Optical theorem for Aharonov-Bohm scattering
Sitenko, Yu A
2011-01-01
Quantum-mechanical scattering off an impermeable magnetic vortex is considered and the optical theorem is derived. The nonvanishing transverse size of the vortex is taken into account, and the Robin boundary condition is imposed on the particle wave function at the edge of the vortex. The persistence of the Fraunhofer diffraction in the short-wavelength limit is shown to be crucial for maintaining the optical theorem in the quasiclassical limit.
Optical zilch and Noether's theorem
Philbin, T G
2013-01-01
A simple conserved quantity for electromagnetic fields in vacuum was discovered by Lipkin in 1964. In recent years this "zilch" has been used as a measure of the chirality of light. The conservation of optical zilch is here shown to be due to a simple symmetry of the standard electromagnetic action. Other investigations of the symmetry underlying zilch conservation have not been based on transformations of the dynamical variables of the standard action (the vector and scalar potentials). The symmetry transformation allows the identification of zilch eigenstates, which are shown to be circularly polarized plane waves. The same symmetry is present for electromagnetism in a homogeneous, dispersive medium, allowing the derivation of the zilch density and flux in such a medium.
Generalized Optical Theorem Detection in Random and Complex Media
Tu, Jing
The problem of detecting changes of a medium or environment based on active, transmit-plus-receive wave sensor data is at the heart of many important applications including radar, surveillance, remote sensing, nondestructive testing, and cancer detection. This is a challenging problem because both the change or target and the surrounding background medium are in general unknown and can be quite complex. This Ph.D. dissertation presents a new wave physics-based approach for the detection of targets or changes in rather arbitrary backgrounds. The proposed methodology is rooted on a fundamental result of wave theory called the optical theorem, which gives real physical energy meaning to the statistics used for detection. This dissertation is composed of two main parts. The first part significantly expands the theory and understanding of the optical theorem for arbitrary probing fields and arbitrary media including nonreciprocal media, active media, as well as time-varying and nonlinear scatterers. The proposed formalism addresses both scalar and full vector electromagnetic fields. The second contribution of this dissertation is the application of the optical theorem to change detection with particular emphasis on random, complex, and active media, including single frequency probing fields and broadband probing fields. The first part of this work focuses on the generalization of the existing theoretical repertoire and interpretation of the scalar and electromagnetic optical theorem. Several fundamental generalizations of the optical theorem are developed. A new theory is developed for the optical theorem for scalar fields in nonhomogeneous media which can be bounded or unbounded. The bounded media context is essential for applications such as intrusion detection and surveillance in enclosed environments such as indoor facilities, caves, tunnels, as well as for nondestructive testing and communication systems based on wave-guiding structures. The developed scalar
Quantum optical ABCD theorem in two-mode case
Institute of Scientific and Technical Information of China (English)
Fan Hong-Yi; Hu Li-Yun
2008-01-01
By introducing the entangled Fresnel operator (EFO) this paper demonstrates that there exists ABCD theorem for two-mode entangled case in quantum optics.The canonical operator method as mapping of ray-transfer ABCD matrix is explicitly shown by EFO's normally ordered expansion through the coherent state representation and the technique of integration within an ordered product of operators.
Optical theorem for multipole sources in wave diffraction theory
Eremin, Yu. A.; Sveshnikov, A. G.
2016-05-01
The optical theorem is generalized to the case of local body excitation by multipole sources. It is found that, to calculate the extinction cross section, it is sufficient to calculate the scattered field derivatives at a single point. It is shown that the Purcell factor, which is a rather important parameter, can be represented in analytic form. The result is generalized to the case of a local scatterer incorporated in a homogeneous halfspace.
The optical theorem for local source excitation of a particle near a plane interface
Eremin, Yuri; Wriedt, Thomas
2015-11-01
Based on classic Maxwell's theory and the Gauss Theorem we extended the Optical Theorem to the case of a penetrable particle excited by a local source deposited near a plane interface. We demonstrate that the derived Extinction Cross-Section involves the total point source radiating cross-section and some definite integrals responsible for the scattering by the interface. The derived extinction cross-section can be employed to estimate the quantum yield and the optical antenna efficiency without computation of the absorption cross-section.
Mitri, F G
2015-09-01
The optical theorem for plane waves is recognized as one of the fundamental theorems in optical, acoustical and quantum wave scattering theory as it relates the extinction cross-section to the forward scattering complex amplitude function. Here, the optical theorem is extended and generalized in a cylindrical coordinates system for the case of 2D beams of arbitrary character as opposed to plane waves of infinite extent. The case of scalar monochromatic acoustical wavefronts is considered, and generalized analytical expressions for the extinction, absorption and scattering cross-sections are derived and extended in the framework of the scalar resonance scattering theory. The analysis reveals the presence of an interference scattering cross-section term describing the interaction between the diffracted Franz waves with the resonance elastic waves. The extended optical theorem in cylindrical coordinates is applicable to any object of arbitrary geometry in 2D located arbitrarily in the beam's path. Related investigations in optics, acoustics and quantum mechanics will benefit from this analysis in the context of wave scattering theory and other phenomena closely connected to it, such as the multiple scattering by a cloud of particles, as well as the resulting radiation force and torque.
On a possible violation of the optical theorem in LHC experiments
Kupczynski, Marian
2013-01-01
The optical theorem allowing the determination of the total cross-section for the hadron-hadron scattering from the imaginary part of the forward elastic scattering amplitude is believed to be an unavoidable consequence of the conservation of the probability and of the unitary S matrix. This is a fundamental theorem which contains not directly measurable imaginary part of the forward elastic scattering amplitude. The impossibility of scattering phenomena without the existence of the elastic channel is considered to be a part of the quantum magic. However if one takes seriously the idea that the hadrons are extended particles one may define a unitary S matrix such that one cannot prove the optical theorem. Moreover the data violating the optical theorem do exist but they are not conclusive due to the uncertainties related to the extrapolation of the differential elastic cross-section to the forward direction. These results were published several years ago but they were forgotten. In this paper we will recall t...
Mitri, F G
2016-04-01
One of the fundamental theorems in (optical, acoustical, quantum, gravitational) wave scattering is the optical theorem for plane waves, which relates the extinction cross-section to the forward scattering complex amplitude function. In this analysis, the optical theorem is extended for the case of 3D-beams of arbitrary character in a cylindrical coordinates system for any angle of incidence and any scattering angle. Generalized analytical expressions for the extinction, absorption, scattering cross-sections and efficiency factors are derived in the framework of the scalar resonance scattering theory for an object of arbitrary shape. The analysis reveals the presence of an interference scattering cross-section term, which describes interference between the diffracted or specularly reflected inelastic (Franz) waves with the resonance elastic waves. Moreover, an alternate expression for the extinction cross-section, which relates the resonance cross-section with the scattering cross-section for an impenetrable object, is obtained, suggesting an improved method for particle characterization. Cross-section expressions are also derived for known acoustical wavefronts centered on the object, defined as the on-axis case. The extended optical theorem in cylindrical coordinates can be applied to evaluate the extinction efficiency from any object of arbitrary geometry placed on or off the axis of the incident beam. Applications in acoustics, optics, and quantum mechanics should benefit from this analysis in the context of wave scattering theory and other phenomena closely connected to it, such as the multiple scattering by many particles, as well as the radiation force and torque.
No-go theorem for passive single-rail linear optical quantum computing.
Wu, Lian-Ao; Walther, Philip; Lidar, Daniel A
2013-01-01
Photonic quantum systems are among the most promising architectures for quantum computers. It is well known that for dual-rail photons effective non-linearities and near-deterministic non-trivial two-qubit gates can be achieved via the measurement process and by introducing ancillary photons. While in principle this opens a legitimate path to scalable linear optical quantum computing, the technical requirements are still very challenging and thus other optical encodings are being actively investigated. One of the alternatives is to use single-rail encoded photons, where entangled states can be deterministically generated. Here we prove that even for such systems universal optical quantum computing using only passive optical elements such as beam splitters and phase shifters is not possible. This no-go theorem proves that photon bunching cannot be passively suppressed even when extra ancilla modes and arbitrary number of photons are used. Our result provides useful guidance for the design of optical quantum computers.
Cancellation of spurious arrivals in Green's function extraction and the generalized optical theorem
Snieder, R.; Van Wijk, K.; Haney, M.; Calvert, R.
2008-01-01
The extraction of the Green's function by cross correlation of waves recorded at two receivers nowadays finds much application. We show that for an arbitrary small scatterer, the cross terms of scattered waves give an unphysical wave with an arrival time that is independent of the source position. This constitutes an apparent inconsistency because theory predicts that such spurious arrivals do not arise, after integration over a complete source aperture. This puzzling inconsistency can be resolved for an arbitrary scatterer by integrating the contribution of all sources in the stationary phase approximation to show that the stationary phase contributions to the source integral cancel the spurious arrival by virtue of the generalized optical theorem. This work constitutes an alternative derivation of this theorem. When the source aperture is incomplete, the spurious arrival is not canceled and could be misinterpreted to be part of the Green's function. We give an example of how spurious arrivals provide information about the medium complementary to that given by the direct and scattered waves; the spurious waves can thus potentially be used to better constrain the medium. ?? 2008 The American Physical Society.
Nieto-Vesperinas, Manuel
2015-01-01
We put forward the physical meaning of the conservation equation for the helicity on scattering of an electromagnetic field with a generally magnetodielectric bi-isotropic dipolar object. This is the optical theorem for the helicity that, as we find, plays a role for this quantity analogous to that of the optical thorem for energy. We discuss its consequences for helicity transfer between molecules and for new detection procedures of circular dichroism based on ellipsometric measurements.
Emitter near an arbitrary body: Purcell effect, optical theorem and the Wheeler-Feynman absorber
Venkatapathi, Murugesan
2012-09-01
The altered spontaneous emission of an emitter near an arbitrary body can be elucidated using an energy balance of the electromagnetic field. From a classical point of view it is trivial to show that the field scattered back from any body should alter the emission of the source. But it is not at all apparent that the total radiative and non-radiative decay in an arbitrary body can add to the vacuum decay rate of the emitter (i.e.) an increase of emission that is just as much as the body absorbs and radiates in all directions. This gives us an opportunity to revisit two other elegant classical ideas of the past, the optical theorem and the Wheeler-Feynman absorber theory of radiation. It also provides us alternative perspectives of Purcell effect and generalizes many of its manifestations, both enhancement and inhibition of emission. When the optical density of states of a body or a material is difficult to resolve (in a complex geometry or a highly inhomogeneous volume) such a generalization offers new directions to solutions.
Emitter near an arbitrary body: Purcell effect, optical theorem and the Wheeler-Feynman absorber
Venkatapathi, Murugesan
2012-01-01
The altered spontaneous emission of an emitter near an arbitrary body can be elucidated using an energy balance of the electromagnetic field. From a classical point of view it is trivial to show that the field scattered back from any body should alter the emission of the source. But it is not at all apparent that the total radiative and non-radiative decay in an arbitrary body can add to the vacuum decay rate of the emitter (i.e.) an increase of emission that is just as much as the body absorbs and radiates in all directions. This gives us an opportunity to revisit two other elegant classical ideas of the past, the optical theorem and the Wheeler-Feynman absorber theory of radiation. It also provides us alternative perspectives of Purcell effect and generalizes many of its manifestations, both enhancement and inhibition of emission. When the optical density of states of a body or a material is difficult to resolve (in a complex geometry or a highly inhomogeneous volume) such a generalization offers new direct...
Avanaki, Mohammad R N; Podoleanu, Adrian Gh; Schofield, John B; Jones, Carole; Sira, Manu; Liu, Yan; Hojjat, Ali
2013-03-10
An optical properties extraction algorithm is developed based on enhanced Huygens-Fresnel light propagation theorem, to extract the scattering coefficient of a specific region in an optical coherence tomography (OCT) image. The aim is to quantitatively analyze the OCT images. The algorithm is evaluated using a set of phantoms with different concentrations of scatterers, designed based on Mie theory. The algorithm is then used to analyze basal cell carcinoma and healthy eyelid tissues, demonstrating distinguishable differences in the scattering coefficient between these tissues. In this study, we have taken advantage of the simplification introduced by the utilization of a dynamic focus OCT system. This eliminates the need to deconvolve the reflectivity profile with the confocal gate profile, as the sensitivity of the OCT system is constant throughout the axial range.
Flatto, Leopold
2009-01-01
Poncelet's theorem is a famous result in algebraic geometry, dating to the early part of the nineteenth century. It concerns closed polygons inscribed in one conic and circumscribed about another. The theorem is of great depth in that it relates to a large and diverse body of mathematics. There are several proofs of the theorem, none of which is elementary. A particularly attractive feature of the theorem, which is easily understood but difficult to prove, is that it serves as a prism through which one can learn and appreciate a lot of beautiful mathematics. This book stresses the modern appro
Heck, Richard G
2011-01-01
Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterlyfundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that a
The quantitative Morse theorem
Loi, Ta Le; Phien, Phan
2013-01-01
In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \\cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.
Vorticity, Stokes' Theorem and the Gauss's Theorem
Narayanan, M.
2004-12-01
Vorticity is a property of the flow of any fluid and moving fluids acquire properties that allow an engineer to describe that particular flow in greater detail. It is important to recognize that mere motion alone does not guarantee that the air or any fluid has vorticity. Vorticity is one of four important quantities that define the kinematic properties of any fluid flow. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. However, the divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into, or away from the region through its boundary. This is also known as Gauss's Theorem. It should also be noted that there are many useful extensions of Gauss's Theorem, including the extension to include surfaces of discontinuity in V. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. Integral (Surface) [(DEL X V)] . dS = Integral (Contour) [V . dx] In this paper, the author outlines and stresses the importance of studying and teaching these mathematical techniques while developing a course in Hydrology and Fluid Mechanics. References Arfken, G. "Gauss's Theorem." 1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985. Morse, P. M. and Feshbach, H. "Gauss's Theorem." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 37-38, 1953. Eric W. Weisstein. "Divergence Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DivergenceTheorem.html
Lagrange Theorem for polygroups
Directory of Open Access Journals (Sweden)
alireza sedighi
2014-12-01
Full Text Available So far?, ?isomorphism theorems in hyperstructure were proved for different structures of polygroups?, ?hyperrings and etc?. ?In this paper?, ?the polygroups properties is studied with the introduction of a suitable equivalence relation?. ?We show that the above relation is strongly regular?. ?Our main purpose in the paper is investigating Lagrang theorem and other expressing of isomorphism theorems for polygroups?.
Levinson, N
1940-01-01
A typical gap theorem of the type discussed in the book deals with a set of exponential functions { \\{e^{{{i\\lambda}_n} x}\\} } on an interval of the real line and explores the conditions under which this set generates the entire L_2 space on this interval. A typical gap theorem deals with functions f on the real line such that many Fourier coefficients of f vanish. The main goal of this book is to investigate relations between density and gap theorems and to study various cases where these theorems hold. The author also shows that density- and gap-type theorems are related to various propertie
The asymmetric sandwich theorem
Simons, Stephen
2011-01-01
We discuss the asymmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As applications, we derive various results on the existence of linear functionals that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duality theorem. Most of the results are about affine functions defined on convex subsets of vector spaces, rather than linear functions defined on vector spaces. We consider both results that use a simple boundedness hypothesis (as in Rockafellar's version of the Fenchel duality theorem) and also results that use Baire's theorem (as in the Robinson-Attouch-Brezis version of the Fenchel duality theorem). This paper also contains some new results about metrizable topological vector spaces that are not necessarily locally convex.
2011-01-01
Arvanitakis established recently a theorem which is a common generalization of Michael's convex selection theorem and Dugundji's extension theorem. In this note we provide a short proof of a more general version of Arvanitakis' result.
Inverting the central limit theorem
Navascues, Miguel; Villanueva, Ignacio
2011-01-01
The central limit theorem states that the sum of N independently distributed n-tuples of real variables (subject to appropriate normalization) tends to a multivariate gaussian distribution for large N. Here we propose to invert this argument: given a set of n correlated gaussian variables, we try to infer information about the structure of the discrete microscopic probability distributions whose convolution generated such a macroscopic behavior. The techniques developed along the article are applied to prove that the classical description of certain macroscopic optical experiments is infinitely more complex than the quantum one.
On the Equivalence of Weyl Theorem and Generalized Weyl Theorem
Institute of Scientific and Technical Information of China (English)
M. BERKANI
2007-01-01
We know that an operator T acting on a Banach space satisfying generalized Weyl's theorem also satisfies Weyl's theo rem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl's theorem, then it also satisfies generalized Weyl's theorem. We give also a similar result for the equivalence of a-Weyl's theorem and generalized a-Weyl's theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.
To string together six theorems of physics by Pythagoras theorem
Cui, H Y
2002-01-01
In this paper, we point out that there are at lest six theorems in physics sharing common virtue of Pythagoras theorem, so that it is possible to string these theorems together with the Pythagoras theorem for physics teaching, the six theorems are Newton's three laws of motion, universal gravitational force, Coulomb's law, and the formula of relativistic dynamics. Knowing the internal relationships between them, which have never been clearly revealed by other author, will benefit the logic of physics teaching.
Trigonometry, Including Snell's Theorem.
Kent, David
1980-01-01
Aspects of the instruction of trigonometry in secondary school mathematics are reviewed. Portions of this document cover basic introductions, a student-developed theorem, the cosine rule, inverse functions, and a sample outdoor activity. (MP)
Trigonometry, Including Snell's Theorem.
Kent, David
1980-01-01
Aspects of the instruction of trigonometry in secondary school mathematics are reviewed. Portions of this document cover basic introductions, a student-developed theorem, the cosine rule, inverse functions, and a sample outdoor activity. (MP)
Coincidence Point Theorems, Intersection Theorems and Saddle Point Theorems on FC-spaces
Institute of Scientific and Technical Information of China (English)
PIAO YONG-JIE; YIN ZHE
2009-01-01
In this paper, we first give the definitions of finitely continuous topological space and FC-subspace generated by some set, and obtain coincidence point theorem, whole intersection theorems and Ky Fan type matching theorems, and finally discuss the existence of saddle point as an application of coincidence point theorem.
