; Kofod, Guggi
2011-04-15
This Letter reports on new methods and a consistent model for voltage tunable optical transmission gratings. Elastomeric gratings were molded from holographically written surface relief gratings in an azobenzene sol-gel material. These were placed on top of a transparent electroactive elastomeric substrate. Two different electro-active substrate elastomers were employed, with a large range of prestretches. A novel finite-deformation theory was found to match the device response excellently, without fitting parameters. The results clearly show that the grating underwent pure-shear deformation, and more surprisingly, that the mechanical properties of the electro-active substrate did not affect device actuation.
ON ABEL-GONTSCHAROFF-GOULD'S POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
He Tianxiao; Leetsch C. Hsu; Peter J. S. Shiue
2003-01-01
In this paper a connective study of Gould's annihilation coefficients and Abel-Gontscharoff polynomials is presented. It is shown that Gould's annihilation coefficients and Abel-Gontscharoff polynomials are actually equivalent to each other under certain linear substitutions for the variables. Moreover, a pair of related expansion formulas involving Gontscharoff's remainder and a new form of it are demonstrated, and also illustrated with several examples.
Rational Convolution Roots of Isobaric Polynomials
Conci, Aura; Li, Huilan; MacHenry, Trueman
2014-01-01
In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first kind and second kind. Hen...
Blind Signature Scheme Based on Chebyshev Polynomials
Maheswara Rao Valluri
2011-01-01
A blind signature scheme is a cryptographic protocol to obtain a valid signature for a message from a signer such that signer’s view of the protocol can’t be linked to the resulting message signature pair. This paper presents blind signature scheme using Chebyshev polynomials. The security of the given scheme depends upon the intractability of the integer factorization problem and discrete logarithms ofChebyshev polynomials.
Positive maps, positive polynomials and entanglement witnesses
Skowronek, Lukasz
2009-01-01
We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
Positive maps, positive polynomials and entanglement witnesses
Energy Technology Data Exchange (ETDEWEB)
Skowronek, Lukasz; Zyczkowski, Karol [Institute of Physics, Jagiellonian University, Krakow (Poland)], E-mail: lukasz.skowronek@uj.edu.pl, E-mail: karol@tatry.if.uj.edu.pl
2009-08-14
We link the study of positive quantum maps, block positive operators and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
Blind Signature Scheme Based on Chebyshev Polynomials
Directory of Open Access Journals (Sweden)
Maheswara Rao Valluri
2011-12-01
Full Text Available A blind signature scheme is a cryptographic protocol to obtain a valid signature for a message from a signer such that signer’s view of the protocol can’t be linked to the resulting message signature pair. This paper presents blind signature scheme using Chebyshev polynomials. The security of the given scheme depends upon the intractability of the integer factorization problem and discrete logarithms ofChebyshev polynomials.
Ferrers Matrices Characterized by the Rook Polynomials
Institute of Scientific and Technical Information of China (English)
MAHai-cheng; HUSheng-biao
2003-01-01
In this paper,we show that there exist precisely W(A) Ferrers matrices F(C1,C2,…,cm)such that the rook polynomials is equal to the rook polynomial of Ferrers matrix F(b1,b2,…,bm), where A={b1,b2-1,…,bm-m+1} is a repeated set,W(A) is weight of A.
Recursive Polynomial Remainder Sequence and its Subresultants
Terui, Akira
2008-01-01
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of init...
Subresultants in Recursive Polynomial Remainder Sequence
Terui, Akira
2008-01-01
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate "recursively" with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresultants to be determinants representing the coefficients in recursive PRS by coefficients of initial po...
Polynomial Subtraction Method for Disconnected Quark Loops
Liu, Quan; Morgan, Ron
2014-01-01
The polynomial subtraction method, a new numerical approach for reducing the noise variance of Lattice QCD disconnected matrix elements calculation, is introduced in this paper. We use the MinRes polynomial expansion of the QCD matrix as the approximation to the matrix inverse and get a significant reduction in the variance calculation. We compare our results with that of the perturbative subtraction and find that the new strategy yields a faster decrease in variance which increases with quark mass.
Local fibred right adjoints are polynomial
DEFF Research Database (Denmark)
Kock, Anders; Kock, Joachim
2013-01-01
For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense......For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense...
Laguerre polynomials method in the valon model
Boroun, G R
2014-01-01
We used the Laguerre polynomials method for determination of the proton structure function in the valon model. We have examined the applicability of the valon model with respect to a very elegant method, where the structure of the proton is determined by expanding valon distributions and valon structure functions on Laguerre polynomials. We compared our results with the experimental data, GJR parameterization and DL model. Having checked, this method gives a good description for the proton structure function in valon model.
Vector-Valued Jack Polynomials from Scratch
Directory of Open Access Journals (Sweden)
Jean-Gabriel Luque
2011-03-01
Full Text Available Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N and studied by one of the authors (C. Dunkl in the specialization r=p=1 (i.e. for the symmetric group. By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.
On λ-Bell polynomials associated with umbral calculus
Kim, T.; Kim, D. S.
2017-01-01
In this paper, we introduce some new λ-Bell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λ-Bell polynomials arising from umbral calculus.
A generalization of the dichromatic polynomial of a graph
1981-01-01
The Subgraph polynomial fo a graph pair (G,H), where H⫅G, is defined. By assigning particular weights to the variables, it is shown that this polynomial reduces to the dichromatic polynomial of G. This idea of a graph pair leads to a dual generalization of the dichromatic polynomial.
Interpolation on Real Algebraic Curves to Polynomial Data
Directory of Open Access Journals (Sweden)
Len Bos
2013-04-01
Full Text Available We discuss a polynomial interpolation problem where the data are of the form of a set of algebraic curves in R^2 on each of which is prescribed a polynomial. The object is then to construct a global bivariate polynomial that agrees with the given polynomials when restricted to the corresponding curves.
Extreme Silica Optical Fibre Gratings
Directory of Open Access Journals (Sweden)
Kevin Cook
2008-10-01
Full Text Available A regenerated optical fibre Bragg grating that survives temperature cycling up to 1,295Ã‚Â°C is demonstrated. A model based on seeded crystallisation or amorphisation is proposed.
MEMS Bragg grating force sensor
DEFF Research Database (Denmark)
Reck, Kasper; Thomsen, Erik Vilain; Hansen, Ole
2011-01-01
We present modeling, design, fabrication and characterization of a new type of all-optical frequency modulated MEMS force sensor based on a mechanically amplified double clamped waveguide beam structure with integrated Bragg grating. The sensor is ideally suited for force measurements in harsh...... environments and for remote and distributed sensing and has a measured sensitivity of -14 nm/N, which is several times higher than what is obtained in conventional fiber Bragg grating force sensors. © 2011 Optical Society of America....
Barillé, R.; Tajalli, P.; Roy, P.; Ahmadi-kandjani, S.; Kucharski, S.; Ortyl, E.
2012-02-01
We propose a new type of hollow-core fiber where the propagation is ensured by a photoinduced self-pattern acting as a surface relief grating (SRG). The SRG is written by launching a suitable laser beam with proper polarization in a capillary glass fiber with the inner surface previously coated with an azopolymer thin film. Such a grating acts as a wavelength/angle dependant reflective mirror and enhances the confinement and the propagation of the light.
Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials
Directory of Open Access Journals (Sweden)
Kim Young-Hee
2009-01-01
Full Text Available The main purpose of this paper is to present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on . We define the - -Euler polynomials and obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. We define -extensions of Apostol type's Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these - -Euler polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Hurwitz type zeta functions of these - -Euler polynomials.
Explicit classes of permutation polynomials of F33m
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over F33m are presented. These polynomials have simple shapes, and they are related to planar functions.
Explicit classes of permutation polynomials of F33m
Institute of Scientific and Technical Information of China (English)
DING CunSheng; XIANG Qing; YUAN Jin; YUAN PingZhi
2009-01-01
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over F33 are pre-sented. These polynomials have simple shapes, and they are related to planar functions.
Certain non-linear differential polynomials sharing a non zero polynomial
Directory of Open Access Journals (Sweden)
Majumder Sujoy
2015-10-01
functions sharing a nonzero polynomial and obtain two results which improves and generalizes the results due to L. Liu [Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl., 56 (2008, 3236-3245.] and P. Sahoo [Uniqueness and weighted value sharing of meromorphic functions, Applied. Math. E-Notes., 11 (2011, 23-32.].
A new class of generalized polynomials associated with Hermite and Bernoulli polynomials
Directory of Open Access Journals (Sweden)
M. A. Pathan
2015-05-01
Full Text Available In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson's polynomials Φ_{n}^{(α}(x,ν of degree n and order α introduced by Derre and Simsek.The concepts of Bernoulli numbers B_n, Bernoulli polynomials B_n(x, generalized Bernoulli numbers B_n(a,b, generalized Bernoulli polynomials B_n(x;a,b,c of Luo et al, Hermite-Bernoulli polynomials {_HB}_n(x,y of Dattoli et al and {_HB}_n^{(α} (x,y of Pathan are generalized to the one {_HB}_n^{(α}(x,y,a,b,c which is called the generalized polynomial depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B_n, B_n(x, B_n(a,b, B_n(x;a,b,c and {}_HB_n^{(α}(x,y;a,b,c are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of generalized Bernoulli numbers and polynomials
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
Miller, David A.; Moseley, James
2012-01-01
In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…
DEFF Research Database (Denmark)
Ribard, Nicolas; Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use the represen......In the paper, we strive to develop an algorithm that simultaneously computes a polynomial control and a polynomial Lyapunov function. This ensures asymptotic stability of the designed feedback system. The above problem is translated to a certificate of positivity. To this end, we use...... the representation of the given control system in Bernstein basis. Subsequently, the control synthesis problem is reduced to finite number of evaluations of a polynomial on vertices of cubes in the space of parameters representing admissible controls and Lyapunov functions....
Twisted Alexander polynomials of hyperbolic knots
Dunfield, Nathan M; Jackson, Nicholas
2011-01-01
We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313,209 hyperbolic knots in S^3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X_0 of the SL(2, C)-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X_0. We use this to help explain some of the patterns observed for knots in S^3, and explore a potential...
Chemical Reaction Networks for Computing Polynomials.
Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc D
2017-01-20
Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: a type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.
Algorithms for Testing Monomials in Multivariate Polynomials
Chen, Zhixiang; Liu, Yang; Schweller, Robert
2010-01-01
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. Second, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear monomials in $\\Pi_m\\Sigma_2\\Pi_t\\times \\Pi_k\\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. The first algorithm extends...
Efficient iterative technique for designing bragg gratings
DEFF Research Database (Denmark)
Plougmann, Nikolai; Kristensen, Martin
2004-01-01
We present a new iterative method for designing Bragg gratings based on the Levenberg-Marquardt method of minimizing a chi-squared merit function. It is effective for designing both weak and strong gratings and is particularly well suited for unchirped gratings.......We present a new iterative method for designing Bragg gratings based on the Levenberg-Marquardt method of minimizing a chi-squared merit function. It is effective for designing both weak and strong gratings and is particularly well suited for unchirped gratings....
Electronically reconfigurable superimposed waveguide long-period gratings
Kulishov, Mykola; Daxhelet, Xavier; Gaidi, Mounir; Chaker, Mohamed
2002-08-01
The perturbation to the refractive index induced by a periodic electric field from two systems of interdigitated electrodes with the electrode-finger period l is analyzed for a waveguide with an electro-optically (EO) active core-cladding. It is shown that the electric field induces two superimposed transmissive refractive-index gratings with different symmetries of their cross-section distributions. One of these gratings has a constant component of an EO-induced refractive index along with its variable component with periodicity l, whereas the second grating possesses only a variable component with periodicity 2l. With the proper waveguide design, the gratings provide interaction between a guided fundamental core mode and two guided cladding modes. Through the externally applied electric potential, these gratings can be independently switched ON and OFF, or they can be activated simultaneously with electronically controlled weighting factors. Coupling coefficients of both gratings are analyzed in terms of their dependence on the electrode duty ratio and dielectric permittivities of the core and cladding. The coupled-wave equations for the superimposed gratings are written and solved. The spectral characteristics are investigated by numerical simulation. It is found that the spectral characteristics are described by a dual-dip transmission spectrum with individual electronic control of the dip depths and positions. Within the concept, a new external potential application scheme is described in which the symmetry of the cross-sectional distribution of the refractive index provides coupling only between the core mode and the cladding modes, preventing interaction of the cladding modes with each another. This simple concept opens opportunities for developing a number of tunable devices for integrated optics by use of the proposed design as a building block.
Uniform trigonometric polynomial B-spline curves
Institute of Scientific and Technical Information of China (English)
吕勇刚; 汪国昭; 杨勋年
2002-01-01
This paper presents a new kind of uniform spline curve, named trigonometric polynomialB-splines, over space Ω = span{sint, cost, tk-3,tk-4,…,t,1} of which k is an arbitrary integerlarger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similarV properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.
Transversals of Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
by rotational constants. Transversals are a certain class of curves for such a family of vector fields that represent the bifurcation states for this family of vector fields. More specifically, transversals are curves that coincide with a homoclinic separatrix for some rotation of the vector field. Given......, an important step was proving that the transversals possessed a certain characteristic. Understanding transversals might be the key to proving other polynomial vector fields are generic, and they are important in understanding bifurcations of polynomial vector fields in general. We consider two important...... examples of rotated families to argue this. There will be discussed several open questions concerning the number of transversals that can appear for a certain degree d of a polynomial vector field, and furthermore how transversals are analyzed with respect to bifurcations around multiple equilibrium points....
Minimal residual method stronger than polynomial preconditioning
Energy Technology Data Exchange (ETDEWEB)
Faber, V.; Joubert, W.; Knill, E. [Los Alamos National Lab., NM (United States)] [and others
1994-12-31
Two popular methods for solving symmetric and nonsymmetric systems of equations are the minimal residual method, implemented by algorithms such as GMRES, and polynomial preconditioning methods. In this study results are given on the convergence rates of these methods for various classes of matrices. It is shown that for some matrices, such as normal matrices, the convergence rates for GMRES and for the optimal polynomial preconditioning are the same, and for other matrices such as the upper triangular Toeplitz matrices, it is at least assured that if one method converges then the other must converge. On the other hand, it is shown that matrices exist for which restarted GMRES always converges but any polynomial preconditioning of corresponding degree makes no progress toward the solution for some initial error. The implications of these results for these and other iterative methods are discussed.
Fast beampattern evaluation by polynomial rooting
Häcker, P.; Uhlich, S.; Yang, B.
2011-07-01
Current automotive radar systems measure the distance, the relative velocity and the direction of objects in their environment. This information enables the car to support the driver. The direction estimation capabilities of a sensor array depend on its beampattern. To find the array configuration leading to the best angle estimation by a global optimization algorithm, a huge amount of beampatterns have to be calculated to detect their maxima. In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations. The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial. We differentiate and root the polynomial to get the extrema of the beampattern. In addition, we show a method to reduce the computational burden even more by decreasing the order of the polynomial.
A complete discrimination system for polynomials
Institute of Scientific and Technical Information of China (English)
杨路; 侯晓荣; 曾振柄
1996-01-01
Given a polynomial with symbolic/literal coefficients,a complete discrimination system is a set of explicit expressions in terms of the coefficients,which is sufficient for determining the numbers and multiplicities of the real and imaginary roots.Though it is of great significance,such a criterion for root-classification has never been given for polynomials with degrees greater than 4.The lack of efficient tools in this aspect extremely prevents computer implementations for Tarski’s and other methods in automated theorem proving.To remedy this defect,a generic algorithm is proposed to produce a complete discrimination system for a polynomial with any degrees.This result has extensive applications in various fields,and its efficiency was demonstrated by computer implementations.
Dominating Sets and Domination Polynomials of Paths
Directory of Open Access Journals (Sweden)
Saeid Alikhani
2009-01-01
Full Text Available Let G=(V,E be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V\\S is adjacent to at least one vertex in S. Let 𝒫ni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn,j=|𝒫nj|. In this paper, we construct 𝒫ni, and obtain a recursive formula for d(Pn,i. Using this recursive formula, we consider the polynomial D(Pn,x=∑i=⌈n/3⌉nd(Pn,ixi, which we call domination polynomial of paths and obtain some properties of this polynomial.
Flexible PCPDTBT:PCBM solar cells with integrated grating structures
DEFF Research Database (Denmark)
Oliveira Hansen, Roana Melina de; Liu, Yinghui; Madsen, Morten;
2013-01-01
spectra of the active layer. This optimized solar cell structure leads to an enhanced absorption in the active layer and thus improved short-circuit currents and power conversion efficiencies in the fabricated devices. Fabrication of the solar cells on thin polyimide substrates which are compatible......We report on development of flexible PCPDTBT:PCBM solar cells with integrated diffraction gratings on the bottom electrodes. The presented results address PCPDTBT:PCBM solar cells in an inverted geometry, which contains implemented grating structures whose pitch is tuned to match the absorption...
Energy Technology Data Exchange (ETDEWEB)
Alessi, D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2016-11-01
Pulse compressors for ultrafast lasers have been identified as a technology gap in the push towards high peak power systems with high average powers for industrial and scientific applications. Gratings for ultrashort (sub-150fs) pulse compressors are metallic and can absorb a significant percentage of laser energy resulting in up to 40% loss as well as thermal issues which degrade on-target performance. We have developed a next generation gold grating technology which we have scaled to the petawatt-size. This resulted in improvements in efficiency, uniformity and processing as compared to previous substrate etched gratings for high average power. This new design has a deposited dielectric material for the grating ridge rather than etching directly into the glass substrate. It has been observed that average powers as low as 1W in a compressor can cause distortions in the on-target beam. We have developed and tested a method of actively cooling diffraction gratings which, in the case of gold gratings, can support a petawatt peak power laser with up to 600W average power. We demonstrated thermo-mechanical modeling of a grating in its use environment and benchmarked with experimental measurement. Multilayer dielectric (MLD) gratings are not yet used for these high peak power, ultrashort pulse durations due to their design challenges. We have designed and fabricated broad bandwidth, low dispersion MLD gratings suitable for delivering 30 fs pulses at high average power. This new grating design requires the use of a novel Out Of Plane (OOP) compressor, which we have modeled, designed, built and tested. This prototype compressor yielded a transmission of 90% for a pulse with 45 nm bandwidth, and free of spatial and angular chirp. In order to evaluate gratings and compressors built in this project we have commissioned a joule-class ultrafast Ti:Sapphire laser system. Combining the grating cooling and MLD technologies developed here could enable petawatt laser systems to
Incomplete Bivariate Fibonacci and Lucas -Polynomials
Directory of Open Access Journals (Sweden)
Dursun Tasci
2012-01-01
Full Text Available We define the incomplete bivariate Fibonacci and Lucas -polynomials. In the case =1, =1, we obtain the incomplete Fibonacci and Lucas -numbers. If =2, =1, we have the incomplete Pell and Pell-Lucas -numbers. On choosing =1, =2, we get the incomplete generalized Jacobsthal number and besides for =1 the incomplete generalized Jacobsthal-Lucas numbers. In the case =1, =1, =1, we have the incomplete Fibonacci and Lucas numbers. If =1, =1, =1, =⌊(−1/(+1⌋, we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas -polynomials are given.
Spread polynomials, rotations and the butterfly effect
Goh, Shuxiang
2009-01-01
The spread between two lines in rational trigonometry replaces the concept of angle, allowing the complete specification of many geometrical and dynamical situations which have traditionally been viewed approximately. This paper investigates the case of powers of a rational spread rotation, and in particular, a curious periodicity in the prime power decomposition of the associated values of the spread polynomials, which are the analogs in rational trigonometry of the Chebyshev polynomials of the first kind. Rational trigonometry over finite fields plays a role, together with non-Euclidean geometries.
