Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
Osipov, Vladimir Al
2010-01-01
Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of tau-functions, we (i) identify a zoo of hierarchical relations satisfied by tau-functions in an abstract infinite-dimensional space, and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasis is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.
Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT
Kieburg, Mario
2012-01-01
The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $\\beta=2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian as well as the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel-Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With help of these results we derive the unquenched $k$-point correlation function of the Hermitian and then non-Hermitian Wilson random matrix in terms of two flavour partition functions only. This representation is due to a Pfaffian factorization drastically...
Yuting eDeng
2013-07-01
Full Text Available 16S rRNA methylase and QepA, a fluoroquinolone efflux pump, are new mechanisms of resistance against aminoglycosides and fluoroquinolone, respectively. One of 16S rRNA methylase genes, rmtB, was found to be associated with qepA, were both located on the same transposable element. In this study, we intended to determine the current prevalence and characteristics of the 16S rRNA methylase genes and qepA, and to study the association between rmtB and qepA. A total of 892 Escherichia coli isolates were collected from various diseased food-producing animals in China from 2004-2008 and screened by PCR for 16S rRNA methylase genes and qepA. About 12.6% (112/892 and 0.1% (1/892 of isolates that were highly resistant to amikacin were positive for rmtB and armA, respectively. The remaining five 16S rRNA methlyase genes were not detected. Thirty-six (4.0% strains carried qepA. About 32.1% of rmtB-positive strains harbored qepA, which was not detected in rmtB-negative strains. Most strains were clonally unrelated, while identical PFGE profiles of rmtB-positive isolates were found in the same farm indicating clonal transmission. Conjugation experiments showed that rmtB was transfered to the recipients, and qepA also cotransfered with rmtB in some cases. The spread of E. coli of food animal origin harboring both rmtB and qepA suggests that surveillance for antimicrobial resistance of animal origin as well as the study of the mechanisms of resistance should be undertaken.
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.
Characteristic polynomials in real Ginibre ensembles
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)
Circular β ensembles, CMV representation, characteristic polynomials
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.
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 ...
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.
Resonance width distribution in RMT: Weak-coupling regime beyond Porter-Thomas
Fyodorov, Yan V.; Savin, Dmitry V.
2015-05-01
We employ the random matrix theory (RMT) framework to revisit the distribution of resonance widths in quantum chaotic systems weakly coupled to the continuum via a finite number M of open channels. In contrast to the standard first-order perturbation theory treatment we do not a priori assume the resonance widths being small compared to the mean level spacing. We show that to the leading order in weak coupling the perturbative χ^2M distribution of the resonance widths (in particular, the Porter-Thomas distribution at M = 1) should be corrected by a factor related to a certain average of the ratio of square roots of the characteristic polynomial (“spectral determinant”) of the underlying RMT Hamiltonian. A simple single-channel expression is obtained that properly approximates the width distribution also at large resonance overlap, where the Porter-Thomas result is no longer applicable.
Circular β ensembles,CMV representation,characteristic polynomials
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 β > 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.
Characteristic polynomials of linear polyacenes and their subspectrality
Gautam Mukherjee; Kakali Datta; Ashok K Mukherjee
2000-02-01
Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal’s triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP’s. Thus it has been shown that the entire eigenspectrum of an -ring LP is included in that of (2 + 1)-ring LP. Correspondence between the eigenspectra of linear chains and LP’s is brought out by a recently developed vertex-alternation and squaring algorithm.
Haar expectations of ratios of random characteristic polynomials
Huckleberry, A; Zirnbauer, M R
2007-01-01
We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n \\otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representatio...
Hjalmar Rosengren
2006-12-01
Full Text Available We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux and number theory (representation of integers as sums of squares.
Characteristic polynomial assignment in F-M model Ⅱ of 2-D systems
唐万生; 亢京力
2004-01-01
The problems of characteristic polynomial assignment in Fornasini-Marchesini (F-M) model Ⅱ of 2-D systems are investigated. The corresponding closed-loop systems described by F-M model Ⅱ are obtained via the state feedback.Using the algebraic geometry method, the characteristic polynomial assignment in the closed-loop systems is discussed. In terms of the theory of algebraic geometry, the problem of characteristic polynomial assignment is transferred to the one whether a rational mapping is onto. Sufficient conditions for almost arbitrary assignment coefficients of characteristic polynomial in F-M model Ⅱ of 2-D systems via state feedback are derived, and they are available for multi-input cases. It also has been shown that this method can be applied to assign the characteristic polynomial with output feedback. The sufficient conditions for almost arbitrary assignment coefficients of characteristic polynomial of multi-input 2-D systems described by F-M model Ⅱ with output feedback are established.
Lin You
2011-01-01
the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curve Cq: v2=up+au+b over the field Fq with q being a power of an odd prime p, Duursma and Sakurai obtained its characteristic polynomial for q=p, a=−1, and b∈Fp. In this paper, we determine the characteristic polynomials of Cq over the finite field Fpn for n=1, 2 and a, b∈Fpn. We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.
Sphere-cone-polynomial special window with good aberration characteristic
Wang Chao; Zhang Xin; Qu He-Meng; Wang Ling-Jie; Wang Yu
2013-01-01
Optical windows with external surfaces shaped to satisfy operational environment needs are known as special windows.A novel special window,a sphere-cone-polynomial (SCP) window,is proposed.The formulas of this window shape are given.An SCP MgF2 window with a fineness ratio of 1.33 is designed as an example.The field-of-regard (FOR) angle is ±75°.From the window system simulation results obtained with the calculated fluid dynamics (CFD) and optical design software,we find that compared to the conventional window forms,the SCP shape can not only introduce relatively less drag in the airflow,but also have the minimal effect on imaging.So the SCP window optical system can achieve a high image quality across a super wide FOR without adding extra aberration correctors.The tolerance analysis results show that the optical performance can be maintained with a reasonable fabricating tolerance to manufacturing errors.
New classes of test polynomials of polynomial algebras
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.
Kirillov O.N.
2007-01-01
Full Text Available Paradoxical effect of small dissipative and gyroscopic forces on the stability of a linear non-conservative system, which manifests itself through the unpredictable at first sight behavior of the critical non-conservative load, is studied. By means of the analysis of bifurcation of multiple roots of the characteristic polynomial of the non-conservative system, the analytical description of this phenomenon is obtained. As mechanical examples two systems possessing friction induced oscillations are considered: a mass sliding over a conveyor belt and a model of a disc brake describing the onset of squeal during the braking of a vehicle.
Roots of the derivative of the Riemann zeta function and of characteristic polynomials
Dueñez, Eduardo; Froehlich, Sara; Hughes, Chris; Mezzadri, Francesco; Phan, Toan
2010-01-01
We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.
Sums of Fourth Powers of Polynomials over a~Finite Field of Characteristic 3
Car, Mireille
2008-01-01
Let $F$ be a finite field with $q$ elements and characteristic $3.$ A sum $$M = M_{1}^4+\\ldots+ M_{s}^4$$ of fourth powers of polynomials $M_1,\\dots, M_{s}$ is a strict one if $ 4\\deg M_i 81$ is congruent to $1$ (mod. $4$), then $P$ is the strict sum of $9$ fourth powers; if $q=81$ or if $q>3$ is congruent to $3$ (mod $4$), then $P$ is the strict sum of $10$ fourth powers. If $q=3,...
Koutras, A; Kostopoulos, G K; Ioannides, A A
2008-03-01
This paper describes the theoretical background of a new data-driven approach to encephalographic single-trial (ST) data analysis. Temporal constrained source extraction using sparse decomposition identifies signal topographies that closely match the shape characteristics of a reference signal, one response for each ST. The correlations between these ST topographies are computed for formal Correlation Matrix Analysis (CMA) based on Random Matrix Theory (RMT). The RMT-CMA provides clusters of similar ST topologies in a completely unsupervised manner. These patterns are then classified into deterministic set and noise using well established RMT results. The efficacy of the method is applied to EEG and MEG data of somatosensory evoked responses (SERs). The results demonstrate that the method can recover brain signals with time course resembling the reference signal and follow changes in strength and/or topography in time by simply stepping the reference signal through time.
Amirhossein Amiri
2012-03-01
Full Text Available Profile monitoring is a new research subject in Statistical Process Control which has been recently considered by many researchers. Profile describes the relationship between a response variable and one or more independent variables. This relationship is often modeled using regression which can be simple linear, multiple linear, polynomial or sometimes nonlinear. To monitor polynomial profiles, some methods have been developed, the best of which is the orthogonal polynomial approach. One of the disadvantages of this method is the large number of control charts which are simultaneously used. In this paper, a new approach has been proposed based on orthogonal polynomial approach, in which only two control charts are used to monitor a kth-order polynomial profile. Simulation findings and average run length (ARL curve analysis imply better performance of the proposed approach compared to the existing approach. Also, using the proposed approach is much easier in practice.
Afanasiev, Ie.
2016-04-01
We consider asymptotics of the correlation functions of characteristic polynomials corresponding to random weighted G(n, p/n) Erdős-Rényi graphs with Gaussian weights in the case of finite p and also when p rightarrow infty . It is shown that for finite p the second correlation function demonstrates a kind of transition: when p 2 there appears an interval (-λ _*(p), λ _*(p)) such that for λ _0 in (-λ _*(p), λ _*(p)) the second correlation function behaves like that for Gaussian unitary ensemble (GUE), while for λ _0 outside the interval the second correlation function is still factorized. For p rightarrow infty there is also a threshold in the behavior of the second correlation function near λ _0 = ± 2: for p ≪ n^{2/3} the second correlation function factorizes, whereas for p ≫ n^{2/3} it behaves like that for GUE. For any rate of p rightarrow infty the asymptotics of correlation functions of any even order for λ _0 in (-2, 2) coincide with that for GUE.
Nonlinear Resistor with Polynomial AV Characteristics and Its Application in Chaotic Oscillator
V. Pospisil
2004-06-01
Full Text Available This paper shows the realization of two terminal devices with anarbitrary polynomial nonlinearity up to the fifth order. The proposeddesign procedure is completely systematic using minimum of components.The very heart of our conception is four-channel four-quadrant analogmultiplier MLT04. The implementation of synthesized nonlinear resistoras a general nonlinearity in chaotic oscillator is also presented andexperimentally verified.
WANG Peng-Zhou; ZHANG Hong-Hao; ZHANG Shun-Li; YAN Wen-Bin; LI Xue-Song
2008-01-01
By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and we further obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace formulae of the Cayley-Hamilton's theorem with all coefficients explicitly given. This implies a byproduct, a complete expression for the determinant of any finite-dimensional matrix in terms of the traces of its successive powers.And we discuss some of their applications to chiral perturbation theory and general relativity.
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
Gonzalez, H.E. [ININ, 52045 Ocoyoacac, Estado de Mexico (Mexico)
2006-07-01
In linear algebra, one can associate an equation to each square matrix: its characteristic equation or secular equation. Starting from this equation, the one characteristic polynomial that codes several important properties of the matrix is obtained: its own values, it determinant and it appearance. The first method to calculate those coefficients of this polynomial were proposed by the french astronomer Urbain Jean Joseph Le Verrier (1811-1877), from then on, many methods have intended to calculate these coefficients. In this work the author proposes a new one method and a bibliographical citation is given where the calculations with others methods that know each other for it, taking like reference the matrix used by Le Verrier are explained. It was concluded that it here proposed, besides being the only mexican method that is knew, has the advantage of being very easy of understanding and of calculating well, in the operations that it carries out, it doesn't use the division and it avoids fractions in matrices whose entrances are whole. This has a great importance for their use in the classroom for their great didactic value and in nuclear reactors and Genetic Engineering. (Author)
An Effective Method of Monitoring the Large-Scale Traffic Pattern Based on RMT and PCA
Jia Liu
2010-01-01
Full Text Available Mechanisms to extract the characteristics of network traffic play a significant role in traffic monitoring, offering helpful information for network management and control. In this paper, a method based on Random Matrix Theory (RMT and Principal Components Analysis (PCA is proposed for monitoring and analyzing large-scale traffic patterns in the Internet. Besides the analysis of the largest eigenvalue in RMT, useful information is also extracted from small eigenvalues by a method based on PCA. And then an appropriate approach is put forward to select some observation points on the base of the eigen analysis. Finally, some experiments about peer-to-peer traffic pattern recognition and backbone aggregate flow estimation are constructed. The simulation results show that using about 10% of nodes as observation points, our method can monitor and extract key information about Internet traffic patterns.
Fyodorov, Yan V.; Doussal, Pierre Le
2016-07-01
We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index H → 0 (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the freezing-duality conjecture (FDC). Here we study the PDF of the position of the maximum x_m through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the β -Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary β >0 and positive integer n in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix 1 from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit n → 0 and to negative Dyson index β → -2, we obtain the moments of x_m and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.
RmtC and RmtF 16S rRNA Methyltransferase in NDM-1–Producing Pseudomonas aeruginosa
Rahman, Mohibur; Prasad, Kashi Nath; Pathak, Ashutosh; Pati, Binod Kumar; Singh, Avinash; Ovejero, Cristina M.; Ahmad, Saheem; Gonzalez-Zorn, Bruno
2015-01-01
We investigated 16S rRNA methyltransferases in 38 bla NDM-1–positive Pseudomonas aeruginosa isolates and found RmtC in 3 isolates, 1 of which also harbored RmtF. The isolates were clonally unrelated; rmtC and rmtF genes were located on a chromosome with the bla NDM-1 gene. Strategies are needed to limit the spread of such isolates.
Jaejoon Kim
2016-08-01
Full Text Available Development of eddy current testing makes possible for the inspection of the conductivity variations, inspection of surface and beneath the surface of conductivity materials and liftoff property to characterize nonconductive materials.This paper deals with impedance analysis and analyzes liftoff distance effect using numerical method on eddy current testing. The focus of this paper is to first characterize impedance values under measurement with ferrite pot core probe in eddy current testing. Pot coreshaping probe used in this work offers a number of advantages, including self-shielding, space efficiency, convenience, good temperature stability and low losses. In addition, utilization of pot core shaped probe makes the edge effect on test sample avoidable. Secondly, a liftoff distance effect using a polynomial model is introduced and can be mitigated on zero of liftoff distance effect for measured data. This research allows to present the possibility of pot core shaped probe utilization and analytical approach for lift off distance effect.