ON RANGE DECOMPOSITION THEOREMS
Institute of Scientific and Technical Information of China (English)
吴利生
1990-01-01
We prove the following theorems.Theorem 1.Suppose f:X→Y is a closed map.X is a ωγ and β space,then Y=Y0∪(∪n=1∞Yn),where f-1(y) is countably compact for each y ∈Y0 and Yn is closed discrete in Y for each n≥1,Theorem 2-3:Suppose f:X→Y is a closed map,X is stratifable space,then Y=Y0 ∪(∪n=1∞Yn),where f-1(y) is compact for each y∈Y0 and Yn is closed discrete in Y for each n≥1.
D'Agostini, G
2005-01-01
It is curious to learn that Enrico Fermi knew how to base probabilistic inference on Bayes theorem, and that some influential notes on statistics for physicists stem from what the author calls elsewhere, but never in these notes, {\\it the Bayes Theorem of Fermi}. The fact is curious because the large majority of living physicists, educated in the second half of last century -- a kind of middle age in the statistical reasoning -- never heard of Bayes theorem during their studies, though they have been constantly using an intuitive reasoning quite Bayesian in spirit. This paper is based on recollections and notes by Jay Orear and on Gauss' ``Theoria motus corporum coelestium'', being the {\\it Princeps mathematicorum} remembered by Orear as source of Fermi's Bayesian reasoning.
Converse Barrier Certificate Theorem
DEFF Research Database (Denmark)
Wisniewski, Rafael; Sloth, Christoffer
2013-01-01
This paper presents a converse barrier certificate theorem for a generic dynamical system.We show that a barrier certificate exists for any safe dynamical system defined on a compact manifold. Other authors have developed a related result, by assuming that the dynamical system has no singular...... points in the considered subset of the state space. In this paper, we redefine the standard notion of safety to comply with generic dynamical systems with multiple singularities. Afterwards, we prove the converse barrier certificate theorem and illustrate the differences between ours and previous work...
Kartavtsev, Alexander
2014-01-01
According to the Goldstone theorem a scalar theory with a spontaneously broken global symmetry contains strictly massless states. In this letter we identify a loophole in the current-algebra proof of the theorem. Therefore, the question whether in models with Mexican hat potential the tangential excitations are strictly massless or are just almost massless as compared to the radial ones remains open. We also argue that mass of the tangential excitations approaches zero even if the symmetry is not spontaneously broken but a combination of the field components invariant under the symmetry transformations acquires a large vacuum expectation value.
Pérez-Espigares, Carlos; Redig, Frank; Giardinà, Cristian
2015-08-01
For non-equilibrium systems of interacting particles and for interacting diffusions in d-dimensions, a novel fluctuation relation is derived. The theorem establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti-Cohen fluctuation theorem related to the time-reversal symmetry. This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables.
Lin, Tai-Chia; Petrovic, Milan S; Hajaiej, Hichem; Chen, Goong
2016-01-01
The virial theorem is a nice property for the linear Schrodinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrodinger equation with power-law nonlinearity, then a similar ratio can be obtained but there seems no way of getting any eigenvalue estimate. It is surprising as far as we are concerned that when the nonlinearity is either square-root or saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue estimate of nonlinear Schrodinger (NLS) equations in R2 with square-root and saturable nonlinearity, respectively. Furthermore, we show here that the eigenvalue estimate can be used to obtain the 2nd order term (which is of order $ln\\Gamma$) of the lower bound of the ground state energy as the coefficient $\\Gamma$ of the nonlinear term tends to infinity.
Virial Theorem and Scale Transformations.
Kleban, Peter
1979-01-01
Discussed is the virial theorem, which is useful in classical, quantum, and statistical mechanics. Two types of derivations of this theorem are presented and the relationship between the two is explored. (BT)
Certified Kruskal's Tree Theorem
Directory of Open Access Journals (Sweden)
Christian Sternagel
2014-07-01
Full Text Available This article presents the first formalization of Kurskal's tree theorem in aproof assistant. The Isabelle/HOL development is along the lines of Nash-Williams' original minimal bad sequence argument for proving the treetheorem. Along the way, proofs of Dickson's lemma and Higman's lemma, as well as some technical details of the formalization are discussed.
DEFF Research Database (Denmark)
Thomassen, Carsten
2004-01-01
We present a short proof of the theorem of Tutte that every planar 3-connected graph has a drawing in the plane such that every vertex which is not on the outer cycle is the barycenter of its neighbors. Moreover, this holds for any prescribed representation of the outer cycle. (C) 2004 Wiley Peri...
DEFF Research Database (Denmark)
Törnquist, Asger Dag; Weiss, W.
2009-01-01
We prove the following descriptive set-theoretic analogue of a theorem of R. 0. Davies: Every σ function f:ℝ × ℝ → ℝ can be represented as a sum of rectangular Σ functions if and only if all reals are constructible. © Instytut Matematyczny PAN, 2009....
Rediscovering Schreinemakers' Theorem.
Bathurst, Bruce
1983-01-01
Schreinemakers' theorem (arrangement of curves around an invariant point), derived from La Chatelier's principle, can be rediscovered by students asked to use the principle when solving a natural problem such as "How does diluting a mineral/fluid alter shape of a pressure/temperature diagram?" Background information and instructional…
Teleman, Nicolae
2011-01-01
$Local^{3}$ Index Theorem means $Local(Local(Local \\;Index \\; Theorem)))$. $Local \\; Index \\; Theorem$ is the Connes-Moscovici local index theorem \\cite{Connes-Moscovici1}, \\cite{Connes-Moscovici2}. The second "Local" refers to the cyclic homology localised to a certain separable subring of the ground algebra, while the last one refers to Alexander-Spanier type cyclic homology. The Connes-Moscovici work is based on the operator $R(A) = \\mathbf{P} - \\mathbf{e}$ associated to the elliptic pseudo-differential operator $A$ on the smooth manifold $M$, where $\\mathbf{P}$, $\\mathbf{e}$ are idempotents, see \\cite{Connes-Moscovici1}, Pg. 353. The operator $R(A)$ has two main merits: it is a smoothing operator and its distributional kernel is situated in an arbitrarily small neighbourhood of the diagonal in $M \\times M$. The operator $R(A)$ has also two setbacks: -i) it is not an idempotent (and therefore it does not have a genuine Connes-Chern character); -ii) even if it were an idempotent, its Connes-Chern character ...
Multivariate irregular sampling theorem
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper,we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result,we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.
Multivariate irregular sampling theorem
Institute of Scientific and Technical Information of China (English)
CHEN GuangGui; FANG GenSun
2009-01-01
In this paper, we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result, we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.
Dalen, D. van
2008-01-01
The following pages make form a new chapter for the book Logic and Structure. This chapter deals with the incompleteness theorem, and contains enough basic material for the treatment of the required notions of computability, representability and the like. This chapter will appear in the next edition
Converse Barrier Certificate Theorems
DEFF Research Database (Denmark)
Wisniewski, Rafael; Sloth, Christoffer
2016-01-01
This paper shows that a barrier certificate exists for any safe dynamical system. Specifically, we prove converse barrier certificate theorems for a class of structurally stable dynamical systems. Other authors have developed a related result by assuming that the dynamical system has neither sing...
Automated Discovery of Inductive Theorems
McCasland, Roy; Bundy, Alan; Serge, Autexier
2007-01-01
Inductive mathematical theorems have, as a rule, historically been quite difficult to prove – both for mathematics students and for auto- mated theorem provers. That said, there has been considerable progress over the past several years, within the automated reasoning community, towards proving some of these theorems. However, little work has been done thus far towards automatically discovering them. In this paper we present our methods of discovering (as well as proving) inductive theorems, ...
The Second Noether Theorem on Time Scales
Malinowska, Agnieszka B.; Natália Martins
2013-01-01
We extend the second Noether theorem to variational problems on time scales. As corollaries we obtain the classical second Noether theorem, the second Noether theorem for the $h$ -calculus and the second Noether theorem for the $q$ -calculus.
An Extension of Sobolev's Theorem
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Sobolev's Theorem is the most fundamental theorem in the theory of Invariant Cubature Formulas (ICFs). In this paper, a quantitative structure is established for the classical ICFs. Enlightened by this structure, the author generalizes the concept of ICFs and extends the Sobolev's Theorem to the case of generalized ICFs. Several illustrative examples are given.
Russell, Alan R.
2004-01-01
Pick's theorem can be used in various ways just like a lemon. This theorem generally finds its way in the syllabus approximately at the middle school level and in fact at times students have even calculated the area of a state considering its outline with the help of the above theorem.
Generalized no-broadcasting theorem.
Barnum, Howard; Barrett, Jonathan; Leifer, Matthew; Wilce, Alexander
2007-12-14
We prove a generalized version of the no-broadcasting theorem, applicable to essentially any nonclassical finite-dimensional probabilistic model satisfying a no-signaling criterion, including ones with "superquantum" correlations. A strengthened version of the quantum no-broadcasting theorem follows, and its proof is significantly simpler than existing proofs of the no-broadcasting theorem.
Virial theorem and hypervirial theorem in a spherical geometry
Energy Technology Data Exchange (ETDEWEB)
Li Yan; Chen Jingling [Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071 (China); Zhang Fulin, E-mail: flzhang@tju.edu.cn, E-mail: chenjl@nankai.edu.cn [Physics Department, School of Science, Tianjin University, Tianjin 300072 (China)
2011-09-09
The virial theorem in the one- and two-dimensional spherical geometry are presented in both classical and quantum mechanics. Choosing a special class of hypervirial operators, the quantum hypervirial relations in the spherical spaces are obtained. With the aid of the Hellmann-Feynman theorem, these relations can be used to formulate a perturbation theorem without wavefunctions, corresponding to the hypervirial-Hellmann-Feynman theorem perturbation theorem of Euclidean geometry. The one-dimensional harmonic oscillator and two-dimensional Coulomb system in the spherical spaces are given as two sample examples to illustrate the perturbation method. (paper)
Experimental studies of the transient fluctuation theorem using liquid crystals
Indian Academy of Sciences (India)
Soma Datta; Arun Roy
2009-05-01
In a thermodynamical process, the dissipation or production of entropy can only be positive or zero, according to the second law of thermodynamics. However, the laws of thermodynamics are applicable to large systems in the thermodynamic limit. Recently a fluctuation theorem, known as the transient fluctuation theorem (TFT), which generalizes the second law of thermodynamics to small systems has been proposed. This theorem has been tested in small systems such as a colloidal particle in an optical trap. We report for the first time an analogous experimental study of TFT in a spatially extended system using liquid crystals.
Taylor, Marika
2016-01-01
The F theorem states that, for a unitary three dimensional quantum field theory, the F quantity defined in terms of the partition function on a three sphere is positive, stationary at fixed point and decreases monotonically along a renormalization group flow. We construct holographic renormalization group flows corresponding to relevant deformations of three-dimensional conformal field theories on spheres, working to quadratic order in the source. For these renormalization group flows, the F quantity at the IR fixed point is always less than F at the UV fixed point, but F increases along the RG flow for deformations by operators of dimension $3/2 < \\Delta < 5/2$. Therefore the strongest version of the F theorem is in general violated.
Indian Academy of Sciences (India)
N V Rao
2003-02-01
The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose is a compact set in the complex plane and 0 belongs to the boundary . Let $\\mathcal{A}(K)$ denote the space of all functions on such that is holomorphic in a neighborhood of and (0) = 0. Also for any given positive integer , let $\\mathcal{A}(m, K)$ denote the space of all such that is holomorphic in a neighborhood of and $f(0) = f'(0) = \\cdots = f^{(m)}(0) = 0$. Then $\\mathcal{A}(m, K)$ is dense in $\\mathcal{A}(K)$ under the supremum norm on provided that there exists a sector $W = \\{re^{i}; 0 ≤ r ≤ , ≤ ≤ \\}$ such that $W \\cap K = \\{0\\}$. (This is the well-known Poincare's external cone condition).} We present various generalizations of this result in the context of higher dimensions replacing holomorphic with harmonic.
1987-03-20
with standard expressions of spherical trigonometry is sinr)0 = cos0 sini//0 (4.37) which is consistent with the results obtained previously with...theorems for discrete transforms. However, sampling questions inlroduce difficult obstacles in the develop- ment of a discrete theory. First, sampling...additional obstacle to discrete represen- tations of the CT. An example of qualitative predication of the shape of silhouettes with the Silhouette-Slice
Gouin, Henri
2015-01-01
Comments on Archimedes' theorem about sphere and cylinder; In his treatise addressed to Dositheus of Pelusium, Archimedes of Syracuse obtained the result of which he was the most proud: a sphere has two-thirds the volume of its circumscribing cylinder. At his request a sculpted sphere and cylinder were placed on his tomb near Syracuse. Usually, it is admitted that to find this formula, Archimedes used a half polygon inscribed in a semicircle; then he performed rotations of these two figures t...
Sandwich classification theorem
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Alexey Stepanov
2015-09-01
Full Text Available The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''. For subgroups FleN of a group G denote by Lat(F,N the set of all subgroups of N , containing F . Let D be a subgroup of G . In this note we study the lattice LL=Lat(D,G and the lattice LL ′ of subgroups of G , normalized by D . We say that LL satisfies sandwich classification theorem if LL splits into a disjoint union of sandwiches Lat(F,N G (F over all subgroups F such that the normal closure of D in F coincides with F . Here N G (F denotes the normalizer of F in G . A similar notion of sandwich classification is introduced for the lattice LL ′ . If D is perfect, i.,e. coincides with its commutator subgroup, then it turns out that sandwich classification theorem for LL and LL ′ are equivalent. We also show how to find basic subroup F of sandwiches for LL ′ and review sandwich classification theorems in algebraic groups over rings.
Transfinite Approximation of Hindman's Theorem
Beiglböck, Mathias
2010-01-01
Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a "transfinite" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.
The Non-Signalling theorem in generalizations of Bell's theorem
Walleczek, J.; Grössing, G.
2014-04-01
Does "epistemic non-signalling" ensure the peaceful coexistence of special relativity and quantum nonlocality? The possibility of an affirmative answer is of great importance to deterministic approaches to quantum mechanics given recent developments towards generalizations of Bell's theorem. By generalizations of Bell's theorem we here mean efforts that seek to demonstrate the impossibility of any deterministic theories to obey the predictions of Bell's theorem, including not only local hidden-variables theories (LHVTs) but, critically, of nonlocal hidden-variables theories (NHVTs) also, such as de Broglie-Bohm theory. Naturally, in light of the well-established experimental findings from quantum physics, whether or not a deterministic approach to quantum mechanics, including an emergent quantum mechanics, is logically possible, depends on compatibility with the predictions of Bell's theorem. With respect to deterministic NHVTs, recent attempts to generalize Bell's theorem have claimed the impossibility of any such approaches to quantum mechanics. The present work offers arguments showing why such efforts towards generalization may fall short of their stated goal. In particular, we challenge the validity of the use of the non-signalling theorem as a conclusive argument in favor of the existence of free randomness, and therefore reject the use of the non-signalling theorem as an argument against the logical possibility of deterministic approaches. We here offer two distinct counter-arguments in support of the possibility of deterministic NHVTs: one argument exposes the circularity of the reasoning which is employed in recent claims, and a second argument is based on the inconclusive metaphysical status of the non-signalling theorem itself. We proceed by presenting an entirely informal treatment of key physical and metaphysical assumptions, and of their interrelationship, in attempts seeking to generalize Bell's theorem on the basis of an ontic, foundational
A Time scales Noether's theorem
Anerot, Baptiste; Cresson, Jacky; Pierret, Frédéric
2016-01-01
We prove a time scales version of the Noether's theorem relating group of symmetries and conservation laws. Our result extends the continuous version of the Noether's theorem as well as the discrete one and corrects a previous statement of Bartosiewicz and Torres in \\cite{BT}.
Abelian theorems for Whittaker transforms
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Richard D. Carmichael
1987-01-01
Full Text Available Initial and final value Abelian theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches 0 or ∞ in absolute value inside a wedge region in the right half plane.
Lopez-Real, Francis
2008-01-01
While the author was searching the web, he came across an article by Michael Keyton of IMSA (Illinois Mathematics and Science Academy) called "Theorems of mystery". The phrase is Keyton's own, and he defines such a theorem as "a result that has considerable structure with minimal hypotheses." The simplest of his 10 examples is one that many…
Directory of Open Access Journals (Sweden)
Coghetto Roland
2015-06-01
Full Text Available Morley’s trisector theorem states that “The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle” [10]. There are many proofs of Morley’s trisector theorem [12, 16, 9, 13, 8, 20, 3, 18]. We follow the proof given by A. Letac in [15].
Geometry of the Adiabatic Theorem
Lobo, Augusto Cesar; Ribeiro, Rafael Antunes; Ribeiro, Clyffe de Assis; Dieguez, Pedro Ruas
2012-01-01
We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.…
Some Theorems on Generalized Basic Hypergeometric Series
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A. D. Wadhwa
1972-07-01
Full Text Available In an earlier paper the author has established two theorems on generalized hypergeometric functions. In each theorem a numerator differs from a denominator by a positive integer. These theorems were further used to prove some theorems on the sums of Kampe de Feriet functions. Here, we have established the theorems which are the basic analogues of the theorems proved in the earlier paper.
Combinatorial Reciprocity Theorems
Beck, Matthias
2012-01-01
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and $P$-partitions. We will see that in each instance we get interesting information out of a counting function when we evaluate it at a \\emph{negative} integer (and so, a priori the counting function does not make sense at this number). Our goals are to convey some of the charm these "alternative" evaluations of counting functions exhibit, and to weave a unifying thread through various combinatorial reciprocity theorems by looking at them through the lens of geometry, which will include some scenic detours through other combinatorial concepts.