Error Minimization of Polynomial Approximation of Delta
Indian Academy of Sciences (India)
Islam Sana; Sadiq Muhammad; Qureshi Muhammad Shahid
2008-09-01
The difference between Universal time (UT) and Dynamical time (TD), known as Delta ( ) is tabulated for the first day of each year in the Astronomical Almanac. During the last four centuries it is found that there are large differences between its values for two consecutive years. Polynomial approximations have been developed to obtain the values of for any time of a year for the period AD 1620 to AD 2000 (Meeu 2000) as no dynamical theories describe the variations in . In this work, a new set of polynomials for is obtained for the period AD 1620 to AD 2007 that is found to produce better results compared to previous attempts.
Knot polynomial identities and quantum group coincidences
Morrison, Scott; Snyder, Noah
2010-01-01
We construct link invariants using the D_2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D_2n planar algebras. We discuss the origins of these coincidences, explaining the role of SO level-rank duality, Kirby-Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves G_2 and does not appear to be related to level-rank duality.
Some Inequalities for the Derivative of Polynomials
Directory of Open Access Journals (Sweden)
Sunil Hans
2014-01-01
Full Text Available If pz=∑υ=0ncυzυ is a polynomial of degree n, having no zeros in z<1, then Aziz (1989 proved maxz=1p′z≤n/2Mα2+Mα+π21/2, where Mα=max1≤k≤npeiα+2kπ/n. In this paper, we consider a class of polynomial Pnμ of degree n, defined as pz=a0+∑υ=μnaυzυ and present certain generalizations of above inequality and some other well-known results.
Polynomial Kernelizations for $\\MINF_1$ and $\\MNP$
Kratsch, Stefan
2009-01-01
The relation of constant-factor approximability to fixed-parameter tractability and kernelization is a long-standing open question. We prove that two large classes of constant-factor approximable problems, namely $\\MINF_1$ and $\\MNP$, including the well-known subclass $\\MSNP$, admit polynomial kernelizations for their natural decision versions. This extends results of Cai and Chen (JCSS 1997), stating that the standard parameterizations of problems in $\\MSNP$ and $\\MINF_1$ are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al. ICALP 2008).
The classification of polynomial basins of infinity
DeMarco, Laura
2011-01-01
We consider the problem of classifying the dynamics of complex polynomials $f: \\mathbb{C} \\to \\mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set of topological conjugacy classes. We give an algorithm for constructing abstract pictographs, and we provide an inductive algorithm for counting topological conjugacy classes with a given pictograph.
High degree interpolation polynomial in Newton form
Tal-Ezer, Hillel
1988-01-01
Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.
The chromatic polynomial and list colorings
DEFF Research Database (Denmark)
Thomassen, Carsten
2009-01-01
We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph.......We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph....
Optical Fiber Grating based Sensors
DEFF Research Database (Denmark)
Michelsen, Susanne
2003-01-01
In this thesis differenct optical fiber gratings are used for sensor purposes. If a fiber with a core concentricity error (CCE) is used, a directional dependent bend sensor can be produced. The CCE direction can be determined by means of diffraction. This makes it possible to produce long......-period gratings in a fiber with a CCE direction parallel or perpendicular to the writing direction. The maximal bending sensitivity is independent on the writing direction, but the detailed bending response is different in the two cases. A temperature and strain sensor, based on a long-period grating and two...... wavelength. It is shown that it is possible to tune and modulate a DFB fiber laser with both strain from a piezoelectric transducer and by temperature through resistive heating of a methal film. Both a chemical deposited silver layer and an electron-beam evaporation technique has been investigated, to find...
Exploiting a Transmission Grating Spectrometer
Energy Technology Data Exchange (ETDEWEB)
Ronald E. Bell
2004-12-08
The availability of compact transmission grating spectrometers now allows an attractive and economical alternative to the more familiar Czerny-Turner configuration for many high-temperature plasma applications. Higher throughput is obtained with short focal length refractive optics and stigmatic imaging. Many more spectra can be obtained with a single spectrometer since smaller, more densely packed optical input fibers can be used. Multiple input slits, along with a bandpass filter, can be used to maximize the number of spectra per detector, providing further economy. Curved slits can correct for the strong image curvature of the short focal length optics. Presented here are the governing grating equations for both standard and high-dispersion transmission gratings, defining dispersion, image curvature, and desired slit curvature, that can be used in the design of improved plasma diagnostics.
Perturbations around the zeros of classical orthogonal polynomials
Sasaki, Ryu
2014-01-01
Starting from degree N solutions of a time dependent Schroedinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,...,N-1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Polynomial Representations for a Wavelet Model of Interest Rates
Directory of Open Access Journals (Sweden)
Dennis G. Llemit
2015-12-01
Full Text Available In this paper, we approximate a non – polynomial function which promises to be an essential tool in interest rates forecasting in the Philippines. We provide two numerical schemes in order to generate polynomial functions that approximate a new wavelet which is a modification of Morlet and Mexican Hat wavelets. The first is the Polynomial Least Squares method which approximates the underlying wavelet according to desired numerical errors. The second is the Chebyshev Polynomial approximation which generates the required function through a sequence of recursive and orthogonal polynomial functions. We seek to determine the lowest order polynomial representations of this wavelet corresponding to a set of error thresholds.
Spectral calibration for convex grating imaging spectrometer
Zhou, Jiankang; Chen, Xinhua; Ji, Yiqun; Chen, Yuheng; Shen, Weimin
2013-12-01
Spectral calibration of imaging spectrometer plays an important role for acquiring target accurate spectrum. There are two spectral calibration types in essence, the wavelength scanning and characteristic line sampling. Only the calibrated pixel is used for the wavelength scanning methods and he spectral response function (SRF) is constructed by the calibrated pixel itself. The different wavelength can be generated by the monochromator. The SRF is constructed by adjacent pixels of the calibrated one for the characteristic line sampling methods. And the pixels are illuminated by the narrow spectrum line and the center wavelength of the spectral line is exactly known. The calibration result comes from scanning method is precise, but it takes much time and data to deal with. The wavelength scanning method cannot be used in field or space environment. The characteristic line sampling method is simple, but the calibration precision is not easy to confirm. The standard spectroscopic lamp is used to calibrate our manufactured convex grating imaging spectrometer which has Offner concentric structure and can supply high resolution and uniform spectral signal. Gaussian fitting algorithm is used to determine the center position and the Full-Width-Half-Maximum（FWHM）of the characteristic spectrum line. The central wavelengths and FWHMs of spectral pixels are calibrated by cubic polynomial fitting. By setting a fitting error thresh hold and abandoning the maximum deviation point, an optimization calculation is achieved. The integrated calibration experiment equipment for spectral calibration is developed to enhance calibration efficiency. The spectral calibration result comes from spectral lamp method are verified by monochromator wavelength scanning calibration technique. The result shows that spectral calibration uncertainty of FWHM and center wavelength are both less than 0.08nm, or 5.2% of spectral FWHM.
Polymer optical fiber bragg grating sensors
DEFF Research Database (Denmark)
Stefani, Alessio; Yuan, Scott Wu; Andresen, Søren
2010-01-01
Fiber-optical accelerometers based on polymer optical fiber Bragg gratings are reported. We have written fiber Bragg gratings for 1550 nm and 850 nm operations, characterized their temperature and strain response, and tested their performance in a prototype accelerometer....
High Efficiency Low Scatter Echelle Grating Project
National Aeronautics and Space Administration — A high efficiency low scatter echelle grating will be developed using a novel technique of multiple diamond shaving cuts. The grating will have mirror surfaces on...
Grating-Coupled Waveguide Cloaking
Institute of Scientific and Technical Information of China (English)
WANG Jia-Fu; QU Shao-Bo; XU Zhuo; MA Hua; WANG Cong-Min; XIA Song; WANG Xin-Hua; ZHOU Hang
2012-01-01
Based on the concept of a grating-coupled waveguide (GCW),a new strategy for realizing EM cloaking is presented.Using metallic grating,incident waves are firstly coupled into the effective waveguide and then decoupled into free space behind,enabling EM waves to pass around the obstacle.Phase compensation in the waveguide keeps the wave-front shape behind the obstacle unchanged.Circular,rectangular and triangular cloaks are presented to verify the robustness of the GCW cloaking.Electric field animations and radar cross section (RCS)comparisons convincingly demonstrate the cloaking effect.
Institute of Scientific and Technical Information of China (English)
Tan Xiaogang; Wei Ping; Li Liping
2009-01-01
To detect higher order polynomial phase signals (HOPPSs), the smoothed-pseudo polynomial Wigner-Ville distribution (SP-PWVD), an improved version of the polynomial Wigner-Ville distribution (PWVD), is pre-sented using a separable kernel. By adjusting the lengths of the functions in the kernel, the balance between resolution retaining and interference suppressing can be adjusted conveniently. The proposed method with merits of interference terms reduction and noise suppression can provide time frequency representation of better readability and more accurate instantaneous frequency (IF) estimation with higher order SP-PWVD. The performance of the SP-PWVD is verified by computer simulations.
Fabrication of Polymer Optical Fibre (POF Gratings
Directory of Open Access Journals (Sweden)
Yanhua Luo
2017-03-01
Full Text Available Gratings inscribed in polymer optical fibre (POF have attracted remarkable interest for many potential applications due to their distinctive properties. This paper overviews the current state of fabrication of POF gratings since their first demonstration in 1999. In particular we summarize and discuss POF materials, POF photosensitivity, techniques and issues of fabricating POF gratings, as well as various types of POF gratings.
A Composite Grating for Moire Interferometry.
1987-07-01
shown in Figure 7 in which two virtual reference gratings of frequencies 2400 and 600 lines/mm were used. This arrangement corresponds to a fringe...fields at the two virtual reference grating frequencies of 2400/600 lines/mm. The light paths of the two virtual reference gratings are controlled by...frequencies were selectively recorded. Figure 10 and 11 shows two moire fringe patterns for virtual reference grating frequencies of 2400 lines/mm and 600
Hybrid grating reflectors: Origin of ultrabroad stopband
DEFF Research Database (Denmark)
Park, Gyeong Cheol; Taghizadeh, Alireza; Chung, Il-Sug
2016-01-01
Hybrid grating (HG) reflectors with a high-refractive-index cap layer added onto a high contrast grating (HCG) provide a high reflectance close to 100% over a broader wavelength range than HCGs. The combination of a cap layer and a grating layer brings a strong Fabry-Perot (FP) resonance as well...
3D Printed Terahertz Focusing Grating Couplers
Jahn, David; Weidenbach, Marcel; Lehr, Jannik; Becker, Leonard; Beltrán-Mejía, Felipe; Busch, Stefan F.; Balzer, Jan C.; Koch, Martin
2017-02-01
We have designed, constructed and characterized a grating that focuses electromagnetic radiation at specific frequencies out of a dielectric waveguide. A simple theoretical model predicts the focusing behaviour of these chirped gratings, along with numerical results that support our assumptions and improved the grating geometry. The leaky waveguide was 3D printed and characterized at 120 GHz demonstrating its potential for manipulating terahertz waves.
The 6 Vertex Model and Schubert Polynomials
Directory of Open Access Journals (Sweden)
Alain Lascoux
2007-02-01
Full Text Available We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.
Dynamic system uncertainty propagation using polynomial chaos
Institute of Scientific and Technical Information of China (English)
Xiong Fenfen; Chen Shishi; Xiong Ying
2014-01-01
The classic polynomial chaos method (PCM), characterized as an intrusive methodology, has been applied to uncertainty propagation (UP) in many dynamic systems. However, the intrusive polynomial chaos method (IPCM) requires tedious modification of the governing equations, which might introduce errors and can be impractical. Alternative to IPCM, the non-intrusive polynomial chaos method (NIPCM) that avoids such modifications has been developed. In spite of the frequent application to dynamic problems, almost all the existing works about NIPCM for dynamic UP fail to elaborate the implementation process in a straightforward way, which is important to readers who are unfamiliar with the mathematics of the polynomial chaos theory. Meanwhile, very few works have compared NIPCM to IPCM in terms of their merits and applicability. Therefore, the mathematic procedure of dynamic UP via both methods considering parametric and initial condition uncertainties are comparatively discussed and studied in the present paper. Comparison of accuracy and efficiency in statistic moment estimation is made by applying the two methods to several dynamic UP problems. The relative merits of both approaches are discussed and summarized. The detailed description and insights gained with the two methods through this work are expected to be helpful to engineering designers in solving dynamic UP problems.
On the Schinzel Identity of Cyclotomic Polynomial
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
@@For integer n>0, let n(x) denote the nth cyclotomic polynomial n(x)=tackrel{01 be an odd square-free number.Aurifeuille and Le Lasseur［1］ proved thatequationn(x)=An2(x)-(-1)n-12)nxBn2(x).equation
Inverse polynomial reconstruction method in DCT domain
Dadkhahi, Hamid; Gotchev, Atanas; Egiazarian, Karen
2012-12-01
The discrete cosine transform (DCT) offers superior energy compaction properties for a large class of functions and has been employed as a standard tool in many signal and image processing applications. However, it suffers from spurious behavior in the vicinity of edge discontinuities in piecewise smooth signals. To leverage the sparse representation provided by the DCT, in this article, we derive a framework for the inverse polynomial reconstruction in the DCT expansion. It yields the expansion of a piecewise smooth signal in terms of polynomial coefficients, obtained from the DCT representation of the same signal. Taking advantage of this framework, we show that it is feasible to recover piecewise smooth signals from a relatively small number of DCT coefficients with high accuracy. Furthermore, automatic methods based on minimum description length principle and cross-validation are devised to select the polynomial orders, as a requirement of the inverse polynomial reconstruction method in practical applications. The developed framework can considerably enhance the performance of the DCT in sparse representation of piecewise smooth signals. Numerical results show that denoising and image approximation algorithms based on the proposed framework indicate significant improvements over wavelet counterparts for this class of signals.
A note on Fibonacci-type polynomials
Amdeberhan, Tewodros
2008-01-01
We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by $G_n=x^kG_{n-1}+G_{n-2}$. The special cases $k=1$ and $k=2$ are found in [4] and [7], respectively.
Algebraic polynomial system solving and applications
Bleylevens, I.W.M.
2010-01-01
The problem of computing the solutions of a system of multivariate polynomial equations can be approached by the Stetter-Möller matrix method which casts the problem into a large eigenvalue problem. This Stetter-Möller matrix method forms the starting point for the development of computational proce
Quantum Hilbert matrices and orthogonal polynomials
DEFF Research Database (Denmark)
Andersen, Jørgen Ellegaard; Berg, Christian
2009-01-01
Using the notion of quantum integers associated with a complex number q≠0 , we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q -Jacobi polynomials when |q|matrices...... of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix....
A recursive algorithm for Zernike polynomials
Davenport, J. W.
1982-01-01
The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.
The GCD property and irreduciable quadratic polynomials
Directory of Open Access Journals (Sweden)
Saroj Malik
1986-01-01
Full Text Available The proof of the following theorem is presented: If D is, respectively, a Krull domain, a Dedekind domain, or a Prüfer domain, then D is correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial in D[X] is a prime element.
The Tutte Polynomial of Some Matroids
Directory of Open Access Journals (Sweden)
Criel Merino
2012-01-01
graphs or matroids. In this work, we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were used to find the formulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.
Orthogonality Relations for Multivariate Krawtchouk Polynomials
Directory of Open Access Journals (Sweden)
Hiroshi Mizukawa
2011-02-01
Full Text Available The orthogonality relations of multivariate Krawtchouk polynomials are discussed. In case of two variables, the necessary and sufficient conditions of orthogonality is given by Grünbaum and Rahman in [SIGMA 6 (2010, 090, 12 pages]. In this study, a simple proof of the necessary and sufficient condition of orthogonality is given for a general case.
UNIQUENESS OF DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS
Institute of Scientific and Technical Information of China (English)
刘永; 祁晓光
2014-01-01
In this article, we investigate the uniqueness problems of difference polynomials of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Shibazaki. Some examples are given to show the results in this article are best possible.
Dynamic system uncertainty propagation using polynomial chaos
Directory of Open Access Journals (Sweden)
Xiong Fenfen
2014-10-01
Full Text Available The classic polynomial chaos method (PCM, characterized as an intrusive methodology, has been applied to uncertainty propagation (UP in many dynamic systems. However, the intrusive polynomial chaos method (IPCM requires tedious modification of the governing equations, which might introduce errors and can be impractical. Alternative to IPCM, the non-intrusive polynomial chaos method (NIPCM that avoids such modifications has been developed. In spite of the frequent application to dynamic problems, almost all the existing works about NIPCM for dynamic UP fail to elaborate the implementation process in a straightforward way, which is important to readers who are unfamiliar with the mathematics of the polynomial chaos theory. Meanwhile, very few works have compared NIPCM to IPCM in terms of their merits and applicability. Therefore, the mathematic procedure of dynamic UP via both methods considering parametric and initial condition uncertainties are comparatively discussed and studied in the present paper. Comparison of accuracy and efficiency in statistic moment estimation is made by applying the two methods to several dynamic UP problems. The relative merits of both approaches are discussed and summarized. The detailed description and insights gained with the two methods through this work are expected to be helpful to engineering designers in solving dynamic UP problems.
Scalar Field Theories with Polynomial Shift Symmetries
Griffin, Tom; Horava, Petr; Yan, Ziqi
2014-01-01
We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz type, focusing on scalar fields that can play the role of Nambu-Goldstone (NG) modes associated with spontaneous symmetry breaking. Such systems allow for an extension of the constant shift symmetry to a shift by a polynomial of degree $P$ in spatial coordinates. These "polynomial shift symmetries" in turn protect the technical naturalness of modes with a higher-order dispersion relation, and lead to a refinement of the proposed classification of infrared Gaussian fixed points available to describe NG modes in nonrelativistic theories. Generic interactions in such theories break the polynomial shift symmetry explicitly to the constant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomial shift symmetry of degree $P$, what are the lowest-dimension operators that preserve this symmetry, and deform the theory into a self-interacting scalar field theory with the shift symmetry of degree $P$? To answer this (essen...
Ideals in Polynomial Near-rings
Institute of Scientific and Technical Information of China (English)
Mark Farag
2002-01-01
In this paper, we further explore the relationship between the ideals of N and those of N[x], where N is a zero-symmetric right near-ring with identity and N[x] is the polynomial near-ring introduced by Bagley in 1993.
Optimization of Cubic Polynomial Functions without Calculus
Taylor, Ronald D., Jr.; Hansen, Ryan
2008-01-01
In algebra and precalculus courses, students are often asked to find extreme values of polynomial functions in the context of solving an applied problem; but without the notion of derivative, something is lost. Either the functions are reduced to quadratics, since students know the formula for the vertex of a parabola, or solutions are…
Polynomial Vector Fields in One Complex Variable
DEFF Research Database (Denmark)
Branner, Bodil
In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....
Piecewise polynomial representations of genomic tracks.
Tarabichi, Maxime; Detours, Vincent; Konopka, Tomasz
2012-01-01
Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/.
Polynomial Asymptotes of the Second Kind
Dobbs, David E.
2011-01-01
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…
Bernoulli Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2013-01-01
Fourier series for Bernoulli polynomials are used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent...
Euler Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2012-01-01
Fourier series for Euler polynomials is used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent series....
Polynomial Structure of Topological String Partition Functions
Zhou, Jie
2015-01-01
We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special K\\"ahler geometry and the ring of almost-holomorphic modular forms defined on modular curves.
Polynomial Eigenvalue Solutions to Minimal Problems in Computer Vision.
Kukelova, Zuzana; Bujnak, Martin; Pajdla, Tomas
2012-07-01
We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Gröbner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.
On an Inequality Concerning the Polar Derivative of a Polynomial
Indian Academy of Sciences (India)
A Aziz; N A Rather
2007-08-01
In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund’s inequality to the polar derivative of a polynomial.
d-Orthogonal Charlier Polynomials and the Weyl Algebra
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Centre de recherches mathematiques, Universite de Montreal, C.P. 6128, succ. Centre-ville, Montreal, Qc, H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: luc.vinet@umontreal.ca, E-mail: zhedanov@yahoo.com [Department of Electronic and Kinetic Properties, Donetsk Institute for Physics and Technology R.Luxemburg str. 72, Donetsk, 83114 (Ukraine)
2011-03-01
It is shown that d-orthogonal Charlier polynomials arise as matrix elements of non unitary automorphisms of the Weyl algebra. The structural formulas that these polynomials obey are derived from this algebraic setting.
Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays
Barry, Paul
2011-01-01
In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.
Irreducibility Results for Compositions of Polynomials in Several Variables
Indian Academy of Sciences (India)
Anca Iuliana Bonciocat; Alexandru Zaharescu
2005-05-01
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.
Representations of Knot Groups and Twisted Alexander Polynomials
Institute of Scientific and Technical Information of China (English)
Xiao Song LIN
2001-01-01
We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but differenttwisted Alexander polynomials are given.
Running gratings in photoconductive materials
DEFF Research Database (Denmark)
Kukhtarev, N. V.; Kukhtareva, T.; Lyuksyutov, S. F.
2005-01-01
gratings at small-contrast approximation and also are applicable for the description of space-charge wave domains. We discuss spatial domain and subharmonic beam formation in bismuth silicon oxide (BSO) crystals in the framework of the small-contrast approximation of STPM. The experimental results...
Field mapping of EEG by unbiased polynomial interpolation.
Ashida, H; Tatsuno, J; Okamoto, J; Maru, E
1984-06-01
The technique for field mapping of EEG using interpolation by an unbiased estimator of two-dimensional isotropic higher-order polynomial is developed. Assuming that the observed EEG contains noise, this method permits free positioning of the electrodes and does not reveal artificial oscillations as large as those of the Gibbs' phenomenon as in the case of the widely used Fourier interpolation. In the example of RMS (root mean square) amplitude at anesthesia, the onset of the fast activities began to appear clearly in the frontal and occipital area. By this method, not only RMS amplitude, but also instantaneous amplitude, such as visual evoked potential and somatosensory evoked potential, can be mapped.
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid
2015-06-07
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L
A 'missing' family of classical orthogonal polynomials
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Centre de Recherches Mathematiques, Universite de Montreal, PO Box 6128, Centre-ville Station, Montreal, Quebec, H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: zhedanov@fti.dn.ua [Donetsk Institute for Physics and Technology, Donetsk 83114 (Ukraine)
2011-02-25
We study a family of 'classical' orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = -1. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q = -1.
Calculation of thermal noise in grating reflectors
Heinert, Daniel; Friedrich, Daniel; Hild, Stefan; Kley, Ernst-Bernhard; Leavey, Sean; Martin, Iain W; Nawrodt, Ronny; Tünnermann, Andreas; Vyatchanin, Sergey P; Yamamoto, Kazuhiro
2013-01-01
Grating reflectors have been repeatedly discussed to improve the noise performance of metrological applications due to the reduction or absence of any coating material. So far, however, no quantitative estimate on the thermal noise of these reflective structures exists. In this work we present a theoretical calculation of a grating reflector's noise. We further apply it to a proposed 3rd generation gravitational wave detector. Depending on the grating geometry, the grating material and the temperature we obtain a thermal noise decrease by up to a factor of ten compared to conventional dielectric mirrors. Thus the use of grating reflectors can substantially improve the noise performance in metrological applications.
On the Lorentz degree of a product of polynomials
Ait-Haddou, Rachid
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.
q-Extensions for the Apostol Type Polynomials
Directory of Open Access Journals (Sweden)
Nazim I. Mahmudov
2014-01-01
Full Text Available The aim of this work is to introduce an extension for q-standard notations. The q-Apostol type polynomials and study some of their properties. Besides, some relations between the mentioned polynomials and some other known polynomials are obtained.
Symmetry identities for 2-variable Apostol type and related polynomials
2015-01-01
In this article, certain symmetry identities for the 2-variable Apostol type polynomials are derived. By taking suitable values of parameters and indices, the symmetry identities for the special cases of the 2-variable Apostol type polynomials are established. Further, the symmetry identities for certain members belonging to the 2-variable Apostol type polynomials are also considered.
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
The Roots of Adjoint Polynomial of the Graphs Contain Triangles
Institute of Scientific and Technical Information of China (English)
YECheng-fu
2004-01-01
We denote h(G,x) as the adjoint polynomial of graph G. In [5], Ma obtained the interpolation properties of the roots of adjoint polynomial of graphs containing triangles. By the properties, we prove the non-zero root of adjoint polynomial of Dn and Fn are single multiple.
Does the polynomial hierarchy collapse if onto functions are invertible?
H. Buhrman; L. Fortnow; M. Koucký; J.D. Rogers; N. Vereshchagin
2010-01-01
The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hi
T. Kim; Choi, J.; Kim, Y. H.; C. S. Ryoo
2010-01-01
In this paper, we give a fermionic p-adic integral representions of Benstein polynomials associated with Euler numbers and polynomials. Finally, we give some interesting identities for the Euler numbers by using the properties of our integral represention.
Grating light modulator for projection display
Institute of Scientific and Technical Information of China (English)
Jiyong Sun; Shanglian Huang; Jie Zhang; Zhihai Zhang; Yong Zhu
2009-01-01
A novel grating light modulator for projection display is introduced. It consists of an upper moveable grat-ing, a bottom mirror, and four supporting posts between them. The moveable grating and the bottom mir-ror compose a phase grating whose phase difference is controlled by the actuating voltage. When the phase difference is 2kπ, the grating light modulator will switch the incident light to zero-order diffraction; when the phase difference is (2k - 1)π, the grating light modulator will diffract light to first-order diffraction. A 16 × 16 modulator array is fabricated by the surface micromachining technology. The device works well when it is actuated by a voltage with 1-kHz frequency and 10-V amplitude. The fabricated grating light modulator can show blackness and brightness when controlled by the voltage. This modulator has potential applications in projection display system.
Curved VPH gratings for novel spectrographs
Clemens, J. Christopher; O'Donoghue, Darragh; Dunlap, Bart H.
2014-07-01
The introduction of volume phase holographic (VPH) gratings into astronomy over a decade ago opened new possibilities for instrument designers. In this paper we describe an extension of VPH grating technology that will have applications in astronomy and beyond: curved VPH gratings. These devices can disperse light while simultaneously correcting aberrations. We have designed and manufactured two different kinds of convex VPH grating prototypes for use in off-axis reflecting spectrographs. One type functions in transmission and the other in reflection, enabling Offnerstyle spectrographs with the high-efficiency and low-cost advantages of VPH gratings. We will discuss the design process and the tools required for modelling these gratings along with the recording layout and process steps required to fabricate them. We will present performance data for the first convex VPH grating produced for an astronomical spectrograph.
Alvarez-Fernandez, Carlos; Manas, Manuel
2009-01-01
We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a Riemann--Hilbert problem.
Soft x-ray blazed transmission grating spectrometer with high resolving power and extended bandpass
Heilmann, Ralf K.; Bruccoleri, Alexander Robert; Schattenburg, Mark
2016-04-01
A number of high priority questions in astrophysics can be addressed by a state-of-the-art soft x-ray grating spectrometer, such as the role of Active Galactic Nuclei in galaxy and star formation, characterization of the Warm-Hot Intergalactic Medium and the “missing baryon” problem, characterization of halos around the Milky Way and nearby galaxies, as well as stellar coronae and surrounding winds and disks. An Explorer-scale, large-area (> 1,000 cm2), high resolving power (R = λ/Δλ > 3,000) soft x-ray grating spectrometer is highly feasible based on Critical-Angle Transmission (CAT) grating technology. Still significantly higher performance can be provided by a CAT grating spectrometer on an X-ray-Surveyor-type mission. CAT gratings combine the advantages of blazed reflection gratings (high efficiency, use of higher diffraction orders) with those of conventional transmission gratings (low mass, relaxed alignment tolerances and temperature requirements, transparent at higher energies) with minimal mission resource requirements. They are high-efficiency blazed transmission gratings that consist of freestanding, ultra-high aspect-ratio grating bars fabricated from silicon-on-insulator (SOI) wafers using advanced anisotropic dry and wet etch techniques. Blazing is achieved through grazing-incidence reflection off the smooth grating bar sidewalls. The reflection properties of silicon are well matched to the soft x-ray band. Nevertheless, CAT gratings with sidewalls made of higher atomic number elements allow extension of the CAT grating principle to higher energies and larger dispersion angles. We show x-ray data from metal-coated CAT gratings and demonstrate efficient blazing to higher energies and larger blaze angles than possible with silicon alone. We also report on measurements of the resolving power of a breadboard CAT grating spectrometer consisting of a Wolter-I slumped-glass focusing mirror pair from Goddard Space Flight Center and CAT gratings, to be
High Resolution of the ECG Signal by Polynomial Approximation
Directory of Open Access Journals (Sweden)
G. Rozinaj
2006-04-01
Full Text Available Averaging techniques as temporal averaging and space averaging have been successfully used in many applications for attenuating interference [6], [7], [8], [9], [10]. In this paper we introduce interference removing of the ECG signal by polynomial approximation, with smoothing discrete dependencies, to make up for averaging methods. The method is suitable for low-level signals of the electrical activity of the heart often less than 10 m V. Most low-level signals arising from PR, ST and TP segments which can be detected eventually and their physiologic meaning can be appreciated. Of special importance for the diagnostic of the electrical activity of the heart is the activity bundle of His between P and R waveforms. We have established an artificial sine wave to ECG signal between P and R wave. The aim focus is to verify the smoothing method by polynomial approximation if the SNR (signal-to-noise ratio is negative (i.e. a signal is lower than noise.
Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.
Lombardo, Rosaria; Beh, Eric J; Kroonenberg, Pieter M
2016-06-01
The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display-the polynomial biplot.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group
Directory of Open Access Journals (Sweden)
Huiyuan Li
2012-10-01
Full Text Available The discrete Fourier analysis on the 30°-60°-90° triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G2, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of m-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.
Polynomial threshold functions and Boolean threshold circuits
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.
2013-01-01
of secondary interest. We show that PTFs on general Boolean domains are tightly connected to depth two threshold circuits. Our main results in regard to this connection are: PTFs of polynomial length and polynomial degree compute exactly the functions computed by THRMAJ circuits. An exponential length lower...... bound for PTFs that holds regardless of degree, thereby extending known lower bounds for THRMAJ circuits. We generalize two-party unbounded error communication complexity to the multi-party number-on-the-forehead setting, and show that communication lower bounds for 3-player protocols would yield size...... lower bounds for THRTHR circuits. We obtain several other results about PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length. We also consider a variant of PTFs over the max-plus algebra. We show that they are connected to PTFs over...
The Medusa Algorithm for Polynomial Matings
DEFF Research Database (Denmark)
Boyd, Suzanne Hruska; Henriksen, Christian
2012-01-01
The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image...... of its Julia set. Whether these approximations converge is answered using Thurston's topological characterization of rational maps. This algorithm was designed by John Hamal Hubbard, and implemented in 1998 by Christian Henriksen and REU students David Farris and Kuon Ju Liu. In this paper we describe...... the algorithm and its implementation, discuss some output from the program (including many pictures) and related questions. Specifically, we include images and a discussion for some shared matings, Lattès examples, and tuning sequences of matings....
Characteristic polynomials in real Ginibre ensembles
Energy Technology Data Exchange (ETDEWEB)
Akemann, G; Phillips, M J [Department of Mathematical Sciences and BURSt Research Centre, Brunel University West London, UB8 3PH Uxbridge (United Kingdom); Sommers, H-J [Fachbereich Physik, Universitaet Duisburg-Essen, 47048 Duisburg (Germany)], E-mail: Gernot.Akemann@brunel.ac.uk, E-mail: Michael.Phillips@brunel.ac.uk, E-mail: H.J.Sommers@uni-due.de
2009-01-09
We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are completely new in the chiral case. They hold for Gaussian ensembles which are partly symmetric, with kernels given in terms of Hermite and Laguerre polynomials respectively, depending on an asymmetry parameter. This allows us to interpolate between the maximally asymmetric real Ginibre and the Gaussian orthogonal ensemble, as well as their chiral counterparts. (fast track communication)
On Polynomial Sized MDP Succinct Policies
Liberatore, P
2011-01-01
Policies of Markov Decision Processes (MDPs) determine the next action to execute from the current state and, possibly, the history (the past states). When the number of states is large, succinct representations are often used to compactly represent both the MDPs and the policies in a reduced amount of space. In this paper, some problems related to the size of succinctly represented policies are analyzed. Namely, it is shown that some MDPs have policies that can only be represented in space super-polynomial in the size of the MDP, unless the polynomial hierarchy collapses. This fact motivates the study of the problem of deciding whether a given MDP has a policy of a given size and reward. Since some algorithms for MDPs work by finding a succinct representation of the value function, the problem of deciding the existence of a succinct representation of a value function of a given size and reward is also considered.
Parabolic refined invariants and Macdonald polynomials
Chuang, Wu-yen; Donagi, Ron; Pantev, Tony
2013-01-01
A string theoretic derivation is given for the conjecture of Hausel, Letellier, and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with G. Pan. Haiman's geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.
Quantum Hurwitz numbers and Macdonald polynomials
Harnad, J
2015-01-01
Parametric families in the centre ${\\bf Z}({\\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda $\\tau$-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of $S_n$ generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.
Eigenvalue conjecture and colored Alexander polynomials
Mironov, A
2016-01-01
We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\\cal R}_i = U_i{\\cal R}_1U_i^{-1} between the i-th and the first generators {\\cal R}_i of the braid group are universally expressible through the eigenvalues of {\\cal R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides a new support to the eigenvalue conjecture, especially for i>2, when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.
Polynomial interpolation methods for viscous flow calculations
Rubin, S. G.; Khosla, P. K.
1977-01-01
Higher-order collocation procedures which result in block-tridiagonal matrix systems are derived from (1) Taylor series expansions and from (2) polynomial interpolation, and the relationships between the two formulations, called respectively Hermite and spline collocation, are investigated. A Hermite block-tridiagonal system for a nonuniform mesh is derived, and the Hermite approach is extended in order to develop a variable-mesh sixth-order block-tridiagonal procedure. It is shown that all results obtained by Hermite development can be recovered by appropriate spline polynomial interpolation. The additional boundary conditions required for these higher-order procedures are also given. Comparative solutions using second-order accurate finite difference and spline and Hermite formulations are presented for the boundary layer on a flat plate, boundary layers with uniform and variable mass transfer, and the viscous incompressible Navier-Stokes equations describing flow in a driven cavity.
Tabulating knot polynomials for arborescent knots
Mironov, A; Morozov, An; Sleptsov, A; Ramadevi, P; Singh, Vivek Kumar
2016-01-01
Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it to arborescent knots in Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials.
Circular β ensembles, CMV representation, characteristic polynomials
Institute of Scientific and Technical Information of China (English)
SU ZhongGen
2009-01-01
In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle, where 0 > 0 is a model parameter. In the special cases β = 1,2 and 4, this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse matrix model, the CMV representation. This representation is new even in the case β = 2; and it has become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.
Effective grating theory for resonance domain surface-relief diffraction gratings.
Golub, Michael A; Friesem, Asher A
2005-06-01
An effective grating model, which generalizes effective-medium theory to the case of resonance domain surface-relief gratings, is presented. In addition to the zero order, it takes into account the first diffraction order, which obeys the Bragg condition. Modeling the surface-relief grating as an effective grating with two diffraction orders provides closed-form analytical relationships between efficiency and grating parameters. The aspect ratio, the grating period, and the required incidence angle that would lead to high diffraction efficiencies are predicted for TE and TM polarization and verified by rigorous numerical calculations.
Sparse DOA estimation with polynomial rooting
Xenaki, Angeliki; Gerstoft, Peter; Fernandez Grande, Efren
2015-01-01
Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve highresolution imaging. Utilizing the dual optimal variables of the CS optimization problem, it is shown with Monte Carlo simulations that the DOAs are accurately reconstructed through polynomial rooting (Root...
Real meromorphic functions and linear differential polynomials
Institute of Scientific and Technical Information of China (English)
LANGLEY; J.; K.
2010-01-01
We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.
A Deterministic and Polynomial Modified Perceptron Algorithm
Directory of Open Access Journals (Sweden)
Olof Barr
2006-01-01
Full Text Available We construct a modified perceptron algorithm that is deterministic, polynomial and also as fast as previous known algorithms. The algorithm runs in time O(mn3lognlog(1/ρ, where m is the number of examples, n the number of dimensions and ρ is approximately the size of the margin. We also construct a non-deterministic modified perceptron algorithm running in timeO(mn2lognlog(1/ρ.
Time-reversal symmetry and random polynomials
Braun, D; Kus, M.; Zyczkowski, K.
1996-01-01
We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. M...
Georeferencing CAMS data: Polynomial rectification and beyond
Yang, Xinghe
The Calibrated Airborne Multispectral Scanner (CAMS) is a sensor used in the commercial remote sensing program at NASA Stennis Space Center. In geographic applications of the CAMS data, accurate geometric rectification is essential for the analysis of the remotely sensed data and for the integration of the data into Geographic Information Systems (GIS). The commonly used rectification techniques such as the polynomial transformation and ortho rectification have been very successful in the field of remote sensing and GIS for most remote sensing data such as Landsat imagery, SPOT imagery and aerial photos. However, due to the geometric nature of the airborne line scanner which has high spatial frequency distortions, the polynomial model and the ortho rectification technique in current commercial software packages such as Erdas Imagine are not adequate for obtaining sufficient geometric accuracy. In this research, the geometric nature, especially the major distortions, of the CAMS data has been described. An analytical step-by-step geometric preprocessing has been utilized to deal with the potential high frequency distortions of the CAMS data. A generic sensor-independent photogrammetric model has been developed for the ortho-rectification of the CAMS data. Three generalized kernel classes and directional elliptical basis have been formulated into a rectification model of summation of multisurface functions, which is a significant extension to the traditional radial basis functions. The preprocessing mechanism has been fully incorporated into the polynomial, the triangle-based finite element analysis as well as the summation of multisurface functions. While the multisurface functions and the finite element analysis have the characteristics of localization, piecewise logic has been applied to the polynomial and photogrammetric methods, which can produce significant accuracy improvement over the global approach. A software module has been implemented with full
Normality and shared values concerning differential polynomials
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let F be a family of functions meromorphic in a domain D, let P be a polynomial with either deg P≥3 or deg P = 2 and P having only one distinct zero, and let b be a finite nonzero complex number. If, each pair of functions f and g in F, P (f)f and P (g)g share b in D, then F is normal in D.
Completeness of the ring of polynomials
DEFF Research Database (Denmark)
Thorup, Anders
2015-01-01
Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥2 variables over an uncountable field k. We prove that R is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we pro...
Piecewise polynomial solutions to linear inverse problems
DEFF Research Database (Denmark)
Hansen, Per Christian; Mosegaard, K.
1996-01-01
We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....
On form factors and Macdonald polynomials
Lashkevich, Michael
2013-01-01
We are developing the algebraic construction for form factors of local operators in the sinh-Gordon theory proposed in [B.Feigin, M.Lashkeivch, 2008]. We show that the operators corresponding to the null vectors in this construction are given by the degenerate Macdonald polynomials with rectangular partitions and the parameters $t=-q$ on the unit circle. We obtain an integral representation for the null vectors and discuss its simple applications.
Moments, positive polynomials and their applications
Lasserre, Jean Bernard
2009-01-01
Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones,
Linear algebra for skew-polynomial matrices
Abramov, Sergei; Bronstein, Manuel
2002-01-01
We describe an algorithm for transforming skew-polynomial matrices over an Ore domain in row-reduced form, and show that this algorithm can be used to perform the standard calculations of linear algebra on such matrices (ranks, kernels, linear dependences, inhomogeneous solving). The main application of our algorithm is to desingularize recurrences and to compute the rational solutions of a large class of linear functional systems. It also turns out to be efficient when applied to ordinary co...