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...
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.
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…
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.
Li Jun-Jie
2012-12-01
Full Text Available Abstract Background Emergence of rmtB-positive Klebsiella pneumoniae carbapenemase (KPC-producing K. pneumoniae (KPC-KP poses a great threat to antimicrobial treatment options. Methods From January 2010 to December 2010, non-duplicate KPC-KP isolates from our hospital were screened for rmtB and multiple other resistance determinants with PCR. Subsequent studies included MIC determination, PFGE, and multilocus sequence typing. Records from patients with KPC-KP isolated were retrospectively reviewed. Comparisons of molecular and clinical characteristics between rmtB-positive and rmtB–negative isolates were systematically performed, as well as the environmental colonization study in ICU wards. Results A total of 84 KPC-KP strains were collected, including 48 rmtB-positive KPC-KP (RPKP and 36 rmtB-negative KPC-KP (RNKP isolates. All KPC-KP isolates were multidrug resistant, with colistin and tigecycline being the most active agents. Compared with RNKP, RPKP displayed a much severer resistance phenotype. Susceptibility rates for amikacin (0% for RPKP versus 88.9% for RNKP, p p p Conclusions RPKP strains have spread widely and gradually replaced RNKP in our hospital. They seemed to show much severer resistance phenotypes compared with RNKP and had a bigger dissemination potential. Prudent use of available active agents combined with good control practices is therefore mandatory.
Narkiewicz, Wŀadysŀaw
1995-01-01
The book deals with certain algebraic and arithmetical questions concerning polynomial mappings in one or several variables. Algebraic properties of the ring Int(R) of polynomials mapping a given ring R into itself are presented in the first part, starting with classical results of Polya, Ostrowski and Skolem. The second part deals with fully invariant sets of polynomial mappings F in one or several variables, i.e. sets X satisfying F(X)=X . This includes in particular a study of cyclic points of such mappings in the case of rings of algebrai integers. The text contains several exercises and a list of open problems.
Chang, T. S.
1974-01-01
A numerical scheme using the method of characteristics to calculate the flow properties and pressures behind decaying shock waves for materials under hypervelocity impact is developed. Time-consuming double interpolation subroutines are replaced by a technique based on orthogonal polynomial least square surface fits. Typical calculated results are given and compared with the double interpolation results. The complete computer program is included.
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-
Bueno, Maria Fernanda C.; Francisco, Gabriela R.; O'Hara, Jessica A.; de Oliveira Garcia, Doroti; Doi, Yohei
2013-01-01
Eight Klebsiella pneumoniae clinical strains with high-level aminoglycoside resistance were collected from eight hospitals in São Paulo State, Brazil, in 2010 and 2011. Three of them produced an RmtD group 16S rRNA methyltransferase, RmtD1 or RmtD2. Five strains were found to produce a novel 16S rRNA methyltransferase, designated RmtG, which shared 57 to 58% amino acid identity with RmtD1 and RmtD2. Seven strains coproduced KPC-2 with or without various CTX-M group extended-spectrum β-lactama...
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.
Oblivious Polynomial Evaluation
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.
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.
A Study on the Randomness of Stock Prices by Using the RMT-Test
Yang, Xin; Tanka-Yamawaki, Mieko
The authors proposed earlier a new method to measure the randomness of long sequences based on the Random Matrix Theory (RMT), which is called as the RMT-test, and proved its effectiveness by distinguishing subtle differences among the randomness of various random numbers generated by physical or algorithmic generators. However, the real application of this method would be on the real data of relatively low randomness. In this article, the RMT-test is applied to measure the randomness of intraday stock price time series. It is found that the stocks having high randomness tend to perform better in the next year, in comparison to the stocks of low randomness.
The Characteristic Polynomials of a Class of Graphical Arrangements%一类图构形的特征多项式
高瑞梅
2015-01-01
超平面构形是奇点理论的一个分支,它是一类具有非孤立奇点的超曲面.超平面构形是处在组合学、代数学、拓扑学、代数几何学等多个学科交汇处的一门年轻的学科,它的巨大魅力在于:能从组合学以及代数学等不同角度去描述它的拓扑不变量.特征多项式作为构形的一个组合不变量,在构形组合、代数、拓扑性质的研究中,起到非常重要的作用.本文利用图论中的顶点着色理论给出一类特殊图构形的特征多项式.%The arrangement of hyperplanes,which is a branch of singularity theory,is a hypersurface with non-isolated singularities. The arrangement of hyperplanes is a new interdisciplinary field which comes from combinatorics, algebra, topology,algebraic geometry and so on. The greatest charm of the theory of hyperplane arrangements is expressing to-pological invariants of the complement space in terms of combinatorics and algebra. The characteristic polynomials are combinatorial invariants for hyperplane arrangements, and play important roles in the studying of the properties of the aspects of combinatorics, algebra and topology. In this paper, the characteristic polynomials of a class of graphical ar-rangements are established by the chromatic theory of the vertices graphs.
Automatic Trading Agent. RMT based Portfolio Theory and Portfolio Selection
Snarska, M; Snarska, Malgorzata; Krzych, Jakub
2006-01-01
Portfolio theory is a very powerful tool in the modern investment theory. It is helpful in estimating risk of an investor's portfolio, which arises from our lack of information, uncertainty and incomplete knowledge of reality, which forbids a perfect prediction of future price changes. Despite of many advantages this tool is not known and is not widely used among investors on Warsaw Stock Exchange. The main reason for abandoning this method is a high level of complexity and immense calculations. The aim of this paper is to introduce an automatic decision - making system, which allows a single investor to use such complex methods of Modern Portfolio Theory (MPT). The key tool in MPT is an analysis of an empirical covariance matrix. This matrix, obtained from historical data is biased by such a high amount of statistical uncertainty, that it can be seen as random. By bringing into practice the ideas of Random Matrix Theory (RMT), the noise is removed or significantly reduced, so the future risk and return are b...
Transversals of Complex Polynomial Vector Fields
Dias, Kealey
, 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......Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied...... a concrete polynomial, it seems to take quite a bit of work to prove that it is generic, i.e. structurally stable. This has been done for a special class of degree d polynomial vector fields having simple equilibrium points at the d roots of unity, d odd. In proving that such vector fields are generic...
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)
STABILITY SYSTEMS VIA HURWITZ POLYNOMIALS
BALTAZAR AGUIRRE HERNÁNDEZ
2017-01-01
Full Text Available To analyze the stability of a linear system of differential equations ẋ = Ax we can study the location of the roots of the characteristic polynomial pA(t associated with the matrix A. We present various criteria - algebraic and geometric - that help us to determine where the roots are located without calculating them directly.
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...
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...
Branched polynomial covering maps
Hansen, Vagn Lundsgaard
1999-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...
Branched polynomial covering maps
Hansen, Vagn Lundsgaard
2002-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...
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.
Identification of near-surface fault structure using radio magnetotelluric (RMT) method
Saputra, Rifki Mega; Widodo
2017-07-01
This study area of Mygdonian Basin in Northern Greece is one of the most active seismic zones in Europe. It is the location where was affected by the largest recent earthquake in 1978 with magnitude 6.5. The geology of this area consist four major units, which are the lower terrace deposit, Holocene deposit, fans, and the metamorphic. Measurement using RMT was intended to get a better understanding of the near-surface fault structure. We used frequency ranges from 16.365 kHz to 863.930 kHz. Due to high frequency and high resistivity in the top layer, the approximation of skin depth is about 35 meters. Data were collected along one profile which had 1,250 meters in length and 50 meters of each station distance, so the RMT soundings gave a total number of 26 data. The RMT model can indicate the surface fault structure.
Weierstrass polynomials for links
Hansen, Vagn Lundsgaard
1997-01-01
There is a natural way of identifying links in3-space with polynomial covering spaces over thecircle. Thereby any link in 3-space can be definedby a Weierstrass polynomial over the circle. Theequivalence relation for covering spaces over thecircle is, however, completely different from...... that for links in 3-space. This paper initiates a study of the connections between polynomial covering spaces over the circle and links in 3-space....
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.
Isolation and characterization of highly replicable hepatitis C virus genotype 1a strain HCV-RMT.
Arai, Masaaki; Tokunaga, Yuko; Takagi, Asako; Tobita, Yoshimi; Hirata, Yuichi; Ishida, Yuji; Tateno, Chise; Kohara, Michinori
2013-01-01
Multiple genotype 1a clones have been reported, including the very first hepatitis C virus (HCV) clone called H77. The replication ability of some of these clones has been confirmed in vitro and in vivo, although this ability is somehow compromised. We now report a newly isolated genotype 1a clone, designated HCV-RMT, which has the ability to replicate efficiently in patients, chimeric mice with humanized liver, and cultured cells. An authentic subgenomic replicon cell line was established from the HCV-RMT sequence with spontaneous introduction of three adaptive mutations, which were later confirmed to be responsible for efficient replication in HuH-7 cells as both subgenomic replicon RNA and viral genome RNA. Following transfection, the HCV-RMT RNA genome with three adaptive mutations was maintained for more than 2 months in HuH-7 cells. One clone selected from the transfected cells had a high copy number, and its supernatant could infect naïve HuH-7 cells. Direct injection of wild-type HCV-RMT RNA into the liver of chimeric mice with humanized liver resulted in vigorous replication, similar to inoculation with the parental patient's serum. A study of virus replication using HCV-RMT derivatives with various combinations of adaptive mutations revealed a clear inversely proportional relationship between in vitro and in vivo replication abilities. Thus, we suggest that HCV-RMT and its derivatives are important tools for HCV genotype 1a research and for determining the mechanism of HCV replication in vitro and in vivo.
Isolation and characterization of highly replicable hepatitis C virus genotype 1a strain HCV-RMT.
Masaaki Arai
Full Text Available Multiple genotype 1a clones have been reported, including the very first hepatitis C virus (HCV clone called H77. The replication ability of some of these clones has been confirmed in vitro and in vivo, although this ability is somehow compromised. We now report a newly isolated genotype 1a clone, designated HCV-RMT, which has the ability to replicate efficiently in patients, chimeric mice with humanized liver, and cultured cells. An authentic subgenomic replicon cell line was established from the HCV-RMT sequence with spontaneous introduction of three adaptive mutations, which were later confirmed to be responsible for efficient replication in HuH-7 cells as both subgenomic replicon RNA and viral genome RNA. Following transfection, the HCV-RMT RNA genome with three adaptive mutations was maintained for more than 2 months in HuH-7 cells. One clone selected from the transfected cells had a high copy number, and its supernatant could infect naïve HuH-7 cells. Direct injection of wild-type HCV-RMT RNA into the liver of chimeric mice with humanized liver resulted in vigorous replication, similar to inoculation with the parental patient's serum. A study of virus replication using HCV-RMT derivatives with various combinations of adaptive mutations revealed a clear inversely proportional relationship between in vitro and in vivo replication abilities. Thus, we suggest that HCV-RMT and its derivatives are important tools for HCV genotype 1a research and for determining the mechanism of HCV replication in vitro and in vivo.
Wavepacket dynamics in energy space, RMT and Quantum-Classical correspondence
Cohen, D; Kottos, T; Cohen, Doron; Izrailev, Felix M.; Kottos, Tsampikos
1999-01-01
We apply random-matrix-theory (RMT) to the analysis of evolution of wavepackets in energy space. We study the crossover from ballistic behavior to saturation, the possibility of having an intermediate diffusive behavior, and the feasibility of strong localization effect. Both theoretical considerations and numerical results are presented. Using quantal-classical correspondence (QCC) considerations we question the validity of the emerging dynamical picture. In particular we claim that the appearance of the intermediate diffusive behavior is possibly an artifact of the RMT strategy.
Polynomial Fibonacci-Hessenberg matrices
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.
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 ...
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…
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…
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.
Nonnegativity of uncertain polynomials
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.
Aspergillus flavus is found colonizing numerous oil seed crops such as corn, peanuts, sorghum, treenuts and cotton worldwide, contaminating them with aflatoxin and other harmful potent toxins. In the phylogenetically related model fungus Aspergillus nidulans, the methyltransferase, RmtA, has been de...
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...
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.
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.
Identification of rat mammary tumor-1 gene (RMT-1), which is highly expressed in rat mammary tumors.
Chiou, S; Yoo, J; Loh, K C; Guzman, R C; Gopinath, G R; Rajkumar, L; Chou, Y C; Yang, J; Popescu, N C; Nandi, S
2001-12-10
Full-term pregnancy early in life results in a permanent reduction in lifetime breast cancer risk in women. Parous rats and mice are also refractory to chemical carcinogenesis. Therefore, investigation of the differences between mammary glands from virgin and parous rats would provide valuable information regarding the protective effects of early full-term pregnancy. In this report, we examined the gene expression patterns in mammary glands from virgin and parous Lewis rats. Using differential display technology, a novel 4.2 kb cDNA, designated rat mammary tumor-1 (RMT-1) was isolated. Northern blot analysis of RMT-1 showed that RMT-1 expression was higher in the pre-pubertal and pubertal stages during rat mammary gland development while it was down-regulated in mammary glands from mature virgin and parous rats. RMT-1 expression was highest in rat mammary cancers compared with either the mammary glands of virgin or parous rats. At the Northern blot sensitivity level, RMT-1 expression was found only in the mammary gland. Northern blot analysis also showed that the expression of this gene was found in 74% of N-methyl-nitrosourea (MNU)-induced mammary cancers while it was not found in MNU-induced cancers from other organs. The examination of the RMT-1 gene structure revealed that it consists of five exons spanning 5.9 kb. Using fluorescence in situ hybridization, the gene was localized on rat chromosome 1 band q 43-51. The present data show that there is a correlation between high RMT-1 expression and rat mammary carcinogenesis or decreased RMT-1 expression and parity associated refractoriness to chemically induced mammary carcinogenesis. However, whether or not RMT-1 gene has a functional role in these processes remains to be investigated.