Towards the Carpenter's Theorem
Argerami, Martin
2008-01-01
Let M be a II_1 factor, A a masa in M and E the unique conditional expectation on A. Under some technical assumptions on the inclusion of A in M, which hold true for any semiregular masa of a separable factor, we show that for every discrete a in the positive part of the unit ball of A it is possible to find a projection p in M such that E(p)=a$. We also show an example of a diffuse operator x in A such that there exists a projection q in M with E(q)=x. These results show a new family of instances of a conjecture by Kadison, the so-called ``Carpenter's Theorem''.
Smorynski, Craig
2017-01-01
This book is about the rise and supposed fall of the mean value theorem. It discusses the evolution of the theorem and the concepts behind it, how the theorem relates to other fundamental results in calculus, and modern re-evaluations of its role in the standard calculus course. The mean value theorem is one of the central results of calculus. It was called “the fundamental theorem of the differential calculus” because of its power to provide simple and rigorous proofs of basic results encountered in a first-year course in calculus. In mathematical terms, the book is a thorough treatment of this theorem and some related results in the field; in historical terms, it is not a history of calculus or mathematics, but a case study in both. MVT: A Most Valuable Theorem is aimed at those who teach calculus, especially those setting out to do so for the first time. It is also accessible to anyone who has finished the first semester of the standard course in the subject and will be of interest to undergraduate mat...
-Dimensional Fractional Lagrange's Inversion Theorem
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F. A. Abd El-Salam
2013-01-01
Full Text Available Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of -dimensional polyadics is derived. A fractional -dimensional Lagrange inversion theorem is proved.
Complex integration and Cauchy's theorem
Watson, GN
2012-01-01
This brief monograph by one of the great mathematicians of the early twentieth century offers a single-volume compilation of propositions employed in proofs of Cauchy's theorem. Developing an arithmetical basis that avoids geometrical intuitions, Watson also provides a brief account of the various applications of the theorem to the evaluation of definite integrals.Author G. N. Watson begins by reviewing various propositions of Poincaré's Analysis Situs, upon which proof of the theorem's most general form depends. Subsequent chapters examine the calculus of residues, calculus optimization, the
Fluctuation theorem: A critical review
Malek Mansour, M.; Baras, F.
2017-10-01
Fluctuation theorem for entropy production is revisited in the framework of stochastic processes. The applicability of the fluctuation theorem to physico-chemical systems and the resulting stochastic thermodynamics were analyzed. Some unexpected limitations are highlighted in the context of jump Markov processes. We have shown that these limitations handicap the ability of the resulting stochastic thermodynamics to correctly describe the state of non-equilibrium systems in terms of the thermodynamic properties of individual processes therein. Finally, we considered the case of diffusion processes and proved that the fluctuation theorem for entropy production becomes irrelevant at the stationary state in the case of one variable systems.
Limit theorem and uniqueness theorem of backward stochastic differential equations
Institute of Scientific and Technical Information of China (English)
JIANG; Long
2006-01-01
This paper establishes a limit theorem for solutions of backward stochastic differential equations (BSDEs). By this limit theorem, this paper proves that, under the standard assumption g(t,y,0)≡0, the generator g of a BSDE can be uniquely determined by the corresponding g-expectation εg; this paper also proves that if a filtration consistent expectation ε can be represented as a g-expectation εg, then the corresponding generator g must be unique.
Cardioids and Morley's Trisector Theorem
J. Brinkhuis (Jan); van de Craats, J.
2017-01-01
textabstractA self-contained account of Morley's own proof of his celebrated trisector theorem is given. This makes this elegant and almost forgotten fragment of analytic Euclidean geometry more accessible to modern readers
Statistics, Causality and Bell's theorem
Gill, Richard D
2012-01-01
Bell's (1964) theorem is popularly supposed to establish the non-locality of quantum physics as a mathematical-physical theory. Building from this, observed violation of Bell's inequality in experiments such as that of Aspect and coworkers (1982) is popularly supposed to provide empirical proof of non-locality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a new proof of a strong (finite sample) version of Bell's theorem which relies only on elementary arithmetic and (counting) probability. This proof underscores the fact that Bell's theorem tells us that quantum theory is incompatible with the conjunction of three cherished and formerly uncontroversial physical principles, nicknamed here locality, realism, and freedom. The first, locality, is obviously connected to causality: causal influences need time to propagate spatially. Less obviously, the other two principles, realism and freedom, are also fo...
The Kolmogorov-Riesz compactness theorem
Hanche-Olsen, Harald
2009-01-01
We show that the Arzela-Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.
The Second Noether Theorem on Time Scales
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Agnieszka B. Malinowska
2013-01-01
Full Text Available We extend the second Noether theorem to variational problems on time scales. As corollaries we obtain the classical second Noether theorem, the second Noether theorem for the h-calculus and the second Noether theorem for the q-calculus.
The second Noether theorem on time scale
Malinowska, Agnieszka B.; Martins, Natália
2014-01-01
We extend the second Noether theorem to variational problems on time scales. Our result provides as corollaries the classical second Noether theorem, the second Noether theorem for the $h$-calculus and the second Noether theorem for the $q$-calculus.
Local virial and tensor theorems.
Cohen, Leon
2011-11-17
We show that for any wave function and potential the local virial theorem can always be satisfied 2K(r) = r·ΔV by choosing a particular expression for the local kinetic energy. In addition, we show that for each choice of local kinetic energy there are an infinite number of quasi-probability distributions which will generate the same expression. We also consider the local tensor virial theorem.
Quantum mechanical version of the classical Liouville theorem
Institute of Scientific and Technical Information of China (English)
Xie Chuan-Mei; Fan Hong-Yi
2013-01-01
In terms of the coherent state evolution in phase space,we present a quantum mechanical version of the classical Liouville theorem.The evolution of the coherent state from | z> to | sz-rz*> corresponds to the motion from a point z (q,p)to another point sz-rz* with |s|2-|r|2 =1.The evolution is governed by the so-called Fresnel operator U(s,r) that was recently proposed in quantum optics theory,which classically corresponds to the matrix optics law and the optical Fresnel transformation,and obeys group product rules.In other words,we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space,which seems to be a combination of quantum statistics and quantum optics.
Fixed-point-like theorems on subspaces
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Bernard Cornet
2004-08-01
Full Text Available We prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Our result generalizes two different kinds of theorems: the fixed-point-like theorem by Hirsch et al. (1990 or Husseini et al. (1990 and the fixed-point theorem by Gale and Mas-Colell (1975 (which generalizes Kakutani's theorem (1941.
Convolution theorems for the linear canonical transform and their applications
Institute of Scientific and Technical Information of China (English)
DENG Bing; TAO Ran; WANG Yue
2006-01-01
As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.
Deviations from Wick's theorem in the canonical ensemble
Schönhammer, K.
2017-07-01
Wick's theorem for the expectation values of products of field operators for a system of noninteracting fermions or bosons plays an important role in the perturbative approach to the quantum many-body problem. A finite-temperature version holds in the framework of the grand canonical ensemble, but not for the canonical ensemble appropriate for systems with fixed particle number such as ultracold quantum gases in optical lattices. Here we present formulas for expectation values of products of field operators in the canonical ensemble using a method in the spirit of Gaudin's proof of Wick's theorem for the grand canonical case. The deviations from Wick's theorem are examined quantitatively for two simple models of noninteracting fermions.
Ferromagnetism beyond Lieb's theorem
Costa, Natanael C.; Mendes-Santos, Tiago; Paiva, Thereza; Santos, Raimundo R. dos; Scalettar, Richard T.
2016-10-01
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction U is turned on, Lieb proved rigorously that at half-filling (ρ =1 ) the ground state has a nonzero spin. In this paper we consider a "CuO2 lattice" (also known as "Lieb lattice," or as a decorated square lattice), in which "d orbitals" occupy the vertices of the squares, while "p orbitals" lie halfway between two d orbitals; both d and p orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of U and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up , originally considered by Lieb, and the inhomogeneous (IH) case, Ud≠Up . For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all U . For the IH system at half-filling, we argue that the case Up≠Ud falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases Up=0 ,Ud=U and Up=U ,Ud=0 . We found that the different environments of d and p sites lead to a ferromagnetic insulator when Ud=0 ; by contrast, Up=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density ρ =1 /3 , strong antiferromagnetic correlations set in, caused by the presence of one fermion on each
Nambu-Goldstone theorem and spin-statistics theorem
Fujikawa, Kazuo
2016-05-01
On December 19-21 in 2001, we organized a yearly workshop at Yukawa Institute for Theoretical Physics in Kyoto on the subject of “Fundamental Problems in Field Theory and their Implications”. Prof. Yoichiro Nambu attended this workshop and explained a necessary modification of the Nambu-Goldstone theorem when applied to non-relativistic systems. At the same workshop, I talked on a path integral formulation of the spin-statistics theorem. The present essay is on this memorable workshop, where I really enjoyed the discussions with Nambu, together with a short comment on the color freedom of quarks.
Fluctuation theorems for quantum processes
Albash, Tameem; Marvian, Milad; Zanardi, Paolo
2013-01-01
We present fluctuation theorems and moment generating function equalities for generalized thermodynamic observables and quantum dynamics described by completely positive trace preserving (CPTP) maps, with and without feedback control. Our results include the quantum Jarzynski equality and Crooks fluctuation theorem, and clarify the special role played by the thermodynamic work and thermal equilibrium states in previous studies. We show that unitality replaces micro-reversibility as the condition for the physicality of the reverse process in our fluctuation theorems. We present an experimental application of our theory to the problem of extracting the system-bath coupling magnitude, which we do for a system of pairs of coupled superconducting flux qubits undergoing quantum annealing.
Improvement of Hartman's linearization theorem
Institute of Scientific and Technical Information of China (English)
SHI; Jinlin(史金麟)
2003-01-01
Hartman's linearization theorem tells us that if matrix A has no zero real part and f(x) isbounded and satisfies Lipchitz condition with small Lipchitzian constant, then there exists a homeomorphismof Rn sending the solutions of nonlinear system x' = Ax + f(x) onto the solutions of linear system x' = Ax.In this paper, some components of the nonlinear item f(x) are permitted to be unbounded and we provethe result of global topological linearization without any special limitation and adding any condition. Thus,Hartman's linearization theorem is improved essentially.
Poutiainen, H. (Hayley)
2015-01-01
Group theory is a mathematical domain where groups and their properties are studied. The evolution of group theory as an area of study is said to be the result of the parallel development of a variety of different studies in mathematics. Sylow’s Theorems were a set of theorems proved around the same time the concept of group theory was being established, in the 1870s. Sylow used permutation groups in his proofs which were then later generalized and shown to hold true for all finite groups. Th...
Louis M. Houston
2012-01-01
Sign data are the signs of signal added to noise. It is well known that a constant signal can be recovered from sign data. In this paper, we show that an integral over variant signal can be recovered from an integral over sign data based on the variant signal. We refer to this as a generalized sign data average. We use this result to derive a Green's theorem for sign data. Green's theorem is important to various seismic processing methods, including seismic migration. Results in this paper ge...
Noether theorems and higher derivatives
Townsend, Paul K
2016-01-01
A simple proof of Noether's first theorem involves the promotion of a constant symmetry parameter $\\epsilon$ to an arbitrary function of time; the Noether charge $Q$ is then the coefficient of $\\dot\\epsilon$ in the variation of the action. Here we examine the validity of this proof for Lagrangian mechanics with arbitrarily-high time derivatives, in which context "higher-level" analogs of Noether's theorem can be similarly proved, and "Noetherian charges" read off from, e.g. the coefficient of $\\ddot \\epsilon$ in the variation of the action. While $Q=0$ implies a restricted gauge invariance, an unrestricted gauge invariance requires zero Noetherian charges too. Some illustrative examples are considered.
Pasicki, Lech
2011-01-01
Many versions of the Stokes theorem are known. More advanced of them require complicated mathematical machinery to be formulated which discourages the users. Our theorem is sufficiently simple to suit the handbooks and yet it is pretty general, as we assume the differential form to be continuous on a compact set F(A) and C1 "inside" while F(A) is built of "bricks" and its inner part is a C1 manifold. There is no problem of orientability and the integrals under consideration are convergent. The proof is based on integration by parts and inner approximation.
The truncated Second Main Theorem and uniqueness theorems
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we first establish a truncated Second Main Theorem for algebraically nondegenerate holomorphic mappings from the complex plane into a complex projective variety V intersecting hypersurfaces. We then prove some uniqueness results for meromorphic mappings. The result of Demailly about a partial solution to the Fujita’s conjecture is used.
A normal form theorem around symplectic leaves
Crainic, M.N.; Marcut, I.T.
2012-01-01
We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization theorem.
Von Laue's theorem and its applications
Wang, Changbiao
2012-01-01
Von Laue's theorem is strictly proved in detail to clarify confusions in textbook and literature. This theorem is used to analyze the classical electron and the static electric field confined in a finite region of space.
The Classical Version of Stokes' Theorem Revisited
DEFF Research Database (Denmark)
Markvorsen, Steen
2005-01-01
Using only fairly simple and elementary considerations - essentially from first year undergraduate mathematics - we prove that the classical Stokes' theorem for any given surface and vector field in $\\mathbb{R}^{3}$ follows from an application of Gauss' divergence theorem to a suitable modification...... of the vector field in a tubular shell around the given surface. The intuitive appeal of the divergence theorem is thus applied to bootstrap a corresponding intuition for Stokes' theorem. The two stated classical theorems are (like the fundamental theorem of calculus) nothing but shadows of the general version...... to above. Our proof that Stokes' theorem follows from Gauss' divergence theorem goes via a well known and often used exercise, which simply relates the concepts of divergence and curl on the local differential level. The rest of the paper uses only integration in $1$, $2$, and $3$ variables together...
Shell theorem for spontaneous emission
DEFF Research Database (Denmark)
Kristensen, Philip Trøst; Mortensen, Jakob Egeberg; Lodahl, Peter;
2013-01-01
and therefore is given exactly by the dipole approximation theory. This surprising result is a spontaneous emission counterpart to the shell theorems of classical mechanics and electrostatics and provides insights into the physics of mesoscopic emitters as well as great simplifications in practical calculations....
JACKSON'S THEOREM FOR COMPACT GROUPS
Institute of Scientific and Technical Information of China (English)
H. Vaezi; S. F. Rzaev
2002-01-01
In this article we consider the generalized shift operator defined by(Sh.f)(g) = ∫Gf (tut-1g)dton compact group G and by help of this operator we define "Spherical" modulus of continuity. So we proveStechkin and Jackson type theorems.
LUROTH'S THEOREM IN DIFFERENTIAL FIELDS
Institute of Scientific and Technical Information of China (English)
GAO Xiaoshan; XU Tao
2002-01-01
In this paper, we present a constructive proof of Liroth's theorem in differentialcase. We also give a method to find the inversion maps for general differential rationalparametric equations. As a consequence, we prove that a differential rational curve alwayshas a set of proper parametric equations.
Angle Defect and Descartes' Theorem
Scott, Paul
2006-01-01
Rene Descartes lived from 1596 to 1650. His contributions to geometry are still remembered today in the terminology "Descartes' plane". This paper discusses a simple theorem of Descartes, which enables students to easily determine the number of vertices of almost every polyhedron. (Contains 1 table and 2 figures.)
GENERALIZED RECIPROCAL THEOREMS AND THEIR APPLICATIONS
Institute of Scientific and Technical Information of China (English)
付宝连
2002-01-01
Generalized reciprocal theorems of non-coupled and coupled systems , which are valid for two deformed bodies with different constitutive relations are established by generalizing the idea of Betti ' s reciprocal theorem. When the constitutive relations of the two deformed bodies are all alike and linear elastic, the generalized reciprocal theorem of non-coupled systems just becomes Betti' s . Meanwhile, the generalized reciprocal theorems are applied to simulate calculations in elasticity.
A definability theorem for first order logic
Butz, C.; Moerdijk, I.
2001-01-01
In this paper we will present a definability theorem for first order logic This theorem is very easy to state and its proof only uses elementary tools To explain the theorem let us first observe that if M is a model of a theory T in a language L then clearly any definable subset S M ie a subset S
Ordering in mechanical geometry theorem proving
Institute of Scientific and Technical Information of China (English)
李洪波￥
1997-01-01
Ordering in mechanical geometry theorem proving is studied from geometric viewpoint and some new ideas are proposed. For Thebault’s theorem which is the most difficult theorem that has ever been proved by Wu’ s method, a very simple proof using Wu’s method under a linear order is discovered.
A note on generalized Weyl's theorem
Zguitti, H.
2006-04-01
We prove that if either T or T* has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every . An application is given for algebraically paranormal operators.
On Brayton and Moser's missing stability theorem
Jeltsema, D.; Scherpen, J. M. A.
2005-01-01
In the early 1960s, Brayton and Moser proved three theorems concerning the stability of nonlinear electrical circuits. The applicability of each theorem depends on three different conditions on the type of admissible nonlinearities in circuit. Roughly speaking, this means that the theorems apply to
A new theorem relating quantum tomogram to the Fresnel operator
Institute of Scientific and Technical Information of China (English)
Xie Chuan-Mei; Fan Hong-Yi
2011-01-01
According to Fan-Hu's formalism (Fan Hong-Yi and Hu Li-Yun 2009 Opt. Commun. 282 3734) that the tomogram of quantum states can be considered as the module-square of the state wave function in the intermediate coordinatemomentum representation which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating quantum tomogram of density operator, i.e., the tomogram of a density operator p is equal to the marginal integration of the classical Weyl correspondence function of F+ρF, where F is the Fresnel operator. Applications of this theorem to evaluating the tomogram of optical chaotic field and squeezed chaotic optical field are presented.