Polynomial approximation of functions in Sobolev spaces
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
Some Orthogonal Polynomials in Four Variables
Directory of Open Access Journals (Sweden)
Charles F. Dunkl
2008-11-01
Full Text Available The symmetric group on 4 letters has the reflection group $D_3$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_3$ and $D_3$. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero-Sutherland model of 4 identical particles on the line.
Algebraic polynomials and moments of stochastic integrals
Langovoy, Mikhail A
2011-01-01
We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.
A dynamic coefficient polynomial predistorter based on direct learning architecture
Institute of Scientific and Technical Information of China (English)
Li Bo; Ge Jianhua; Ai Bo
2008-01-01
A dynamic coefficient polynomial predistorter based on direct learning architecture is proposed. Compared to the existing polynomial predistorter, on the one hand, the proposed predistorter based on the direct learning architecture is more robust to initial conditions of the tap coefficients than that based on indirect learning architecture; on the other hand, by using two polynomial coefficient combinations, different polynomial coefficient combination can be selected when the input signal amplitude changes, which effectively decreases the estimate error. This paper introduces the direct learning architecture and gives the dynamic coefficient polynomial expression. A simplified nonlinear recursive least-squares (RLS) algorithm for polynomial coefficient estimation is also derived in detail. Computer simulations show that the proposed predistorter can attain 31dB, 28dB and 40dB spectrum suppression gain when our method is applied to the traveling wave tube amplifier (TWTA), solid state power amplifier (SSPA) and polynomial power amplifier (PA) model, respectively.
Characteristic polynomials of pseudo-Anosov maps
Birman, Joan; Kawamuro, Keiko
2010-01-01
We study the relationship between three different approaches to the action of a pseudo-Anosov mapping class $[F]$ on a surface: the original theorem of Thurston, its algorithmic proof by Bestvina-Handel, and related investigations of Penner-Harer. Bestvina and Handel represent $[F]$ as a suitably chosen homotopy equivalence $f: G\\to G$ of a finite graph, with an associated transition matrix $T$ whose largest eigenvalue is the dilatation of $[F]$. Extending a skew-symmetric form introduced by Penner and Harer to the setting of Bestvina and Handel, we show that the characteristic polynomial of $T$ is a monic and palindromic or anti-palindromic polynomial, possibly multiplied by a power of $x$. Moreover, it factors as a product of three polynomials. One of them reflects the action of $[F]$ on a certain symplectic space, the second one relates to the degeneracies of the skew-symmetric form, and the third one reflects the restriction of $f$ to the vertices of $G$. We give an application to the problem of deciding ...
Simplified Storm Surge Simulations Using Bernstein Polynomials
Beisiegel, Nicole; Behrens, Jörn
2016-04-01
Storm surge simulations are vital for forecasting, hazard assessment and eventually improving our understanding of Earth system processes. Discontinuous Galerkin (DG) methods have recently been explored in that context, because they are locally mass-conservative and in combination with suitable robust nodal filtering techniques (slope limiters) positivity-preserving and well-balanced for the still water state at rest. These filters manipulate interpolation point values in every time step in order to retain the desirable properties of the scheme. In particular, DG methods are able to represent prognostic variables such as the fluid height at high-order accuracy inside each element (triangle). For simulations that include wetting and drying, however, the high-order accuracy will destabilize the numerical model because point values on quadrature points may become negative during the computation if they do not coincide with interpolation points. This is why the model that we are presenting utilizes Bernstein polynomials as basis functions to model the wetting and drying. This has the advantage that negative pointvalues away from interpolation points are prevented, the model is stabilized and no additional time step restriction is introduced. Numerical tests show that the model is capable of simulating simplified storm surges. Furthermore, a comparison of model results with third-order Bernstein polynomials with results using traditional nodal Lagrange polynomials reveals an improvement in numerical convergence.
Deterministic Polynomial Factoring and Association Schemes
Arora, Manuel; Karpinski, Marek; Saxena, Nitin
2012-01-01
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n-1) has a `large' r-smooth divisor s, then we find a nontrivial factor of f(x) in deterministic poly(n^r,log q) time; assuming GRH and that s > sqrt{n/(2^r)}. Thus, for r = O(1) our algorithm is polynomial time. Further, for r > loglog n there are infinitely many prime degrees n for which our algorithm is applicable and better than the best known; assuming GRH. Our methods build on the algebraic-combinatorial framework of m-schemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the m-scheme on n points, implicitly appearing in our factoring algorithm, has an exceptional structure; leading us to the improved time ...
The bivariate Rogers Szegö polynomials
Chen, William Y. C.; Saad, Husam L.; Sun, Lisa H.
2007-06-01
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials hn(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y|q) involves a 3phi2 sum and the Rogers formula involves a 2phi1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegö polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
The bivariate Rogers-Szegoe polynomials
Energy Technology Data Exchange (ETDEWEB)
Chen, William Y C [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China); Saad, Husam L [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China); Sun, Lisa H [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China)
2007-06-08
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegoe polynomials h{sub n}(x, y vertical bar q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials H{sub n}(x; a vertical bar q) due to Askey, Rahman and Suslov. Mehler's formula for h{sub n}(x, y vertical bar q) involves a {sub 3}{phi}{sub 2} sum and the Rogers formula involves a {sub 2}{phi}{sub 1} sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegoe polynomials h{sub n}(x vertical bar q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for h{sub n}(x, y vertical bar q). Finally, we give a change of base formula for H{sub n}(x; a vertical bar q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
Nested Canalyzing, Unate Cascade, and Polynomial Functions.
Jarrah, Abdul Salam; Raposa, Blessilda; Laubenbacher, Reinhard
2007-09-15
This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.
Heilmann, Ralf K.; Bruccoleri, Alexander R.; Kolodziejczak, Jeffery; Gaskin, Jessica A.; O'Dell, Stephen L.; Bhatia, Ritwik; Schattenburg, Mark L.
2016-07-01
A number of high priority subjects in astrophysics can be addressed by a state-of-the-art soft x-ray grating spectrometer, such as the role of Active Galactic Nuclei in galaxy and star formation, characterization of the Warm-Hot Intergalactic Medium and the missing baryon problem, characterization of halos around the Milky Way and nearby galaxies, as well as stellar coronae and surrounding winds and disks. An Explorer-scale, largearea (> 1,000 cm2), high resolving power (R =λ/Δλ > 3,000) soft x-ray grating spectrometer is highly feasible based on Critical-Angle Transmission (CAT) grating technology, even for telescopes with angular resolution of 5-10 arcsec. Still, significantly higher performance can be provided by a CAT grating spectrometer on an X-ray- Surveyor-type mission. CAT gratings combine the advantages of blazed reflection gratings (high efficiency, use of higher diffraction orders) with those of conventional transmission gratings (lowmass, relaxed alignment tolerances and temperature requirements, transparent at higher energies) with minimalmission resource requirements. They are high-efficiency blazed transmission gratings that consist of freestanding, ultra-high aspect-ratio grating bars fabricated from silicon-on-insulator (SOI) wafers using advanced anisotropic dry and wet etch techniques. Blazing is achieved through grazing-incidence reflection off the smooth grating bar sidewalls. The reflection properties of silicon are well matched to the soft x-ray band, and existing silicon CAT gratings can exceed 30% absolute diffraction efficiency, with clear paths for further improvement. Nevertheless, CAT gratings with sidewalls made of higher atomic number elements allow extension of the CAT grating principle to higher energies and larger dispersion angles, thus enabling higher resolving power at shorter wavelengths. We show x-ray data from CAT gratings coated with a thin layer of platinum using atomic layer deposition, and demonstrate efficient
Directory of Open Access Journals (Sweden)
Michael Basin
2011-04-01
Full Text Available In this paper, the mean-square filtering problem for polynomial system states confused with white Poisson noises over polynomial observations is studied proceeding from the general expression for the stochastic Ito differentials of the mean-square estimate and the error variance. In contrast to the previously obtained results, the paper deals with the general case of nonlinear polynomial states and observations with white Poisson noises. As a result, the Ito differentials for the mean-square estimate and error variance corresponding to the stated filtering problem are first derived. The procedure for obtaining an approximate closed-form finite-dimensional system of the filtering equations for any polynomial state over observations with any polynomial drift is then established. In the example, the obtained closed-form filter is applied to solve the third order sensor filtering problem for a quadratic state, assuming a conditionally Poisson initial condition for the extended third order state vector. The simulation results show that the designed filter yields a reliable and rapidly converging estimate.
Toan Dang, Phuc; Nguyen, Truong Khang; Le, Khai Q.
2017-01-01
We revisit previous studies of metallic gratings for optical absorption enhancement in an organic solar cell with a thin active layer. Our device geometry is designed for a real solar cell with full of functional layers. Various metallic gratings calibrated to generate periodic scatterers and low reflectors for broadband light account for increases in short circuit current density of up to 47% when compared to its flat counterpart. We found that the tapered grating has greater performance than the regular rectangular grating for transverse magnetic (TM) polarization while the latter shows better performance for transverse electric (TE) polarization. The overall metallic grating induced absorption enhancement was found at all angles of incidence. The best configuration was realized for the tapered grating-based solar cell at 25° of inclination.
The spectral combination characteristic of grating and the bi-grating diffraction imaging effect
Institute of Scientific and Technical Information of China (English)
ZHANG WeiPing; HE XiaoRong
2007-01-01
This paper reports on a new property of grating,namely spectral combination,and on bi-grating diffraction imaging that is based on spectral combination.The spectral combination characteristic of a grating is the capability of combining multiple light beams of different wavelengths incident from specific angles into a single beam.The bi-grating diffraction imaging is the formation of the image of an object with two gratings: the first grating disperses the multi-color light beams from the object and the second combines the dispersed light beams to form the image.We gave the conditions necessary for obtaining the spectral combination.We also presented the equations that relate the two gratings' spatial frequencies,diffraction orders and positions necessary for obtaining the bi-grating diffraction imaging.
Monolithic integrated optic fiber Bragg grating sensor interrogator
Mendoza, Edgar A.; Esterkin, Yan; Kempen, Cornelia; Sun, Songjian
2010-04-01
Fiber Bragg gratings (FBGs) are a mature sensing technology that has gained rapid acceptance in civil, aerospace, chemical and petrochemical, medicine, aviation and automotive industries. Fiber Bragg grating sensors can be use for a variety of measurements including strain, stress, vibration, acoustics, acceleration, pressure, temperature, moisture, and corrosion distributed at multiple locations within the structure using a single fiber element. The most prominent advantages of FBGs are: small size and light weight, multiple FBG transducers on a single fiber, and immunity to radio frequency interference. A major disadvantage of FBG technology is that conventional state-of-the-art fiber Bragg grating interrogation systems are typically bulky, heavy, and costly bench top instruments that are assembled from off-the-shelf fiber optic and optical components integrated with a signal electronics board into an instrument console. Based on the need for a compact FBG interrogation system, this paper describes recent progress towards the development of a miniature fiber Bragg grating sensor interrogator (FBG-TransceiverTM) system based on multi-channel monolithic integrated optic sensor microchip technology. The integrated optic microchip technology enables the monolithic integration of all of the functionalities, both passive and active, of conventional bench top FBG sensor interrogators systems, packaged in a miniaturized, low power operation, 2-cm x 5-cm small form factor (SFF) package suitable for the long-term structural health monitoring in applications where size, weight, and power are critical for operation.
Graphene-ribbon-coupled tunable enhanced transmission through metallic grating
Peng, Yu-Xiang; He, Meng-Dong; Li, Ze-Jun; Wang, Kai-Jun; Li, Shui; Li, Jian-Bo; Liu, Jian-Qiang; Long, Mengqiu; Hu, Wei-Da; Chen, Xiaoshuang
2017-01-01
We report the tunable enhanced transmission of light through a hybrid metal-graphene structure, in which a graphene ribbon array is situated over a metallic grating. The graphene ribbon is employed to make the graphene-insulator-metal waveguide of finite length as a Fabry-Perot (F-P) cavity. When the slit of metallic grating is opened at the position with a maximal magnetic field in F-P resonant cavity, the transmission of light through metallic grating is greatly enhanced since the strongly localized magnetic field is effectively coupled to the slit. The transmission spectrum and the enhancement factor can be adjusted by changing geometrical parameters including the width and the length of slit, the width of graphene ribbon and the period of metallic grating. The transmission peaks exhibit a broad tuning range with a small change in the Fermi energy level of graphene. Moreover, the enhancement factor of transmission peak can be manipulated by the Fermi energy level and the carrier mobility of graphene, and an enhancement factor of 154 is obtained. The findings expand our understanding of hybrid metal-graphene plasmons and have potential applications in building active plasmonic devices.
Time-reversal symmetry and random polynomials
Braun, D; Zyczkowski, K
1996-01-01
We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. Moreover, a class of time-reverse invariant quantum systems is shown, for which spectra display fluctuations characteristic of orthogonal ensemble, while eigenvectors confer to predictions of unitary ensemble.
Conditional Density Approximations with Mixtures of Polynomials
DEFF Research Database (Denmark)
Varando, Gherardo; López-Cruz, Pedro L.; Nielsen, Thomas Dyhre
2015-01-01
Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce...... is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated...
A Matricial Algorithm for Polynomial Refinement
King, Emily J
2011-01-01
In order to have a multiresolution analysis, the scaling function must be refinable. That is, it must be the linear combination of 2-dilation, $\\mathbb{Z}$-translates of itself. Refinable functions used in connection with wavelets are typically compactly supported. In 2002, David Larson posed the question, "Are all polynomials (of a single variable) finitely refinable?" That summer the author proved that the answer indeed was true using basic linear algebra. The result was presented in a number of talks but had not been typed up until now. The purpose of this short note is to record that particular proof.
Polynomial identities for ternary intermolecular recombination
Bremner, Murray R
2010-01-01
The operation of binary intermolecular recombination, originating in the theory of DNA computing, permits a natural generalization to n-ary operations which perform simultaneous recombination of n molecules. In the case n = 3, we use computer algebra to determine the polynomial identities of degree <= 9 satisfied by this trilinear nonassociative operation. Our approach requires computing a basis for the nullspace of a large integer matrix, and for this we compare two methods: (i) the row canonical form, and (ii) the Hermite normal form with lattice basis reduction. In the conclusion, we formulate some conjectures for the general case of n-ary intermolecular recombination.
Pade interpolation by F-polynomials and transfinite diameter
Coman, Dan
2011-01-01
We define $F$-polynomials as linear combinations of dilations by some frequencies of an entire function $F$. In this paper we use Pade interpolation of holomorphic functions in the unit disk by $F$-polynomials to obtain explicitly approximating $F$-polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set $K\\subset\\mathbb C$ then optimal choices for the frequencies of interpolating polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of $K$. In case of the Laplace transforms of measures on $K$, we show that the coefficients of interpolating polynomials stay bounded provided that the frequencies are Fekete points. Finally, we give a sufficient condition for measures on the unit circle which ensures that the sums of the absolute values of the coefficients of interpolating polynomials stay bounded.
Guts of surfaces and the colored Jones polynomial
Futer, David; Purcell, Jessica S
2011-01-01
This work derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A-adequacy), we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and only if a particular coefficient vanishes. Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our approach is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses (A-adequacy), we show that the checkerboard knot su...
Computing the Tutte Polynomial in Vertex-Exponential Time
DEFF Research Database (Denmark)
Björklund, Andreas; Husfeldt, Thore; Kaski, Petteri;
2008-01-01
The deletion–contraction algorithm is perhapsthe most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models...... algorithm that computes the Tutte polynomial—and hence, all the aforementioned invariants and more—of an arbitrary graph in time within a polynomial factor of the number of connected vertex sets. The algorithm actually evaluates a multivariate generalization of the Tutte polynomial by making use...... of an identity due to Fortuin and Kasteleyn. We also provide a polynomial-space variant of the algorithm and give an analogous result for Chung and Graham's cover polynomial....
On the Tutte-Krushkal-Renardy polynomial for cell complexes
Bajo, Carlos; Chmutov, Sergei
2012-01-01
Recently V.Krushkal and D.Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A.Duval, C.Klivans, and J.Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decomposition of a sphere, this modified polynomial satisfies the same duality identity as before. We find that evaluating the Tutte-Krushkal-Renardy along a certain line is the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial.
Multiple Twisted -Euler Numbers and Polynomials Associated with -Adic -Integrals
Directory of Open Access Journals (Sweden)
Jang Lee-Chae
2008-01-01
Full Text Available By using -adic -integrals on , we define multiple twisted -Euler numbers and polynomials. We also find Witt's type formula for multiple twisted -Euler numbers and discuss some characterizations of multiple twisted -Euler Zeta functions. In particular, we construct multiple twisted Barnes' type -Euler polynomials and multiple twisted Barnes' type -Euler Zeta functions. Finally, we define multiple twisted Dirichlet's type -Euler numbers and polynomials, and give Witt's type formula for them.
Representations of the Schroedinger group and matrix orthogonal polynomials
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Centre de recherches mathematiques, Universite de Montreal, CP 6128, succ. Centre-ville, Montreal, QC H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: luc.vinet@umontreal.ca, E-mail: zhedanov@fti.dn.ua [Donetsk Institute for Physics and Technology, Donetsk 83114 (Ukraine)
2011-09-02
The representations of the Schroedinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states. These representations are seen to involve matrix orthogonal polynomials in a discrete variable that have Charlier and Meixner polynomials as building blocks. The underlying Lie-theoretic framework allows for a systematic derivation of the structural formulas (recurrence relations, difference equations, Rodrigues' formula, etc) that these matrix orthogonal polynomials satisfy. (paper)
THE λ-GR(O)BNER BASES UNDER POLYNOMIAL COMPOSITION
Institute of Scientific and Technical Information of China (English)
Jinwang LIU; Dongmei LI; Xiaosong CHEN
2007-01-01
Polynomial composition is the operation of replacing variables in a polynomial with other polynomials. λ-Gr(o)bner basis is an especial Gr(o)bner basis. The main problem in the paper is: when does composition commute with λ-Gr(o)bner basis computation? We shall answer better the above question. This has a natural application in the computation of λ-Gr(o)bner bases.
Chromatic polynomials of graphs from Kac-Moody algebras
Venkatesh, R.; Viswanath, Sankaran
2013-01-01
We give a new interpretation of the chromatic polynomial of a simple graph G in terms of the Kac-Moody Lie algebra with Dynkin diagram G. We show that the chromatic polynomial is essentially the q-Kostant partition function of this Lie algebra evaluated on the sum of the simple roots. Applying the Peterson recurrence formula for root multiplicities, we obtain a new realization of the chromatic polynomial as a weighted sum of paths in the bond lattice of G.
On an Inequality of Pual Turan Concerning Polynomials-II
Institute of Scientific and Technical Information of China (English)
Abdullah Mir∗
2015-01-01
Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z)=nP(z)+(α−z)P′(z) denote the polar derivative of the polynomial P(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a poly-nomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.
Darboux polynomials for Lotka-Volterra systems in three dimensions
Christodoulides, Yiannis T
2008-01-01
We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding cofactors. More precisely, we show that a Darboux polynomial of degree greater than one is reducible. In fact, it is a product of linear Darboux polynomials and first integrals.
Some Results on the Polar Derivative of a Polynomial
Institute of Scientific and Technical Information of China (English)
Abdullah Mir; Bilal Dar
2014-01-01
Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z)=nP(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect toα. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.
Univariate Niho Bent Functions from o-Polynomials
Budaghyan, Lilya; Kholosha, Alexander; Carlet, Claude; Helleseth, Tor
2014-01-01
In this paper, we discover that any univariate Niho bent function is a sum of functions having the form of Leander-Kholosha bent functions with extra coefficients of the power terms. This allows immediately, knowing the terms of an o-polynomial, to obtain the powers of the additive terms in the polynomial representing corresponding bent function. However, the coefficients are calculated ambiguously. The explicit form is given for the bent functions obtained from quadratic and cubic o-polynomi...