Jun Li; Ming-Xiang Zou; Hai-Chen Wang; Qing-Ya Dou; Yong-Mei Hu; Qun Yan; Wen-En Liu
2016-01-01
Background:Klebsiellapneumoniae carbapenemase (KPC)-producing K.pneumoniae bacteria,which cause serious disease outbreaks worldwide,was rarely detected in Xiangya Hospital,prior to an outbreak that occurred from August 4,2014,to March 17,2015.The aim of this study was to analyze the epidemiology and molecular characteristics of the K.pneumoniae strains isolated during the outbreak.Methods:Nonduplicate carbapenem-resistant K.pneumoniae isolates were screened for blaKPC-2 and multiple other resistance determinants using polymerase chain reaction.Subsequent studies included pulsed-field gel electrophoresis (PFGE),multilocus sequence typing,analysis ofplasmids,and genetic organization of blaKPC-2 locus.Results:Seventeen blaKPC-2-positive K.pneumoniae were identified.A wide range of resistant determinants was detected.Most isolates (88.2％) coharbored blaKPC-2 and rmtB in addition to other resistance genes,including blasHV4,blaTEM4,and aac(3)-Ⅱa.The blaKPC-2 and rmtB genes were located on the conjugative IncFIB-type plasmid.Genetic organization of blaKPC-2 locus in most strains was consistent with that of the plasmid pKP048.Four types (A1,A2,A3,and B) were detected by PFGE,and Type A1,an ST11,was the predominant PFGE type.A novel K.pneumoniae sequence type (ST 1883) related to ST ll was discovered.Conclusions:These isolates in our study appeared to be clonal and ST11 K.pneumoniae was the predominant clone attributed to the outbreak.Coharbing of blaKPC-2 and rmtB,which were located on a transferable plasmid,in clinical K.pneumoniae isolates may lead to the emergence of a new pattern of drug resistance.
Collective Correlations of Brodmann Areas fMRI Study with RMT-Denoising
Burda, Zdzislaw; Kornelsen, Jennifer; Nowak, Maciej A.; Porebski, Bartosz; Sboto-Frankenstein, Uta; Tomanek, Boguslaw; Tyburczyk, Jacek
2013-01-01
We study collective behavior of Brodmann regions of human cerebral cortex using functional Magnetic Resonance Imaging (fMRI) and Random Matrix Theory (RMT). The raw fMRI data is mapped onto the cortex regions corresponding to the Brodmann areas with the aid of the Talairach coordinates. Principal Component Analysis (PCA) of the Pearson correlation matrix for 41 different Brodmann regions is carried out to determine their collective activity in the idle state and in the active state stimulated...
Gutierrez, Belen; Escudero, Jose A; San Millan, Alvaro; Hidalgo, Laura; Carrilero, Laura; Ovejero, Cristina M; Santos-Lopez, Alfonso; Thomas-Lopez, Daniel; Gonzalez-Zorn, Bruno
2012-05-01
Arm/Rmt methyltransferases have emerged recently in pathogenic bacteria as enzymes that confer high-level resistance to 4,6-disubstituted aminoglycosides through methylation of the G1405 residue in the 16S rRNA (like ArmA and RmtA to -E). In prokaryotes, nucleotide methylations are the most common type of rRNA modification, and they are introduced posttranscriptionally by a variety of site-specific housekeeping enzymes to optimize ribosomal function. Here we show that while the aminoglycoside resistance methyltransferase RmtC methylates G1405, it impedes methylation of the housekeeping methyltransferase RsmF at position C1407, a nucleotide that, like G1405, forms part of the aminoglycoside binding pocket of the 16S rRNA. To understand the origin and consequences of this phenomenon, we constructed a series of in-frame knockout and knock-in mutants of Escherichia coli, corresponding to the genotypes rsmF(+), ΔrsmF, rsmF(+) rmtC(+), and ΔrsmF rmtC(+). When analyzed for the antimicrobial resistance pattern, the ΔrsmF bacteria had a decreased susceptibility to aminoglycosides, including 4,6- and 4,5-deoxystreptamine aminoglycosides, showing that the housekeeping methylation at C1407 is involved in intrinsic aminoglycoside susceptibility in E. coli. Competition experiments between the isogenic E. coli strains showed that, contrary to expectation, acquisition of rmtC does not entail a fitness cost for the bacterium. Finally, matrix-assisted laser desorption ionization (MALDI) mass spectrometry allowed us to determine that RmtC methylates the G1405 residue not only in presence but also in the absence of aminoglycoside antibiotics. Thus, the coupling between housekeeping and acquired methyltransferases subverts the methylation architecture of the 16S rRNA but elicits Arm/Rmt methyltransferases to be selected and retained, posing an important threat to the usefulness of aminoglycosides worldwide.
DARBOUX POLYNOMIALS AND NON-ALGEBRAIC INTEGRABILITY OF THE L SYSTEM
无
2009-01-01
In this paper, we characterize all of the Darboux polynomials of the L system and prove that the system is not algebraically integrable, using the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.
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
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.
Mehta, Suman; Bastani, Mehrdad; Malehmir, Alireza; Wang, Shunguo; Pedersen, Laust
2014-05-01
The Radio-MagnetoTelluric (RMT) method uses the electromagnetic signal from distant radio transmitters in the frequency range 15 to 250 kHz. RMT applications in near-surface studies have already been well established. Two components of electric field and three components of magnetic field are measured. These measured components are related to each other via transfer functions which contain detailed information about the variation of electrical resistivity of the subsurface. The present study is carried out in the frame of TRUST (TRansparent Underground STructure) project supported by several research and public organizations as well as industry. The study area is located close to central Stockholm in Sweden where the Swedish traffic authority has planned to construct a 21-km long motorway to bypass the city. In order to reduce the impact on natural and cultural environments, 18 km of the motorway will be located in tunnels. The main objective of this study is thus to identify potential fracture zones and faults as well as the general geological settings. The proposed path of the tunnel partly passes under the Lake Mälaren at a depth of about 60 m. Thus a challenge was posed on the applicability of RMT method in shallow water environments. Successful applications of RMT measurements using the Uppsala University's EnviroMT system on land encouraged us to modify the system to acquire data over lake water especially in urban areas. Pioneered by the Geological Survey of Sweden (SGU), RMT data were collected over the Lake Mälaren in spring 2012. The prototype acquisition system did not only turn out to be appropriate for such a challenging environment, but it was also much more efficient as compared with land surveys. Fifty two lines including 1160 stations with an average spacing of 15 m were covered in three days. Cultural noise associated with the city-related environment had to be identified and filtered out before inversion could be carried out. Reliable estimates
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.…
On Generalized Bell Polynomials
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.
Complexity of Ising Polynomials
Kotek, Tomer
2011-01-01
This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weight values. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by L. A. Goldberg, M. Jerrum and M. Patersonin 2003. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied in by D. Andr\\'{e}n and K. Markstr\\"{o}m in 2009. We consider the complexity of Z(G;t,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well-known to be closely related to the Potts model in the absence of an external field. We show that Z(G;\\x,\\y,\\z) is #P-hard to evaluate at all points in $mathbb{Q}^3$, except those in ...
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 ...
ON PROPERTIES OF DIFFERENCE POLYNOMIALS
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).
Computing the Alexander Polynomial Numerically
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....
Explicit Formulas for Meixner Polynomials
Dmitry V. Kruchinin
2015-01-01
Full Text Available Using notions of composita and composition of generating functions, we show an easy way to obtain explicit formulas for some current polynomials. Particularly, we consider the Meixner polynomials of the first and second kinds.
Chromatic polynomials for simplicial complexes
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...
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…
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
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.
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…
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.
RMT focal plane sensitivity to seismic network geometry and faulting style
Johnson, Kendra L.; Hayes, Gavin P.; Herrmann, Robert B.; Benz, Harley M.; McNamara, Dan E.; Bergman, Eric
2016-07-01
Modern tectonic studies often use regional moment tensors (RMTs) to interpret the seismotectonic framework of an earthquake or earthquake sequence; however, despite extensive use, little existing work addresses RMT parameter uncertainty. Here, we quantify how network geometry and faulting style affect RMT sensitivity. We examine how data-model fits change with fault plane geometry (strike and dip) for varying station configurations. We calculate the relative data fit for incrementally varying geometries about a best-fitting solution, applying our workflow to real and synthetic seismograms for both real and hypothetical station distributions and earthquakes. Initially, we conduct purely observational tests, computing RMTs from synthetic seismograms for hypothetical earthquakes and a series of well-behaved network geometries. We then incorporate real data and station distributions from the International Maule Aftershock Deployment (IMAD), which recorded aftershocks of the 2010 MW 8.8 Maule earthquake, and a set of regional stations capturing the ongoing earthquake sequence in Oklahoma and southern Kansas. We consider RMTs computed under three scenarios: (1) real seismic records selected for high data quality; (2) synthetic seismic records with noise computed for the observed source-station pairings and (3) synthetic seismic records with noise computed for all possible station-source pairings. To assess RMT sensitivity for each test, we observe the `fit falloff', which portrays how relative fit changes when strike or dip varies incrementally; we then derive the ranges of acceptable strikes and dips by identifying the span of solutions with relative fits larger than 90 per cent of the best fit. For the azimuthally incomplete IMAD network, Scenario 3 best constrains fault geometry, with average ranges of 45° and 31° for strike and dip, respectively. In Oklahoma, Scenario 3 best constrains fault dip with an average range of 46°; however, strike is best constrained by
The RMT method for describing many-electron atoms in intense short laser pulses
Lysaght, M. A.; Moore, L. R.; Nikolopoulos, L. A. A.; Parker, J. S.; van der Hart, H. W.; Taylor, K. T.
2012-11-01
We describe how we have developed an ab initio R-Matrix incorporating Time (RMT) method to provide an accurate description of the single ionization of a general many-electron atom exposed to short intense laser pulses. The new method implements the "division-of-space" concept central to R-matrix theory and takes over the sophisticated time-propagation algorithms of the HELIUM code. We have tested the accuracy of the new method by calculating multiphoton ionization rates of He and Ne and have found excellent agreement with other highly accurate and well-established methods.
Complex Polynomial Vector Fields
Dias, Kealey
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...... 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....
Complex Polynomial Vector Fields
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
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...
Stability properties of autonomous homogeneous polynomial differential systems
Samardzija, Nikola
A geometrical approach is used to derive a generalized characteristic value problem for dynamic systems described by homogeneous polynomials. It is shown that a nonlinear homogeneous polynomial system possesses eigenvectors and eigenvalues, quantities normally associated with a linear system. These quantities are then employed in studying stability properties. The necessary and sufficient conditions for all forms of stabilities characteristic of a two-dimensional system are provided. This result, together with the classical theorem of Frommer, completes a stability analysis for a two-dimensional homogeneous polynomial system.
Geometry dependence of RMT-based methods to extract the low-energy constants Sigma and F
Lehner, Christoph; Hashimoto, Shoji; Wettig, Tilo
2011-01-01
The lowest-order low-energy constants $\\Sigma$ and $F$ of chiral pertubation theory can be extracted from lattice data using methods based on the equivalence of random matrix theory (RMT) and QCD in the epsilon regime. We discuss how the choice of the lattice geometry affects such methods. In particular, we show how to minimize systematic deviations from RMT by an optimal choice of the lattice geometry in the case of two light quark flavors. We illustrate our findings by determining $\\Sigma$ and $F$ from lattice configurations with two dynamical overlap fermions generated by JLQCD, using two different lattice geometries.
Complex Polynomial Vector Fields
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...
Multivariable q-Racah polynomials
Van Diejen, J F
1996-01-01
The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) q-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit q\\rightarrow 1 leading to multivariable Racah type polynomials. Of special interest is the situation that q lies on the unit circle; in that case it is found that there exists a natural parameter domain for which the discrete orthogonality measure (which is complex in general) becomes real-valued and positive. We investigate the properties of a finite-dimensional discrete integral transform for functions over the grid, whose ...
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...
Witt Rings and Permutation Polynomials
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.
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.
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.
Yu, Ting; He, Tao; Yao, Hong; Zhang, Jin-Bao; Li, Xiao-Na; Zhang, Rong-Ming; Wang, Gui-Qin
2015-09-01
The aim of this study is to understand the prevalence and molecular characterization of 16S rRNA methylase gene, rmtB, among Escherichia coli strains isolated from bovine mastitis in China. A total of 245 E. coli isolates were collected from bovine mastitis in China between 2013 and 2014 and were screened for 16S rRNA methylase genes (armA, rmtA, rmtB, rmtC, rmtD, rmtE, and npmA) by polymerase chain reaction. About 5.3% (13/245) of the isolates carried the rmtB gene; the isolates were highly resistant to amikacin. Thirteen rmtB-positive strains were analyzed for the presence of extended-spectrum β-lactamase genes (bla(TEM), bla(CTX-M), bla(OXA), and bla(SHV)). All the isolates harbored both bla(TEM-1) and bla(CTX-M-15) genes and two of the isolates were also positive for bla(OXA-1). Pulsed-field gel electrophoresis (PFGE) analysis indicated that the nine rmtB-positive strains belonging to ST10 from one farm showed the similar PFGE pattern, indicating a clonal expansion in this farm. S1-PFGE and Southern blotting showed that 12 isolates harbored the rmtB gene in plasmids of two different sizes (≈45 kb [n=10] and ≈48 kb [n=2]), while only 1 strain harbored the rmtB gene in the chromosome. These plasmids were transferable by conjugation studies, and two isolates from two respective farms carried the same size of plasmid, suggesting that the horizontal transmission of plasmids also contributed to the spread of rmtB gene. This is the first report of prevalence of the 16S rRNA methylase gene rmtB among E. coli isolated from bovine mastitis in China, and rmtB-carrying E. coli may pose a threat to the treatment of bovine mastitis.
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
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.
Collective Correlations of Brodmann Areas fMRI Study with RMT-Denoising
Burda, Z.; Kornelsen, J.; Nowak, M. A.; Porebski, B.; Sboto-Frankenstein, U.; Tomanek, B.; Tyburczyk, J.