The de Finetti theorem for test spaces
Barrett, Jonathan; Leifer, Matthew
2009-03-01
We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptions may fail, leading to probabilistic models that do not have a de Finetti theorem.
The de Finetti theorem for test spaces
Energy Technology Data Exchange (ETDEWEB)
Barrett, Jonathan [H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL (United Kingdom); Leifer, Matthew [Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1 (Canada)], E-mail: j.barrett@bristol.ac.uk, E-mail: matt@mattleifer.info
2009-03-15
We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptions may fail, leading to probabilistic models that do not have a de Finetti theorem.
Multiwavelet sampling theorem in Sobolev spaces
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (H s (R) , H-s (R)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ H s (R) with s > 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L 2 (R), whose Sobolev exponents are greater than 1 /2.
The relativistic virial theorem and scale invariance
Gaite, Jose
2013-01-01
The virial theorem is related to the dilatation properties of bound states. This is realized, in particular, by the Landau-Lifshitz formulation of the relativistic virial theorem, in terms of the trace of the energy-momentum tensor. We construct a Hamiltonian formulation of dilatations in which the relativistic virial theorem naturally arises as the condition of stability against dilatations. A bound state becomes scale invariant in the ultrarelativistic limit, in which its energy vanishes. However, for very relativistic bound states, scale invariance is broken by quantum effects and the virial theorem must include the energy-momentum tensor trace anomaly. This quantum field theory virial theorem is directly related to the Callan-Symanzik equations. The virial theorem is applied to QED and then to QCD, focusing on the bag model of hadrons. In massless QCD, according to the virial theorem, 3/4 of a hadron mass corresponds to quarks and gluons and 1/4 to the trace anomaly.
Navier Stokes Theorem in Hydrology
Narayanan, M.
2005-12-01
In a paper presented at the 2004 AGU International Conference, the author outlined and stressed the importance of studying and teaching certain important mathematical techniques while developing a course in Hydrology and Fluid Mechanics. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. In an article entitled "Corrections to Fluid Dynamics" R. F. Streater, (Open Systems and Information Dynamics, 10, 3-30, 2003.) proposes a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a non-linear discrete-time Markov chain for random fields on a lattice. In the continuum limit he obtains a non-linear coupled parabolic system of field equations, showing a correction to the Navier-Stokes equations. In 2001, David Hoff published an article in Journees Equations aux derivees partielles. (Art. No. 7, 9 p.). His paper is entitled : Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions. In his paper, David Hoff proves the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time
Expectation Value in Bell's Theorem
Wang, Zheng-Chuan
2006-01-01
We will demonstrate in this paper that Bell's theorem (Bell's inequality) does not really conflict with quantum mechanics, the controversy between them originates from the different definitions for the expectation value using the probability distribution in Bell's inequality and the expectation value in quantum mechanics. We can not use quantum mechanical expectation value measured in experiments to show the violation of Bell's inequality and then further deny the local hidden-variables theor...
ANDRAGOGY OF DEVELOPMENT: BASIC THEOREMS
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A. G. Teslinov
2016-01-01
Full Text Available The article presents criticism of the state of scientific knowledge about adult education and provides the reasons for the choice of directions of its development.Methods. The approach to the substantiation of directions of development of andragogy includes aspectual analysis of scientific rhetoric of adult education; summarizing the symptoms and causes of the problems of educational practice examples of education managers; the analysis of the status of andragogy as a scientific paradigm; a conceptual analysis of the key theses of the modern synthesis of andragogy and the provisions for developmental adult education.Results and scientific novelty. Four theorems are formulated that specify the complete set of propositions about a developmental approach to adult education. These theorems are presented as a scientific hypothesis about the features of the approach. The theorems are proved, and the substantiation of the conditions of emergence of the adult education of educational properties is described. The idea of adult education as a developing culture is in the centre of reasoning. It is shown that the assertions of theorems form the conceptual core of the scientific branches in adult education – andragogy of development. The effect of the practical interpretation of its provisions is disclosed.Practical significance. Disclosed meanings and recommendations may be oriented to developers of educational systems and media for adults while creating the developmental components. These references will help to overcome the evident trend information of adult education to the "pulling" them up to continually outdated standards, and to give it the look of a truly developing technology.
An improvement of Papadakis' theorem
Institute of Scientific and Technical Information of China (English)
ZHANG Zhihua; MU Lehua; ZHANG Peixuan
2004-01-01
There exist many orthonormal wavelets which cannot be derived by multiresolution analysis (MRA) with a single scaling function.In 2000,Papadakis announced that any orthonormal wavelet is derived by a generalized MRA with countable scaling functions at most.We improve Papadakis' theorem and find that for any othonormal wavelet,the least number of the corresponding scaling functions is just the essential supremum of the dimension function of the orthonormal wavelet.Moreover,we construct directly the fewest scaling functions.
Pythagoras Theorem and Relativistic Kinematics
Mulaj, Zenun; Dhoqina, Polikron
2010-01-01
In two inertial frames that move in a particular direction, may be registered a light signal that propagates in an angle with this direction. Applying Pythagoras theorem and principles of STR in both systems, we can derive all relativistic kinematics relations like the relativity of simultaneity of events, of the time interval, of the length of objects, of the velocity of the material point, Lorentz transformations, Doppler effect and stellar aberration.
Compactness theorems of fuzzy semantics
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The relationship among diverse fuzzy semantics vs. the corresponding logic consequence operators has been analyzed systematically. The results that compactness and logical compactness of fuzzy semantics are equivalent to compactness and continuity of the logic consequence operator induced by the semantics respectively have been proved under certain conditions. A general compactness theorem of fuzzy semantics have been established which says that every fuzzy semantics defined on a free algebra with members corresponding to continuous functions is compact.
Software Reliability through Theorem Proving
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S.G.K. Murthy
2009-05-01
Full Text Available Improving software reliability of mission-critical systems is widely recognised as one of the major challenges. Early detection of errors in software requirements, designs and implementation, need rigorous verification and validation techniques. Several techniques comprising static and dynamic testing approaches are used to improve reliability of mission critical software; however it is hard to balance development time and budget with software reliability. Particularly using dynamic testing techniques, it is hard to ensure software reliability, as exhaustive testing is not possible. On the other hand, formal verification techniques utilise mathematical logic to prove correctness of the software based on given specifications, which in turn improves the reliability of the software. Theorem proving is a powerful formal verification technique that enhances the software reliability for missioncritical aerospace applications. This paper discusses the issues related to software reliability and theorem proving used to enhance software reliability through formal verification technique, based on the experiences with STeP tool, using the conventional and internationally accepted methodologies, models, theorem proving techniques available in the tool without proposing a new model.Defence Science Journal, 2009, 59(3, pp.314-317, DOI:http://dx.doi.org/10.14429/dsj.59.1527
The classical version of Stokes' Theorem revisited
DEFF Research Database (Denmark)
Markvorsen, Steen
2008-01-01
Using only fairly simple and elementary considerations - essentially from first year undergraduate mathematics - we show how the classical Stokes' theorem for any given surface and vector field in $\\mathbb{R}^{3}$ follows from an application of Gauss' divergence theorem to a suitable modification...... of the vector field in a tubular shell around the given surface. The two stated classical theorems are (like the fundamental theorem of calculus) nothing but shadows of the general version of Stokes' theorem for differential forms on manifolds. The main points in the present paper, however, is firstly...... exercise, which simply relates the concepts of divergence and curl on the local differential level. The rest of the paper uses only integration in $1$, $2$, and $3$ variables together with a 'fattening' technique for surfaces and the inverse function theorem....
On Krasnoselskii's Cone Fixed Point Theorem
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Man Kam Kwong
2008-04-01
Full Text Available In recent years, the Krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types. In the first part of this paper, we revisit the Krasnoselskii theorem, in a more topological perspective, and show that it can be deduced in an elementary way from the classical Brouwer-Schauder theorem. This viewpoint also leads to a topology-theoretic generalization of the theorem. In the second part of the paper, we extend the cone theorem in a different direction using the notion of retraction and show that a stronger form of the often cited Leggett-Williams theorem is a special case of this extension.
The Helmholtz theorem and retarded fields
Heras, Ricardo
2016-11-01
Textbooks frequently use the Helmholtz theorem to derive expressions for electrostatic and magnetostatic fields but they do not usually apply this theorem to derive expressions for time-dependent electric and magnetic fields, even when there is no formal objection to doing so because the proof of the theorem does not involve time derivatives but only spatial derivatives. Here we address the question as to whether the Helmholtz theorem is useful in deriving expressions for the fields of Maxwell’s equations. We show that when this theorem is applied to Maxwell’s equations we obtain instantaneous expressions of the electric and magnetic fields, which are formally correct but of little practical usefulness. We then discuss two generalizations of the theorem which are shown to be useful in deriving the retarded fields.
The Helmholtz theorem and retarded fields
Heras, Ricardo
2016-01-01
Textbooks frequently use the Helmholtz theorem to derive expressions for the electrostatic and magnetostatic fields but they do not usually apply this theorem to derive expressions for the time-dependent electric and magnetic fields, even when there is no formal objection to doing so because the proof of the theorem does not involve time derivatives but only spatial derivatives. Here we address the question as to whether the Helmholtz theorem is useful to derive expressions for the fields of Maxwell's equations. We show that when this theorem is applied to Maxwell's equations we obtain instantaneous expressions of the electric and magnetic fields, which are formally correct but of little practical usefulness. We then discuss two generalizations of the theorem which are shown to be useful to derive the retarded fields.
Symbolic logic and mechanical theorem proving
Chang, Chin-Liang
1969-01-01
This book contains an introduction to symbolic logic and a thorough discussion of mechanical theorem proving and its applications. The book consists of three major parts. Chapters 2 and 3 constitute an introduction to symbolic logic. Chapters 4-9 introduce several techniques in mechanical theorem proving, and Chapters 10 an 11 show how theorem proving can be applied to various areas such as question answering, problem solving, program analysis, and program synthesis.
Expanding the Interaction Equivalency Theorem
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Brenda Cecilia Padilla Rodriguez
2015-06-01
Full Text Available Although interaction is recognised as a key element for learning, its incorporation in online courses can be challenging. The interaction equivalency theorem provides guidelines: Meaningful learning can be supported as long as one of three types of interactions (learner-content, learner-teacher and learner-learner is present at a high level. This study sought to apply this theorem to the corporate sector, and to expand it to include other indicators of course effectiveness: satisfaction, knowledge transfer, business results and return on expectations. A large Mexican organisation participated in this research, with 146 learners, 30 teachers and 3 academic assistants. Three versions of an online course were designed, each emphasising a different type of interaction. Data were collected through surveys, exams, observations, activity logs, think aloud protocols and sales records. All course versions yielded high levels of effectiveness, in terms of satisfaction, learning and return on expectations. Yet, course design did not dictate the types of interactions in which students engaged within the courses. Findings suggest that the interaction equivalency theorem can be reformulated as follows: In corporate settings, an online course can be effective in terms of satisfaction, learning, knowledge transfer, business results and return on expectations, as long as (a at least one of three types of interaction (learner-content, learner-teacher or learner-learner features prominently in the design of the course, and (b course delivery is consistent with the chosen type of interaction. Focusing on only one type of interaction carries a high risk of confusion, disengagement or missed learning opportunities, which can be managed by incorporating other forms of interactions.
Comparison theorems in Riemannian geometry
Cheeger, Jeff
2008-01-01
The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem-the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most re
A parametrization of the abstract Ramsey theorem
Mijares, Jose G
2010-01-01
We give a parametrization with perfect subsets of $2^{\\infty}$ of the abstract Ramsey theorem (see \\cite{todo}) Our main tool is an extension of the parametrized version of the combinatorial forcing developed in \\cite{nash} and \\cite{todo}, used in \\cite{mij} to the obtain a parametrization of the abstract Ellentuck theorem. As one of the consequences, we obtain a parametrized version of the Hales-Jewett theorem. Finally, we conclude that the family of perfectly ${\\cal S}$-Ramsey subsets of $2^{\\infty}\\times {\\cal R}$ is closed under the Souslin operation. {\\bf Key words and phrases}: Ramsey theorem, Ramsey space, parametrization.
A generalised Sylvester-Gallai Theorem
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L. M. Pretorius
2007-09-01
Full Text Available We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:Let S be a ﬁnite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points A and B in S sharing a colour k, there is a third point in S which has (inter alia the colour different from k and is collinear with A and B. Then all the points in S are collinear.This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.
A History of the Central Limit Theorem
Fischer, Hans
2011-01-01
This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications. Making this theorem an autonomous mathematical
A generalized preimage theorem in global analysis
Institute of Scientific and Technical Information of China (English)
MA; Jipu
2001-01-01
［1］Ma Jipu, (1.2) inverses of operators between Banach spaces and conjugacy theorem, Chinese Annals of Math., B, 1999, 20(1): 57.［2］Ma Jipu, Rank theorem of operators between Banach spaces, Science in China, Ser. A, 2000, 43(1): 1.［3］Ma Jipu, Local conjugacy theorem, rank theorems in advenced calculus and a generalized principle constructing Banach manifolds, Science in China, Ser. A, 2000, 43(12): 1233.［4］Zeidler, A. E., Nonlinear Function Analysis and Its Applications, IV: Applications to Mathematical Physics, New York: Springer-Verlag, 1988.
Directory of Open Access Journals (Sweden)
Sol Swords
2011-10-01
Full Text Available Interactive theorem proving requires a lot of human guidance. Proving a property involves (1 figuring out why it holds, then (2 coaxing the theorem prover into believing it. Both steps can take a long time. We explain how to use GL, a framework for proving finite ACL2 theorems with BDD- or SAT-based reasoning. This approach makes it unnecessary to deeply understand why a property is true, and automates the process of admitting it as a theorem. We use GL at Centaur Technology to verify execution units for x86 integer, MMX, SSE, and floating-point arithmetic.
The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces
Huang, Jianhua
2005-12-01
In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679-690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results.
An elementary derivation of the quantum virial theorem from Hellmann-Feynman theorem
İpekoğlu, Y.; Turgut, S.
2016-07-01
A simple proof of the quantum virial theorem that can be used in undergraduate courses is given. The proof proceeds by first showing that the energy eigenvalues of a Hamiltonian remain invariant under a scale transformation. Then invoking the Hellmann-Feynman theorem produces the final statement of the virial theorem.
Vallès, Jean
2012-01-01
Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective. More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2,..., A2n is well inscribed in a configuration Ln of n lines if each line of the configuration contains exactly two points among A1, A2, ..., A2n. Then we prove : "Let Ln be a configuration of n lines and D a smooth conic in the complex projective plane. If it exists one polygon with 2n sides well inscribed in Ln and circumscribed around D then there are infinitely many such polygons. In particular a general point in Ln is a vertex of such a polygon." We propose an elementary proof based on Fr\\'egier's involution. We begin by recalling some facts about these involutions. Then we explore the following question : When does the product of involutions correspond to an involution? It leads to Pascal theorem, to its dual version proved by Brianchon, and to its generalization proved by M\\"obius.
THE EXISTENCE THEOREM OF OPTIMAL GROWTH MODEL
Institute of Scientific and Technical Information of China (English)
Gong Liutang; Peng Xianze
2005-01-01
This paper proves a general existence theorem of optimal growth theory. This theorem is neither restricted to the case of a constant technology progress, nor stated in terms of mathematical conditions which have no direct economic interpretation and moreover, are difficult to apply.
Euler and the Fundamental Theorem of Algebra.
Duham, William
1991-01-01
The complexity of the proof of the Fundamental Theorem of Algebra makes it inaccessible to lower level students. Described are more understandable attempts of proving the theorem and a historical account of Euler's efforts that relates the progression of the mathematical process used and indicates some of the pitfalls encountered. (MDH)
A note on the tolerated Tverberg theorem
2016-01-01
In this paper we give an asymptotically tight bound for the tolerated Tverberg Theorem when the dimension and the size of the partition are fixed. To achieve this we study certain partitions of order-type homogeneous sets and use a generalization of the Erd\\H{o}s-Szekeres theorem.
A New Fixed Point Theorem and Applications
Directory of Open Access Journals (Sweden)
Min Fang
2013-01-01
Full Text Available A new fixed point theorem is established under the setting of a generalized finitely continuous topological space (GFC-space without the convexity structure. As applications, a weak KKM theorem and a minimax inequalities of Ky Fan type are also obtained under suitable conditions. Our results are different from known results in the literature.
Double soft theorem for perturbative gravity
Saha, Arnab Priya
2016-09-01
Following up on the recent work of Cachazo, He and Yuan [1], we derive the double soft graviton theorem in perturbative gravity. We show that the double soft theorem derived using CHY formula precisely matches with the perturbative computation involving Feynman diagrams. In particular, we find how certain delicate limits of Feynman diagrams play an important role in obtaining this equivalence.
On the Hausdorff-Young theorem
Nasserddine, W
2005-01-01
Let $G_{mn}=ax + b$ be the matricial group of a local field. The Hausdorff-Young theorem for $G_{11}$ was proved by Eymard-Terp in 1978. We will establish here the Hausdorff-Young theorem for $G_{nn}$ for all $n \\in \\mathbb{N}$.
Abel's theorem in the noncommutative case
Leitenberger, Frank
2004-03-01
We define noncommutative binary forms. Using the typical representation of Hermite we prove the fundamental theorem of algebra and we derive a noncommutative Cardano formula for cubic forms. We define quantized elliptic and hyperelliptic differentials of the first kind. Following Abel we prove Abel's theorem.
A New Type of Singularity Theorem
Senovilla, José M M
2007-01-01
A new type of singularity theorem, based on spatial averages of physical quantities, is presented and discussed. Alternatively, the results inform us of when a spacetime can be singularity-free. This theorem provides a decisive observational difference between singular and non-singular, globally hyperbolic, open cosmological models.