Local polynomial Whittle estimation covering non-stationary fractional processes
DEFF Research Database (Denmark)
Nielsen, Frank
This paper extends the local polynomial Whittle estimator of Andrews & Sun (2004) to fractionally integrated processes covering stationary and non-stationary regions. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the local polynomial Whittle estimator ...... study illustrates the performance of the proposed estimator compared to the classical local Whittle estimator and the local polynomial Whittle estimator. The empirical justi.cation of the proposed estimator is shown through an analysis of credit spreads....
Multivariate polynomial interpolation on Lissajous-Chebyshev nodes
Dencker, Peter; Erb, Wolfgang
2015-01-01
In this article, we study multivariate polynomial interpolation and quadrature rules on non-tensor product node sets related to Lissajous curves and Chebyshev varieties. After classifying multivariate Lissajous curves and the interpolation nodes linked to these curves, we derive a discrete orthogonality structure on these node sets. Using this orthogonality structure, we obtain unique polynomial interpolation in appropriately defined spaces of multivariate Chebyshev polynomials. Our results g...
Conventional modeling of the multilayer perceptron using polynomial basis functions
Chen, Mu-Song; Manry, Michael T.
1993-01-01
A technique for modeling the multilayer perceptron (MLP) neural network, in which input and hidden units are represented by polynomial basis functions (PBFs), is presented. The MLP output is expressed as a linear combination of the PBFs and can therefore be expressed as a polynomial function of its inputs. Thus, the MLP is isomorphic to conventional polynomial discriminant classifiers or Volterra filters. The modeling technique was successfully applied to several trained MLP networks.
Return times of polynomials as meta-Fibonacci numbers
Emerson, Nathaniel D.
2004-01-01
We consider generalized closest return times of a complex polynomial of degree at least two. Most previous studies on this subject have focused on the properties of polynomials with particular return times, especially the Fibonacci numbers. We study the general form of these closest return times. The main result of this paper is that these closest return times are meta-Fibonacci numbers. This result applies to the return times of a principal nest of a polynomial. Furthermore, we show that an ...
Polynomial analysis of canopy spectra and biochemical component content inversion
Institute of Scientific and Technical Information of China (English)
YAN Chunyan; LIU Qiang; NIU Zheng; WANG Jihua; HUANG Wenjiang; LIU Liangyun
2005-01-01
A polynomial expression model was developed in this paper to describe directional canopy spectra, and the decomposition of the polynomial expression was used as a tool for retrieving biochemical component content from canopy multi-angle spectra. First, the basic formula of the polynomial expression was introduced and the physical meaning of its terms and coefficients was discussed. Based on this analysis, a complete polynomial expression model and its decomposition method were given. By decomposing the canopy spectra simulated with SAILH model, it shows that the polynomial expression can not only fit well the canopy spectra, but also show the contribution of every order scattering to the whole reflectance. Taking the first scattering coefficients a10 and a01 for example, the test results show that the polynomial coefficients reflect very well the hot spot phenomenon and the effects of viewing angles, LAI and leaf inclination angle on canopy spectra. By coupling the polynomial expression with leaf model PROSPECT, a canopy biochemical component content inversion model was given. In the simulated test, the canopy multi-angle spectra were simulated by two different models, SAILH and 4-SCALE respectively, then the biochemical component content was retrieved by inverting the coupled polynomial expression + PROSPECT model. Results of the simulated test are promising, and when applying the algorithm to measured corn canopy multi-angle spectra, we also get relatively accurate chlorophyll content. It shows that the polynomial analysis provides a new method to get biochemical component content independent of any specific canopy model.
A Bivariate Analogue to the Composed Product of Polynomials
Institute of Scientific and Technical Information of China (English)
Donald Mills; Kent M. Neuerburg
2003-01-01
The concept of a composed product for univariate polynomials has been explored extensively by Brawley, Brown, Carlitz, Gao,Mills, et al. Starting with these fundamental ideas andutilizing fractional power series representation(in particular, the Puiseux expansion) of bivariate polynomials, we generalize the univariate results. We define a bivariate composed sum,composed multiplication,and composed product (based on function composition). Further, we investigate the algebraic structure of certain classes of bivariate polynomials under these operations. We also generalize a result of Brawley and Carlitz concerningthe decomposition of polynomials into irreducibles.
Pseudorandom Numbers and Hash Functions from Iterations of Multivariate Polynomials
Ostafe, Alina
2009-01-01
Dynamical systems generated by iterations of multivariate polynomials with slow degree growth have proved to admit good estimates of exponential sums along their orbits which in turn lead to rather stronger bounds on the discrepancy for pseudorandom vectors generated by these iterations. Here we add new arguments to our original approach and also extend some of our recent constructions and results to more general orbits of polynomial iterations which may involve distinct polynomials as well. Using this construction we design a new class of hash functions from iterations of polynomials and use our estimates to motivate their "mixing" properties.
Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems
Liu, Jiang; Zhao, Hengjun
2011-01-01
The notion of Lyapunov function plays a key role in design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives of certain functions along the system's vector field. Furthermore, we present a complete method to automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is complete in the sense that it is able to discover all polynomial RLFs by enumerating all polynomial templates for any PDS.
An Analytic Formula for the A_2 Jack Polynomials
Directory of Open Access Journals (Sweden)
Vladimir V. Mangazeev
2007-01-01
Full Text Available In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003, 451-482] on separation of variables (SoV for the $A_n$ Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995, 27-34] where the integral representations for the $A_2$ Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the $A_2$ Jack polynomials in terms of generalised hypergeometric functions.
ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS
Institute of Scientific and Technical Information of China (English)
ZHU Xiao-feng; LI Xiu-chun
2006-01-01
Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.
Multi-indexed Jacobi polynomials and Maya diagrams
Takemura, Kouichi
2014-11-01
Multi-indexed Jacobi polynomials are defined by the Wronskian of four types of eigenfunctions of the Pöschl-Teller Hamiltonian. We give a correspondence between multi-indexed Jacobi polynomials and pairs of Maya diagrams, and we show that any multi-indexed Jacobi polynomial is essentially equal to some multi-indexed Jacobi polynomial of two types of eigenfunction. As an application, we show a Wronskian-type formula of some special eigenstates of the deformed Pöschl-Teller Hamiltonian.
Casoratian Identities for the Wilson and Askey-Wilson Polynomials
Odake, Satoru
2013-01-01
Infinitely many Casoratian identities are derived for the Wilson and Askey-Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical systems. Similar identities hold for various reduced form polynomials of the Wilson and Askey-Wilson polynomials, e.g. the continuous q-Jacobi, continuous (dual) (q-)Hahn, Meixner-Pollaczek, Al-Salam-Chihara, continuous (big) q-Hermite, etc.
Structure relations for monic orthogonal polynomials in two discrete variables
Rodal, J.; Area, I.; Godoy, E.
2008-04-01
In this paper, extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given, by using an appropriate vector-matrix notation. Three-term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type. Structure relations, difference representations as well as lowering and raising operators are obtained. Finally, expressions for all matrix coefficients appearing in these finite-type relations are explicitly presented for a finite set of Hahn and Kravchuk orthogonal polynomials.
The distribution of zeros of general q-polynomials
Álvarez-Nodarse, R.; Buendía, E.; Dehesa, J. S.
1997-10-01
A general system of q-orthogonal polynomials is defined by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analogue of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation. This result is applied to specific classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey and Wilson, Al Salam - Carlitz and the celebrated little and big q-Jacobi.
Unimodularity of zeros of self-inversive polynomials
Lalin, Matilde N
2012-01-01
We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang and Lal\\'in and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lal\\'in and Rogers on computational evidence.
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2017-01-01
This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The programming environment has also evolved, and the books examples are changed accordingly. The applications section is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semi-definite programming form, ready to be solved with algorithms freely available, like those from the libraries SeDuMi, CVX and Pos3Poly. A new chapter discusses applications in super-resolution theory, where Bounded Real Lemma for trigonometric polynomials is an important tool. This revision is written to be more appealing and easier to use for new readers. < Features updated information on LMI...
Orthogonal Polynomials and $S$-curves
Rakhmanov, E A
2011-01-01
This paper is devoted to a study of $S$-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property ($S$-property). Such curves have many applications. In particular, they play a fundamental role in the theory of complex (non-hermitian) orthogonal polynomials. One of the main theorems on zero distribution of such polynomials asserts that the limit zero distribution is presented by an equilibrium measure of an $S$-curve associated with the problem if such a curve exists. These curves are also the starting point of the matrix Riemann-Hilbert approach to srtong asymptotics. Other approaches to the problem of strong asymptotics (differential equations, Riemann surfaces) are also related to $S$-curves or may be interpreted this way. Existence problem $S$-curve in a given class of curves in presence of a nontrivial external field presents certain challenge. We formulate and prove a version of existence theorem for the case whe...
On Factorization of Generalized Macdonald Polynomials
Kononov, Ya
2016-01-01
A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from $W_\\infty$ -- is that they factorize at the peculiar two-parametric topological locus in the space of time-variables, what is known as the hook formula for quantum dimensions of representations of $U_q(SL_N)$ and plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMP), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time-variables, we discover a weak factorization -- on a one- (rather than four-) parametric slice of the topological locus, what is already a very non-trivial property, calling for proof and better understanding.
On factorization of generalized Macdonald polynomials
Energy Technology Data Exchange (ETDEWEB)
Kononov, Ya. [Landau Institute for Theoretical Physics, Chernogolovka (Russian Federation); HSE, Math Department, Moscow (Russian Federation); Morozov, A. [ITEP, Moscow (Russian Federation); Institute for Information Transmission Problems, Moscow (Russian Federation); National Research Nuclear University MEPhI, Moscow (Russian Federation)
2016-08-15
A remarkable feature of Schur functions - the common eigenfunctions of cut-and-join operators from W{sub ∞} - is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U{sub q}(SL{sub N}) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorization - on a one- (rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding. (orig.)
Tabulating knot polynomials for arborescent knots
Mironov, A.; Morozov, A.; Morozov, A.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A.
2017-02-01
Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys. B 899 395–413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials.
On factorization of generalized Macdonald polynomials
Kononov, Ya.; Morozov, A.
2016-08-01
A remarkable feature of Schur functions—the common eigenfunctions of cut-and-join operators from W_∞ —is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U_q(SL_N) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorization—on a one- (rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding.
Seizure prediction using polynomial SVM classification.
Zisheng Zhang; Parhi, Keshab K
2015-08-01
This paper presents a novel patient-specific algorithm for prediction of seizures in epileptic patients with low hardware complexity and low power consumption. In the proposed approach, we first compute the spectrogram of the input fragmented EEG signals from a few electrodes. Each fragmented data clip is ten minutes in duration. Band powers, relative spectral powers and ratios of spectral powers are extracted as features. The features are then subjected to electrode selection and feature selection using classification and regression tree. The baseline experiment uses all features from selected electrodes and these features are then subjected to a radial basis function kernel support vector machine (RBF-SVM) classifier. The proposed method further selects a small number features from the selected electrodes and train a polynomial support vector machine (SVM) classifier with degree of 2 on these features. Prediction performances are compared between the baseline experiment and the proposed method. The algorithm is tested using intra-cranial EEG (iEEG) from the American Epilepsy Society Seizure Prediction Challenge database. The baseline experiment using a large number of features and RBF-SVM achieves a 100% sensitivity and an average AUC of 0.9985, while the proposed algorithm using only a small number of features and polynomial SVM with degree of 2 can achieve a sensitivity of 100.0%, an average area under curve (AUC) of 0.9795. For both experiments, only 10% of the available training data are used for training.
Alshal, Mohamed A.; Allam, Nageh K.
2016-11-01
A design for a highly efficient modified grating crystalline silicon (c-Si) thin film solar cell is demonstrated and analyzed using the two-dimensional (2-D) finite element method. The suggested grating has a double-sided pyramidal structure. The incorporation of the modified grating in a c-Si thin film solar cell offers a promising route to harvest light into the few micrometers active layer. Furthermore, a layer of silicon nitride is used as an antireflection coating (ARC). Additionally, the light trapping through the suggested design is significantly enhanced by the asymmetry of the top and bottom pyramids. The effects of the thickness of the active layer and facet angle of the pyramid on the spectral absorption, ultimate efficiency ( η), and short-circuit current density ( J sc) are investigated. The numerical results showed 87.9% efficiency improvement over the conventional thin film c-Si solar cell counterpart without gratings.
The spectral combination characteristic of grating and the bi-grating diffraction imaging effect
Institute of Scientific and Technical Information of China (English)
2007-01-01
This paper reports on a new property of grating, namely spectral combination, and on bi-grating diffraction imaging that is based on spectral combination. The spectral combination characteristic of a grating is the capability of combining multiple light beams of different wavelengths incident from specific angles into a single beam. The bi-grating diffraction imaging is the formation of the image of an object with two gratings: the first grating disperses the multi-color light beams from the object and the second combines the dispersed light beams to form the image. We gave the conditions necessary for obtaining the spectral combination. We also presented the equations that relate the two gratings’ spatial frequencies, diffraction orders and positions necessary for obtaining the bi-grating diffraction imaging.
Soto, Dan; Le Helloco, Antoine; Clanet, Cristophe; Quere, David; Varanasi, Kripa
2016-11-01
A drop thrown against a mesh can pass through its holes if impacting with enough inertia. As a result, although part of the droplet may remain on one side of the sieve, the rest will end up grated through the other side. This inexpensive method to break up millimetric droplets into micrometric ones may be of particular interest in a wide variety of applications: enhancing evaporation of droplets launched from the top of an evaporative cooling tower or preventing drift of pesticides sprayed above crops by increasing their initial size and atomizing them at the very last moment with a mesh. In order to understand how much liquid will be grated we propose in this presentation to start first by studying a simpler situation: a drop impacting a plate pierced with a single off centered hole. The study of the role of natural parameters such as the radius drop and speed or the hole position, size and thickness allows us to discuss then the more general situation of a plate pierced with multiple holes: the mesh.
Alignment free characterization of 2D gratings
Madsen, Morten Hannibal; Hansen, Poul-Erik; Jørgensen, Jan Friis
2015-01-01
Fast characterization of 2-dimensional gratings is demonstrated using a Fourier lens optical system and a differential optimization algorithm. It is shown that both the grating specific parameters such as the basis vectors and the angle between them and the alignment of the sample, such as the rotation of the sample around the x-, y-, and z-axis, can be deduced from a single measurement. More specifically, the lattice vectors and the angle between them have been measured, while the corrections of the alignment parameters are used to improve the quality of the measurement, and hence reduce the measurement uncertainty. Alignment free characterization is demonstrated on both a 2D hexagonal grating with a period of 700 nm and a checkerboard grating with a pitch of 3000 nm. The method can also be used for both automatic alignment and in-line characterization of gratings.
A Parallel Algorithm for Finding Roots of a Complex Polynomial
Institute of Scientific and Technical Information of China (English)
程锦松
1990-01-01
A distribution theory of the roots of a polynomial and a parallel algorithm for finding roots of a complex polynomial based on that theory are developed in this paper.With high parallelism,the algorithm is an improvement over the Wilf algorithm[3].
Weighted Approximation for Jackson-Matsuoka Polynomials on the Sphere
Directory of Open Access Journals (Sweden)
Guo Feng
2012-01-01
Full Text Available We consider the best approximation by Jackson-Matsuoka polynomials in the weighted Lp space on the unit sphere of Rd. Using the relation between K-functionals and modulus of smoothness on the sphere, we obtain the direct and inverse estimate of approximation by these polynomials for the h-spherical harmonics.
On Continued Fraction Expansion of Real Roots of Polynomial Systems
DEFF Research Database (Denmark)
Mantzaflaris, Angelos; Mourrain, Bernard; Tsigaridas, Elias
2011-01-01
We elaborate on a correspondence between the coefficients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions that use only integer arithmetic (in contrast to the Bernstein basis) and are...
Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials
Institute of Scientific and Technical Information of China (English)
Lin Sen XIE
2005-01-01
We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness ωγφ(f, t) where φ is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.
Polynomial coordinates and their behavior in higher dimensions
Berson, J.J.T.
2004-01-01
A coordinate is an element of a polynomial ring which is the first component of some automorphism of this ring. Understanding the structure of coordinates is still one of the major problems in the theory of polynomial automorphisms. It is already known, that in two variables over a field every coord
A second addition formula for continuous q-ultraspherical polynomials
Koornwinder, T.H.
2005-01-01
This paper provides the details of Remark 5.4 in the author's paper "Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group", SIAM J. Math. Anal. 24 (1993), 795-813. In formula (5.9) of the 1993 paper a two-parameter class of Askey-Wilson polynomials was expanded as a finit
Mutations of Laurent Polynomials and Flat Families with Toric Fibers
Ilten, Nathan Owen
2012-01-01
We give a general criterion for two toric varieties to appear as fibers in a flat family over the projective line. We apply this to show that certain birational transformations mapping a Laurent polynomial to another Laurent polynomial correspond to deformations between the associated toric varieties.
Strict Positivstellens\\" atze for matrix polynomials with scalar constraints
Cimpric, Jaka
2010-01-01
We extend Krivine's strict positivstellensatz for usual (real multivariate) polynomials to symmetric matrix polynomials with scalar constraints. The proof is an elementary computation with Schur complements. Analogous extensions of Schm\\" udgen's and Putinar's strict positivstellensatz were recently proved by Hol and Scherer using methods from optimization theory.
Animating Nested Taylor Polynomials to Approximate a Function
Mazzone, Eric F.; Piper, Bruce R.
2010-01-01
The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…
POLYNOMIAL RECURRENCE FOR L（E）VY PROCESSES
Institute of Scientific and Technical Information of China (English)
ZHAO MINZHI; YING JIANGANG
2004-01-01
In this paper, the authors study the ω-transience and ω-recurrence for Lévy processes with any weight function ω, give a relation between ω-recurrence and the last exit times. As a special case, the polynomial recurrence and polynomial transience are also studied.
Approximation to Continuous Functions by a Kind of Interpolation Polynomials
Institute of Scientific and Technical Information of China (English)
Yuan Xue-gang; Wang De-hui
2001-01-01
In this paper, an interpolation polynomial operator Fn (f; l, x ) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈ Cb[1,1] (0≤b≤l) Fn(f; l,x) converges to f(x) uniformly, where l is an odd number.
Lm Extremal Polynomials Associated with Generalized Jacobi Weights
Institute of Scientific and Technical Information of China (English)
Ying-guang Shi
2003-01-01
Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turan quadrature formula is given.
On Period of the Sequence of Fibonacci Polynomials Modulo
Directory of Open Access Journals (Sweden)
İnci Gültekin
2013-01-01
Full Text Available It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo . It is found that order of cyclic group generated with matrix is equal to the period of these sequences.
A polynomial f(R) inflation model
Huang, Qing-Guo
2014-01-01
Motivated by the ultraviolet complete quantum theory of gravity, for example string theory, we investigate a polynomial $f(R)$ inflation model in detail. We calculate the spectral index and tensor-to-scalar ratio in the $f(R)$ inflation model with the form of $f(R)=R+{R^2\\over 6M^2}+{\\lambda_n\\over 2n}{R^n\\over (3M^2)^{n-1}}$. If the dimensionless coupling $\\lambda_n$ is much smaller than one, there are two regions for achieving the slow-roll inflation if $\\lambda_n>0$ and only one region if $\\lambda_n<0$. Compared to Planck 2013, numerically we find that $|\\lambda_n|\\lesssim 10^{-2n+2.6}$ in the region I and $0<\\lambda_n\\lesssim 10^{-3n+5.6}$ in the region II.
Irreducible polynomials with prescribed sums of coefficients
Tuxanidy, Aleksandr; Wang, Qiang
2016-01-01
Let $q$ be a power of a prime, let $\\mathbb{F}_q$ be the finite field with $q$ elements and let $n \\geq 2$. For a polynomial $h(x) \\in \\mathbb{F}_q[x]$ of degree $n \\in \\mathbb{N}$ and a subset $W \\subseteq [0,n] := \\{0, 1, \\ldots, n\\}$, we define the sum-of-digits function $$S_W(h) = \\sum_{w \\in W}[x^{w}] h(x)$$ to be the sum of all the coefficients of $x^w$ in $h(x)$ with $w \\in W$. In the case when $q = 2$, we prove, except for a few genuine exceptions, that for any $c \\in \\mathbb{F}_2$ an...