We study collective behavior of Brodmann regions of human cerebral cortex using functional Magnetic Resonance Imaging (fMRI) and Random Matrix Theory (RMT). The raw fMRI data is mapped onto the cortex regions corresponding to the Brodmann areas with the aid of the Talairach coordinates. Principal Component Analysis (PCA) of the Pearson correlation matrix for 41 different Brodmann regions is carried out to determine their collective activity in the idle state and in the active state stimulated by tapping. The collective brain activity is identified through the statistical analysis of the eigenvectors to the largest eigenvalues of the Pearson correlation matrix. The leading eigenvectors have a large participation ratio. This indicates that several Broadmann regions collectively give rise to the brain activity associated with these eigenvectors. We apply random matrix theory to interpret the underlying multivariate data.
Cohen, D; Cohen, Doron; Heller, Eric J.
2000-01-01
We consider a classically chaotic system that is described by an Hamiltonian ${\\cal H}(Q,P;x)$ where x is a constant parameter. Our main interest is in the case of a gas-particle inside a cavity, where $x$ controls a deformation of the boundary or the position of a `piston'. The quantum-eigenstates of the system are $|n(x)>$. We describe how the parametric kernel $P(n|m)= ||^2$ evolves as a function of $\\delta x=x{-}x_0$. We explore both the perturbative and the non-perturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as of random-waves and random-matrix-theory (RMT) considerations.
Optimization over polynomials: Selected topics
M. Laurent (Monique); S.Y. Jang; Y.R. Kim; D.-W. Lee; I. Yie
2014-01-01
htmlabstractMinimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra
Parallel Construction of Irreducible Polynomials
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) ...
Polynomial methods in combinatorics
Guth, Larry
2016-01-01
This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book. Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accompl...
Complex Polynomial Vector Fields
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.
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...
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.
Yi, Huiyue
2014-01-01
Estimating the number of signals is a fundamental problem in many scientific and engineering fields. As a well-known estimator based on random matrix theory (RMT), the RMT estimator estimates the number of signals via sequentially testing the likelihood of an eigenvalue as arising from a signal or from noise. However, the RMT estimator tends to down-estimate the number of signals as some signals will be buried in the interaction term among eigenvalues. In order to overcome this problem, we fo...
Lee, Shiann-Jong; Liang, Wen-Tzong; Cheng, Hui-Wen; Tu, Feng-Shan; Ma, Kuo-Fong; Tsuruoka, Hiroshi; Kawakatsu, Hitoshi; Huang, Bor-Shouh; Liu, Chun-Chi
2014-01-01
We have developed a real-time moment tensor monitoring system (RMT) which takes advantage of a grid-based moment tensor inversion technique and real-time broad-band seismic recordings to automatically monitor earthquake activities in the vicinity of Taiwan. The centroid moment tensor (CMT) inversion technique and a grid search scheme are applied to obtain the information of earthquake source parameters, including the event origin time, hypocentral location, moment magnitude and focal mechanism. All of these source parameters can be determined simultaneously within 117 s after the occurrence of an earthquake. The monitoring area involves the entire Taiwan Island and the offshore region, which covers the area of 119.3°E to 123.0°E and 21.0°N to 26.0°N, with a depth from 6 to 136 km. A 3-D grid system is implemented in the monitoring area with a uniform horizontal interval of 0.1° and a vertical interval of 10 km. The inversion procedure is based on a 1-D Green's function database calculated by the frequency-wavenumber (fk) method. We compare our results with the Central Weather Bureau (CWB) catalogue data for earthquakes occurred between 2010 and 2012. The average differences between event origin time and hypocentral location are less than 2 s and 10 km, respectively. The focal mechanisms determined by RMT are also comparable with the Broadband Array in Taiwan for Seismology (BATS) CMT solutions. These results indicate that the RMT system is realizable and efficient to monitor local seismic activities. In addition, the time needed to obtain all the point source parameters is reduced substantially compared to routine earthquake reports. By connecting RMT with a real-time online earthquake simulation (ROS) system, all the source parameters will be forwarded to the ROS to make the real-time earthquake simulation feasible. The RMT has operated offline (2010-2011) and online (since January 2012 to present) at the Institute of Earth Sciences (IES), Academia Sinica
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.
Polynomial Regressions and Nonsense Inference
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.
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
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.
On chromatic and flow polynomial unique graphs
Duan, Yinghua; Wu, Haidong; Yu, Qinglin
2008-01-01
... research on graphs uniquely determined by their chromatic polynomials and more recently on their Tutte polynomials, but rather spotty research on graphs uniquely determined by their flow polynomials or the combination of both chromatic and flow polynomials. This article is an initiation of investigation on graphs uniquely determin...
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
Uniqueness and Zeros of -Shift Difference Polynomials
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.
Akemann, G; Shifrin, L; Wettig, T
2007-01-01
For QCD at non-zero chemical potential $\\mu$, the Dirac eigenvalues are scattered in the complex plane. We define a notion of ordering for individual eigenvalues in this case and derive the distributions of individual eigenvalues from random matrix theory (RMT). We distinguish two cases depending on the parameter $\\alpha=\\mu^2 F^2 V$, where $V$ is the volume and $F$ is the familiar low-energy constant of chiral perturbation theory. For small $\\alpha$, we use a Fredholm determinant expansion and observe that already the first few terms give an excellent approximation. For large $\\alpha$, all spectral correlations are rotationally invariant, and exact results can be derived. We compare the RMT predictions to lattice data and in both cases find excellent agreement in the topological sectors $\
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
Mannitol-induced drought stress on calli of Trigonellafoenum-graecum L. Var. RMt-303.
Pant, Naveen C; Agarrwal, Ruchi; Agrawal, Sanjeev
2014-11-01
Different explants of fenugreek, T. foenum-graecum L. (Var. RMt-303), were compared for their callus induction and subsequent shoot regeneration capabilities on Murashige and Skoog media supplemented with different phytohormones in varying concentration. The highest percentage of callus induction frequency was observed in 1 ppm benzylaminopurine (BAP). Maximum shoots were induced on media supplemented with 0.5 ppm BAP using leaf and stem tissues as explants. However, root tissues showed only callusing with no subsequent shooting. Cotyledonary node responded better than hypocotyls in terms of shoot induction on media supplemented with thidiazuron (0.1 ppm). The callus was subjected to drought stress as simulated by reduced water potential of growth media due to addition of mannitol. Calli could withstand -2 MPa water potential till 30 days indicating that the drought stress tolerance mechanisms are functional in this variety. Chlorophyll a and b and total chlorophyll, proline and total phenolic contents, total peroxidase and catalase activities increased under stress conditions suggesting the tolerance of callus to drought stress. However, ascorbate peroxidase, guaiacol peroxidase activities were found to decrease slightly. Malondialdehyde and H2O2 contents were found to decrease while only a slight disturbance was found in membrane stability index. These results underline the mechanisms that are crucial for drought stress tolerance in fenugreek.
R-Matrix incorporating Time (RMT) method for H2+ in short and intense laser fields
Broin, Cathal Ó
2015-01-01
In this work we develop an approach for a molecular hydrogen ion (H2+) in the Born-Oppenheimer approximation while exposed to intense short-pulse radiation. Our starting point is the R-Matrix incorporating Time (RMT) formulation for atomic hydrogen [L. A. A. Nikolopoulos et al, Phys. Rev. A 78, 063420 (2008)] which has proven to be successful at treating multi-electron atomic systems efficiently and to high accuracy [L. R. Moore et al J. Mod. Opt. 58,1132, (2011)]. The present study on H2+ has been performed with a similar objective of developing an ab initio method for solving the Time-dependent Schr\\"odinger Equation (TDSE) for multi-electron diatomic molecules exposed to an external time-dependent potential field. The theoretical formulation is developed in detail for the molecular hydrogen ion where all the multi-electron and inter-nuclei complications are absent. As in the atomic case, the configuration space of the electron's coordinates are separated artificially over two regions; the inner (I) and out...
On the Complexity of the Interlace Polynomial
Bläser, Markus
2007-01-01
We consider the two-variable interlace polynomial introduced by Arratia, Bollob\\'as and Sorkin. For this graph polynomial we derive two graph transformations yielding point-to-point reductions similar to the thickening transformation in the context of the Tutte polynomial. This enables us to prove that the two-variable interlace polynomial is #P-hard to evaluate at every algebraic point of R^2, except at one line, where it is trivially polynomial time computable, and four lines and two points, where the complexity is still open. As a consequence, three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are #P-hard to evaluate almost everywhere, too. For the independent set polynomial, our graph transformations allow us to prove that it is even hard to approximate at every algebraic point except at -1 and 0.
陈琳; 张俊丰; 刘健华; 陈杖榴; 曾振灵
2010-01-01
为研究16S rRNA甲基化酶rmtB在猪肠道菌中的传播方式,用脉冲场凝胶电泳(PFGE)分别对来源于A、B猪场的48株rmtB阳性大肠埃希氏菌进行基因分型;用膜接合试验研究rmtB基因的水平传播方式;用微量稀释法对rmtB阳性菌及其接合子进行药敏试验.有45株rmtB阳性大肠埃希氏菌(45/48)能进行PFGE分型,可分为28种不同的PFGE基因型.其中来源于A猪场的20株菌可分为12种基因型,来源于B猪场的25株菌可分为16种基因型;46株rmtB阳性菌可以通过接合试验将rmtB基因及其介导的耐药性传递给受体菌E.coli C600和E.coli 488 Rifr,接合频率从4.6×10-13～3.0×10-6不等,接合子对卡那霉素、阿米卡星、妥布霉素、西梭米星、萘替米星、庆大霉素6种二脱氧链霉胺类抗生素均高度耐药(MIC≥512μg/mL).结果表明,rmtB基因位于接合性质粒上,该基因在两猪场的大肠埃希氏菌中既存在克隆传播,又存在水平传播,以水平传播为主.
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.
Xia, Jing; Sun, Jian; Cheng, Ke; Li, Liang; Fang, Liang-Xing; Zou, Meng-Ting; Liao, Xiao-Ping; Liu, Ya-Hong
2016-05-30
A total of 963 non-duplicate Escherichia coli strains isolated from food-producing animals between 2002 and 2012 were screened for the presence of the 16S rRNA methyltransferase genes. Among the positive isolates, resistance determinants to extended spectrum β-lactamases, plasmid-mediated quinolone resistance genes as well as floR and fosA/A3/C2 were detected using PCR analysis. These isolates were further subjected to antimicrobial susceptibility testing, molecular typing, PCR-based plasmid replicon typing and plasmid analysis. Of the 963 E. coli isolates, 173 (18.0%), 3 (0.3%) and 2 (0.2%) were rmtB-, armA- and rmtE-positive strains, respectively. All the 16S rRNA methyltransferase gene-positive isolates were multidrug resistant and over 90% of them carried one or more type of resistance gene. IncF (especially IncFII) and non-typeable plasmids played the main role in the dissemination of rmtB, followed by the IncN plasmids. Plasmids that harbored rmtB ranged in size from 20kb to 340kb EcoRI-RFLP testing of the 109 rmtB-positive plasmids from different years and different origins suggested that horizontal (among diverse animals) and vertical transfer of IncF, non-typeable and IncN-type plasmids were responsible for the spread of rmtB gene. In summary, our findings highlight that rmtB was the most prevalent 16S rRNA methyltransferase gene, which present persistent spread in food-producing animals in China and a diverse group of plasmids was responsible for rmtB dissemination.
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.
Chromatic polynomials of random graphs
Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc
2010-04-01
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
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.
Free arrangements and coefficients of characteristic polynomials
Abe, Takuro
2011-01-01
Ziegler showed that free arrangements have free restricted multiarrangements (multirestrictions). After Ziegler's work, several results concerning "reverse direction", namely characterizing freeness of an arrangement via that of multirestriction, have appeared. In this paper, we prove that the second Betti number of the arrangement plays a crucial role.
Dynamic normal forms and dynamic characteristic polynomial
Frandsen, Gudmund Skovbjerg; Sankowski, Piotr
2011-01-01
with relative error 2−b in additional O(nlog2nlogb) time. Furthermore, it can be used to dynamically maintain the singular value decomposition (SVD) of a generic matrix. Together with the algorithm, the hardness of the problem is studied. For the symmetric case, we present an Ω(n2) lower bound for rank...
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 operator orderings
Hamdi, Adel; 10.1063/1.3372526
2010-01-01
An alternative and combinatorial proof is given for a connection between a system of Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56 (1986), J. Math. Phys. 28, 509 (1987)] and proved by Koornwinder [J. Phys. Phys. 30(4), 1989]. In the same vein two results announced by Bender and Dunne [J. Math. Phys. 29 (8), 1988] connecting a special one-parameter class of Hermitian operator orderings and the continuous Hahn polynomials are also proved.
Abelian avalanches and Tutte polynomials
Gabrielov, Andrei
1993-04-01
We introduce a class of deterministic lattice models of failure, Abelian avalanche (AA) models, with continuous phase variables, similar to discrete Abelian sandpile (ASP) models. We investigate analytically the structure of the phase space and statistical properties of avalanches in these models. We show that the distributions of avalanches in AA and ASP models with the same redistribution matrix and loading rate are identical. For an AA model on a graph, statistics of avalanches is linked to Tutte polynomials associated with this graph and its subgraphs. In the general case, statistics of avalanches is linked to an analog of a Tutte polynomial defined for any symmetric matrix.
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).
Polynomial Invariants for Arbitrary Rank D Weakly-Colored Stranded Graphs
Avohou, Remi Cocou
2016-03-01
Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollobás-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we deepen the analysis of the polynomial invariant defined on rank 3 weakly-colored stranded graphs introduced in arXiv:1301.1987. We successfully find in dimension D≥3 a modified Euler characteristic with D-2 parameters. Using this modified invariant, we extend the rank 3 weakly-colored graph polynomial, and its main properties, on rank 4 and then on arbitrary rank D weakly-colored stranded graphs.
Symbolic computation of Appell polynomials using Maple
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.