Interpolation theorems on weighted Lorentz martingale spaces
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper several interpolation theorems on martingale Lorentz spaces are given.The proofs are based on the atomic decompositions of martingale Hardy spaces over weighted measure spaces.Applying the interpolation theorems,we obtain some inequalities on martingale transform operator.
The Euler Line and Nine-Point-Circle Theorems.
Eccles, Frank M.
1999-01-01
Introduces the Euler line theorem and the nine-point-circle theorem which emphasize transformations and the power of functions in a geometric concept. Presents definitions and proofs of theorems. (ASK)
Pointwise ergodic theorems beyond amenable groups
Bowen, Lewis
2011-01-01
We prove pointwise and maximal ergodic theorems for probability measure preserving actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type III_r for some r >0. Our approach is based on the following two principles. First, it is possible to generalize the ergodic theory of measure-preserving actions of amenable groups to include probability-measure-preserving amenable equivalence relations. Second, it is possible to reduce the proof of ergodic theorems for actions of a general group to the proof of ergodic theorems in an associated measure-preserving amenable equivalence relation, provided the group admits an amenable action with the properties stated above. The general ergodic theorems established here are used in a sequel paper to prove mean and pointwise ergodic theorems for arbitrary Gromov-hyperbolic groups.
Generalized fluctuation theorems for classical systems
Agarwal, G S
2015-01-01
Fluctuation theorems have a very special place in the study of non equilibrium dynamics of physical systems. The form in which it is used most extensively is the Gallavoti-Cohen Fluctuation Theorem which is in terms of the distribution of the work $p(W)/p(-W)=\\exp(\\alpha W)$. We derive the general form of the fluctuation theorems for an arbitrary Gaussian Markov process and find conditions when the parameter $\\alpha$ becomes a universal parameter $1/kT$. As an application we consider fluctuation theorems for classical cyclotron motion of an electron in a parabolic potential. The motion of the electron is described by four coupled Langevin equations and thus is non-trivial. The generalized theorems are equally valid for non-equilibrium steady states.
Uniqueness theorems in linear elasticity
Knops, Robin John
1971-01-01
The classical result for uniqueness in elasticity theory is due to Kirchhoff. It states that the standard mixed boundary value problem for a homogeneous isotropic linear elastic material in equilibrium and occupying a bounded three-dimensional region of space possesses at most one solution in the classical sense, provided the Lame and shear moduli, A and J1 respectively, obey the inequalities (3 A + 2 J1) > 0 and J1>O. In linear elastodynamics the analogous result, due to Neumann, is that the initial-mixed boundary value problem possesses at most one solution provided the elastic moduli satisfy the same set of inequalities as in Kirchhoffs theorem. Most standard textbooks on the linear theory of elasticity mention only these two classical criteria for uniqueness and neglect altogether the abundant literature which has appeared since the original publications of Kirchhoff. To remedy this deficiency it seems appropriate to attempt a coherent description ofthe various contributions made to the study of uniquenes...
Singlet and triplet instability theorems
Yamada, Tomonori; Hirata, So
2015-09-01
A useful definition of orbital degeneracy—form-degeneracy—is introduced, which is distinct from the usual energy-degeneracy: Two canonical spatial orbitals are form-degenerate when the energy expectation value in the restricted Hartree-Fock (RHF) wave function is unaltered upon a two-electron excitation from one of these orbitals to the other. Form-degenerate orbitals tend to have isomorphic electron densities and occur in the highest-occupied and lowest-unoccupied molecular orbitals (HOMOs and LUMOs) of strongly correlated systems. Here, we present a mathematical proof of the existence of a triplet instability in a real or complex RHF wave function of a finite system in the space of real or complex unrestricted Hartree-Fock wave functions when HOMO and LUMO are energy- or form-degenerate. We also show that a singlet instability always exists in a real RHF wave function of a finite system in the space of complex RHF wave functions, when HOMO and LUMO are form-degenerate, but have nonidentical electron densities, or are energy-degenerate. These theorems provide Hartree-Fock-theory-based explanations of Hund's rule, a singlet instability in Jahn-Teller systems, biradicaloid electronic structures, and a triplet instability during some covalent bond breaking. They also suggest (but not guarantee) the spontaneous formation of a spin density wave (SDW) in a metallic solid. The stability theory underlying these theorems extended to a continuous orbital-energy spectrum proves the existence of an oscillating (nonspiral) SDW instability in one- and three-dimensional homogeneous electron gases, but only at low densities or for strong interactions.
Posterior Probability and Fluctuation Theorem in Stochastic Processes
Ohkubo, Jun
2009-12-01
A generalization of fluctuation theorems in stochastic processes is proposed. The new theorem is written in terms of posterior probabilities, which are introduced via Bayes’ theorem. In conventional fluctuation theorems, a forward path and its time reversal play an important role, so that a microscopically reversible condition is essential. In contrast, the microscopically reversible condition is not necessary in the new theorem. It is shown that the new theorem recovers various theorems and relations previously known, such as the Gallavotti-Cohen-type fluctuation theorem, the Jarzynski equality, and the Hatano-Sasa relation, when suitable assumptions are employed.
The pointwise Hellmann-Feynman theorem
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David Carfì
2010-02-01
Full Text Available In this paper we study from a topological point of view the Hellmann-Feynman theorem of Quantum Mechanics. The goal of the paper is twofold: On one hand we emphasize the role of the strong topology in the classic version of the theorem in Hilbert spaces, for what concerns the kind of convergence required on the space of continuous linear endomorphisms, which contains the space of (continuous observables.On the other hand we state and prove a new pointwise version of the classic Hellmann-Feynman theorem. This new version is not yet present in the literature and follows the idea of A. Bohm concerning the topology which is desiderable to use in Quantum Mechanics. It is indeed out of question that this non-trivial new version of the Hellmann-Feynman theorem is the ideal one - for what concerns the continuous observables on Hilbert spaces, both from a theoretical point of view, since it is the strongest version obtainable in this context - we recall that the pointwise topology is the coarsest one compatible with the linear structure of the space of continuous observables -, and from a practical point of view, because the pointwise topology is the easiest to use among topologies: it brings back the problems to the Hilbert space topology. Moreover, we desire to remark that this basic theorem of Quantum Mechanics, in his most desiderable form, is deeply interlaced with two cornerstones of Functional Analysis: the Banach-Steinhaus theorem and the Baire theorem.
Haag's theorem in renormalised quantum field theories
Klaczynski, Lutz
2016-01-01
We review a package of no-go results in axiomatic quantum field theory with Haag's theorem at its centre. Since the concept of operator-valued distributions in this framework comes very close to what we believe canonical quantum fields are about, these results are of consequence to quantum field theory: they suggest the seeming absurdity that this highly victorious theory is incapable of describing interactions. We single out unitarity of the interaction picture's intertwiner as the most salient provision of Haag's theorem and critique canonical perturbation theory to argue that renormalisation bypasses Haag's theorem by violating this very assumption.
Existence theorems for ordinary differential equations
Murray, Francis J
2007-01-01
Theorems stating the existence of an object-such as the solution to a problem or equation-are known as existence theorems. This text examines fundamental and general existence theorems, along with the Picard iterants, and applies them to properties of solutions and linear differential equations.The authors assume a basic knowledge of real function theory, and for certain specialized results, of elementary functions of a complex variable. They do not consider the elementary methods for solving certain special differential equations, nor advanced specialized topics; within these restrictions, th
Effective randomness, strong reductions and Demuth's theorem
Bienvenu, Laurent
2011-01-01
We study generalizations of Demuth's Theorem, which states that the image of a Martin-L\\"of random real under a tt-reduction is either computable or Turing equivalent to a Martin-L\\"of random real. We show that Demuth's Theorem holds for Schnorr randomness and computable randomness (answering a question of Franklin), but that it cannot be strengthened by replacing the Turing equivalence in the statement of the theorem with wtt-equivalence. We also provide some additional results about the Turing and tt-degrees of reals that are random with respect to some computable measure.
Security Theorems via Model Theory
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Joshua Guttman
2009-11-01
Full Text Available A model-theoretic approach can establish security theorems for cryptographic protocols. Formulas expressing authentication and non-disclosure properties of protocols have a special form. They are quantified implications for all xs . (phi implies for some ys . psi. Models (interpretations for these formulas are *skeletons*, partially ordered structures consisting of a number of local protocol behaviors. *Realized* skeletons contain enough local sessions to explain all the behavior, when combined with some possible adversary behaviors. We show two results. (1 If phi is the antecedent of a security goal, then there is a skeleton A_phi such that, for every skeleton B, phi is satisfied in B iff there is a homomorphism from A_phi to B. (2 A protocol enforces for all xs . (phi implies for some ys . psi iff every realized homomorphic image of A_phi satisfies psi. Hence, to verify a security goal, one can use the Cryptographic Protocol Shapes Analyzer CPSA (TACAS, 2007 to identify minimal realized skeletons, or "shapes," that are homomorphic images of A_phi. If psi holds in each of these shapes, then the goal holds.
Index theorems for quantum graphs
Fulling, S A; Wilson, J H
2007-01-01
In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a "Laplacian") with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e., conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H =A*A, and relating the constant term to the index of A. An independent consideration provides an index f...
Applications of the ergodic iteration theorem
Zapletal, J.
2010-01-01
I prove several natural preservation theorems for the countable support iteration. This solves a question of Roslanowski regarding the preservation of localization properties and greatly simplifies the proofs in the area.
The virial theorem for nonlinear problems
Energy Technology Data Exchange (ETDEWEB)
Amore, Paolo [Facultad de Ciencias, Universidad de Colima, Bernal DIaz del Castillo 340, Colima (Mexico); Fernandez, Francisco M [INIFTA (UNLP, CCT La Plata-CONICET), Division Quimica Teorica, Blvd 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina)], E-mail: paolo.amore@gmail.com, E-mail: fernande@quimica.unlp.edu.ar
2009-09-15
We show that the virial theorem provides a useful simple tool for approximating nonlinear problems. In particular, we consider conservative nonlinear oscillators and obtain the same main result derived earlier from the expansion in Chebyshev polynomials. (letters and comments)
Affine and Projective Tree Metric Theorems
Harel, Matan; Pachter, Lior
2011-01-01
The tree metric theorem provides a combinatorial four point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A similar (but weaker) four point condition characterizes dissimilarity maps derived from circular split systems (Kalmanson metrics). The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics and Robinsonian matrices, and the theorems that relate them. As with tree metric...
Transformation groups and the virial theorem
Kampen, N.G. van
1972-01-01
A generalization of Noether's result for classical mechanics is given, which shows that the virial theorem is related to an invariance property of the Lagrange function. Two examples are discussed in detail.
Dimensional analysis beyond the Pi theorem
Zohuri, Bahman
2017-01-01
Dimensional Analysis and Physical Similarity are well understood subjects, and the general concepts of dynamical similarity are explained in this book. Our exposition is essentially different from those available in the literature, although it follows the general ideas known as Pi Theorem. There are many excellent books that one can refer to; however, dimensional analysis goes beyond Pi theorem, which is also known as Buckingham’s Pi Theorem. Many techniques via self-similar solutions can bound solutions to problems that seem intractable. A time-developing phenomenon is called self-similar if the spatial distributions of its properties at different points in time can be obtained from one another by a similarity transformation, and identifying one of the independent variables as time. However, this is where Dimensional Analysis goes beyond Pi Theorem into self-similarity, which has represented progress for researchers. In recent years there has been a surge of interest in self-similar solutions of the First ...
Two No-Go Theorems on Superconductivity
Tada, Yasuhiro
2016-01-01
We study lattice superconductors such as attractive Hubbard models. As is well known, Bloch's theorem asserts absence of persistent current in ground states and equilibrium states for general fermion systems. While the statement of the theorem is true, we can show that the theorem cannot exclude possibility of a surface persistent current. Such a current can be stabilized by boundary magnetic fields which do not penetrate into the bulk region of a superconductor, provided emergence of massive photons, i.e., Meissner effect. Therefore, we can expect that a surface persistent current is realized for a ground/equilibrium state in the sense of stability against local perturbations. We also apply Elitzur's theorem to superconductors at finite temperatures. As a result, we prove absence of symmetry breaking of the global U(1) phase of electrons for almost all gauge fixings. These observations suggest that the nature of superconductivity is the emergence of massive photons rather than the symmetry breaking of the U(...
Rank theorems of operators between Banach spaces
Institute of Scientific and Technical Information of China (English)
2000-01-01
Let E and F be Banach spaces, and B( E, F) all of bounded linear operators on E into F. Let T0 ∈ B( E, F) with an outer inverse T0# ∈ B( F, E). Then a characteristic condition of S= (I + T0# ( T- T0))-1 T0# with T∈ B( E, F) and || T0# ( T- T0) || < 1, being a generalized inverse of T, is presented, and hence, a rank theorem of operators on E into F is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.
Rank theorems of operators between Banach spaces
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Let E and F be Banach spaces, and B(E,F) all of bounded linear operators on E into F. Let T0∈B(E,F) with an outer inverse T#0∈B(F,E). Then a characteristic condition of S=(I+T#0(T-T0))-1T#0 with T∈B(E,F) and ‖T#0(T-T0)‖<1, being a generalized inverse of T, is presented, and hence, a rank theorem of operators on E into F is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.
A generalized preimage theorem in global analysis
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The concept of locally fine point and generalized regular valueof a C1 map between Banach spaces were carried over C1 map between Banach manifolds. Hence the preimage theorem, a principle constructing Banach manifolds in global analysis, is generalized.
Sahoo- and Wayment-Type Integral Mean Value Theorems
Tiryaki, Aydin; Cakmak, Devrim
2010-01-01
In this article, by using Rolle's theorem, we establish some results related to the mean value theorem for integrals. Our results are different from the set of integral mean value theorems which are given by Wayment ["An integral mean value theorem", Math. Gazette 54 (1970), pp. 300-301] and Sahoo ["Some results related to the integral mean value…
A New GLKKM Theorem and Its Application to Abstract Economies
Institute of Scientific and Technical Information of China (English)
WEN Kai-ting
2012-01-01
In this paper,a new GLKKM theorem in L-convex spaces is established.As applications,a new fixed point theorem and a maximal element theorem are obtained in Lconvex spaces.Finally,equilibrium existence theorems for abstract economies and qualitative games in L-convex spaces are yielded.
Sahoo- and Wayment-Type Integral Mean Value Theorems
Tiryaki, Aydin; Cakmak, Devrim
2010-01-01
In this article, by using Rolle's theorem, we establish some results related to the mean value theorem for integrals. Our results are different from the set of integral mean value theorems which are given by Wayment ["An integral mean value theorem", Math. Gazette 54 (1970), pp. 300-301] and Sahoo ["Some results related to the integral mean value…
Double Soft Theorem for Perturbative Gravity
Saha, Arnab Priya
2016-01-01
Following up on the recent work of Cachazo, He and Yuan \\cite{arXiv:1503.04816 [hep-th]}, we derive the double soft graviton theorem in perturbative gravity. We show that the double soft theorem derived using CHY formula precisely matches with the perturbative computation involving Feynman diagrams. In particular, we find how certain delicate limits of Feynman diagrams play an important role in obtaining this equivalence.
Transversality theorems for the weak topology
2011-01-01
In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is $(a)$-regular. Here we first discuss the Thom transversality theorem for the weak topology and then give a similiar kind of result for the weak topology, under very weak hypotheses. Recently sever...
The large deviations theorem and ergodicity
Energy Technology Data Exchange (ETDEWEB)
Gu Rongbao [School of Finance, Nanjing University of Finance and Economics, Nanjing 210046 (China)
2007-12-15
In this paper, some relationships between stochastic and topological properties of dynamical systems are studied. For a continuous map f from a compact metric space X into itself, we show that if f satisfies the large deviations theorem then it is topologically ergodic. Moreover, we introduce the topologically strong ergodicity, and prove that if f is a topologically strongly ergodic map satisfying the large deviations theorem then it is sensitively dependent on initial conditions.
Herbrand's theorem and non-Euclidean geometry
Beeson, Michael; Boutry, Pierre; Narboux, Julien
2014-01-01
International audience; We use Herbrand's theorem to give a new proof that Eu- clid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non- Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.
Mental Constructions for The Group Isomorphism Theorem
Directory of Open Access Journals (Sweden)
Arturo Mena-Lorca
2016-03-01
Full Text Available The group isomorphism theorem is an important subject in any abstract algebra undergraduate course; nevertheless, research shows that it is seldom understood by students. We use APOS theory and propose a genetic decomposition that separates it into two statements: the first one for sets and the second with added structure. We administered a questionnaire to students from top Chilean universities and selected some of these students for interviews to gather information about the viability of our genetic decomposition. The students interviewed were divided in two groups based on their familiarity with equivalence relations and partitions. Students who were able to draw on their intuition of partitions were able to reconstruct the group theorem from the set theorem, while those who stayed on the purely algebraic side could not. Since our approach to learning this theorem was successful, it may be worthwhile to gather data while teaching it the way we propose here in order to check how much the learning of the group isomorphism theorem is improved. This approach could be expanded to other group homomorphism theorems provided further analysis is conducted: going from the general (e.g., sets to the particular (e.g., groups might not always the best strategy, but in some cases we may just be turning to more familiar settings.
A novel sampling theorem on the sphere
McEwen, J D
2011-01-01
We develop a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension. The fundamental property of any sampling theorem is the number of samples required to represent a band-limited signal. To represent exactly a signal on the sphere band-limited at L, all sampling theorems on the sphere require O(L^2) samples. However, our sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere and an asymptotically identical, but smaller, number of samples than the Gauss-Legendre sampling theorem. The complexity of our algorithms scale as O(L^3), however, the continual use of fast Fourier transforms reduces the constant prefactor associated with the asymptotic scaling considerably, resulting in algorithms that are fast. Furthermore, we do not require any precomputation and our algorithms apply to both scalar and spin functions on the sphere without any change in computational comple...