Digital terrain modeling with the Chebyshev polynomials
Florinsky, I V
2015-01-01
Mathematical problems of digital terrain analysis include interpolation of digital elevation models (DEMs), DEM generalization and denoising, and computation of morphometric variables by calculation of partial derivatives of elevation. Traditionally, these procedures are based on numerical treatments of two-variable discrete functions of elevation. We developed a spectral analytical method and algorithm based on high-order orthogonal expansions using the Chebyshev polynomials of the first kind with the subsequent Fejer summation. The method and algorithm are intended for DEM analytical treatment, such as, DEM global approximation, denoising, and generalization as well as computation of morphometric variables by analytical calculation of partial derivatives. To test the method and algorithm, we used a DEM of the Northern Andes including 230,880 points (the elevation matrix 480 $\\times$ 481). DEMs were reconstructed with 480, 240, 120, 60, and 30 expansion coefficients. The first and second partial derivatives ...
The evolution of piecewise polynomial wave functions
Andrews, Mark
2017-01-01
For a non-relativistic particle, we consider the evolution of wave functions that consist of polynomial segments, usually joined smoothly together. These spline wave functions are compact (that is, they are initially zero outside a finite region), but they immediately extend over all available space as they evolve. The simplest splines are the square and triangular wave functions in one dimension, but very complicated splines have been used in physics. In general the evolution of such spline wave functions can be expressed in terms of antiderivatives of the propagator; in the case of a free particle or an oscillator, all the evolutions are expressed exactly in terms of Fresnel integrals. Some extensions of these methods to two and three dimensions are discussed.
Inequalities for a Polynomial and its Derivative
Indian Academy of Sciences (India)
V K Jain
2000-05-01
For an arbitrary entire function and any > 0, let $M(f, r):=\\max_{|z|=r}|f(z)|$. It is known that if is a polynomial of degree having no zeros in the open unit disc, and $m:=\\min_{|z|=1}|p(z)|$, then $$M(p',1)≤\\frac{n}{2}\\{M(p,1)-m\\},$$ $$M(p, R)≤\\left(\\frac{R^n+1}{2}\\right)M(p, 1)-m\\left(\\frac{R^n-1}{2}\\right), R>> 1.$$ It is also known that if has all its zeros in the closed unit disc, then $$M(p', 1)≥\\frac{n}{2}\\{M(p, 1)+m\\}.$$ The present paper contains certain generalizations of these inequalities
Study on the Grey Polynomial Geometric Programming
Institute of Scientific and Technical Information of China (English)
LUODang
2005-01-01
In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this model can be regarded as interval grey numbers. When the model contains grey numbers, it is hard for common programming method to solve them. By combining the common programming model with the grey system theory,and using some analysis strategies, a model of grey polynomial geometric programming, a model of 8 positioned geometric programming and their quasi-optimum solution or optimum solution are put forward. At the same time, we also developed an algorithm for the problem.This approach brings a new way for the application research of geometric programming. An example at the end of this paper shows the rationality and feasibility of the algorithm.
A polynomial algorithm for abstract maximum flow
Energy Technology Data Exchange (ETDEWEB)
McCormick, S.T. [Univ. of British Columbia, Vancouver, British Columbia (Canada)
1996-12-31
Ford and Fulkerson`s original 1956 max flow/min cut paper formulated max flow in terms of flows on paths, rather than the more familiar flows on arcs. In 1974 Hoffman pointed out that Ford and Fulkerson`s original proof was quite abstract, and applied to a wide range of max flow-like problems. In this abstract model we have capacitated elements, and linearly ordered subsets of elements called paths. When two paths share an element ({open_quote}cross{close_quote}), then there must be a path that is a subset of the first path up to the cross, and a subset of the second path after the cross. (Hoffman`s generalization of) Ford and Fulkerson`s proof showed that the max flow/min cut theorem still holds under this weak assumption. However, this proof is non-constructive. To get an algorithm, we assume that we have an oracle whose input is an arbitrary subset of elements, and whose output is either a path contained in that subset, or the statement that no such path exists. We then use complementary slackness to show how to augment any feasible set of path flows to a set with a strictly larger total flow value using a polynomial number of calls to the oracle. Then standard scaling techniques yield an overall polynomial algorithm for finding both a max flow and a min cut. Hoffman`s paper actually considers a sort of supermodular objective on the path flows, which allows him to include transportation problems and thus rain-cost flow in his frame-work. We also discuss extending our algorithm to this more general case.
Structured matrix based methods for approximate polynomial GCD
Boito, Paola
2011-01-01
Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. .
Higher order branching of periodic orbits from polynomial isochrones
Directory of Open Access Journals (Sweden)
B. Toni
1999-09-01
Full Text Available We discuss the higher order local bifurcations of limit cycles from polynomial isochrones (linearizable centers when the linearizing transformation is explicitly known and yields a polynomial perturbation one-form. Using a method based on the relative cohomology decomposition of polynomial one-forms complemented with a step reduction process, we give an explicit formula for the overall upper bound of branch points of limit cycles in an arbitrary $n$ degree polynomial perturbation of the linear isochrone, and provide an algorithmic procedure to compute the upper bound at successive orders. We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and show that at most nine branch points of limit cycles can bifurcate in a cubic polynomial perturbation. Moreover, perturbations with exactly two, three, four, six, and nine local families of limit cycles may be constructed.
Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials
Gómez-Ullate, David; Milson, Robert
2012-01-01
The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A generalization of the classical Heine-Mehler formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros. We also describe the location and the asymptotic behaviour of the exceptional zeros, which converge for large n to fixed values.
The lowest degree $0,1$-polynomial divisible by cyclotomic polynomial
Reddy, A Satyanarayana
2011-01-01
Let $n$ be an even positive integer with at most three distinct prime factors and let $\\ze_n$ be a primitive $n$-th root of unity. In this study, we made an attempt to find the lowest-degree $0,1$-polynomial $f(x) \\in \\Q[x]$ having at least three terms such that $f(\\ze_n)$ is a minimal vanishing sum of the $n$-th roots of unity.
Etessami, Kousha; Yannakakis, Mihalis
2012-01-01
We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic max(min) polynomial equations, referred to as maxPPSs (and minPPSs, respectively), in time polynomial in both the encoding size of the system of equations and in log(1/epsilon), where epsilon > 0 is the desired additive error bound of the solution. (The model of computation is the standard Turing machine model.) We establish this result using a generalization of Newton's method which applies to maxPPSs and minPPSs, even though the underlying functions are only piecewise-differentiable. This generalizes our recent work which provided a P-time algorithm for purely probabilistic PPSs. These equations form the Bellman optimality equations for several important classes of infinite-state Markov Decision Processes (MDPs). Thus, as a corollary, we obtain the first polynomial time algorithms for computing to within arbitrary desired precision the optimal value vector for several classes of infinite-state...
21 CFR 133.147 - Grated American cheese food.
2010-04-01
... 21 Food and Drugs 2 2010-04-01 2010-04-01 false Grated American cheese food. 133.147 Section 133... Cheese and Related Products § 133.147 Grated American cheese food. (a)(1) Grated American cheese food is... granular mixture. (2) Grated American cheese food contains not less than 23 percent of milkfat,...
Fiber Bragg Grating Sensors for Harsh Environments
Directory of Open Access Journals (Sweden)
Stephen J. Mihailov
2012-02-01
Full Text Available Because of their small size, passive nature, immunity to electromagnetic interference, and capability to directly measure physical parameters such as temperature and strain, fiber Bragg grating sensors have developed beyond a laboratory curiosity and are becoming a mainstream sensing technology. Recently, high temperature stable gratings based on regeneration techniques and femtosecond infrared laser processing have shown promise for use in extreme environments such as high temperature, pressure or ionizing radiation. Such gratings are ideally suited for energy production applications where there is a requirement for advanced energy system instrumentation and controls that are operable in harsh environments. This paper will present a review of some of the more recent developments.
Large size metallic glass gratings by embossing
Ma, J.; Yi, J.; Zhao, D. Q.; Pan, M. X.; Wang, W. H.
2012-09-01
Bulk metallic glasses have excellent thermoforming ability in their wide supercooled liquid region. We show that large-size metallic glass grating (˜8 × 8 mm2) with fine periodicity and ultra smooth surface feature can be readily fabricated by hot embossing. The method for fabrication of gratings is proved to be much cheaper, and requires low pressure and short time (less than 30 s). The metallic glass gratings exhibit comparable optical properties such as rainbow-like spectrum when shone by fluorescent lamp light.
Heilmann, Ralf K.; Bruccoleri, Alexander; Schattenburg, Mark; Kolodziejczak, jeffery; Gaskin, Jessica; O'Dell, Stephen L.
2017-01-01
A number of high priority subjects in astrophysics are addressed by a state-of-the-art soft x-ray grating spectrometer, e.g. the role of Active Galactic Nuclei in galaxy and star formation, characterization of the WHIM and the “missing baryon” problem, characterization of halos around the Milky Way and nearby galaxies, and stellar coronae and surrounding winds and disks. An Explorer-scale, large-area (A > 1,000 cm2), high resolving power (R > 3,000) soft x-ray grating spectrometer is highly feasible based on Critical-Angle Transmission (CAT) grating technology, even for telescopes with angular resolution of 5-10 arcsec. Significantly higher performance could be provided by a CAT grating spectrometer on an X-ray-Surveyor-type mission (A > 4,000 cm2, R > 5,000). CAT gratings combine advantages of blazed reflection gratings (high efficiency, use of higher orders) with those of transmission gratings (low mass, relaxed alignment tolerances and temperature requirements, transparent at higher energies) with minimal mission resource requirements. Blazing is achieved through grazing-incidence reflection off the smooth silicon grating bar sidewalls. Silicon is well matched to the soft x-ray band, and 30% absolute diffraction efficiency has been acheived with clear paths for further improvement. CAT gratings with sidewalls made of high-Z elements allow extension of blazing to higher energies and larger dispersion angles, enabling higher resolving power at shorter wavelengths. X-ray data from CAT gratings coated with a thin layer of platinum using atomic layer deposition demonstrate efficient blazing to higher energies and much larger blaze angles than possible with silicon alone. Measurements of the resolving power of a breadboard CAT grating spectrometer consisting of a Wolter-I slumped-glass focusing optic from GSFC and CAT gratings, taken at the MSFC Stray Light Facility, have demonstrated resolving power > 10,000. Thus currently fabricated CAT gratings are compatible
Influence of grating thickness in low-contrast subwavelength grating concentrating lenses
Ye, Mao; Yi, Ya Sha
2016-07-01
Conventional subwavelength grating concentrating lenses are designed based on calculated phase overlap, wherein the phase change is fixed by the grating thickness, bar-width, and airgap, and therefore the focus. We found that certain concentration effects can still be maintained by changing the grating thickness with the same bar-widths and airgap dimensions. Following that, we discovered the existence of the grating thickness threshold; light concentration intensity spikes upon exceeding this limit. However, the light concentration property does not change continuously with respect to a steady increase in grating thickness. This observation indicates that there exists a concentration mode self-interference effect along the light propagation direction inside the gratings. Our results may provide guidance in designing and fabricating microlenses in a potentially more easy and controllable manner. Such approaches can be utilized in various integrated nanophotonics applications ranging from optical cavities and read/write heads to concentrating photovoltaics.
Access Platforms for Offshore Wind Turbines Using Gratings
DEFF Research Database (Denmark)
Andersen, Thomas Lykke; Rasmussen, Michael R.
2008-01-01
The paper deals with forces generated by a stationary jet on different types of gratings and a solid plate. The force reduction factors for the different gratings compared to the solid plate mainly depend on the porosity of the gratings, but the geometry of the grating is also of some importance........ The derived reduction factors are expected to be applicable to design of offshore wind turbine access platforms with gratings where slamming also is an important factor....
Long period fiber gratings induced by mechanical resonance
Shahal, Shir; Duadi, Hamootal; Fridman, Moti
2015-01-01
We present a simple, and robust method for writing long period fiber gratings with low polarization dependent losses. Our method is based on utilizing mechanical vibrations of the tapered fiber while pooling it. Our method enables real-time tunability of the periodicity, efficiency and length of the grating. We also demonstrate complex grating by writing multiple gratings simultaneously. Finally, we utilized the formation of the gratings in different fiber diameters to investigate the Young's modulus of the fiber.
Liquid crystal on subwavelength metal gratings
Energy Technology Data Exchange (ETDEWEB)
Palto, S. P.; Barnik, M. I.; Artemov, V. V.; Shtykov, N. M.; Geivandov, A. R.; Yudin, S. G.; Gorkunov, M. V. [Shubnikov Institute of Crystallography of Russian Academy of Sciences, Leninsky pr. 59, 119333 Moscow (Russian Federation)
2015-06-14
Optical and electrooptical properties of a system consisting of subwavelength metal gratings and nematic liquid crystal layer are studied. Aluminium gratings that also act as interdigitated electrodes are produced by focused ion beam lithography. It is found that a liquid crystal layer strongly influences both the resonance and light polarization properties characteristic of the gratings. Enhanced transmittance is observed not only for the TM-polarized light in the near infrared spectral range but also for the TE-polarized light in the visible range. Although the electrodes are separated by nanosized slits, and the electric field is strongly localized near the surface, a pronounced electrooptical effect is registered. The effect is explained in terms of local reorientation of liquid crystal molecules at the grating surface and propagation of the orientational deformation from the surface into the bulk of the liquid crystal layer.
Theory of photorefractive dynamic grating formulation
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The photorefractive holographic recording and two-beam coupling are both dynamic grating formulation process. The interference light intensity of the two coherent beams induces a phase grating though photo-induced refractive index variation and the phase grating changing the intensities of the two beams through beam-coupling take place at the same time. By solving simultaneously the band transport equations and wave-coupled equations, and using the light intensity modulation as the main variable, the analytic solution is obtained, which is valid for any light intensity modulation and constant light excitation efficiency. Here all the mechanics of drift, diffusion and photovoltaic effect are considered. The result shows that the modulation of the dynamic grating varies more slowly compared with that of the linear modulation approximation.
Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces
Escobar-Ruiz, M. A.; Miller, Willard, Jr.
2016-07-01
2nd-order conformal superintegrable systems in n dimensions are Laplace equations on a manifold with an added scalar potential and 2n-1 independent 2nd order conformal symmetry operators. They encode all the information about Helmholtz (eigenvalue) superintegrable systems in an efficient manner: there is a 1-1 correspondence between Laplace superintegrable systems and Stäckel equivalence classes of Helmholtz superintegrable systems. In this paper we focus on superintegrable systems in two-dimensions, n = 2, where there are 44 Helmholtz systems, corresponding to 12 Laplace systems. For each Laplace equation we determine the possible two-variate polynomial subspaces that are invariant under the action of the Laplace operator, thus leading to families of polynomial eigenfunctions. We also study the behavior of the polynomial invariant subspaces under a Stäckel transform. The principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions. The hidden gl 3-algebraic structure is exhibited for the exact and quasi-exact systems. For physically meaningful solutions, the orthogonality properties and normalizability of the polynomials are presented as well. Finally, for all Helmholtz superintegrable solvable systems we give a unified construction of one-dimensional (1D) and two-dimensional (2D) quasi-exactly solvable potentials possessing polynomial solutions, and a construction of new 2D PT-symmetric potentials is established.
A comparison of high-order polynomial and wave-based methods for Helmholtz problems
Lieu, Alice; Gabard, Gwénaël; Bériot, Hadrien
2016-09-01
The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.
Hydraulic Capacity of an ADA Compliant Street Drain Grate
Energy Technology Data Exchange (ETDEWEB)
Lottes, Steven A. [Argonne National Lab. (ANL), Argonne, IL (United States); Bojanowski, Cezary [Argonne National Lab. (ANL), Argonne, IL (United States)
2015-09-01
Resurfacing of urban roads with concurrent repairs and replacement of sections of curb and sidewalk may require pedestrian ramps that are compliant with the American Disabilities Act (ADA), and when street drains are in close proximity to the walkway, ADA compliant street grates may also be required. The Minnesota Department of Transportation ADA Operations Unit identified a foundry with an available grate that meets ADA requirements. Argonne National Laboratory’s Transportation Research and Analysis Computing Center used full scale three dimensional computational fluid dynamics to determine the performance of the ADA compliant grate and compared it to that of a standard vane grate. Analysis of a parametric set of cases was carried out, including variation in longitudinal, gutter, and cross street slopes and the water spread from the curb. The performance of the grates was characterized by the fraction of the total volume flow approaching the grate from the upstream that was captured by the grate and diverted into the catch basin. The fraction of the total flow entering over the grate from the side and the fraction of flow directly over a grate diverted into the catch basin were also quantities of interest that aid in understanding the differences in performance of the grates. The ADA compliant grate performance lagged that of the vane grate, increasingly so as upstream Reynolds number increased. The major factor leading to the performance difference between the two grates was the fraction of flow directly over the grates that is captured by the grates.
vs. a polynomial chaos-based MCMC
Siripatana, Adil
2014-08-01
Bayesian Inference of Manning\\'s n coefficient in a Storm Surge Model Framework: comparison between Kalman lter and polynomial based method Adil Siripatana Conventional coastal ocean models solve the shallow water equations, which describe the conservation of mass and momentum when the horizontal length scale is much greater than the vertical length scale. In this case vertical pressure gradients in the momentum equations are nearly hydrostatic. The outputs of coastal ocean models are thus sensitive to the bottom stress terms de ned through the formulation of Manning\\'s n coefficients. This thesis considers the Bayesian inference problem of the Manning\\'s n coefficient in the context of storm surge based on the coastal ocean ADCIRC model. In the first part of the thesis, we apply an ensemble-based Kalman filter, the singular evolutive interpolated Kalman (SEIK) filter to estimate both a constant Manning\\'s n coefficient and a 2-D parameterized Manning\\'s coefficient on one ideal and one of more realistic domain using observation system simulation experiments (OSSEs). We study the sensitivity of the system to the ensemble size. we also access the benefits from using an in ation factor on the filter performance. To study the limitation of the Guassian restricted assumption on the SEIK lter, 5 we also implemented in the second part of this thesis a Markov Chain Monte Carlo (MCMC) method based on a Generalized Polynomial chaos (gPc) approach for the estimation of the 1-D and 2-D Mannning\\'s n coe cient. The gPc is used to build a surrogate model that imitate the ADCIRC model in order to make the computational cost of implementing the MCMC with the ADCIRC model reasonable. We evaluate the performance of the MCMC-gPc approach and study its robustness to di erent OSSEs scenario. we also compare its estimates with those resulting from SEIK in term of parameter estimates and full distributions. we present a full analysis of the solution of these two methods, of the
Spatial heterodyne interferometry with polarization gratings.
Kudenov, Michael W; Miskiewicz, Matthew N; Escuti, Michael J; Dereniak, Eustace L
2012-11-01
The implementation of a polarization-based spatial heterodyne interferometer (SHI) is described. While a conventional SHI uses a Michelson interferometer and diffraction gratings, our SHI exploits mechanically robust Wollaston prisms and polarization gratings. A theoretical model for the polarization SHI is provided and validated with data from our proof of concept experiments. This device is expected to provide a compact monolithic sensor for subangstrom resolution spectroscopy in remote sensing, biomedical imaging, and machine vision applications.
Sampled phase-shift fiber Bragg gratings
Institute of Scientific and Technical Information of China (English)
Xu Wang(王旭); Chongxiu Yu(余重秀); Zhihui Yu(于志辉); Qiang Wu(吴强)
2004-01-01
A phase-shift fiber Bragg grating (FBG) with sampling is proposed to generate a multi-channel bandpass filter in the background of multi-channel stopbands. The sampled noire fiber gratings are analyzed by Fourier transform theory first, and then simulation and experiment are performed, the results show that transmission peaks are opened in every reflective channel, the spectrum shape of every channel is identical.It can be used to fabricate multi-wavelength distributed feedback (DFB) fiber laser.
Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials
Horozov, Emil
2016-05-01
We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.
Combinatorial theory of Macdonald polynomials I: Proof of Haglund's formula
Haglund, J.; Haiman, M.; Loehr, N
2005-01-01
Haglund recently proposed a combinatorial interpretation of the modified Macdonald polynomials H̃μ. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of H̃μ. As corollaries, we obtain the cocharge formula of Lascoux and Schützenberger for Hall–Littlewood polynomials, a formula of Sahi and Knop for Jack's symmetric functions, a generalization of this result to the integral Macdonald polynomials Jμ, a formula for H̃μ in terms of Lascoux–Leclerc–Th...
Representations of non-negative polynomials via critical ideals
Hiep, Dang Tuan
2011-01-01
This paper studies the representations of a non-negative polynomial $f$ on a non-compact semi-algebraic set $K$ modulo its critical ideal. Under the assumptions that the semi-algebraic set $K$ is regular and $f$ satisfies the boundary Hessian conditions (BHC) at each zero of $f$ in $K$, we show that $f$ can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if $f\\ge 0$ on $K$. In particular, we focus on the polynomial ring $\\mathbb R[x]$.
Polynomial system solving for decoding linear codes and algebraic cryptanalysis
2009-01-01
This thesis is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The ...
Generalized Freud's equation and level densities with polynomial potential
Boobna, Akshat; Ghosh, Saugata
2013-08-01
We study orthogonal polynomials with weight $\\exp[-NV(x)]$, where $V(x)=\\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\\mu}=h_{\\mu}/h_{\\mu -1}$, where $h_{\\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\\rightarrow\\infty$ are derived.
Non-Hermitian oscillator Hamiltonians and multiple Charlier polynomials
Energy Technology Data Exchange (ETDEWEB)
Miki, Hiroshi, E-mail: miki@amp.i.kyoto-u.ac.jp [Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-Ku, Kyoto 606 8501 (Japan); Vinet, Luc, E-mail: luc.vinet@umontreal.ca [Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: zhedanov@fti.dn.ua [Donetsk Institute for Physics and Technology, Donetsk 83 114 (Ukraine)
2011-12-05
A set of r non-Hermitian oscillator Hamiltonians in r dimensions is shown to be simultaneously diagonalizable. Their spectra are real and the common eigenstates are expressed in terms of multiple Charlier polynomials. An algebraic interpretation of these polynomials is thus achieved and the model is used to derive some of their properties. -- Highlights: ► A set of r non-Hermitian oscillator Hamiltonians in r dimensions is presented. ► Their spectra are real. ► The common eigenstates are expressed in terms of multiple Charlier polynomials (MCP). ► This “integrable” model allows to interpret structural formulas of the MCPs.
Chromatic Polynomials Of Some (m,l-Hyperwheels
Directory of Open Access Journals (Sweden)
Julian A. Allagan
2014-03-01
Full Text Available In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete non-uniform $(m,l-$hyperwheels and non-uniform $(m,l-$hyperfans. These hypergraphs, constructed through a ``join" graph operation, are some generalizations of the well-known wheel and fan graphs, respectively. Further, we revisit some results concerning these graphs and present their chromatic polynomials in a standard form that involves the Stirling numbers of the second kind.
Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences
Chen, William Y C; Zhou, Elaine L F
2010-01-01
The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x+1), which leads to the log-concavity of P(x+c) for any $c\\geq 1$ due to Llamas and Mart\\'{\\i}nez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients.
Polynomials in Control Theory Parametrized by Their Roots
Directory of Open Access Journals (Sweden)
Baltazar Aguirre-Hernández
2012-01-01
Full Text Available The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem.
An eigenvalue problem for the associated Askey-Wilson polynomials
Bruder, Andrea; Suslov, Sergei K
2012-01-01
To derive an eigenvalue problem for the associated Askey-Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey-Wilson polynomials introduced by Ismail and Rahman. The Askey-Wilson operator, applied in each variable separately, maps this function to the ordinary Askey-Wilson polynomials with different sets of parameters. A third Askey-Wilson operator is found with the help of a computer algebra program which links the two, and an eigenvalue problem is stated.
q-Bernoulli numbers and q-Bernoulli polynomials revisited
Directory of Open Access Journals (Sweden)
Kim Taekyun
2011-01-01
Full Text Available Abstract This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010, some incorrect properties are revised. It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994 (see Equation 9, some new generating functions for the q-Bernoulli numbers and polynomials are shown. Mathematics Subject Classification (2000 11B68, 11S40, 11S80
Learning Read-constant Polynomials of Constant Degree modulo Composites
DEFF Research Database (Denmark)
Chattopadhyay, Arkadev; Gavaldá, Richard; Hansen, Kristoffer Arnsfelt;
2011-01-01
Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class \\textACC0ACC0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class...... is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable...
Polynomial complexity algorithm for Max-Cut problem
Katkov, Mikhail
2010-01-01
The standard NP-complete max-cut problem is reformulated as a binary quadratic program xQx s.t x^2=1. This problem is further reformulated as global minimum of quartic polynomial (xQ'x - z)^2 + \\sum_i (x_i^2-1)^2+ \\alpha z^2, for some \\alpha. The global minimum is found by polynomial complexity semi-definite program. Numerical examples and code is provided. The resulting algorithm solves arbitrary max-cut problem in polynomial time, therefore P=NP.
Lower bounds for polynomials using geometric programming
Ghasemi, Mehdi
2011-01-01
We make use of a result of Hurwitz and Reznick, and a consequence of this result due to Fidalgo and Kovacec, to determine a new sufficient condition for a polynomial $f\\in\\mathbb{R}[X_1,...,X_n]$ of even degree to be a sum of squares. This result generalizes a result of Lasserre and a result of Fidalgo and Kovacec, and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound $f_{gp}$ for $f$, and we explain how $f_{gp}$ can be computed using geometric programming. The lower bound $f_{gp}$ is generally not as good as the lower bound $f_{sos}$ introduced by Lasserre and Parrilo and Sturmfels, which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of $f_{gp}$ is much faster. The computation is simplest when the highest degree term of $f$ has the form $\\sum_{i=1}^n a_iX_i^{2d}$, $a_i>0$, $i=1,...,n$. The lower bounds for $f$ established in [6] are obtained by evaluating the objective function ...
Integral Inequalities for Self-Reciprocal Polynomials
Indian Academy of Sciences (India)
Horst Alzer
2010-04-01
Let $n≥ 1$ be an integer and let $\\mathcal{P}_n$ be the class of polynomials of degree at most satisfying $z^nP(1/z)=P(z)$ for all $z\\in C$. Moreover, let be an integer with $1≤ r≤ n$. Then we have for all $P\\in\\mathcal{P}_n$: $$_n(r)\\int^{2}_0|P(e^{it})|^2dt≤\\int^{2}_0|P^{(r)}(e^{it})|^2dt≤_n(r)\\int^{2}_0|P(e^{it})|^2dt$$ with the best possible factors \\begin{equation*}_n(r)=\\begin{cases}\\prod^{r-1}_{j=0}\\left(\\frac{n}{2}-j\\right)^2, < \\text{if is even},\\\\ \\frac{1}{2}\\left[\\prod^{r-1}_{j=0}\\left(\\frac{n+1}{2}-j\\right)^2+\\prod^{r-1}_{j=0}\\left(\\frac{n-1}{2}-j\\right)^2\\right], < \\text{if is odd},\\end{cases}\\end{equation*} and \\begin{equation*}_n(r)=\\frac{1}{2}\\prod\\limits^{r-1}_{j=0}(n-j)^2.\\end{equation*} This refines and extends a result due to Aziz and Zargar (1997).
Polynomial Linear Programming with Gaussian Belief Propagation
Bickson, Danny; Shental, Ori; Dolev, Danny
2008-01-01
Interior-point methods are state-of-the-art algorithms for solving linear programming (LP) problems with polynomial complexity. Specifically, the Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where $n$ is the number of unknown variables. Karmarkar's celebrated algorithm is known to be an instance of the log-barrier method using the Newton iteration. The main computational overhead of this method is in inverting the Hessian matrix of the Newton iteration. In this contribution, we propose the application of the Gaussian belief propagation (GaBP) algorithm as part of an efficient and distributed LP solver that exploits the sparse and symmetric structure of the Hessian matrix and avoids the need for direct matrix inversion. This approach shifts the computation from realm of linear algebra to that of probabilistic inference on graphical models, thus applying GaBP as an efficient inference engine. Our construction is general and can be used for any interior-point algorithm which uses the Newt...
Polynomial super-gl(n) algebras
Jarvis, P D
2003-01-01
We introduce a class of finite dimensional nonlinear superalgebras $L = L_{\\bar{0}} + L_{\\bar{1}}$ providing gradings of $L_{\\bar{0}} = gl(n) \\simeq sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of degree $>1$) in the $gl(n)$ generators. Specifically, we investigate `type I' super-$gl(n)$ algebras, having odd generators transforming in a single irreducible representation of $gl(n)$ together with its contragredient. Admissible structure constants are discussed in terms of available $gl(n)$ couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the $n$-dimensional defining representation, with odd generators $Q_{a}, \\bar{Q}{}^{b}$, and even generators ${E^{a}}_{b}$, $a,b = 1,...,n$, a three parameter family of quadratic super-$gl(n)$ algebras (deformations of $sl(n/1)$) is defined. In general, additional covariant Serre-type conditions are imposed, in order that the Jacobi identities be fulf...
Polynomial-time solutions to image segmentation
Energy Technology Data Exchange (ETDEWEB)
Asano, Tetsuo [Osaka Electro-Communication Univ., Neyagawa (Japan); Chen, D.Z. [Notre Dame, South Bend, IN (United States); Katoh, Naoki [Kobe Univ. of Commerce (Japan)
1996-12-31
Separating an object in an image from its background is a central problem (called segmentation) in pattern recognition and computer vision. In this paper, we study the complexity of the segmentation problem, assuming that the object forms a connected region in an intensity image. We show that the optimization problem of separating a connected region in an n-pixel grid is NP-hard under the interclass variance, a criterion that is used in discriminant analysis. More importantly, we consider the basic case in which the object is separated by two x-monotone curves (i.e., the object itself is x-monotone), and present polynomial-time algorithms for computing exact and approximate optimal segmentation. Our main algorithm for exact optimal segmentation by two x-monotone curves runs in O(n{sup 2}) time; this algorithm is based on several techniques such as a parametric optimization formulation, a hand-probing algorithm for the convex hull of an unknown point set, and dynamic programming using fast matrix searching. Our efficient approximation scheme obtains an {epsilon}-approximate solution in O({epsilon}{sup -1} n log L) time, where {epsilon} is any fixed constant with 1 > {epsilon} > 0, and L is the total sum of the absolute values of brightness levels of the image.
Polynomial Time Algorithms for Minimum Energy Scheduling
Baptiste, Philippe; Durr, Christoph
2009-01-01
The aim of power management policies is to reduce the amount of energy consumed by computer systems while maintaining satisfactory level of performance. One common method for saving energy is to simply suspend the system during the idle times. No energy is consumed in the suspend mode. However, the process of waking up the system itself requires a certain fixed amount of energy, and thus suspending the system is beneficial only if the idle time is long enough to compensate for this additional energy expenditure. In the specific problem studied in the paper, we have a set of jobs with release times and deadlines that need to be executed on a single processor. Preemptions are allowed. The processor requires energy L to be woken up and, when it is on, it uses one unit of energy per one unit of time. It has been an open problem whether a schedule minimizing the overall energy consumption can be computed in polynomial time. We solve this problem in positive, by providing an O(n^5)-time algorithm. In addition we pr...
Polynomial Method for PLL Controller Optimization
Directory of Open Access Journals (Sweden)
Tsung-Yu Chiou
2011-06-01
Full Text Available The Phase-Locked Loop (PLL is a key component of modern electronic communication and control systems. PLL is designed to extract signals from transmission channels. It plays an important role in systems where it is required to estimate the phase of a received signal, such as carrier tracking from global positioning system satellites. In order to robustly provide centimeter-level accuracy, it is crucial for the PLL to estimate the instantaneous phase of an incoming signal which is usually buried in random noise or some type of interference. This paper presents an approach that utilizes the recent development in the semi-definite programming and sum-of-squares field. A Lyapunov function will be searched as the certificate of the pull-in range of the PLL system. Moreover, a polynomial design procedure is proposed to further refine the controller parameters for system response away from the equilibrium point. Several simulation results as well as an experiment result are provided to show the effectiveness of this approach.
Constructing Polynomial Spectral Models for Stars
Rix, Hans-Walter; Conroy, Charlie; Hogg, David W
2016-01-01
Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these $\\mathcal{N}\\sim \\,$10-40 model labels to observed spectra has been deemed unfeasible, because the number of ab initio spectral model grid calculations scales exponentially with $\\mathcal{N}$. We suggest instead the construction of a polynomial spectral model (PSM) of order $\\mathcal{O}$ for the model flux at each wavelength. Building this approximation requires a minimum of only ${\\mathcal{N}+\\mathcal{O}\\choose\\mathcal{O}}$ calculations: e.g. a quadratic spectral model ($\\mathcal{O}=\\,$2), which can then fit $\\mathcal{N}=\\,$20 labels simultaneously, can be constructed from as few as 231 ab initio spectral model calculations; in practice, a somewhat larger number ($\\sim\\,$300-1000) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation to ab initio spectral models only over a limited portion of label space, which will vary case by case. Y...
Analytical and numerical study on grating depth effects in grating coupled waveguide sensors
DEFF Research Database (Denmark)
Horvath, R.; Wilcox, L.C.; Pedersen, H.C.;
2005-01-01
The in-coupling process for grating-coupled planar optical waveguide sensors is investigated in the case of TE waves. A simple analytical model based on the Rayleigh-Fourier-Kiselev method is applied to take into account the depth of the grating coupler, which is usually neglected in the modeling...
Classification of complex polynomial vector fields in one complex variable
DEFF Research Database (Denmark)
Branner, Bodil; Dias, Kealey
2010-01-01
, the main result of the paper. This result is an extension and refinement of Douady et al. (Champs de vecteurs polynomiaux sur C. Unpublished manuscript) classification of the structurally stable polynomial vector fields. We further review some general concepts for completeness and show that vector fields......This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing...... the topology, and a set of analytic invariants, describing the geometry. Conversely, given admissible combinatorial and analytic data sets, we show using surgery the existence of a unique monic and centred polynomial vector field realizing the given invariants. This is the content of the Structure Theorem...
Sparse Volterra and Polynomial Regression Models: Recoverability and Estimation
Kekatos, Vassilis
2011-01-01
Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but unfortunately cannot be met by least-squares based or kernel regression methods. To this end, compressed sampling (CS) approaches, already successful in linear regression settings, can offer a viable alternative. The viability of CS for sparse Volterra and polynomial models is the core theme of this work. A common sparse regression task is initially posed for the two models. Building on (weighted) Lasso-based schemes, an adaptive RLS-type algorithm is developed for sparse polynomial regressions. The identifiability of polynomial models is critically challenged by dimensionality. However, following the CS principle, when these models are sparse, they could be recovered by far fewer measurements. ...
Force prediction in cold rolling mills by polynomial methods
Directory of Open Access Journals (Sweden)
Nicu ROMAN
2007-12-01
Full Text Available A method for steel and aluminium strip thickness control is provided including a new technique for predictive rolling force estimation method by statistic model based on polynomial techniques.
Generalized Rayleigh and Jacobi Processes and Exceptional Orthogonal Polynomials
Chou, C.-I.; Ho, C.-L.
2013-09-01
We present four types of infinitely many exactly solvable Fokker-Planck equations, which are related to the newly discovered exceptional orthogonal polynomials. They represent the deformed versions of the Rayleigh process and the Jacobi process.
Log-concavity of the genus polynomials of Ringel Ladders
Directory of Open Access Journals (Sweden)
Jonathan L Gross
2015-10-01
Full Text Available A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polyno- mials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distribution of a symmetric ladder. Analysis of the resulting genus polynomial involves the use of Chebyshev polynomials. This paper continues our quest to affirm the quarter-century-old conjecture that all graphs have log-concave genus polynomials.
Maximum likelihood polynomial regression for robust speech recognition
Institute of Scientific and Technical Information of China (English)
LU Yong; WU Zhenyang
2011-01-01
The linear hypothesis is the main disadvantage of maximum likelihood linear re- gression （MLLR）. This paper applies the polynomial regression method to model adaptation and establishes a nonlinear model adaptation algorithm using maximum likelihood polyno
Certified counting of roots of random univariate polynomials
Cleveland, Joseph; Hauenstein, Jonathan D; Haywood, Ian; Mehta, Dhagash; Morse, Anthony; Robol, Leonardo; Schlenk, Taylor
2014-01-01
A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polynomials with two certification methods, namely Smale's $\\alpha$-theory and one based on Gerschgorin's theorem, for showing that a given numerical approximation is in the quadratic convergence region of Newton's method of some exact solution. With this combination, we can certifiably count the number of real roots of random polynomials. We quantify the difference between the two certification procedures and list the salient features of both of them. After benchmarking on random polynomials where the coefficients are drawn from the Gaussian distribution, we obtain novel experimental results for the Cauchy distribution case.
Kauffman polynomials of some links and invariants of 3-manifolds
Institute of Scientific and Technical Information of China (English)
李起升
2002-01-01
Kauffman bracket polynomials of the so-called generalized tree-like links are studied. An algorithm of Witten type invariants, which was defined by Blanchet and Habegger et al. of more general 3-manifolds is given.
Estimation in the polynomial errors-in-variables model
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
Estimators are presented for the coefficients of the polynomial errors-in-variables (EV) model when replicated observations are taken at some experimental points. These estimators are shown to be strongly consistent under mild conditions.
Guts of surfaces and the colored Jones polynomial
Futer, David; Purcell, Jessica
2013-01-01
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber for the knot; in particular, the surface is a fiber if and only if a particular coefficient vanishes. We also relate hyperbolic volume to colored Jones polynomials. Our method is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses, we show that these surfaces are essential, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to relate the pieces of the JSJ decomposition of the complement to the combinatorics of certain surface spines (state graphs). Since state graphs have p...
Pure states, positive matrix polynomials and sums of hermitian squares
Klep, Igor
2009-01-01
Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all real n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and the cartesian product of S with the real projective (t-1)-space. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for non-symmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.
A complete algorithm to find exact minimal polynomial by approximations
Qin, Xiaolin; Chen, Jingwei; Zhang, Jingzhong
2010-01-01
We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on obtaining an exact rational number from its approximation. The algorithm is applicable for finding exact minimal polynomial of an algebraic number by its approximate root. This also enables us to provide an efficient method of converting the rational approximation representation to the minimal polynomial representation, and devise a simple algorithm to factor multivariate polynomials with rational coefficients. Compared with the subsistent methods, our method combines advantage of high efficiency in numerical computation, and exact, stable results in symbolic computation. we also discuss some applications to some transcendental numbers by approximations. Moreover, the Digits of our algorithm is far less than the LLL-lattice basis reduction technique in theory. In this paper, we...
Quantization of gauge fields, graph polynomials and graph homology
Energy Technology Data Exchange (ETDEWEB)
Kreimer, Dirk, E-mail: kreimer@physik.hu-berlin.de [Humboldt University, 10099 Berlin (Germany); Sars, Matthias [Humboldt University, 10099 Berlin (Germany); Suijlekom, Walter D. van [Radboud University Nijmegen, 6525 AJ Nijmegen (Netherlands)
2013-09-15
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology. -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial.
Symmetric polynomials in information theory: Entropy and subentropy
Energy Technology Data Exchange (ETDEWEB)
Jozsa, Richard; Mitchison, Graeme [DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
2015-06-15
Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q.
Fast computation of interlace polynomials on graphs of bounded treewidth
Bläser, Markus
2009-01-01
We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to compute the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. Our algorithm uses 2^{4.5k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel. It is tailor-made for the interlace polynomial and uses linear algebra arguments concerning adjacency matrices of graphs. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n^4, where f(k) is doubly exponential in k.