矿井主要通风机性能曲线的最优多项式拟合%Optimal polynomial fitting for mine main fan characteristic curves
吴奉亮
2011-01-01
为确定主要通风机性能曲线多项式拟合函数的最优阶数,将回归系数的显著性作为最优阶数的判定依据.分析了基于最小二乘法的任意阶数主扇性能曲线拟合模型;指出了回归多项式的各项系数应与风压显著相关才是合理的,在合理基础之上阶数最大的即为最优;采用F检验法计算了回归多项式各系数的显著性.对于有m个样点的测定数据,通过m-1次拟合,可确定多项式的最优阶数.编制了相应的程序,实现了最佳阶数回归函数求解与曲线可视化,结合一包含13个测点的实例显示了这种方法的优越性.%For determinating the optimal power of polynomial fitting function of main fan' s characteristic curve, the significance of regression coefficient was taken as the criterion of optimum power. Based on the least square method, the fitting model of main fan' s characteristic curve with arbitrary power number was analyzed. The reasonable and optimal power of the regression functe were pointed out. F test was used to calculation significance of each coefficient. Where the count of sample data is n, fitting times of n - 1 was needed to calculate the optimum power. Program was coded to calculate the regression function and to visualize the curve. Finally, one example including thirteen group of sample datas was employed to display the advantage of this method.
The geometry of the critically-periodic curves in the space of cubic polynomials
DeMarco, Laura
2012-01-01
We provide an algorithm for computing the Euler characteristic of the curves $S_p$ in the space of cubic polynomials, consisting of all polynomials with a periodic critical point of period $p$. The curves were introduced in [Milnor, Bonifant-Kiwi-Milnor], and the algorithm applies the main results of [DeMarco-Pilgrim]. The output is shown for periods $p \\leq 26$.
Global Polynomial Kernel Hazard Estimation
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...
Global Polynomial Kernel Hazard Estimation
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...
On Modular Counting with Polynomials
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
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.
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
Polynomial Regressions and Nonsense Inference
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
Herman's condition and Siegel disks of polynomials
Chéritat, Arnaud
2011-01-01
Herman proved the presence of critical points on the boundary of Siegel disks of unicritical polynomials under some diophantine condition now called the Herman condition. We extend this result to polynomials with two critical points.
Simple G-graded algebras and their polynomial identities
Aljadeff, Eli
2011-01-01
Let G be any group and F an algebraically closed field of characteristic zero. We show that any two G-graded finite dimensional G-simple algebras over F are G-graded isomorphic if and only if the satisfy the same G-graded polynomial identities. This result was proved by Koshlukov and Zaicev in case G is abelian.
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
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.
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.
On Star-Varieties with Almost Polynomial Growth
A. Giambruno; S. Mishchenko
2001-01-01
Let 2 be a variety of algebras with involution * over a field of characteristic zero and cn(V, *) the corresponding sequence of *-codimensions. Here,we characterize those varieties V such that cn (V, *) is polynomially bounded. We prove that V is such a variety if and only if G2,M V, where G2 and M are two explicit finite-dimensional algebras with involution previously constructed. It follows that G2 and M generate the only two varieties of algebras with involution with almost polynomial growth and there is no variety with intermediate growth.
General Eulerian Numbers and Eulerian Polynomials
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.
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.
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.
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.
Factorization of Polynomials and GCD Computations for Finding Universal Denominators
Abramov, S. A.; Gheffar, A.; Khmelnov, D. E.
We discuss the algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, compute a polynomial U(x) ∈ k[x] (a universal denominator) such that the denominator of each of rational solutions (if exist) of the given equation divides U(x). We consider two types of such algorithms. One of them is based on constructing a set of irreducible polynomials that are candidates for divisors of denominators of rational solutions, and on finding a bound for the exponent of each of these candidates (the full factorization of polynomials is used). The second one is related to earlier algorithms for finding universal denominators, where the computation of gcd was used instead of the full factorization. The algorithms are applicable to scalar equations of arbitrary orders as well as to systems of first-order equations.
Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
Akemann, G; Phillips, M J [Department of Mathematical Sciences and BURSt Research Centre, Brunel University West London, Uxbridge UB8 3PH (United Kingdom); Kieburg, M, E-mail: gernot.akemann@brunel.ac.u [Fakultaet fuer Physik, Universitaet Duisburg-Essen, 47048 Duisburg (Germany)
2010-09-17
We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as the expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble, we recover the SOP of Forrester and Nagao in terms of Hermite polynomials.
Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
Akemann, G; Phillips, M J
2010-01-01
We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials...
Bipartition Polynomials, the Ising Model, and Domination in Graphs
Dod Markus
2015-05-01
Full Text Available This paper introduces a trivariate graph polynomial that is a common generalization of the domination polynomial, the Ising polynomial, the matching polynomial, and the cut polynomial of a graph. This new graph polynomial, called the bipartition polynomial, permits a variety of interesting representations, for instance as a sum ranging over all spanning forests. As a consequence, the bipartition polynomial is a powerful tool for proving properties of other graph polynomials and graph invariants. We apply this approach to show that, analogously to the Tutte polynomial, the Ising polynomial introduced by Andrén and Markström in [3], can be represented as a sum over spanning forests.
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
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.
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.
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...
Mikael Erlstroem, Jan; Persson, Lena
2014-05-01
The Silurian bedrock on the island of Gotland in the central Baltic Sea is composed of a series of stacked carbonate platforms. Up to 10 cycles of shallow marine back reef, reef and fore reef limestone overlie and interfinger with shallow shelf marl and marlstone deposits. Three main regional, up to 70 m thick limestone dominated bodies with high electrical resistivity are clearly marked in the electromagnetic map. The subsurface structural framework of the different lithofacies is at presently less well known. There is therefore a great interest to develop and test the applicability of different geophysical methods for this purpose. Especially as the limestone bodies constitute important groundwater aquifers and raw material to the lime and steel industry. The primary aim of this investigation is to assess the application of the RMT method regarding obtaining information on the thickness of the limestone body and the subsurface distribution of different lithofacies on central Gotland. RMT is an electromagnetic method that make use of distant radio transmitters in the frequency range between 10-250 kHz. A significant difference in resistivity between the limestone (1 000-10 000 Ωm) and the marlstone (
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
Tutte polynomial of the Apollonian network
Liao, Yunhua; Hou, Yaoping; Shen, Xiaoling
2014-10-01
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 combinatorics and statistical physics. The computation of this invariant for a graph is, in general, NP-hard. The aim of this paper is to compute the Tutte polynomial of the Apollonian network. Based on the well-known duality property of the Tutte polynomial, we extend the subgraph-decomposition method. In particular, we do not calculate the Tutte polynomial of the Apollonian network directly, instead we calculate the Tutte polynomial of the Apollonian dual graph. By using the close relation between the Apollonian dual graph and the Hanoi graph, we express the Tutte polynomial of the Apollonian dual graph in terms of that of the Hanoi graph. As an application, we also give the number of spanning trees of the Apollonian network.
Stable piecewise polynomial vector fields
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.
Fixed-point bifurcation analysis in biological models using interval polynomials theory.
Rigatos, Gerasimos G
2014-06-01
The paper proposes a systematic method for fixed-point bifurcation analysis in circadian cells and similar biological models using interval polynomials theory. The stages for performing fixed-point bifurcation analysis in such biological systems comprise (i) the computation of fixed points as functions of the bifurcation parameter and (ii) the evaluation of the type of stability for each fixed point through the computation of the eigenvalues of the Jacobian matrix that is associated with the system's nonlinear dynamics model. Stage (ii) requires the computation of the roots of the characteristic polynomial of the Jacobian matrix. This problem is nontrivial since the coefficients of the characteristic polynomial are functions of the bifurcation parameter and the latter varies within intervals. To obtain a clear view about the values of the roots of the characteristic polynomial and about the stability features they provide to the system, the use of interval polynomials theory and particularly of Kharitonov's stability theorem is proposed. In this approach, the study of the stability of a characteristic polynomial with coefficients that vary in intervals is equivalent to the study of the stability of four polynomials with crisp coefficients computed from the boundaries of the aforementioned intervals. The efficiency of the proposed approach for the analysis of fixed-point bifurcations in nonlinear models of biological neurons is tested through numerical and simulation experiments.
Baldi, Antonio; Bertolino, Filippo
2013-10-01
It is well known that displacement components estimated using digital image correlation are affected by a systematic error due to the polynomial interpolation required by the numerical algorithm. The magnitude of bias depends on the characteristics of the speckle pattern (i.e., the frequency content of the image), on the fractional part of displacements and on the type of polynomial used for intensity interpolation. In literature, B-Spline polynomials are pointed out as being able to introduce the smaller errors, whereas bilinear and cubic interpolants generally give the worst results. However, the small bias of B-Spline polynomials is partially counterbalanced by a somewhat larger execution time. We will try to improve the accuracy of lower order polynomials by a posteriori correcting their results so as to obtain a faster and more accurate analysis.
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'.
Chromatic Polynomials of Mixed Hypercycles
Allagan Julian A.
2014-08-01
Full Text Available We color the vertices of each of the edges of a C-hypergraph (or cohypergraph in such a way that at least two vertices receive the same color and in every proper coloring of a B-hypergraph (or bihypergraph, we forbid the cases when the vertices of any of its edges are colored with the same color (monochromatic or when they are all colored with distinct colors (rainbow. In this paper, we determined explicit formulae for the chromatic polynomials of C-hypercycles and B-hypercycles
Orthogonal polynomials and random matrices
Deift, Percy
2000-01-01
This volume expands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. The central question was the following: Why do very general ensembles of random n {\\times} n matrices exhibit universal behavior as n {\\rightarrow} {\\infty}? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems.
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 ...
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.
Exceptional polynomials and SUSY quantum mechanics
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.
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
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.
On the verification of polynomial system solvers
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.
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.
Polynomial chaotic inflation in supergravity revisited
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.
Chromatic polynomials, Potts models and all that
Sokal, Alan D.
2000-04-01
The q-state Potts model can be defined on an arbitrary finite graph, and its partition function encodes much important information about that graph, including its chromatic polynomial, flow polynomial and reliability polynomial. The complex zeros of the Potts partition function are of interest both to statistical mechanicians and to combinatorists. I give a pedagogical introduction to all these problems, and then sketch two recent results: (a) Construction of a countable family of planar graphs whose chromatic zeros are dense in the whole complex q-plane except possibly for the disc | q-1|chromatic polynomial (or antiferromagnetic Potts-model partition function) in terms of the graph's maximum degree.
Control to Facet for Polynomial Systems
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...
The q-Laguerre matrix polynomials.
Salem, Ahmed
2016-01-01
The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.
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.
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.
Aspects of the Tutte polynomial
Ok, Seongmin
This thesis studies various aspects of the Tutte polynomial, especially focusing on the Merino-Welsh conjecture. We write T(G;x,y) for the Tutte polynomial of a graph G with variables x and y. In 1999, Merino and Welsh conjectured that if G is a loopless 2-connected graph, then T(G;1,1) ≤ max{T(G;2......-Welsh conjecture. Assume the graph G is loopless, bridgeless and has n vertices and m edges. If m ≤ 1.066 n then T(G;1,1) ≤ T(G;2,0). If m ≥ 4(n-1) then T(G;1,1) ≤ T(G;0,2). I improve in this thesis Thomassen's result as follows: If m ≤ 1.29(n-1) then T(G;1,1) ≤ T(G;2,0). If m ≥ 3.58(n-1) and G is 3-edge...
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.
Algorithms in Solving Polynomial Inequalities
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.
Zarubica, Tamara; Baker, Matthew R; Wright, H Tonie; Rife, Jason P
2011-02-01
Bacterial resistance to 4,6-type aminoglycoside antibiotics, which target the ribosome, has been traced to the ArmA/RmtA family of rRNA methyltransferases. These plasmid-encoded enzymes transfer a methyl group from S-adenosyl-L-methionine to N7 of the buried G1405 in the aminoglycoside binding site of 16S rRNA of the 30S ribosomal subunit. ArmA methylates mature 30S subunits but not 16S rRNA, 50S, or 70S ribosomal subunits or isolated Helix 44 of the 30S subunit. To more fully characterize this family of enzymes, we have investigated the substrate requirements of ArmA and to a lesser extent its ortholog RmtA. We determined the Mg+² dependence of ArmA activity toward the 30S ribosomal subunits and found that the enzyme recognizes both low Mg+² (translationally inactive) and high Mg+² (translationally active) forms of this substrate. We tested the effects of LiCl pretreatment of the 30S subunits, initiation factor 3 (IF3), and gentamicin/kasugamycin resistance methyltransferase (KsgA) on ArmA activity and determined whether in vivo derived pre-30S ribosomal subunits are ArmA methylation substrates. ArmA failed to methylate the 30S subunits generated from LiCl washes above 0.75 M, despite the apparent retention of ribosomal proteins and a fully mature 16S rRNA. From our experiments, we conclude that ArmA is most active toward the 30S ribosomal subunits that are at or very near full maturity, but that it can also recognize more than one form of the 30S subunit.
Weighted Polynomial Approximation for Automated Detection of Inspiratory Flow Limitation
Sheng-Cheng Huang
2017-01-01
Full Text Available Inspiratory flow limitation (IFL is a critical symptom of sleep breathing disorders. A characteristic flattened flow-time curve indicates the presence of highest resistance flow limitation. This study involved investigating a real-time algorithm for detecting IFL during sleep. Three categories of inspiratory flow shape were collected from previous studies for use as a development set. Of these, 16 cases were labeled as non-IFL and 78 as IFL which were further categorized into minor level (20 cases and severe level (58 cases of obstruction. In this study, algorithms using polynomial functions were proposed for extracting the features of IFL. Methods using first- to third-order polynomial approximations were applied to calculate the fitting curve to obtain the mean absolute error. The proposed algorithm is described by the weighted third-order (w.3rd-order polynomial function. For validation, a total of 1,093 inspiratory breaths were acquired as a test set. The accuracy levels of the classifications produced by the presented feature detection methods were analyzed, and the performance levels were compared using a misclassification cobweb. According to the results, the algorithm using the w.3rd-order polynomial approximation achieved an accuracy of 94.14% for IFL classification. We concluded that this algorithm achieved effective automatic IFL detection during sleep.