Khare, A; Paranjape, M B; Khare, Avinash; MacKenzie, R; Paranjape, M B
1994-01-01
The Coleman-Hill theorem prohibits the appearance of radiative corrections to the topological mass (more precisely, to the parity-odd part of the vacuum polarization tensor at zero momentum) in a wide class of abelian gauge theories in 2+1 dimensions. We re-express the theorem in terms of the effective action rather than in terms of the vacuum polarization tensor. The theorem so restated becomes somewhat stronger: a known exception to the theorem, spontaneously broken scalar Chern-Simons electrodynamics, obeys the new non-renormalization theorem. Whereas the vacuum polarization {\\sl does} receive a one-loop, parity-odd correction, this does not translate to a radiative contribution to the Chern-Simons term in the effective action. We also point out a new situation, involving scalar fields and parity-odd couplings, which was overlooked in the original analysis, where the conditions of the theorem are satisfied and where the topological mass does, in fact, get a radiative correction.
The modified Poynting theorem and the concept of mutual energy
Zhao, Shuang-ren; Yang, Kang; Yang, Xingang; Yang, Xintie
2015-01-01
The Poynting theorem is generalized to the modified Poynting theorem. In the modified Poynting theorem the electromagnetic field is superimposition of different electromagnetic fields including the field of retarded potential and advanced potential. The media epsilon (permittivity) and mu (permeability) can also be different in the different fields. The concept of mutual energy is introduced which is the difference between the total energy and self-energy. Using the modified Poynting theorem with the concept of the mutual energy the modified mutual energy theorem is derived. Applying time-offset transform and time integral to the modified mutual energy theorem, the time-correlation modified mutual energy theorem is obtained. Assume there are only two fields which are retarded potential, and there is only one media, the modified time-correlation energy theorem becomes the time-correlation energy theorem, which is also referred as the time-correlation reciprocity theorem. Assume there are two electromagnetic fi...
Institute of Scientific and Technical Information of China (English)
Lei DENG; Ming Ge YANG
2006-01-01
Some new coincidence theorems involving admissible set-valued mappings are proved in general noncompact topological spaces. As applications, some new minimax inequalities, section theorem, best approximation theorem, existence theorems of weighted Nash equilibria and Pareto equilibria for multiobjective games are given in general topological spaces.
Limit Theorems For Closed Queuing Networks With Excess Of Servers
Tsitsiashvili, G.
2013-01-01
In this paper limit theorems for closed queuing networks with excess of servers are formulated and proved. First theorem is a variant of the central limit theorem and is proved using classical results of V.I. Romanovskiy for discrete Markov chains. Second theorem considers a convergence to chi square distribution. These theorems are mainly based on an assumption of servers excess in queuing nodes.
Cosine directions using Rao-Blackwell Theorem and Hausdorff metric in Quasars
Bell, Byron E
2015-01-01
This analysis will determine the equations of the cosine directions for all flux of the optical spectrum in quasars. Studies on Hausdorff metric will greatly enhance our understanding of quasars distances. This study will complete steps in the classification of quasars by finding the minimum variance of flux by using the RaoBlackwell Theorem. The papers of C. R. Rao and D. Blackwell will be examined to clarify more of the above theorem. Keywords: Theory of Flux, SDSS, Quasars, Redshift (z), Population Perimeters, Regression Analysis
Quantum Effects on the Deflection of Light and the Gauss-Bonnet Theorem
Jusufi, Kimet
2016-01-01
In this letter we apply the Gauss--Bonnet theorem to calculate the deflection angle by a quantum corrected Schwarzschild black hole in the weak limit approximation. In particular, we calculate the light deflection by two types of quantum corrected black holes: the renormalization group improved Schwarzschild solution and the quantum corrected Schwarzschild solution in Bohmian quantum mechanics. We start from the corresponding optical metrics to use then the Gauss--Bonnet theorem and calculate the Gaussian curvature in both cases. Finally, we calculate the leading terms of the deflection angle and show that quantum corrections modifies the deflection angle in both solutions.
The Hellmann–Feynman theorem, the comparison theorem, and the envelope theory
Directory of Open Access Journals (Sweden)
Claude Semay
2015-01-01
Full Text Available The envelope theory is a convenient method to compute approximate solutions for bound state equations in quantum mechanics. It is shown that these approximate solutions obey a kind of Hellmann–Feynman theorem, and that the comparison theorem can be applied to these approximate solutions for two ordered Hamiltonians.
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Applications of locally fine property for operators are further developed. Let E and F be Banach spaces and f: be C1 nonlinear map, where U (x0) is an open set containing point x0∈E. With the locally fine property for Frechet derivatives f′(x) and generalized rank theorem for f′(x), a local conjugacy theorem, i.e. a characteristic condition for f being conjugate to f′(x0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.
Institute of Scientific and Technical Information of China (English)
马吉溥
2000-01-01
Applications of locally fine property for operators are further developed. Let E and F be Banach spaces and f: U( x0) E—→F be C1 nonlinear map, where U (x0) is an open set containing point x0∈ E. With the locally fine property for Frechet derivatives f’ (x) and generalized rank theorem for f ’( x), a local conjugacy theorem, i. e. a characteristic condition for f being conjugate to f (x0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.
Topological interpretation of the Luttinger theorem
Seki, Kazuhiro; Yunoki, Seiji
2017-08-01
Based solely on the analytical properties of the single-particle Green's function of fermions at finite temperatures, we show that the generalized Luttinger theorem inherently possesses topological aspects. The topological interpretation of the generalized Luttinger theorem can be introduced because (i) the Luttinger volume is represented as the winding number of the single-particle Green's function and, thus, (ii) the deviation of the theorem, expressed with a ratio between the interacting and noninteracting single-particle Green's functions, is also represented as the winding number of this ratio. The formulation based on the winding number naturally leads to two types of the generalized Luttinger theorem. Exploring two examples of single-band translationally invariant interacting electrons, i.e., simple metal and Mott insulator, we show that the first type falls into the original statement for Fermi liquids given by Luttinger, where poles of the single-particle Green's function appear at the chemical potential, while the second type corresponds to the extended one for nonmetallic cases with no Fermi surface such as insulators and superconductors generalized by Dzyaloshinskii, where zeros of the single-particle Green's function appear at the chemical potential. This formulation also allows us to derive a sufficient condition for the validity of the Luttinger theorem of the first type by applying the Rouche's theorem in complex analysis as an inequality. Moreover, we can rigorously prove in a nonperturbative manner, without assuming any detail of a microscopic Hamiltonian, that the generalized Luttinger theorem of both types is valid for generic interacting fermions as long as the particle-hole symmetry is preserved. Finally, we show that the winding number of the single-particle Green's function can also be associated with the distribution function of quasiparticles, and therefore the number of quasiparticles is equal to the Luttinger volume. This implies that
WEYL'S TYPE THEOREMS AND HYPERCYCLIC OPERATORS
Institute of Scientific and Technical Information of China (English)
M.H. M. Rashid
2012-01-01
For a bounded operator T acting on an infinite dimensional separable Hilbert space H,we prove the following assertions: (i) If T or T* ∈ SC,then generalized aBrowder's theorem holds for f(T) for every f ∈ Hol(σ(T)).(ii) If T or T* ∈ HC has topological uniform descent at all λ ∈ iso(σ(T)),then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)).(iii) If T ∈ HC has topological uniform descent at all λ ∈ E(T),then T satisfies generalized Weyl's theorem.(iv) Let T ∈ HC.If T satisfies the growth condition Gd(d ≥ 1),then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)).(v) If T ∈ SC,then,f(σSBF-+ (T)) =σSBF-+ (f(T)) for all f ∈ Hol(σ(T)).(vi) Let T be a-isoloid such that T* ∈ HC.If T - λI has finite ascent at every λ ∈ Ea(T)and if F is of finite rank on H such that TF =FT,then T + F obeys generalized a-Weyl's theorem.
Anti-Bell - Refutation of Bell's theorem
Barukčić, Ilija
2012-12-01
In general, Albert Einstein as one of "the founding fathers of quantum mechanics" had some problems to accept especially the Copenhagen dominated interpretation of quantum mechanics. Einstein's dissatisfaction with Copenhagen's interpretation of quantum mechanics, the absence of locality and causality within the Copenhagen dominated quantum mechanics lead to the well known Einstein, Podolsky and Rosen thought experiment. According to Einstein et al., the Copenhagen dominated quantum mechanics cannot be regarded as a complete physical theory. The Einstein, Podolsky and Rosen thought experiment was the origin of J. S. Bell's publication in 1964; known as Bell's theorem. Meanwhile, some dramatic violations of Bell's inequality (by so called Bell test experiments) have been reported which is taken as an empirical evidence against local realism and causality at quantum level and as positive evidence in favor of the Copenhagen dominated quantum mechanics. Thus far, Quantum mechanics is still regarded as a "strictly" non-local theory. The purpose of this publication is to refute Bell's original theorem. Thus far, if we accept Bell's theorem as correct, we must accept that +0> = +1. We can derive a logical contradiction out of Bell's theorem, Bell's theorem is refuted.
Generalized fluctuation theorems for classical systems
Agarwal, G. S.; Dattagupta, Sushanta
2015-11-01
The fluctuation theorem has a very special place in the study of nonequilibrium dynamics of physical systems. The form in which it is used most extensively is the Gallavoti-Cohen fluctuation theorem which is in terms of the distribution of the work p (W )/p (-W )=exp(α W ) . We derive the general form of the fluctuation theorems for an arbitrary multidimensional Gaussian Markov process. Interestingly, the parameter α is by no means universal, hitherto taken for granted in the case of linear Gaussian processes. As a matter of fact, conditions under which α does become a universal parameter 1 /K T are found to be rather restrictive. As an application we consider fluctuation theorems for classical cyclotron motion of an electron in a parabolic potential. The motion of the electron is described by four coupled Langevin equations and thus is nontrivial. The generalized theorems are equally valid for nonequilibrium steady states and could be especially important in the presence of anisotropic diffusion.
Ergodic theorem, ergodic theory, and statistical mechanics.
Moore, Calvin C
2015-02-17
This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject--namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.
Adiabatic Theorem for Quantum Spin Systems
Bachmann, S.; De Roeck, W.; Fraas, M.
2017-08-01
The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g., in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation ɛ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this Letter, we prove a version of the adiabatic theorem for gapped ground states of interacting quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo's linear response formula for a broad class of gapped interacting systems. We predict that the density of nonadiabatic excitations is exponentially small in the driving rate and the scaling of the exponent depends on the dimension.
Bayes' theorem: scientific assessment of experience
Directory of Open Access Journals (Sweden)
Lex Rutten
2010-10-01
Full Text Available Homeopathy is based on experience and this is a scientific procedure if we follow Bayes' theorem. Unfortunately this is not the case at the moment. Symptoms are added to our materia medica based on absolute occurrence, while Bayes theorem tells us that this should be based on relative occurrence. Bayes theorem can be applied on prospective research, but also on retrospective research and consensus based on a large number of cases. Confirmation bias is an important source of false data in experience based systems like homeopathy. Homeopathic doctors should become more aware of this and longer follow-up of cases could remedy this. The existing system of adding symptoms to our materia medica is obsolete.
Some Limit Theorems in Geometric Processes
Institute of Scientific and Technical Information of China (English)
Yeh Lam; Yao-hui Zheng; Yuan-lin Zhang
2003-01-01
Geometric process (GP) was introduced by Lam[4,5], it is defined as a stochastic process {Xn, n =1, 2,...} for which there exists a real number a > 0, such that {an-1Xn, n = 1, 2,...} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for Sn with a > 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.
Causality, Bell's theorem, and Ontic Definiteness
Henson, Joe
2011-01-01
Bell's theorem shows that the reasonable relativistic causal principle known as "local causality" is not compatible with the predictions of quantum mechanics. It is not possible maintain a satisfying causal principle of this type while dropping any of the better-known assumptions of Bell's theorem. However, another assumption of Bell's theorem is the use of classical logic. One part of this assumption is the principle of "ontic definiteness", that is, that it must in principle be possible to assign definite truth values to all propositions treated in the theory. Once the logical setting is clarified somewhat, it can be seen that rejecting this principle does not in any way undermine the type of causal principle used by Bell. Without ontic definiteness, the deterministic causal condition known as Einstein Locality succeeds in banning superluminal influence (including signalling) whilst allowing correlations that violate Bell's inequalities. Objections to altering logic, and the consequences for operational and...
Spectral mapping theorems a bluffer's guide
Harte, Robin
2014-01-01
Written by an author who was at the forefront of developments in multi-variable spectral theory during the seventies and the eighties, this guide sets out to describe in detail the spectral mapping theorem in one, several and many variables. The basic algebraic systems – semigroups, rings and linear algebras – are summarised, and then topological-algebraic systems, including Banach algebras, to set up the basic language of algebra and analysis. Spectral Mapping Theorems is written in an easy-to-read and engaging manner and will be useful for both the beginner and expert. It will be of great importance to researchers and postgraduates studying spectral theory.
Jarzynski's theorem for lattice gauge theory
Caselle, Michele; Nada, Alessandro; Panero, Marco; Toniato, Arianna
2016-01-01
Jarzynski's theorem is a well-known equality in statistical mechanics, which relates fluctuations in the work performed during a non-equilibrium transformation of a system, to the free-energy difference between two equilibrium states. In this article, we extend Jarzynski's theorem to lattice gauge theory, and present examples of applications for two challenging computational problems, namely the calculation of interface free energies and the determination of the equation of state. We conclude with a discussion of further applications of interest in QCD and in other strongly coupled gauge theories, in particular for the Schroedinger functional and for simulations at finite density using reweighting techniques.
Central Limit Theorem for Nonlinear Hawkes Processes
Zhu, Lingjiong
2012-01-01
Hawkes process is a self-exciting point process with clustering effect whose jump rate depends on its entire past history. It has wide applications in neuroscience, finance and many other fields. Linear Hawkes process has an immigration-birth representation and can be computed more or less explicitly. It has been extensively studied in the past and the limit theorems are well understood. On the contrary, nonlinear Hawkes process lacks the immigration-birth representation and is much harder to analyze. In this paper, we obtain a functional central limit theorem for nonlinear Hawkes process.
Limit theorems for fragmentation processes with immigration
Knobloch, Robert
2012-01-01
In this paper we extend two limit theorems which were recently obtained for fragmentation processes to such processes with immigration. More precisely, in the setting with immigration we consider a limit theorem for the process counted with a random characteristic as well as the asymptotic behaviour of an empirical measure associated with the stopping line corresponding to the first blocks, in their respective line of descent, that are smaller than a given size. In addition, we determine the asymptotic decay rate of the size of the largest block in a homogeneous fragmentation process with immigration. The techniques used to proves these results are based on submartingale arguments.
A Noether Theorem for Markov Processes
Baez, John C
2012-01-01
Noether's theorem links the symmetries of a quantum system with its conserved quantities, and is a cornerstone of quantum mechanics. Here we prove a version of Noether's theorem for Markov processes. In quantum mechanics, an observable commutes with the Hamiltonian if and only if its expected value remains constant in time for every state. For Markov processes that no longer holds, but an observable commutes with the Hamiltonian if and only if both its expected value and standard deviation are constant in time for every state.
General Common Fixed Point Theorems and Applications
Directory of Open Access Journals (Sweden)
Shyam Lal Singh
2012-01-01
Full Text Available The main result is a common fixed point theorem for a pair of multivalued maps on a complete metric space extending a recent result of Đorić and Lazović (2011 for a multivalued map on a metric space satisfying Ćirić-Suzuki-type-generalized contraction. Further, as a special case, we obtain a generalization of an important common fixed point theorem of Ćirić (1974. Existence of a common solution for a class of functional equations arising in dynamic programming is also discussed.
Pauli and the spin-statistics theorem
Duck, Ian M
1997-01-01
This book makes broadly accessible an understandable proof of the infamous spin-statistics theorem. This widely known but little-understood theorem is intended to explain the fact that electrons obey the Pauli exclusion principle. This fact, in turn, explains the periodic table of the elements and their chemical properties. Therefore, this one simply stated fact is responsible for many of the principal features of our universe, from chemistry to solid state physics to nuclear physics to the life cycle of stars.In spite of its fundamental importance, it is only a slight exaggeration to say that
Generalizations of the Abstract Boundary singularity theorem
Whale, Ben E; Scott, Susan M
2015-01-01
The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from $C^1$ curves to locally Lipschitz curves.
Fincham, W H A
2013-01-01
Optics: Ninth Edition Optics: Ninth Edition covers the work necessary for the specialization in such subjects as ophthalmic optics, optical instruments and lens design. The text includes topics such as the propagation and behavior of light; reflection and refraction - their laws and how different media affect them; lenses - thick and thin, cylindrical and subcylindrical; photometry; dispersion and color; interference; and polarization. Also included are topics such as diffraction and holography; the limitation of beams in optical systems and its effects; and lens systems. The book is recommen
A Dual of the Compression-Expansion Fixed Point Theorems
Directory of Open Access Journals (Sweden)
Henderson Johnny
2007-01-01
Full Text Available This paper presents a dual of the fixed point theorems of compression and expansion of functional type as well as the original Leggett-Williams fixed point theorem. The multi-valued situation is also discussed.
A Note on a Broken-Cycle Theorem for Hypergraphs
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Trinks Martin
2014-08-01
Full Text Available Whitney’s Broken-cycle Theorem states the chromatic polynomial of a graph as a sum over special edge subsets. We give a definition of cycles in hypergraphs that preserves the statement of the theorem there
A duality theorem of crossed coproduct for Hopf algebras
Institute of Scientific and Technical Information of China (English)
王栓宏
1995-01-01
A duality theorem for Hopf crossed coproduct is proved. This theorem plays a role similar to that appearing in the work of Koppinen (which generalized the corresponding results of group grraded ring).
Adiabatic limits,vanishing theorems and the noncommutative residue
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper,we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems by taking adiabatic limits.As an application,we give a Kastler-Kalau-Walze type theorem for foliations.