Cubic Polynomials with Rational Roots and Critical Points
Gupta, Shiv K.; Szymanski, Waclaw
2010-01-01
If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.
Discrete Darboux transformation for discrete polynomials of hypergeometric type
Bangerezako, Gaspard
1998-03-01
The Darboux transformation, well known in second-order differential operator theory, is applied to the difference equations satisfied by the discrete hypergeometric polynomials (Charlier, Meixner-Kravchuk, Hahn).
An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem
Ephremidze, Lasha
2010-01-01
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
An operator approach to the Al-Salam-Carlitz polynomials
Chen, William Y. C.; Saad, Husam L.; Sun, Lisa H.
2010-04-01
We present an operator approach to Rogers-type formulas and Mehler's formula for the Al-Salam-Carlitz polynomials Un(x,y,a;q). By using the q-exponential operator, we obtain a Rogers-type formula, which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to Al-Salam and Carlitz ["Some orthogonal q-polynomials," Math. Nachr. 30, 47 (1965)]. By means of the Cauchy companion augmentation operator, we obtain an equivalent form of Mehler's formula. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials, which are extensions of the formulas for the Rogers-Szegö polynomials.
SOME EXTREMAL PROPERTIES OF THE INTEGRAL OF LEGENDRE POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
史应光; 王子玉
2001-01-01
Some extremal properties of the integral of Legendre polynomials are given, which are of independent interest. Meanwhile they show that a conjecture of P. Erdos[1] is plausible and maybe provides some means to prove this conjecture.
APPROXIMATION BY GENERALIZED MKZ-OPERATORS IN POLYNOMIAL WEIGHTED SPACES
Institute of Scientific and Technical Information of China (English)
Lucyna Rempulska; Mariola Skorupka
2007-01-01
We prove some approximation properties of generalized Meyer-K(o)nig and Zeller operators for differentiable functions in polynomial weighted spaces. The results extend some results proved in [ 1-3,7-16].
Asymptotic expansions of Feynman integrals of exponentials with polynomial exponent
Kravtseva, A. K.; Smolyanov, O. G.; Shavgulidze, E. T.
2016-10-01
In the paper, an asymptotic expansion of path integrals of functionals having exponential form with polynomials in the exponent is constructed. The definition of the path integral in the sense of analytic continuation is considered.
The Differential Dimension Polynomial for Characterizable Differential Ideals
Lange-Hegermann, Markus
2014-01-01
We generalize the differential dimension polynomial from prime differential ideals to characterizable differential ideals. Its computation is algorithmic, its degree and leading coefficient remain differential birational invariants, and it decides equality of characterizable differential ideals contained in each other.
Raising and Lowering Operators for Askey-Wilson Polynomials
Directory of Open Access Journals (Sweden)
Siddhartha Sahi
2007-01-01
Full Text Available In this paper we describe two pairs of raising/lowering operators for Askey-Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the ''classical'' properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.
Fast Parallel Computation of Polynomials Using Few Processors
DEFF Research Database (Denmark)
Valiant, Leslie G.; Skyum, Sven; Berkowitz, S.;
1983-01-01
It is shown that any multivariate polynomial of degree $d$ that can be computed sequentially in $C$ steps can be computed in parallel in $O((\\log d)(\\log C + \\log d))$ steps using only $(Cd)^{O(1)} $ processors.......It is shown that any multivariate polynomial of degree $d$ that can be computed sequentially in $C$ steps can be computed in parallel in $O((\\log d)(\\log C + \\log d))$ steps using only $(Cd)^{O(1)} $ processors....
Fast parallel computation of polynomials using few processors
DEFF Research Database (Denmark)
Valiant, Leslie; Skyum, Sven
1981-01-01
It is shown that any multivariate polynomial that can be computed sequentially in C steps and has degree d can be computed in parallel in 0((log d) (log C + log d)) steps using only (Cd)0(1) processors.......It is shown that any multivariate polynomial that can be computed sequentially in C steps and has degree d can be computed in parallel in 0((log d) (log C + log d)) steps using only (Cd)0(1) processors....
Asymptotic analysis of the Nörlund and Stirling polynomials
Directory of Open Access Journals (Sweden)
Mark Daniel Ward
2012-04-01
Full Text Available We provide a full asymptotic analysis of the N{\\"o}rlund polynomials and Stirling polynomials. We give a general asymptotic expansion---to any desired degree of accuracy---when the parameter is not an integer. We use singularity analysis, Hankel contours, and transfer theory. This investigation was motivated by a need for such a complete asymptotic description, with parameter 1/2, during this author's recent solution of Wilf's 3rd (previously Unsolved Problem.
Nonstandard decision methods for the solvability of real polynomial equations
Institute of Scientific and Technical Information of China (English)
曾广兴
1999-01-01
For a multivariate polynomial equation with coefficients in a computable ordered field, two criteria of this equation having real solutions are given. Based on the criteria, decision methods for the existence of real zeros and the semidefiniteness of binary polynomials are provided. With the aid of computers, these methods are used to solve several examples. The technique is to extend the original field involved in the question to a computable non-Archimedean ordered field containing infinitesimal elements.
First extension groups of Verma modules and $R$-polynomials
Abe, Noriyuki
2010-01-01
We study the first extension groups between Verma modules. There was a conjecture which claims that the dimensions of the higher extension groups between Verma modules are the coefficients of $R$-polynomials defined by Kazhdan-Lusztig. This conjecture was known as the Gabber-Joseph conjecture (although Gebber and Joseph did not state.) However, Boe gives a counterexample to this conjecture. In this paper, we study how far are the dimensions of extension groups from the coefficients of $R$-polynomials.
Multivariate Permutation Polynomial Systems and Nonlinear Pseudorandom Number Generators
Ostafe, Alina
2009-01-01
In this paper we study a class of dynamical systems generated by iterations of multivariate permutation polynomial systems which lead to polynomial growth of the degrees of these iterations. Using these estimates and the same techniques studied previously for inversive generators, we bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates on average over all initial values than in the general case and thus can be of use for pseudorandom number generation.
Solving Heat and Wave-Like Equations Using He's Polynomials
Directory of Open Access Journals (Sweden)
Syed Tauseef Mohyud-Din
2009-01-01
Full Text Available We use He's polynomials which are calculated form homotopy perturbation method (HPM for solving heat and wave-like equations. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that suggested technique solves nonlinear problems without using Adomian's polynomials is a clear advantage of this algorithm over the decomposition method.
JACOBI POLYNOMIALS USED TO INVERT THE LAPLACE TRANSFORM
Institute of Scientific and Technical Information of China (English)
A.Al-Shuaibi; F.Al-Rawjih
2004-01-01
Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.
CONVERGENCE ARTE FOR INTERATES OF q-BERNSTEIN POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein when n →∞ and convergence rate of Bn(f,q;x) when f ∈ Cn-1[0, 1], q →∞ are also presented.
Border Basis for Polynomial System Solving and Optimization
Trébuchet, Philippe; Mourrain, Bernard; Abril Bucero, Marta
2016-01-01
International audience; We describe the software package borderbasix dedicated to the computation of border bases and the solutions of polynomial equations. We present the main ingredients of the border basis algorithm and the other methods implemented in this package: numerical solutions from multiplication matrices, real radical computation, polynomial optimization. The implementation parameterized by the coefficient type and the choice function provides a versatile family of tools for poly...
On Some Inequalities Concerning Rate of Growth of Polynomials
Institute of Scientific and Technical Information of China (English)
Abdullah Mir; Imtiaz Hussain; Q. M. Dawood
2014-01-01
In this paper we consider a class of polynomials P(z)=a0+∑nv=t avzv, t≥1, not vanishing in|z|
Recognition of Arabic Sign Language Alphabet Using Polynomial Classifiers
2005-01-01
Building an accurate automatic sign language recognition system is of great importance in facilitating efficient communication with deaf people. In this paper, we propose the use of polynomial classifiers as a classification engine for the recognition of Arabic sign language (ArSL) alphabet. Polynomial classifiers have several advantages over other classifiers in that they do not require iterative training, and that they are highly computationally scalable with the number of classes. Based on...
Invariant hyperplanes and Darboux integrability of polynomial vector fields
Zhang Xia
2002-01-01
This paper is composed of two parts. In the first part, we provide an upper bound for the number of invariant hyperplanes of the polynomial vector fields in n variables. This result generalizes those given in Artes et al (1998 Pac. J. Math. 184 207-30) and Llibre and Rodriguez (2000 Bull. Sci. Math. 124 599-619). The second part gives an extension of the Darboux theory of integrability to polynomial vector fields on algebraic varieties.
ON THE COEFFICIENTS OF A POLYNOMIAL WITH RESTRICTED ZEROS
Institute of Scientific and Technical Information of China (English)
A.Aziz; W.M.Shah
2009-01-01
LetP(Z)=αn Zn + αn-1Zn-1 +…+α0 be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coeffi-cients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).
The Terminal Hosoya Polynomial of Some Families of Composite Graphs
Directory of Open Access Journals (Sweden)
Emeric Deutsch
2014-01-01
Full Text Available Let G be a connected graph and let Ω(G be the set of pendent vertices of G. The terminal Hosoya polynomial of G is defined as TH(G,t∶=∑x,y∈Ω(G:x≠ytdG(x,y, where dG(x,y denotes the distance between the pendent vertices x and y. In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs.
Difference oscillator in terms of the Meixner polynomials
Atakishiyev, Natig M.; Jafarov, Elchin I.; Nagiyev, Shakir M.; Wolf, Kurt B.
1998-07-01
We discuss a difference model of the linear harmonic oscillator based on the Meixner polynomials. As limit and special cases, it contains difference oscillator models in terms of the Kravchuk and Charlier polynomials, as well as the wavefunctions of the linear harmonic oscillator in quantum mechanics. We show that the dynamical group is SU(1,1) and construct explicitly the corresponding coherent state. The reproducing kernel for the wavefunctions of the Meixner model is also found.
Higher-Order Singular Systems and Polynomial Matrices
2005-01-01
There is a one-to-one correspondence between the set of quadruples of matrices defining singular linear time-invariant dynamical systems and a subset of the set of polynomial matrices. This correspondence preserves the equivalence relations introduced in both sets (feedback-similarity and strict equivalence): two quadruples of matrices are feedback-equivalent if, and only if, the polynomial matrices associated to them are also strictly equivalent. Los sistemas lineales singulares...
The Fractional Differential Polynomial Neural Network for Approximation of Functions
Directory of Open Access Journals (Sweden)
Rabha W. Ibrahim
2013-09-01
Full Text Available In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a generalization of input patterns. This method can be employed on data to describe modelling of complex systems. Furthermore, the total information is calculated by using the fractional Poisson process.
Higher order derivatives of R-Jacobi polynomials
Das, Sourav; Swaminathan, A.
2016-06-01
In this work, the R-Jacobi polynomials defined on the nonnegative real axis related to F-distribution are considered. Using their Sturm-Liouville system higher order derivatives are constructed. Orthogonality property of these higher ordered R-Jacobi polynomials are obtained besides their normal form, self-adjoint form and hypergeometric representation. Interesting results on the Interpolation formula and Gaussian quadrature formulae are obtained with numerical examples.
Polynomial Interpolation and Sums of Powers of Integers
Cereceda, José Luis
2017-01-01
In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P[subscript k](n) and Q[subscript k](n), such that P[subscript k](n) = Q[subscript k](n) = f[subscript k](n) for n = 1, 2,… , k, where f[subscript k](1), f[subscript k](2),… , f[subscript k](k) are k arbitrarily chosen…
Twisted exponential sums of polynomials in one variable
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
The twisted T-adic exponential sums associated to a polynomial in one variable are studied.An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums.This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.
Polynomial Estimates for c-functions on Reductive Symmetric Spaces
DEFF Research Database (Denmark)
van den Ban, Erik; Schlichtkrull, Henrik
2012-01-01
The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions.......The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions....
A novel single-order diffraction grating: Random position rectangle grating
Zuhua, Yang; Qiangqiang, Zhang; Jing, Wang; Quanping, Fan; Yuwei, Liu; Lai, Wei; Leifeng, Cao
2016-05-01
Spectral diagnosis of radiation from laser plasma interaction and monochromation of radiation source are hot and important topics recently. Grating is one of the primary optical elements to do this job. Although easy to fabricate, traditional diffraction grating suffers from multi-order diffraction contamination. On the other hand, sinusoidal amplitude grating has the nonharmonic diffraction property, but it is too difficult to fabricate, especially for x-ray application. A novel nonharmonic diffraction grating named random position rectangle grating (RPRG) is proposed in this paper. Theoretical analysis and experiment results show that the RPRG is both higher order diffraction suppressing and not difficult to fabricate. Additionally, it is highly efficient; its first order absolute theoretical diffraction efficiency reaches 4.1%. Our result shows that RPRG is a novel tool for radiation diagnosis and monochromation. Project supported by the National Natural Science Foundation of China (Grant No. 11375160) and the National Science Instruments Major Project of China (Grant No. 2012YQ130125).
DEFF Research Database (Denmark)
Taghizadeh, Alireza
This thesis deals with theoretical investigations of a newly proposed grating structure, referred to as hybrid grating (HG) as well as vertical cavity lasers based on the grating reflectors. The HG consists of a near-subwavelength grating layer and an unpatterned high-refractive-index cap layer...... behind these reflector and resonator properties are studied thoroughly. A HG structure comprising a III-V cap layer with a gain material and a Si grating layer enables the realization of a compact vertical cavity laser integrated on Si platform, which has a superior thermal property and fabrication...... directions, which is analogous to electronic quantum wells in conduction or valence bands. Several interesting configurations of heterostructures have been investigated and their potential in fundamental physics study and applications are discussed. For numerical and theoretical studies, a three...
Moniem, T. A.
2016-05-01
This article presents a methodology for an integrated Bragg grating using an alloy of GaAs, AlGaAs, and InGaAs with a controllable refractive index to obtain an adaptive Bragg grating suitable for many applications on optical processing and adaptive control systems, such as limitation and filtering. The refractive index of a Bragg grating is controlled by using an external electric field for controlling periodic modulation of the refractive index of the active waveguide region. The designed Bragg grating has refractive indices programmed by using that external electric field. This article presents two approaches for designing the controllable refractive indices active region of a Bragg grating. The first approach is based on the modification of a planar micro-strip structure of the iGaAs traveling wave as the active region, and the second is based on the modification of self-assembled InAs/GaAs quantum dots of an alloy from GaAs and InGaAs with a GaP traveling wave. The overall design and results are discussed through numerical simulation by using the finite-difference time-domain, plane wave expansion, and opto-wave simulation methods to confirm its operation and feasibility.
Factorization of colored knot polynomials at roots of unity
Kononov, Ya.; Morozov, A.
2015-07-01
HOMFLY polynomials are the Wilson-loop averages in Chern-Simons theory and depend on four variables: the closed line (knot) in 3d space-time, representation R of the gauge group SU (N) and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m = 1, HOMFLY polynomials in symmetric representations [ r ] satisfy recursion identity: Hr+m =Hr ṡHm for any A =qN, which is a generalization of the property Hr = H1r for special polynomials at m = 1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2 = e 2 πi / | R |, turns equal to the special polynomial with A substituted by A| R |, provided R is a single-hook representations (including arbitrary symmetric) - what provides a q - A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots - existence of such universal relations means that these variables are still not unconstrained.
Factorization of colored knot polynomials at roots of unity
Directory of Open Access Journals (Sweden)
Ya. Kononov
2015-07-01
Full Text Available HOMFLY polynomials are the Wilson-loop averages in Chern–Simons theory and depend on four variables: the closed line (knot in 3d space–time, representation R of the gauge group SU(N and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: Hr+m=Hr⋅Hm for any A=qN, which is a generalization of the property Hr=H1r for special polynomials at m=1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2=e2πi/|R|, turns equal to the special polynomial with A substituted by A|R|, provided R is a single-hook representations (including arbitrary symmetric – what provides a q−A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots – existence of such universal relations means that these variables are still not unconstrained.
A robust polynomial fitting approach for contact angle measurements.
Atefi, Ehsan; Mann, J Adin; Tavana, Hossein
2013-05-14
Polynomial fitting to drop profile offers an alternative to well-established drop shape techniques for contact angle measurements from sessile drops without a need for liquid physical properties. Here, we evaluate the accuracy of contact angles resulting from fitting polynomials of various orders to drop profiles in a Cartesian coordinate system, over a wide range of contact angles. We develop a differentiator mask to automatically find a range of required number of pixels from a drop profile over which a stable contact angle is obtained. The polynomial order that results in the longest stable regime and returns the lowest standard error and the highest correlation coefficient is selected to determine drop contact angles. We find that, unlike previous reports, a single polynomial order cannot be used to accurately estimate a wide range of contact angles and that a larger order polynomial is needed for drops with larger contact angles. Our method returns contact angles with an accuracy of contact angles in a wide range with a fourth-order polynomial. We show that this approach returns dynamic contact angles with less than 0.7° error as compared to ADSA-P, for the solid-liquid systems tested. This new approach is a powerful alternative to drop shape techniques for estimating contact angles of drops regardless of drop symmetry and without a need for liquid properties.
On Zeros of Self-Reciprocal Random Algebraic Polynomials
Directory of Open Access Journals (Sweden)
K. Farahmand
2008-01-01
Full Text Available This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(ÃŽÂ¸=Ã¢ÂˆÂ‘j=0NÃ¢ÂˆÂ’1{ÃŽÂ±NÃ¢ÂˆÂ’jcos(j+1/2ÃŽÂ¸+ÃŽÂ²NÃ¢ÂˆÂ’jsin(j+1/2ÃŽÂ¸}, where ÃŽÂ±j and ÃŽÂ²j, j=0,1,2,Ã¢Â€Â¦, NÃ¢ÂˆÂ’1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+ÃŽÂ¸/2. We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.
Orthogonality of Hermite polynomials in superspace and Mehler type formulae
Coulembier, Kevin; Sommen, Frank
2010-01-01
In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview of the results for O(m) and \\cG, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proven. Next, Hermite polynomials in a full superspace with O(m) x Sp(2n) symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schroedinger operators in superspace. Subsequently, a Mehler fo...
Accurate estimation of solvation free energy using polynomial fitting techniques.
Shyu, Conrad; Ytreberg, F Marty
2011-01-15
This report details an approach to improve the accuracy of free energy difference estimates using thermodynamic integration data (slope of the free energy with respect to the switching variable λ) and its application to calculating solvation free energy. The central idea is to utilize polynomial fitting schemes to approximate the thermodynamic integration data to improve the accuracy of the free energy difference estimates. Previously, we introduced the use of polynomial regression technique to fit thermodynamic integration data (Shyu and Ytreberg, J Comput Chem, 2009, 30, 2297). In this report we introduce polynomial and spline interpolation techniques. Two systems with analytically solvable relative free energies are used to test the accuracy of the interpolation approach. We also use both interpolation and regression methods to determine a small molecule solvation free energy. Our simulations show that, using such polynomial techniques and nonequidistant λ values, the solvation free energy can be estimated with high accuracy without using soft-core scaling and separate simulations for Lennard-Jones and partial charges. The results from our study suggest that these polynomial techniques, especially with use of nonequidistant λ values, improve the accuracy for ΔF estimates without demanding additional simulations. We also provide general guidelines for use of polynomial fitting to estimate free energy. To allow researchers to immediately utilize these methods, free software and documentation is provided via http://www.phys.uidaho.edu/ytreberg/software.
Universal Racah matrices and adjoint knot polynomials: Arborescent knots
Mironov, A.; Morozov, A.
2016-04-01
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SUN) and Kauffman (SON) polynomials. For E8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the "eigenvalue conjecture", which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6 × 6 case. The adjoint polynomials do not distinguish between mutants and therefore are not very efficient in knot theory, however, universal polynomials in higher representations can probably be better in this respect.