Weighted Polynomial Approximation for Automated Detection of Inspiratory Flow Limitation.
Huang, Sheng-Cheng; Jan, Hao-Yu; Fu, Tieh-Cheng; Lin, Wen-Chen; Lin, Geng-Hong; Lin, Wen-Chi; Tsai, Cheng-Lun; Lin, Kang-Ping
2017-01-01
Inspiratory flow limitation (IFL) is a critical symptom of sleep breathing disorders. A characteristic flattened flow-time curve indicates the presence of highest resistance flow limitation. This study involved investigating a real-time algorithm for detecting IFL during sleep. Three categories of inspiratory flow shape were collected from previous studies for use as a development set. Of these, 16 cases were labeled as non-IFL and 78 as IFL which were further categorized into minor level (20 cases) and severe level (58 cases) of obstruction. In this study, algorithms using polynomial functions were proposed for extracting the features of IFL. Methods using first- to third-order polynomial approximations were applied to calculate the fitting curve to obtain the mean absolute error. The proposed algorithm is described by the weighted third-order (w.3rd-order) polynomial function. For validation, a total of 1,093 inspiratory breaths were acquired as a test set. The accuracy levels of the classifications produced by the presented feature detection methods were analyzed, and the performance levels were compared using a misclassification cobweb. According to the results, the algorithm using the w.3rd-order polynomial approximation achieved an accuracy of 94.14% for IFL classification. We concluded that this algorithm achieved effective automatic IFL detection during sleep.
Mehta, Suman; Bastani, Mehrdad; Malehmir, Alireza; Pedersen, Laust B.
2017-04-01
The resolution and sensitivity of water-borne boat-towed multi-frequency radio-magnetotelluric (RMT) data for delineating zones of weaknesses in bedrock are examined in this study. 2D modeling of RMT data along 40 profiles in joint transverse electric (TE) and transverse magnetic (TM) as well as determinant mode was used for this purpose. The RMT data were acquired over two water passages from the Lake Mälaren near the city of Stockholm where one of the largest underground infrastructure projects, a multi-lane tunnel, in Europe is currently being developed. Comparison with available borehole coring, refraction seismic and bathymetric data was used to scrutinize the RMT resistivity models. A low-resistivity zone observed in the middle of all the profiles is suggested to be from fracture/fault zones striking in the same direction as the water passages. Drilling observations confirm the presence of brittle structures in the bedrock, which manifest themselves as zones of low-resistivity and low-velocity in the RMT and refraction seismic data, respectively. Nevertheless, RMT is an inductive electromagnetic method hence the presence of conductive lake sediments may shield detecting the underlying fractured bedrock. The loss of resolution at depth implies that the structures within the bedrock under the lake sediments cannot reliably be delineated. To support this, a synthetic data analysis was carried out providing useful information on how to improve and plan the lake measurements for future studies. Synthetic modeling results for example suggested that frequencies as low as 3 kHz would be required to reliably resolve the bedrock and fracture zone within it in the study area. The modeling further illustrated the advantage of a fresh water layer that acts as a near-surface homogeneous medium eliminating the static shift effects. While boat-towed RMT data provided substantial information about the subsurface geology, the acquisition system should be upgraded to enable
BOUNDS FOR THE ZEROS OF POLYNOMIALS
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.
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
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
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…
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.
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.
ON FIRST INTEGRALS OF POLYNOMIAL AUTONOMOUS SYSTEMS
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.
Large degree asymptotics of generalized Bessel polynomials
J.L. López; N.M. Temme (Nico)
2011-01-01
textabstractAsymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the
On Polynomial Functions over Finite Commutative Rings
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
Connections between the matching and chromatic polynomials
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.
Application of polynomial preconditioners to conservation laws
Geurts, Bernardus J.; van Buuren, R.; Lu, H.
2000-01-01
Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of
Sums of Powers of Fibonacci Polynomials
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.
A Note on Solvable Polynomial Algebras
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.
The topology of Julia sets for polynomials
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.
Percolation critical polynomial as a graph invariant
Scullard, Christian R.
2012-10-01
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how this generalized critical polynomial can be viewed as a graph invariant, similar to the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the recursive deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction pc=0.52440572⋯, which differs from the numerical value, pc=0.52440503(5), by only 6.9×10-7.
Tutte Polynomial of Multi-Bridge Graphs
Julian A. Allagan
2013-10-01
Full Text Available In this paper, using a well-known recursion for computing the Tutte polynomial of any graph, we found explicit formulae for the Tutte polynomials of any multi-bridge graph and some $2-$tree graphs. Further, several recursive formulae for other graphs such as the fan and the wheel graphs are also discussed.
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 $n$.
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
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.
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.
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.
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).
Fuzzy Morphological Polynomial Image Representation
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.
Polynomial weights and code constructions
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....
Kopeliovich, Vladimir B
2016-01-01
The angular dependence of the cumulative particles production off nuclei near the kinematical boundary for multistep process is defined by characteristic polynomials in angular variables $J_N^2(z_N^\\theta)$, where $\\theta$ is the polar angle defining the momentum of the final (cumulative) particle, $z_N^\\theta = cos (\\theta/N)$, the integer $N$ being the multiplicity of the process (the number of interactions). Physical argumentation, exploring the small phase space method, leads to the appearance of equations for these polynomials $J_N^2[cos(\\pi/N)]=0$. The recurrent relations between polynomials with different $N$ are obtained, and their connection with known in mathematics Chebyshev polynomials of 2-d kind is established. As a result of this equality, differential cross section of the cumulative particle production has characteristic behaviour $d\\sigma \\sim 1/ \\sqrt {\\pi - \\theta}$ at $\\theta \\sim \\pi$ (the backward focusing effect). Such behaviour takes place for any multiplicity of the interaction, begin...
Polynomials with Palindromic and Unimodal Coeﬃ cients
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.
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.
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...
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...
Tutte polynomial in functional magnetic resonance imaging
García-Castillón, Marlly V.
2015-09-01
Methods of graph theory are applied to the processing of functional magnetic resonance images. Specifically the Tutte polynomial is used to analyze such kind of images. Functional Magnetic Resonance Imaging provide us connectivity networks in the brain which are represented by graphs and the Tutte polynomial will be applied. The problem of computing the Tutte polynomial for a given graph is #P-hard even for planar graphs. For a practical application the maple packages "GraphTheory" and "SpecialGraphs" will be used. We will consider certain diagram which is depicting functional connectivity, specifically between frontal and posterior areas, in autism during an inferential text comprehension task. The Tutte polynomial for the resulting neural networks will be computed and some numerical invariants for such network will be obtained. Our results show that the Tutte polynomial is a powerful tool to analyze and characterize the networks obtained from functional magnetic resonance imaging.
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.
HIGHER ORDER MULTIVARIABLE NORLUND EULER-BERNOULLI POLYNOMIALS
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.
Generalized Gegenbauer Koornwinder's type polynomials change of bases
AlQudah, Mohammad; AlMheidat, Maalee
2017-07-01
In this paper we characterize the generalized Gegenbauer polynomials using Bernstein basis, and derive the matrix of transformation of the generalized Gegenbauer polynomial basis form into the Bernstein polynomial basis and vice versa.
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,...
Graph Polynomials: From Recursive Definitions To Subset Expansion Formulas
Godlin, Benny; Makowsky, Johann A
2008-01-01
Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logic-based framework which gives a precise meaning to recursive definitions of graph polynomials. We then prove that in this framework every recursive definition of a graph polynomial can be converted into a subset expansion formula.
Christiane Quesne
2009-08-01
Full Text Available New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same characteristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are constructed. In the linear case, they contain (ν+1th-degree polynomials with ν = 0,1,2,..., which are shown to be X1-Laguerre or X1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of (ν+2th-degree Laguerre-type polynomials and a single one of (ν+2th-degree Jacobi-type polynomials with ν = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.
Zhang Xue-qing
2010-06-01
Full Text Available Abstract Background Recently, production of 16S rRNA methylases by Gram-negative bacilli has emerged as a novel mechanism for high-level resistance to aminoglycosides by these organisms in a variety of geographic locations. Therefore, the spread of high-level aminoglycoside resistance determinants has become a great concern. Methods Between January 2006 and July 2008, 680 distinct Escherichia coli clinical isolates were collected from a teaching hospital in Wenzhou, China. PCR and DNA sequencing were used to identify 16S rRNA methylase and extended-spectrum β-lactamase (ESBL genes, including armA and rmtB, and in situ hybridization was performed to determine the location of 16S rRNA methylase genes. Conjugation experiments were subsequently performed to determine whether aminoglycoside resistance was transferable from the E. coli isolates via 16S rRNA methylase-bearing plasmids. Homology of the isolates harboring 16S rRNA methylase genes was determined using pulse-field gel electrophoresis (PFGE. Results Among the 680 E. coli isolates, 357 (52.5%, 346 (50.9% and 44 (6.5% isolates were resistant to gentamicin, tobramycin and amikacin, respectively. Thirty-seven of 44 amikacin-resistant isolates harbored 16S rRNA methylase genes, with 36 of 37 harboring the rmtB gene and only one harboring armA. The positive rates of 16S rRNA methylase genes among all isolates and amikacin-resistant isolates were 5.4% (37/680 and 84.1% (37/44, respectively. Thirty-one isolates harboring 16S rRNA methylase genes also produced ESBLs. In addition, high-level aminoglycoside resistance could be transferred by conjugation from four rmtB-positive donors. The plasmids of incompatibility groups IncF, IncK and IncN were detected in 34, 3 and 3 isolates, respectively. Upstream regions of the armA gene contained ISCR1 and tnpU, the latter a putative transposase gene,. Another putative transposase gene, tnpD, was located within a region downstream of armA. Moreover, a
Polynomial Interpolation in the Elliptic Curve Cryptosystem
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.
Polynomial threshold functions and Boolean threshold circuits
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...
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.
Exponential Polynomial Approximation with Unrestricted Upper Density
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.
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.
On Calculation of Adomian Polynomials by MATLAB
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.
ECG data compression using Jacobi polynomials.
Tchiotsop, Daniel; Wolf, Didier; Louis-Dorr, Valérie; Husson, René
2007-01-01
Data compression is a frequent signal processing operation applied to ECG. We present here a method of ECG data compression utilizing Jacobi polynomials. ECG signals are first divided into blocks that match with cardiac cycles before being decomposed in Jacobi polynomials bases. Gauss quadratures mechanism for numerical integration is used to compute Jacobi transforms coefficients. Coefficients of small values are discarded in the reconstruction stage. For experimental purposes, we chose height families of Jacobi polynomials. Various segmentation approaches were considered. We elaborated an efficient strategy to cancel boundary effects. We obtained interesting results compared with ECG compression by wavelet decomposition methods. Some propositions are suggested to improve the results.
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.
Empowering Polynomial Theory Conjectures with Spreadsheets
Chris Petersdinh
2017-06-01
Full Text Available Polynomial functions and their properties are fundamental to algebra, calculus, and mathematical modeling. Students who do not have a strong understanding of the relationship between factoring and solving equations can have difficulty with optimization problems in calculus and solving application problems in any field. Understanding function transformations is important in trigonometry, the idea of the general antiderivative, and describing the geometry of a problem mathematically. This paper presents spreadsheet activities designed to bolster students' conceptualization of the factorization theorem for polynomials, complex zeros of polynomials, and function transformations. These activities were designed to use a constructivist approach involving student experimentation and conjectures.
A Polynomial Preconditioner for the CMRH Algorithm
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 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...
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.
More on rotations as spin matrix polynomials
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.
Cerdeira, Louise; Fernandes, Miriam R; Francisco, Gabriela R; Bueno, Maria Fernanda C; Ienne, Susan; Souza, Tiago A; de Oliveira Garcia, Doroti; Lincopan, Nilton
2016-11-03
We report here the draft genome sequence of a Klebsiella pneumoniae strain 1194/11, belonging to the hospital-associated sequence type 340 (ST340; clonal complex CC258), isolated from a catheter tip culture from a pediatric patient. The multidrug-resistant strain coproduced the 16S rRNA methyltransferase rRNA RmtG and β-lactamases KPC-2 and CTX-M-15.
Twisted Polynomials and Forgery Attacks on GCM
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...
Handbook on semidefinite, conic and polynomial optimization
Anjos, Miguel F
2012-01-01
This book offers the reader a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization and polynomial optimization. It covers theory, algorithms, software and applications.
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.
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.
Local Polynomial Estimation of Distribution Functions
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.
Hermite polynomials and quasi-classical asymptotics
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.
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...
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.
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.
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...
Ferrers Matrices Characterized by the Rook Polynomials
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.
Blind Signature Scheme Based on Chebyshev Polynomials
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.
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.
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...
On Certain Divisibility Property of Polynomials
Caceres, Luis F
2010-01-01
We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover, we give a characterization for D-rings that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations.
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
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.
ON ABEL-GONTSCHAROFF-GOULD'S POLYNOMIALS
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.
Local fibred right adjoints are polynomial
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
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.
Norm convergence of continuous-time polynomial multiple ergodic averages
Austin, Tim
2011-01-01
For a jointly measurable probability-preserving action \\tau:\\bbR^D\\curvearrowright (X,\\mu) and a tuple of polynomial maps p_i:\\bbR\\to \\bbR^D, i=1,2,...,k, the multiple ergodic averages \\frac{1}{T}\\int_0^T (f_1\\circ \\tau^{p_1(t)})(f_2\\circ\\tau^{p_2(t)})... (f_k\\circ\\tau^{p_k(t)})\\,\\d t converge in L^2(\\mu) as T \\to \\infty for any f_1,f_2,...,f_k \\in L^\\infty(\\mu). This confirms the continuous-time analog of the conjectured norm convergence of discrete polynomial multiple ergodic averages, which in is its original formulation remains open in most cases. A proof of convergence can be given based on the idea of passing up to a sated extension of (X,\\mu,\\tau) in order to find simple characteristic factors, similarly to the recent development of this idea for the study of related discrete-time averages, together with a new inductive scheme on tuples of polynomials. The new induction scheme becomes available upon changing the time variable in the above integral by some fractional power, and provides an alternative t...
A proof of a conjecture by Schweizer on the Drinfeld modular polynomial ΦT (X, Y )
Bassa, Alp; Beelen, Peter
2011-01-01
In this paper we prove a conjecture by Schweizer on the reduction of the Drinfeld modular polynomial ΦT (X, Y ) modulo T − 1. The proof mainly involves manipulations of binomial coefficients in characteristic p....
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
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.
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.
Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials
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
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
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.
Zeros of Jones polynomials for families of knots and links
Chang, S.-C.; Shrock, R.
2001-12-01
We calculate Jones polynomials VL( t) for several families of alternating knots and links by computing the Tutte polynomials T( G, x, y) for the associated graphs G and then obtaining VL( t) as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is also given of the calculation of Jones polynomials for non-alternating links.
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…
Certain non-linear differential polynomials sharing a non zero polynomial
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
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…
Rogiers, Bart
2015-04-01
Since a few years, an increasing number of contributed R packages is becoming available, in the field of hydrology. Hydrological time series analysis packages, lumped conceptual rainfall-runoff models, distributed hydrological models, weather generators, and different calibration and uncertainty estimation methods are all available. Also a few packages are available for solving partial differential equations. Subsurface hydrological modelling is however still seldomly performed in R, or with codes interfaced with R, despite the fact that excellent geostatistical packages, model calibration/inversion options and state-of-the-art visualization libraries are available. Moreover, other popular scientific programming languages like matlab and python have packages for pre- and post-processing files of MODFLOW (Harbaugh 2005) and MT3DMS (Zheng 2010) models. To fill this gap, we present here the development versions of the RMODFLOW and RMT3DMS packages, which allow pre- and post-processing MODFLOW and MT3DMS input and output files from within R. File reading and writing functions are currently available for different packages, and plotting functions are foreseen making use of the ggplot2 package (plotting system based on the grammar of graphics; Wickham 2009). The S3 generic-function object oriented programming style is used for this. An example is provided, making modifications to an existing model, and visualization of the model output. References Harbaugh, A. (2005). MODFLOW-2005: The US Geological Survey Modular Ground-water Model--the Ground-water Flow Process, U.S. Geological Survey Techniques and Methods 6-A16 (p. 253). Wickham, H. (2009). ggplot2: elegant graphics for data analysis. Springer New York, 2009. Zheng, C. (2010). MT3DMS v5.3, a modular three-dimensional multispecies transport model for simulation of advection, dispersion and chemical reactions of contaminants in groundwater systems. Supplemental User's Guide. (p. 56).
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....
Linear representations of SU(2) described by using Kravchuk polynomials
Cotfas, Nicolae
2016-01-01
We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU(2). Our approach offers a deeper insight into the structure of the l...
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...
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.
Uniform trigonometric polynomial B-spline curves
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.
A complete discrimination system for polynomials
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.
Transversals of Complex Polynomial Vector Fields
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......Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied...... 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....
Quantum chaotic dynamics and random polynomials
Bogomolny, E.; Bohigas, O.; Leboeuf, P.
1995-11-01
The distribution of roots of polynomials of high degree with random coefficients is investigated which, among others, appear naturally in the context of `quantum chaotic dynamics`. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, the particular case of self-inverse random polynomials is studied, and it is shown that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials. (author). 32 refs.
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.
New development in theory of Laguerre polynomials
Guseinov, I I
2012-01-01
The new complete orthonormal sets of -Laguerre type polynomials (-LTP,) are suggested. Using Schr\\"odinger equation for complete orthonormal sets of -exponential type orbitals (-ETO) introduced by the author, it is shown that the origin of these polynomials is the centrally symmetric potential which contains the core attraction potential and the quantum frictional potential of the field produced by the particle itself. The quantum frictional forces are the analog of radiation damping or frictional forces suggested by Lorentz in classical electrodynamics. The new -LTP are complete without the inclusion of the continuum states of hydrogen like atoms. It is shown that the nonstandard and standard conventions of -LTP and their weight functions are the same. As an application, the sets of infinite expansion formulas in terms of -LTP and L-Generalized Laguerre polynomials (L-GLP) for atomic nuclear attraction integrals of Slater type orbitals (STO) and Coulomb-Yukawa like correlated interaction potentials (CIP) wit...
Polynomial solution of quantum Grassmann matrices
Tierz, Miguel
2017-05-01
We study a model of quantum mechanical fermions with matrix-like index structure (with indices N and L) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with q-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of L and arbitrary N. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given L, the number of states of different energy is quadratic in N, which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps N and L, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic L and N, in terms of a single generalized Hermite polynomial.
Minimal residual method stronger than polynomial preconditioning
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.
Dominating Sets and Domination Polynomials of Paths
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.
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.
Error Minimization of Polynomial Approximation of Delta
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.
Some Inequalities for the Derivative of Polynomials
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.
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.
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).
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.
Incomplete Bivariate Fibonacci and Lucas -Polynomials
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.
The Potts model and the Tutte polynomial
Welsh, D. J. A.; Merino, C.
2000-03-01
This is an invited survey on the relation between the partition function of the Potts model and the Tutte polynomial. On the assumption that the Potts model is more familiar we have concentrated on the latter and its interpretations. In particular we highlight the connections with Abelian sandpiles, counting problems on random graphs, error correcting codes, and the Ehrhart polynomial of a zonotope. Where possible we use the mean field and square lattice as illustrations. We also discuss in some detail the complexity issues involved.
The chromatic polynomial and list colorings
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....
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
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.
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.
Park, Yeon-Joon; Yu, Jin Kyung; Kim, Sang-Il; Lee, Kyungwon; Arakawa, Yoshichika
2009-01-01
A new plasmid-mediated fluoroquinolone efflux pump gene, qepA, is known to be associated with the rmtB gene, which confers high-level resistance to aminoglycosides. We investigated the qepA gene in 573 AmpC-producing Enterobacteriaceae including one Citrobacter freundii known to harbor rmtB. Of them, two clonally unrelated E. aerogenes harbored qepA. Both isolates co-harbored rmtB, qnrS1, qepA, and bla(LAP-1) on an IncFI type plasmid. The qepA was flanked by two copies of IS26 containing ISCR3C, tnpA, tnpR, bla(TEM), and rmtB. The qnrS1 and bla(LAP-1) were located upstream of qepA. All the resistance determinants (qepA, qnrS1, rmtB, and bla(LAP-1)) were co-transferred to E. coli J53 by filter mating from both isolates. Although the prevalence of qepA is currently low, considering the presence of ISCR3C and the possibility of co-selection and co-transferability of plasmids, more active surveillance for these multi-drug resistant bacteria and prudent use of antimicrobials are needed.
The 6 Vertex Model and Schubert Polynomials
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.
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...
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.
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…
Ideals in Polynomial Near-rings
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.
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/.
Euler Polynomials, Fourier Series and Zeta Numbers
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 Vector Fields in One Complex Variable
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....
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…
On the Schinzel Identity of Cyclotomic Polynomial
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
Quantum Hilbert matrices and orthogonal polynomials
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....
On Arithmetic-Geometric-Mean Polynomials
Griffiths, Martin; MacHale, Des
2017-01-01
We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…
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.
The approach of moments for polynomial equations
M. Laurent (Monique); P. Rostalski
2012-01-01
htmlabstractIn this chapter we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of
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
Dynamic system uncertainty propagation using polynomial chaos
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.
Modeling Power Amplifiers using Memory Polynomials
Kokkeler, Andre B.J.
2005-01-01
In this paper we present measured in- and output data of a power amplifier (PA). We compare this data with an AM-AM and AM-PM model. We conclude that a more sophisticated PA model is needed to cope with severe memory effects. We suggest to use memory polynomials and introduce two approaches to
UNIQUENESS OF DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS
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.
Orthogonality Relations for Multivariate Krawtchouk Polynomials
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.
The Tutte Polynomial of Some Matroids
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.
The GCD property and irreduciable quadratic polynomials
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.
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
Feynman graph polynomials and iterative algorithms
Bogner, Christian [Johannes Gutenberg-Universitaet, Mainz (Germany)
2009-07-01
I briefly report on recent work with Stefan Weinzierl, where we have proven a theorem, stating that the Laurent coefficients of scalar Feynman integrals are periods in the sense of Kontsevich and Zagier, if they are evaluated at kinematical invariants taking rational values in Euclidean momentum space. Our proof uses the (extended) sector decomposition algorithm by Binoth and Heinrich. Our result is related to the appearance of multiple zeta values in coefficients of Feynman integrals which has recently been investigated by Francis Brown, using another iterative algorithm. Both of these algorithms apply to the Feynman parametric representation of the integral and perform iterative manipulations of the polynomials in the integrand, which originate from the Symanzik polynomials. Motivated by the success of these methods I give a brief review on some more and some less well-known combinatorial properties of Symanzik polynomials. I focus on their accessibility to generalized theorems of the matrix-tree type and their relation to the multivariate Tutte polynomial.
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.
Dynamic system uncertainty propagation using polynomial chaos
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.
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…
Euler Polynomials, Fourier Series and Zeta Numbers
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....
Bernoulli Polynomials, Fourier Series and Zeta Numbers
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 ...
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.
Intrinsic Diophantine approximation on general polynomial surfaces
Tiljeset, Morten Hein
2017-01-01
We study the Hausdorff measure and dimension of the set of intrinsically simultaneously -approximable points on a curve, surface, etc, given as a graph of integer polynomials. We obtain complete answers to these questions for algebraically “nice” manifolds. This generalizes earlier work done...
Irreducibility Results for Compositions of Polynomials in Several Variables
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.
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.
d-Orthogonal Charlier Polynomials and the Weyl Algebra
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.
Density of Real Zeros of the Tutte Polynomial
Ok, Seongmin; Perrett, Thomas
2017-01-01
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...
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
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.
Representations of Knot Groups and Twisted Alexander Polynomials
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.
Khan, Waseem A; Haroon, Hiba
2016-01-01
In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite-Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.
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
基于标准特征多项式的横侧向飞控系统设计%Lateral flight control system design based on standard characteristic polynomial
吕旸; 万小朋; 李爱军; 徐丽娜
2012-01-01
With regard to the parameter regulation problem with 2D freedom PID controllers which have multiple parameters, a parameter regulation and optimization method is put forward based on differential evolution(DE) algorithm, which can converge to the equilibrium point with global convergence in probability. The relation between 2D freedom PID controller parameters and the suppression and tracking characteristics is studied, and parameter optimization results using the DE algorithm are compared with those using the genetic algorithm and the PSO algorithm respectively. Simulation shows that the DE algorithm can regulate 2D freedom PID controllers very well with excellent tracking and suppression characteristics.%采用基于标准特征多项式的飞控系统设计方法设计了某型飞机横侧向通道倾斜姿态保持和航向保持控制律,并进行了仿真研究.控制律的设计过程和仿真结果表明,该方法可以确切地给出系统的过渡过程时间和超调量,避免了任意选择初始参数值的盲目性和经验性,大大简化了飞控系统设计的工作量.
A 'missing' family of classical orthogonal polynomials
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.
Lettington, Matthew C
2012-01-01
We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a result of Lettington's and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.
q-Extensions for the Apostol Type Polynomials
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.
The Roots of Adjoint Polynomial of the Graphs Contain Triangles
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
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…
Heat polynomial analogs for higher order evolution equations
G. N. Hile
2001-05-01
Full Text Available Polynomial solutions analogous to the heat polynomials are demonstrated for higher order linear homogeneous evolution equations with coefficients depending on the time variable. Further parallels with the heat polynomials are established when the equation is parabolic with constant coefficients and only highest order terms.
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.
Exact Potts/Tutte polynomials for polygon chain graphs
Shrock, Robert
2011-04-01
We present exact calculations of Potts model partition functions and the equivalent Tutte polynomials for polygon chain graphs with open and cyclic boundary conditions. Special cases of the results that yield flow and reliability polynomials are discussed. We also analyze special cases of the Tutte polynomials that determine various quantities of graph-theoretic interest.
Tutte polynomial of a small-world Farey graph
Liao, Yunhua; Hou, Yaoping; Shen, Xiaoling
2013-11-01
In this paper, we find recursive formulas for the Tutte polynomials of a family of small-world Farey graphs, which are modular, and has an exponential degree hierarchy. As applications of the recursive formula, the exact expressions for the chromatic polynomial and the reliability polynomial of Fare graphs are derived and the number of connected spanning subgraphs is also obtained.
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…
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.
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.
On the Complex Zeros of Some Families of Orthogonal Polynomials
Eugenia N. Petropoulou
2010-01-01
Full Text Available The complex zeros of the orthogonal Laguerre polynomials (( for <−, ultraspherical polynomials (( for <−, Jacobi polynomials (,( for <−, <−, +<−2(+1, orthonormal Al-Salam-Carlitz II polynomials ((; for <0, 0<<1, and -Laguerre polynomials ((; for <−, 0<<1 are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group
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.
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.
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.
Polynomial threshold functions and Boolean threshold circuits
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
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....
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.
Polynomial chaos representation of databases on manifolds
Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)
2017-04-15
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
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.
杜智涛; 付宏; 任晓刚
2014-01-01
This article first established a polynomial regression model based on a large sample network questionnaire survey, and then discussed the structure and characteristics of online political participant of China from the following four aspects: demographic characteristics, basic characteristics of using network, characteristics of online political participative behavior and attitude about the effect of online political participation. It is found that these four aspects have the following characteristics respectively. The online political participants are mainly consisted of those rational youth who have highly educated background, high incomes and high comprehensive quality. The occupational prestige in the realistic society does not match with the network voice, sometimes it even means the least network voice. Social media has become the main tool and platform for online political participation, especially microblogs which play a u-nique role in online rights protection. The main goal of civic online political participation is self-actualization.%基于大样本的网络问卷调查构建了多项式回归模型，从人口统计特征、网络基本使用特征、网络政治参与行为特征和对网络政治参与效果的态度四个方面，对中国网络政治参与主体的结构与特征进行了探讨。这些特征主要是：参与主体为年纪轻、高学历、高收入、高素质、赋有理性的群体；现实社会中职业声望与网络话语权并不完全匹配，甚至是对立的；社会化媒体成为网络政治参与的主要工具，特别是微博在网络维权上发挥着独特的作用；公民网络政治参与的主要目的是自我实现。
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,
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.
Piecewise polynomial solutions to linear inverse problems
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....
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...
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...
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...
A Deterministic and Polynomial Modified Perceptron Algorithm
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.
Real meromorphic functions and linear differential polynomials
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.
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.
Normality and shared values concerning differential polynomials
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
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...
Some Orthogonal Polynomials in Four Variables
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
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.
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
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.
Maximal aggregation of polynomial dynamical systems.
Cardelli, Luca; Tribastone, Mirco; Tschaikowski, Max; Vandin, Andrea
2017-09-19
Ordinary differential equations (ODEs) with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science, but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models. Here we propose an aggregation technique that rests on two notions of equivalence relating ODE variables whenever they have the same solution (backward criterion) or if a self-consistent system can be written for describing the evolution of sums of variables in the same equivalence class (forward criterion). A key feature of our proposal is to encode a polynomial ODE system into a finitary structure akin to a formal chemical reaction network. This enables the development of a discrete algorithm to efficiently compute the largest equivalence, building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements. The physical interpretability of the aggregation is shown on polynomial ODE systems for biochemical reaction networks, gene regulatory networks, and evolutionary game theory.
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.
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 ...
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.
Commutators with idempotent values on multilinear polynomials in prime rings
2017-02-01
Let $R$ be a prime ring of characteristic different from 2, $C$ its extended centroid, $d$ a nonzero derivation of $R$, $f (x_1,\\ldots , x_n)$ a multilinear polynomial over $C$, $\\varrho$ a nonzero right ideal of $R$ and $m > 1$ a fixed integer such that $$([d(f (r_1, \\ldots , r_n)), f (r_1, \\ldots , _n)])^m = [d(f (r_1, \\ldots , r_n)), f(r_1, \\ldots , r_n)]$$ for all $r_1, \\ldots , r_n \\in\\varrho$. Then either $[f(x_1, \\ldots , x_n), x_{n+1}]x_{n+2}$ is an identity for $\\varrho$ or $d(\\varrho)\\varrho = 0$.
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.
Conditional Density Approximations with Mixtures of Polynomials
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...
Tetromino tilings and the Tutte polynomial
Lykke Jacobsen, Jesper
2007-02-01
We consider tiling rectangles of size 4m × 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial ZG(Q, v) (known also to physicists as the partition function of the Q-state Potts model) on an (m - 1) × (n - 1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.
Tetromino tilings and the Tutte polynomial
Jacobsen, Jesper Lykke [Laboratoire de Physique Theorique et Modeles Statistiques, Universite Paris-Sud, Bat. 100, 91405 Orsay (France); Service de Physique Theorique, CEA Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette (France)
2007-02-16
We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial Z{sub G}(Q, v) (known also to physicists as the partition function of the Q-state Potts model) on an (m - 1) x (n - 1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.
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...
Parallel multigrid smoothing: polynomial versus Gauss-Seidel
Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray
2003-07-01
Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.
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.
On an Inequality of Pual Turan Concerning Polynomials-II
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.
Some Results on the Polar Derivative of a Polynomial
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.
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 ...
On a method of approximation by Jacobi polynomials
Dubey,R.K.; Pandey, R. K.
2005-01-01
Convolution structure for Jacobi series allows end point summability of Fourier-Jacobi expansions to lead an approximation of function by a linear combination of Jacobi polynomials. Thus, using Ces$\\grave a$ro summability of some orders $>1$ at $x=1,$ we prove a result of approximation of functions on $[-1,1]$ by operators involving Jacobi polynomials. Precisely, we pick up functions from a Lebesgue integrable space and then study its representation by Jacobi polynomials und...
Local polynomial Whittle estimation covering non-stationary fractional processes
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....
Tridiagonal Operators and Zeros of Polynomials in Two Variables
Chrysi G. Kokologiannaki
2016-01-01
Full Text Available The aim of this paper is to connect the zeros of polynomials in two variables with the eigenvalues of a self-adjoint operator. This is done by use of a functional-analytic method. The polynomials in two variables are assumed to satisfy a five-term recurrence relation, similar to the three-term recurrence relation that the classical orthogonal polynomials satisfy.
On Lehmer's Conjecture for Polynomials and for Elliptic Curves
Silverman, Joseph H
2010-01-01
A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of the form 1+X+...+X^n for some n > \\epsilon*deg(f). We also formulate and prove an analogous statement for elliptic curves.
Representations of the Schroedinger group and matrix orthogonal polynomials
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)
Multiple Twisted -Euler Numbers and Polynomials Associated with -Adic -Integrals
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.
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...
A lattice Boltzmann method based on generalized polynomials
Coelho, Rodrigo C V; Doria, Mauro M
2015-01-01
We propose a lattice Boltzmann method based on the expansion of the equilibrium distribution function in powers of generalized orthonormal polynomials which are weighted by the equilibrium distribution function itself. The D-dimensional Euclidean space Hermite polynomials correspond to the particular weight of a gaussian function. The proposed polynomials give a general method to obtain an expansion of the equilibrium distribution function in powers of the ratio between the displacement velocity and the local scale velocity of the fluid.
THE λ-GR(O)BNER BASES UNDER POLYNOMIAL COMPOSITION
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.
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.
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.
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.
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.
An Analytic Formula for the A_2 Jack Polynomials
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
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.
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.
Polynomial analysis of canopy spectra and biochemical component content inversion
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.
Tutte Polynomial of Pseudofractal Scale-Free Web
Peng, Junhao; Xiong, Jian; Xu, Guoai
2015-06-01
The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both Combinatorics and Statistical physics. It contains various numerical invariants and polynomial invariants, such as the number of spanning trees, the number of spanning forests, the number of acyclic orientations, the reliability polynomial, chromatic polynomial and flow polynomial. In this paper, we study and obtain a recursive formula for the Tutte polynomial of pseudofractal scale-free web (PSFW), and thus logarithmic complexity algorithm to calculate the Tutte polynomial of the PSFW is obtained, although it is NP-hard for general graph. By solving the recurrence relations derived from the Tutte polynomial, the rigorous solution for the number of spanning trees of the PSFW is obtained. Therefore, an alternative approach to determine explicitly the number of spanning trees of the PSFW is given. Furthermore, we analyze the all-terminal reliability of the PSFW and compare the results with those of the Sierpinski gasket which has the same number of nodes and edges as the PSFW. In contrast with the well-known conclusion that inhomogeneous networks (e.g., scale-free networks) are more robust than homogeneous networks (i.e., networks in which each node has approximately the same number of links) with respect to random deletion of nodes, the Sierpinski gasket (which is a homogeneous network), as our results show, is more robust than the PSFW (which is an inhomogeneous network) with respect to random edge failures.
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.
Arithmetic matroids, the Tutte polynomial and toric arrangements
D’Adderio, Michele; Moci, Luca
2013-01-01
.... Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's...
Quantum algorithms for virtual Jones polynomials via Thistlethwaite theorems
Vélez, Mario; Ospina, Juan
2010-04-01
Recently a quantum algorithm for the Jones polynomial of virtual links was proposed by Kauffman and Dye via the implementation of the virtual braid group in anyonic topological quantum computation when the virtual crossings are considered as generalized swap gates. Also recently, a mathematical method for the computation of the Jones polynomial of a given virtual link in terms of the relative Tuttle polynomial of its face (Tait) graph with some suitable variable substitutions was proposed by Diao and Hetyei. The method of Diao and Hetyei is offered as an alternative to the ribbon graph approach according to which the Tutte polynomial of a given virtual link is computed in terms of the Bollobás- Riordan polynomial of the corresponding ribbon graph. The method of Diao and Hetyei can be considered as an extension of the celebrated Thistlethwaite theorem according to which invariant polynomials for knots and links are derived from invariant polynomials for graphs. Starting from these ideas we propose a quantum algorithm for the Jones polynomial of a given virtual link in terms of the generalized Tutte polynomials by exploiting the Thistlethwaite theorem and the Kauffman algorithm . Our method is claimed as the quantum version of the Diao-Hetyei method. Possible supersymmetric implementations of our algortihm are discussed jointly with its formulations using topological quantum lambda calculus.
Using Tutte polynomials to characterize sexual contact networks
Cadavid Muñoz, Juan José
2014-06-01
Tutte polynomials are used to characterize the dynamic and topology of the sexual contact networks, in which pathogens are transmitted as an epidemic. Tutte polynomials provide an algebraic characterization of the sexual contact networks and allow the projection of spread control strategies for sexual transmission diseases. With the usage of Tutte polynomials, it allows obtaining algebraic expressions for the basic reproductive number of different pathogenic agents. Computations are done using the computer algebra software Maple, and it's GraphTheory Package. The topological complexity of a contact network is represented by the algebraic complexity of the correspondent polynomial. The change in the topology of the contact network is represented as a change in the algebraic form of the associated polynomial. With the usage of the Tutte polynomials, the number of spanning trees for each contact network can be obtained. From the obtained results in the polynomial form, it can be said that Tutte polynomials are of great importance for designing and implementing control measures for slowing down the propagation of sexual transmitted pathologies. As a future research line, the analysis of weighted sexual contact networks using weighted Tutte polynomials is considered.
A Bivariate Analogue to the Composed Product of Polynomials
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.
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.
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.
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
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.
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...
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...
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. [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.)
The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link
Oblomkov, Alexei; Shende, Vivek
2012-01-01
The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a nontrivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreduci...
A Parallel Algorithm for Finding Roots of a Complex Polynomial
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].
Approximation to Continuous Functions by a Kind of Interpolation Polynomials
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
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.
POLYNOMIAL RECURRENCE FOR L（E）VY PROCESSES
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.
q-ultraspherical polynomials for q a root of unity
Spiridonov, V P
1996-01-01
Properties of the q-ultraspherical polynomials for q being a primitive root of unity are derived using a formalism of the so_q(3) algebra. The orthogonality condition for these polynomials provides a new class of trigonometric identities representing discrete finite-dimensional analogs of q-beta integrals of Ramanujan.
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.
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.
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
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…
On Continued Fraction Expansion of Real Roots of Polynomial Systems
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...
Weighted Approximation for Jackson-Matsuoka Polynomials on the Sphere
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.
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
Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials
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.
On Period of the Sequence of Fibonacci Polynomials Modulo
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.
Computing the Tutte Polynomial in Vertex-Exponential Time
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...
Inequalities for a Polynomial and its Derivative
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
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.
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.
Study on the Grey Polynomial Geometric Programming
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.
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 ...
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...
A polynomial algorithm for abstract maximum flow
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. .
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.
Towards synthesis of polynomial controllers - a multi-affine approach
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....
Higher order branching of periodic orbits from polynomial isochrones
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.
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...
Lam, Hak-Keung
2016-01-01
This book presents recent research on the stability analysis of polynomial-fuzzy-model-based control systems where the concept of partially/imperfectly matched premises and membership-function dependent analysis are considered. The membership-function-dependent analysis offers a new research direction for fuzzy-model-based control systems by taking into account the characteristic and information of the membership functions in the stability analysis. The book presents on a research level the most recent and advanced research results, promotes the research of polynomial-fuzzy-model-based control systems, and provides theoretical support and point a research direction to postgraduate students and fellow researchers. Each chapter provides numerical examples to verify the analysis results, demonstrate the effectiveness of the proposed polynomial fuzzy control schemes, and explain the design procedure. The book is comprehensively written enclosing detailed derivation steps and mathematical derivations also for read...
Yu, J G; Zhang, Ch; Lefebvre, J E
2014-08-01
Wave propagation in multilayered piezoelectric structures has received much attention in past forty years. But the research objects of previous research works are only for semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes an extension of the orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2-D) piezoelectric structure, namely, a multilayered piezoelectric bar with a rectangular cross-section. Through numerical comparison with the available reference results for a purely elastic multilayered rectangular bar, the validity of the extended polynomial series approach is illustrated. The dispersion curves and electric potential distributions of various multilayered piezoelectric rectangular bars are calculated to reveal their wave propagation characteristics.
Graefe, Eva-Maria; Korsch, Hans Jürgen; Rush, Alexander
2016-04-01
Bosonic quantum conversion systems can be modeled by many-particle single-mode Hamiltonians describing a conversion of m molecules of type A into n molecules of type B and vice versa. These Hamiltonians are analyzed in terms of generators of a polynomially deformed su(2) algebra. In the mean-field limit of large particle numbers, these systems become classical and their Hamiltonian dynamics can again be described by polynomial deformations of a Lie algebra, where quantum commutators are replaced by Poisson brackets. The Casimir operator restricts the motion to Kummer shapes, deformed Bloch spheres with cusp singularities depending on m and n . It is demonstrated that the many-particle eigenvalues can be recovered from the mean-field dynamics using a WKB-type quantization condition. The many-particle state densities can be semiclassically approximated by the time periods of periodic orbits, which show characteristic steps and singularities related to the fixed points, whose bifurcation properties are analyzed.
Robinson-Schensted-Knuth Correspondence and Weak Polynomial Identities of M1,1(E)
Onofrio Mario Di Vincenzo; Roberto La Scala
2005-01-01
In this paper, it is proved that the ideal Iω of the weak polynomial identities of the superalgebra M1,1(E) is generated by the p roper polynomials [x1,x2, x3] and [x2,x1][xa,x1][x4, x1]. This is proved for any infinite field F of characteristic different a basis and the dimension of any multihomogeneous component of the quotient algebra B/(B ∩ Iw). We also compute the Hilbert series of this algebra. One of the main tools of this paper is a variant we found of the Robinson-Schensted-Knuth correspondence defined for single semistandard tableaux of double shape.