Lp-inverse theorem for modified beta operators
Directory of Open Access Journals (Sweden)
V. K. Jain
2003-04-01
Full Text Available We obtain a converse theorem for the linear combinations of modified beta operators whose weight function is the Baskakov operators. To prove our inverse theorem, we use the technique of linear approximating method, namely, Steklov mean.
Application of the residue theorem to bilateral hypergeometric series
Directory of Open Access Journals (Sweden)
Wenchang Chu
2007-12-01
Full Text Available The application of the residue theorem to bilateral hypergeometric series identities is systematically reviewed by exemplifying three classes of summation theorems due to Dougall (1907, Jackson (1949, 1952 and Slater-Lakin (1953.
An existence theorem for Volterra integrodifferential equations with infinite delay
Directory of Open Access Journals (Sweden)
Ferenc Izsak
2003-01-01
Full Text Available Using Schauder's fixed point theorem, we prove an existence theorem for Volterra integrodifferential equations with infinite delay. As an appplication, we consider an $n$ species Lotka-Volterra competitive system.
Central Limit Theorem for Coloured Hard Dimers
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Maria Simonetta Bernabei
2010-01-01
Full Text Available We study the central limit theorem for a class of coloured graphs. This means that we investigate the limit behavior of certain random variables whose values are combinatorial parameters associated to these graphs. The techniques used at arriving this result comprise combinatorics, generating functions, and conditional expectations.
Generalization of the Hellmann-Feynman theorem
Energy Technology Data Exchange (ETDEWEB)
Esteve, J.G., E-mail: esteve@unizar.e [Departamento de Fisica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza (Spain); Instituto de Biocomputacion y Fisica de Sistemas complejos (BIFI), Universidad de Zaragoza, 50009 Zaragoza (Spain); Falceto, Fernando [Departamento de Fisica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza (Spain); Instituto de Biocomputacion y Fisica de Sistemas complejos (BIFI), Universidad de Zaragoza, 50009 Zaragoza (Spain); Garcia Canal, C. [Laboratorio de Fisica Teorica, Departamento de Fisica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata and IFLP-CONICET (Argentina)
2010-01-25
The well-known Hellmann-Feynman theorem of quantum mechanics connected with the derivative of the eigenvalues with respect to a parameter upon which the Hamiltonian depends, is generalized to include cases in which the domain of definition of the Hamiltonian of the system also depends on that parameter.
Student Research Project: Goursat's Other Theorem
Petrillo, Joseph
2009-01-01
In an elementary undergraduate abstract algebra or group theory course, a student is introduced to a variety of methods for constructing and deconstructing groups. What seems to be missing from contemporary texts and syllabi is a theorem, first proved by Edouard Jean-Baptiste Goursat (1858-1936) in 1889, which completely describes the subgroups of…
A simpler derivation of the coding theorem
Lomnitz, Yuval
2012-01-01
A simple proof for the Shannon coding theorem, using only the Markov inequality, is presented. The technique is useful for didactic purposes, since it does not require many preliminaries and the information density and mutual information follow naturally in the proof. It may also be applicable to situations where typicality is not natural.
Generalizations of Brandl's theorem on Engel length
Quek, S. G.; Wong, K. B.; Wong, P. C.
2013-04-01
Let n Engel cycle generated by g and h. The length of the Engel cycle is m-n. A group G is said to have Engel length r, if all the length of the Engel cycles in G divides r. In this paper we discuss the Brandl's theorem on Engel length and give some of its generalizations.
Ptolemy's Theorem and Familiar Trigonometric Identities.
Bidwell, James K.
1993-01-01
Integrates the sum, difference, and multiple angle identities into an examination of Ptolemy's Theorem, which states that the sum of the products of the lengths of the opposite sides of a quadrilateral inscribed in a circle is equal to the product of the lengths of the diagonals. (MDH)
The central limit theorem and chaos
Institute of Scientific and Technical Information of China (English)
NIU Ying-xuan
2009-01-01
Let X be a compact metric space and f : X → X be a continuous map. This paper studies some relationships between stochastic and topological properties of dynamical systems.It is shown that if f satisfies the central limit theorem, then f is topologically ergodic and f is sensitively dependent on initial conditions if and only if f is neither minimal nor equicontinuous.
JACKSON‘S THEOREM FOR COMPACT GROUPS
Institute of Scientific and Technical Information of China (English)
H.Vaezi; S.F.Rzaev
2002-01-01
In this article we consider the generalized shift operator defined by (Shuf)(g)=∫Gf(tut-1g)dt on compact group G and by help of this operator we define “Spherical” modulus of continuity.So we prove Stechkin and Jackson type theorems.
A cosmological no-hair theorem
Chambers, C M; Chris M Chambers; Ian G Moss
1994-01-01
A generalisation of Price's theorem is given for application to Inflationary Cosmologies. Namely, we show that on a Schwarzschild--de Sitter background there are no static solutions to the wave or gravitational perturbation equations for modes with angular momentum greater than their intrinsic spin.
Multiplier theorems for special Hermite expansions on
Institute of Scientific and Technical Information of China (English)
张震球; 郑维行
2000-01-01
The weak type (1,1) estimate for special Hermite expansions on Cn is proved by using the Calderon-Zygmund decomposition. Then the multiplier theorem in Lp(1 < p < ω ) is obtained. The special Hermite expansions in twisted Hardy space are also considered. As an application, the multipli-ers for a certain kind of Laguerre expansions are given in Lp space.
Lagrange’s Four-Square Theorem
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Watase Yasushige
2015-02-01
Full Text Available This article provides a formalized proof of the so-called “the four-square theorem”, namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in [14], [1] or [23].
SOME REFINEMENTS OF ENESTROM-KAKEYA THEOREM
Institute of Scientific and Technical Information of China (English)
A.Aziz; B.A.Zargar
2007-01-01
In this paper we present certain interesting refinements of a well-known Enestrom-Kakeya theorem in the theory of distribution of zeros of polynomials which among other things also improve upon some results of Aziz and Mohammad, Govil and Rehman and others.
The Viner-Wong Envelope Theorem.
Silberberg, Eugene
1999-01-01
Observes that the envelope theorem, a fundamental tool in duality analysis, is still a puzzle to many people. Argues that the essence of a solution proposed by Paul Samuelson (1947) is also unclear to many people, but can be communicated with a simple cost diagram. Presents and explains the proposed diagram. (DSK)
Some Generalizations of Jungck's Fixed Point Theorem
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J. R. Morales
2012-01-01
Full Text Available We are going to generalize the Jungck's fixed point theorem for commuting mappings by mean of the concepts of altering distance functions and compatible pair of mappings, as well as, by using contractive inequalities of integral type and contractive inequalities depending on another function.
A non-differentiable Noether's theorem
Cresson, Jacky; Greff, Isabelle
2011-02-01
In the framework of the nondifferentiable embedding of Lagrangian systems, defined by Cresson and Greff [non-dierentiable embedding of lagrangian systems and partial dierential equations. Preprint Max-Plank-Institut für Mathematik in den Naturwissenschaften, Leipzig 16, 26 (2010)], we prove a Noether's theorem based on the lifting of one-parameter groups of diffeomorphisms.
Fixed Point Theorems for Asymptotically Contractive Multimappings
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M. Djedidi
2012-01-01
Full Text Available We present fixed point theorems for a nonexpansive set-valued mapping from a closed convex subset of a reflexive Banach space into itself under some asymptotic contraction assumptions. Some existence results of coincidence points and eigenvalues for multimappings are given.
Agreement Theorems in Dynamic-Epistemic Logic
Degremont, Cedric; Roy, Oliver
2012-01-01
This paper introduces Agreement Theorems to dynamic-epistemic logic. We show first that common belief of posteriors is sufficient for agreement in epistemic-plausibility models, under common and well-founded priors. We do not restrict ourselves to the finite case, showing that in countable structure
An extension theorem for conformal gauge singularities
Tod, Paul
2007-01-01
We analyse conformal gauge, or isotropic, singularities in cosmological models in general relativity. Using the calculus of tractors, we find conditions in terms of tractor curvature for a local extension of the conformal structure through a cosmological singularity and prove a local extension theorem.
A Generalized Krein-Rutman Theorem
Zhang, Lei
2016-01-01
A generalized Krein-Rutman theorem for a strongly positive bounded linear operator whose spectral radius is larger than essential spectral radius is established: the spectral radius of the operator is an algebraically simple eigenvalue with strongly positive eigenvector and other eigenvalues are less than the spectral radius.
Stokes' theorem, volume growth and parabolicity
Valtorta, Daniele
2010-01-01
We present some new Stokes'type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.
Generalized Friedland's theorem for C0-semigroups
Cichon, Dariusz; Jung, Il Bong; Stochel, Jan
2008-07-01
Friedland's characterization of bounded normal operators is shown to hold for infinitesimal generators of C0-semigroups. New criteria for normality of bounded operators are furnished in terms of Hamburger moment problem. All this is achieved with the help of the celebrated Ando's theorem on paranormal operators.
Answering Junior Ant's "Why" for Pythagoras' Theorem
Pask, Colin
2002-01-01
A seemingly simple question in a cartoon about Pythagoras' Theorem is shown to lead to questions about the nature of mathematical proof and the profound relationship between mathematics and science. It is suggested that an analysis of the issues involved could provide a good vehicle for classroom discussions or projects for senior students.…
The Fundamental Theorems of Interval Analysis
van Emden, M. H.; Moa, B.
2007-01-01
Expressions are not functions. Confusing the two concepts or failing to define the function that is computed by an expression weakens the rigour of interval arithmetic. We give such a definition and continue with the required re-statements and proofs of the fundamental theorems of interval arithmetic and interval analysis. Revision Feb. 10, 2009: added reference to and acknowledgement of P. Taylor.
Automated theorem proving theory and practice
Newborn, Monty
2001-01-01
As the 21st century begins, the power of our magical new tool and partner, the computer, is increasing at an astonishing rate. Computers that perform billions of operations per second are now commonplace. Multiprocessors with thousands of little computers - relatively little! -can now carry out parallel computations and solve problems in seconds that only a few years ago took days or months. Chess-playing programs are on an even footing with the world's best players. IBM's Deep Blue defeated world champion Garry Kasparov in a match several years ago. Increasingly computers are expected to be more intelligent, to reason, to be able to draw conclusions from given facts, or abstractly, to prove theorems-the subject of this book. Specifically, this book is about two theorem-proving programs, THEO and HERBY. The first four chapters contain introductory material about automated theorem proving and the two programs. This includes material on the language used to express theorems, predicate calculus, and the rules of...
Nash-Williams’ cycle-decomposition theorem
DEFF Research Database (Denmark)
Thomassen, Carsten
2016-01-01
We give an elementary proof of the theorem of Nash-Williams that a graph has an edge-decomposition into cycles if and only if it does not contain an odd cut. We also prove that every bridgeless graph has a collection of cycles covering each edge at least once and at most 7 times. The two results...
Tennis Rackets and the Parallel Axis Theorem
Christie, Derek
2014-01-01
This simple experiment uses an unusual graph straightening exercise to confirm the parallel axis theorem for an irregular object. Along the way, it estimates experimental values for g and the moment of inertia of a tennis racket. We use Excel to find a 95% confidence interval for the true values.
Average sampling theorems for shift invariant subspaces
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The sampling theorem is one of the most powerful results in signal analysis. In this paper, we study the average sampling on shift invariant subspaces, e.g. wavelet subspaces. We show that if a subspace satisfies certain conditions, then every function in the subspace is uniquely determined and can be reconstructed by its local averages near certain sampling points. Examples are given.
On the Hardy-Littlewood maximal theorem
Directory of Open Access Journals (Sweden)
Shinji Yamashita
1982-01-01
Full Text Available The Hardy-Littlewood maximal theorem is extended to functions of class PL in the sense of E. F. Beckenbach and T. Radó, with a more precise expression of the absolute constant in the inequality. As applications we deduce some results on hyperbolic Hardy classes in terms of the non-Euclidean hyperbolic distance in the unit disk.
Crum's Theorem for `Discrete' Quantum Mechanics
Odake, Satoru; Sasaki, Ryu
2009-01-01
In one-dimensional quantum mechanics, or the Sturm-Liouville theory, Crum's theorem describes the relationship between the original and the associated Hamiltonian systems, which are iso-spectral except for the lowest energy state. Its counterpart in `discrete' quantum mechanics is formulated algebraically, elucidating the basic structure of the discrete quantum mechanics, whose Schr\\"odinger equation is a difference equation.
1/4-pinched contact sphere theorem
DEFF Research Database (Denmark)
Ge, Jian; Huang, Yang
2016-01-01
Given a closed contact 3-manifold with a compatible Riemannian metric, we show that if the sectional curvature is 1/4-pinched, then the contact structure is universally tight. This result improves the Contact Sphere Theorem in [EKM12], where a 4/9-pinching constant was imposed. Some tightness res...
Green's Theorem for Generalized Fractional Derivatives
Odzijewicz, Tatiana; Torres, Delfim F M
2012-01-01
We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo.
On Viviani's Theorem and Its Extensions
Abboud, Elias
2010-01-01
Viviani's theorem states that the sum of distances from any point inside an equilateral triangle to its sides is constant. Here, in an extension of this result, we show, using linear programming, that any convex polygon can be divided into parallel line segments on which the sum of the distances to the sides of the polygon is constant. Let us say…
A non-archimedean Montel's theorem
Favre, Charles; Trucco, Eugenio
2011-01-01
We prove a version of Montel's theorem for analytic functions over a non-archimedean complete valued field. We propose a definition of normal family in this context, and give applications of our results to the dynamics of non-archimedean entire functions.
Norton's theorem for batch routing queueing networks
Bause, Falko; Boucherie, Richard J.; Buchholz, Peter
2001-01-01
This paper shows that the aggregation and decomposition result known as Norton’s theorem for queueing networks can be extended to a general class of batch routing queueing networks with product-form solution that allows for multiple components to simultaneously release and receive (batches of) custo
Two Theorems on Calculating the Relative Entropy of Entanglement
Institute of Scientific and Technical Information of China (English)
WU Sheng-Jun; ZHANG Yong-De; WU Qiang
2001-01-01
We present two theorems on calculating the relative entropy of entanglement. Theorem 1 is an extension of Vedral and Plenio's theorem (Phys. Rev. A 57 (1998) 1619) for pure states, which is useful for calculating the relative entropy of entanglement for all pure states as well as for a class of mixed states. Theorem 2 gives the relative entropy of entanglement for any bipartite state whose tripartite purification has two separable reduced bipartite states.
Nonempty intersection theorems in topological spaces with applications
Institute of Scientific and Technical Information of China (English)
Min FANG; Nan-jing HUANG
2009-01-01
In this paper,we establish some new nonempty intersection theorems for generalized L-KKM mappings and prove some new fixed point theorems for set-valued mappings under suitable conditions in topological spaces.As applications,an existence theorem for an equilibrium problem with lower and upper bounds and two existence theorems for a quasi-equilibrium problem with lower and upper bounds are obtained in topological spaces.Our results generalize some known results in the literature.
Ehrenfest theorem, Galilean invariance and nonlinear Schr"odinger equations
Kälbermann, G
2003-01-01
Galilean invariant Schr"odinger equations possessing nonlinear terms coupling the amplitude and the phase of the wave function can violate the Ehrenfest theorem. An example of this kind is provided. The example leads to the proof of the theorem: A Galilean invariant Schr"odinger equation derived from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for any linear or nonlinear lagrangian.
Virial theorem and universality in a unitary fermi gas.
Thomas, J E; Kinast, J; Turlapov, A
2005-09-16
Unitary Fermi gases, where the scattering length is large compared to the interparticle spacing, can have universal properties, which are independent of the details of the interparticle interactions when the range of the scattering potential is negligible. We prepare an optically trapped, unitary Fermi gas of 6Li, tuned just above the center of a broad Feshbach resonance. In agreement with the universal hypothesis, we observe that this strongly interacting many-body system obeys the virial theorem for an ideal gas over a wide range of temperatures. Based on this result, we suggest a simple volume thermometry method for unitary gases. We also show that the observed breathing mode frequency, which is close to the unitary hydrodynamic value over a wide range of temperature, is consistent with a universal hydrodynamic gas with nearly isentropic dynamics.
Four Theorems on the Psychometric Function
May, Keith A.; Solomon, Joshua A.
2013-01-01
In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, . This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull “slope” parameter, , can be approximated by , where is the of the Weibull function that fits best to the cumulative noise distribution, and depends on the transducer. We derive general expressions for and , from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when , . We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4–0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian. PMID:24124456
Applications of square-related theorems
Srinivasan, V. K.
2014-04-01
The square centre of a given square is the point of intersection of its two diagonals. When two squares of different side lengths share the same square centre, there are in general four diagonals that go through the same square centre. The Two Squares Theorem developed in this paper summarizes some nice theoretical conclusions that can be obtained when two squares of different side lengths share the same square centre. These results provide the theoretical basis for two of the constructions given in the book of H.S. Hall and F.H. Stevens , 'A Shorter School Geometry, Part 1, Metric Edition'. In page 134 of this book, the authors present, in exercise 4, a practical construction which leads to a verification of the Pythagorean theorem. Subsequently in Theorems 29 and 30, the authors present the standard proofs of the Pythagorean theorem and its converse. In page 140, the authors present, in exercise 15, what amounts to a geometric construction, whose verification involves a simple algebraic identity. Both the constructions are of great importance and can be replicated by using the standard equipment provided in a 'geometry toolbox' carried by students in high schools. The author hopes that the results proved in this paper, in conjunction with the two constructions from the above-mentioned book, would provide high school students an appreciation of the celebrated theorem of Pythagoras. The diagrams that accompany this document are based on the free software GeoGebra. The author formally acknowledges his indebtedness to the creators of this free software at the end of this document.
Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics
von Neumann, John
2010-01-01
It is shown how to resolve the apparent contradiction between the macroscopic approach of phase space and the validity of the uncertainty relations. The main notions of statistical mechanics are re-interpreted in a quantum-mechanical way, the ergodic theorem and the H-theorem are formulated and proven (without "assumptions of disorder"), followed by a discussion of the physical meaning of the mathematical conditions characterizing their domain of validity.
A remark on Kov\\'acs' vanishing theorem
Fujino, Osamu
2012-01-01
We give an alternative proof of Kov\\'acs' vanishing theorem. Our proof is based on the standard arguments of the minimal model theory. We do not need the notion of Du Bois pairs. We reduce Kov\\'acs' vanishing theorem to the well-known relative Kawamata--Viehweg--Nadel vanishing theorem.
A Simple Geometrical Derivation of the Spatial Averaging Theorem.
Whitaker, Stephen
1985-01-01
The connection between single phase transport phenomena and multiphase transport phenomena is easily accomplished by means of the spatial averaging theorem. Although different routes to the theorem have been used, this paper provides a route to the averaging theorem that can be used in undergraduate classes. (JN)
The Interpretability of Inconsistency: Feferman's Theorem and Related Results
Visser, Albert
2014-01-01
This paper is an exposition of Feferman's Theorem concerning the interpretability of inconsistency and of further insights directly connected to this result. Feferman's Theorem is a strengthening of the Second Incompleteness Theorem. It says, in metaphorical paraphrase, that it is not just the case
ON COLLECTIVELY FIXED POINT THEOREMS ON FC-SPACES
Institute of Scientific and Technical Information of China (English)
Yongjie Piao
2010-01-01
Based on a KKM type theorem on FC-space,some new fixed point theorems for Fan-Browder type are established,and then some collectively fixed point theorems for a family of Φ-maps defined on product space of FC-spaees are given.These results generalize and improve many corresponding results.
Fincham, W H A
2013-01-01
Optics: Eighth Edition covers the work necessary for the specialization in such subjects as ophthalmic optics, optical instruments and lens design. The text includes topics such as the propagation and behavior of light; reflection and refraction - their laws and how different media affect them; lenses - thick and thin, cylindrical and subcylindrical; photometry; dispersion and color; interference; and polarization. Also included are topics such as diffraction and holography; the limitation of beams in optical systems and its effects; and lens systems. The book is recommended for engineering st
Causarum Investigatio and the Two Bell's Theorems of John Bell
Wiseman, Howard M
2015-01-01
"Bell's theorem" can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the joint assumptions of Locality and Predetermination. His 1976 theorem is their incompatibility with the single property of Local Causality. This is contrary to Bell's own later assertions, that his 1964 theorem began with the assumption of Local Causality, even if not by that name. Although the two Bell's theorems are logically equivalent, their assumptions are not. Hence, the earlier and later theorems suggest quite different conclusions, embraced by operationalists and realists, respectively. The key issue is whether Locality or Local Causality is the appropriate notion emanating from Relativistic Causality, and this rests on one's basic notion of causation. For operationalists the appropriate notion is what is here called the Principle of Agent-Causation, while for realists it is Reichenbach's Principle of common cause. By...
Multideviations: The hidden structure of Bell's theorems
Fogel, Brandon
2015-01-01
Specification of the strongest possible Bell inequalities for arbitrarily complicated physical scenarios -- any number of observers choosing between any number of observables with any number of possible outcomes -- is currently an open problem. Here I provide a new set of tools, which I refer to as "multideviations", for finding and analyzing these inequalities for the fully general case. In Part I, I introduce the multideviation framework and then use it to prove an important theorem: the Bell distributions can be generated from the set of joint distributions over all observables by deeming specific degrees of freedom unobservable. In Part II, I show how the theorem provides a new method for finding tight Bell inequalities. I then specify a set of new tight Bell inequalities for arbitrary event spaces -- the "even/odd" inequalities -- which have a straightforward interpretation when expressed in terms of multideviations. The even/odd inequalities concern degrees of freedom that are independent of those invol...
Lesovik, G B; Lebedev, A V; Sadovskyy, I A; Suslov, M V; Vinokur, V M
2016-09-12
Remarkable progress of quantum information theory (QIT) allowed to formulate mathematical theorems for conditions that data-transmitting or data-processing occurs with a non-negative entropy gain. However, relation of these results formulated in terms of entropy gain in quantum channels to temporal evolution of real physical systems is not thoroughly understood. Here we build on the mathematical formalism provided by QIT to formulate the quantum H-theorem in terms of physical observables. We discuss the manifestation of the second law of thermodynamics in quantum physics and uncover special situations where the second law can be violated. We further demonstrate that the typical evolution of energy-isolated quantum systems occurs with non-diminishing entropy.
Generalized Sampling Theorem for Bandpass Signals
Directory of Open Access Journals (Sweden)
Prokes Ales
2006-01-01
Full Text Available The reconstruction of an unknown continuously defined function from the samples of the responses of linear time-invariant (LTI systems sampled by the th Nyquist rate is the aim of the generalized sampling. Papoulis (1977 provided an elegant solution for the case where is a band-limited function with finite energy and the sampling rate is equal to times cutoff frequency. In this paper, the scope of the Papoulis theory is extended to the case of bandpass signals. In the first part, a generalized sampling theorem (GST for bandpass signals is presented. The second part deals with utilizing this theorem for signal recovery from nonuniform samples, and an efficient way of computing images of reconstructing functions for signal recovery is discussed.
Generalized Sampling Theorem for Bandpass Signals
Prokes, Ales
2006-12-01
The reconstruction of an unknown continuously defined function[InlineEquation not available: see fulltext.] from the samples of the responses of[InlineEquation not available: see fulltext.] linear time-invariant (LTI) systems sampled by the[InlineEquation not available: see fulltext.]th Nyquist rate is the aim of the generalized sampling. Papoulis (1977) provided an elegant solution for the case where[InlineEquation not available: see fulltext.] is a band-limited function with finite energy and the sampling rate is equal to[InlineEquation not available: see fulltext.] times cutoff frequency. In this paper, the scope of the Papoulis theory is extended to the case of bandpass signals. In the first part, a generalized sampling theorem (GST) for bandpass signals is presented. The second part deals with utilizing this theorem for signal recovery from nonuniform samples, and an efficient way of computing images of reconstructing functions for signal recovery is discussed.
An Equivalent Gauge and the Equivalence Theorem
Wulzer, Andrea
2014-01-01
I describe a novel covariant formulation of massive gauge theories in which the longitudinal polarization vectors do not grow with the energy. Therefore in the present formalism, differently from the ordinary one, the energy and coupling power-counting is completely transparent at the level of individual Feynman diagrams, with obvious advantages both at the conceptual and practical level. Since power-counting is transparent, the high-energy limit of the amplitudes involving longitudinal particles is immediately taken, and the Equivalence Theorem is easily demonstrated at all orders in perturbation theory. Since the formalism makes the Equivalence Theorem self-evident, and because it is based on a suitable choice of the gauge, we can call it an "Equivalent Gauge".
Noether theorem for {mu}-symmetries
Energy Technology Data Exchange (ETDEWEB)
Cicogna, Giampaolo [Dipartimento di Fisica, Universita di Pisa and INFN, Sezione di Pisa, Largo B Pontecorvo 3, 50127 Pisa (Italy); Gaeta, Giuseppe [Dipartimento di Matematica, Universita di Milano, via Saldini 50, 20133 Milano (Italy)
2007-09-28
We give a version of Noether theorem adapted to the framework of {mu}-symmetries; this extends to such case recent work by Muriel, Romero and Olver in the framework of {lambda}-symmetries, and connects {mu}-symmetries of a Lagrangian to a suitably modified conservation law. In some cases this '{mu}-conservation law' actually reduces to a standard one; we also note a relation between {mu}-symmetries and conditional invariants. We also consider the case where the variational principle is itself formulated as requiring vanishing variation under {mu}-prolonged variation fields, leading to modified Euler-Lagrange equations. In this setting, {mu}-symmetries of the Lagrangian correspond to standard conservation laws as in the standard Noether theorem. We finally propose some applications and examples.
Parameterized quantum field theory without Haag's theorem
Seidewitz, Ed
2015-01-01
Under the normal assumptions of quantum field theory, Haag's theorem states that any field unitarily equivalent to a free field must itself be a free field. Unfortunately, the derivation of the Dyson series perturbation expansion relies on the use of the interaction picture, in which the interacting field is unitarily equivalent to the free field but must still account for interactions. Thus, the traditional perturbative derivation of the scattering matrix in quantum field theory is mathematically ill defined. Nevertheless, perturbative quantum field theory is currently the only practical approach for addressing scattering for realistic interactions, and it has been spectacularly successful in making empirical predictions. This paper explains this success by showing that quantum field theory can be formulated, using an invariant, fifth path parameter in addition to the usual four position parameters, in such a way that Haag's theorem no longer applies, but such that the Dyson perturbation expansion for the sc...
De Finetti theorems for easy quantum groups
Banica, Teodor; Speicher, Roland
2009-01-01
We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite, quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S_n, O_n, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of K\\"ostler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.
A noncommutative extended de Finetti theorem
Köstler, Claus
2008-01-01
The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and K\\"ummerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors.
Thermodynamics of biochemical networks and duality theorems.
De Martino, Daniele
2013-05-01
One interesting yet difficult computational issue has recently been posed in biophysics in regard to the implementation of thermodynamic constraints to complex networks. Biochemical networks of enzymes inside cells are among the most efficient, robust, differentiated, and flexible free-energy transducers in nature. How is the second law of thermodynamics encoded for these complex networks? In this article it is demonstrated that for chemical reaction networks in the steady state the exclusion (presence) of closed reaction cycles makes possible (impossible) the definition of a chemical potential vector. Interestingly, this statement is encoded in one of the key results in combinatorial optimization, i.e., the Gordan theorem of the alternatives. From a computational viewpoint, the theorem reveals that calculating a reaction's free energy and identifying infeasible loops in flux states are dual problems whose solutions are mutually exclusive, and this opens the way for efficient and scalable methods to perform the energy balance analysis of large-scale biochemical networks.
Lesovik, G. B.; Lebedev, A. V.; Sadovskyy, I. A.; Suslov, M. V.; Vinokur, V. M.
2016-09-01
Remarkable progress of quantum information theory (QIT) allowed to formulate mathematical theorems for conditions that data-transmitting or data-processing occurs with a non-negative entropy gain. However, relation of these results formulated in terms of entropy gain in quantum channels to temporal evolution of real physical systems is not thoroughly understood. Here we build on the mathematical formalism provided by QIT to formulate the quantum H-theorem in terms of physical observables. We discuss the manifestation of the second law of thermodynamics in quantum physics and uncover special situations where the second law can be violated. We further demonstrate that the typical evolution of energy-isolated quantum systems occurs with non-diminishing entropy.
Locomotion in complex fluids: Integral theorems
Lauga, Eric
2014-01-01
The biological fluids encountered by self-propelled cells display complex microstructures and rheology. We consider here the general problem of low-Reynolds number locomotion in a complex fluid. {Building on classical work on the transport of particles in viscoelastic fluids,} we demonstrate how to mathematically derive three integral theorems relating the arbitrary motion of an isolated organism to its swimming kinematics {in a non-Newtonian fluid}. These theorems correspond to three situations of interest, namely (1) squirming motion in a linear viscoelastic fluid, (2) arbitrary surface deformation in a weakly non-Newtonian fluid, and (3) small-amplitude deformation in an arbitrarily non-Newtonian fluid. Our final results, valid for a wide-class of {swimmer geometry,} surface kinematics and constitutive models, at most require mathematical knowledge of a series of Newtonian flow problems, and will be useful to quantity the locomotion of biological and synthetic swimmers in complex environments.
Wigner-Eckart theorem for induced symmetries
Energy Technology Data Exchange (ETDEWEB)
Klein, D.J. (Texas A and M University, Galveston (USA). Department of Marine Sciences); Seligman, T.H. (Universidad Nacional Autonoma de Mexico, Mexico City. Inst. de Fisica)
1982-01-01
A unified treatment is given for all group-theoretic problems arising from the evaluation of matrix elements involving operators and states of induced symmetries. To achieve this general treatment two group-theoretic theorems are proven, the first characterizing recoupling coefficients between different symmetry adaptation schemes, and the second making a double coset factorization of a group algebraic matrix basis element. A number of problems previously discussed in the literature, including the conventional Wigner-Eckart theorem and more recent double coset expansions of matrix elements, are realized as special cases in the present treatment. These results entail two new types of recoupling coefficients, namely DC coefficients and 3-symmetry symbols, so that some of their properties are indicated.
Oseledec multiplicative ergodic theorem for laminations
Nguyên, Viêt-Anh
2017-01-01
Given a n-dimensional lamination endowed with a Riemannian metric, the author introduces the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, the author defines the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. He also proves an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere which is holonomy invariant. Moreover, in the case of differentiable cocycles the author establishes effective integral estimates for the Lyapunov exponents. These results find applications in the geometric and dynamical theory of laminations. They are also applicable to (not necessarily closed) laminations with singularities. Interesting holonomy properties of a generic leaf of a foliation are obtained...
Fluctuation theorem for constrained equilibrium systems
Gilbert, Thomas; Dorfman, J. Robert
2006-02-01
We discuss the fluctuation properties of equilibrium chaotic systems with constraints such as isokinetic and Nosé-Hoover thermostats. Although the dynamics of these systems does not typically preserve phase-space volumes, the average phase-space contraction rate vanishes, so that the stationary states are smooth. Nevertheless, finite-time averages of the phase-space contraction rate have nontrivial fluctuations which we show satisfy a simple version of the Gallavotti-Cohen fluctuation theorem, complementary to the usual fluctuation theorem for nonequilibrium stationary states and appropriate to constrained equilibrium states. Moreover, we show that these fluctuations are distributed according to a Gaussian curve for long enough times. Three different systems are considered here: namely, (i) a fluid composed of particles interacting with Lennard-Jones potentials, (ii) a harmonic oscillator with Nosé-Hoover thermostatting, and (iii) a simple hyperbolic two-dimensional map.
Carnot's theorem and Szil\\'ard engine
Shu, Liangsuo; Huang, Suyi; Jin, Shiping
2016-01-01
In this work, the relationship between Carnot engine and Szil\\'ard engine was discussed. By defining the available information about the temperature difference between two heat reservoirs, the Carnot engine was found to have a same physical essence with Szil\\'ard engine: lossless conversion of available information. Thus, a generalized Carnot's theorem for wider scope of application can be described as "all the available information is 100% coded into work".
New electromagnetic memories and soft photon theorems
Mao, Pujian; Ouyang, Hao; Wu, Jun-Bao; Wu, Xiaoning
2017-06-01
In this paper, we present a new type of electromagnetic memory. It is a "magnetic" type, or B mode, radiation memory effect. Rather than a residual velocity, we find a position displacement of a charged particle induced by the B mode radiation with memory. We find two types of electromagnetic displacement (ordinary and null). We also show that the null electromagnetic memory formulas are nothing but a Fourier transformation of soft photon theorems.
Volume Integral Theorem for Exotic Matter
Nandi, Kamal Kanti; Zhang, Yuan-Zhong; Kumar, K. B. Vijaya
2004-01-01
We answer an important question in general relativity about the volume integral theorem for exotic matter by suggesting an exact integral quantifier for matter violating Averaged Null Energy Condition (ANEC). It is checked against some well known static, spherically symmetric traversable wormhole solutions of general relativity with a sign reversed kinetic term minimally coupled scalar field. The improved quantifier is consistent with the principle that traversable wormholes can be supported ...
Theorems for Asymptotic Safety of Gauge Theories
Bond, Andrew D
2016-01-01
We classify the weakly interacting fixed points of general gauge theories coupled to matter and explain how the competition between gauge and matter fluctuations gives rise to a rich spectrum of high- and low-energy fixed points. The pivotal role played by Yukawa couplings is emphasized. Necessary and sufficient conditions for asymptotic safety of gauge theories are also derived, in conjunction with strict no go theorems. Implications for phase diagrams of gauge theories and physics beyond the Standard Model are indicated.
Bezout's theorem and Cohen-Macaulay modules
1999-01-01
We define very proper intersections of modules and projective subschemes. It turns out that equidimensional locally Cohen-Macaulay modules intersect very properly if and only if they intersect properly. We prove a Bezout theorem for modules which meet very properly. Furthermore, we show for equidimensional subschemes $X$ and $Y$: If they intersect properly in an arithmetically Cohen-Macaulay subscheme of positive dimension then $X$ and $Y$ are arithmetically Cohen-Macaulay. The module version...
Hyperplane arrangements and Lefschetz's hyperplane section theorem
Yoshinaga, Masahiko
2005-01-01
The Lefschetz hyperplane section theorem asserts that a complex affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to give an explicit description of attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real numbers. The cells and attaching maps are described in combinatorial terms of chambers. We also discuss the cellular chain complex with coefficient...
Hildebrandt's theorem for the essential spectrum
Directory of Open Access Journals (Sweden)
Janko Bračič
2015-01-01
Full Text Available We prove a variant of Hildebrandt's theorem which asserts that the convex hull of the essential spectrum of an operator \\(A\\ on a complex Hilbert space is equal to the intersection of the essential numerical ranges of operators which are similar to \\(A\\. As a consequence, it is given a necessary and sufficient condition for zero not being in the convex hull of the essential spectrum of \\(A\\.
Simultaneous DOA estimation based on Kolmogorov's theorem
Nájar Martón, Montserrat; Lagunas Hernandez, Miguel A.
1993-01-01
The design of a new architecture for signal processing, based on the Kolmogorov's theorem (1957), is addressed. This architecture is applied to solve the problem of source separation. Particularly, an adaptive algorithm is proposed to separate simultaneously all the unknown impinging sources on an aperture of sensors. The implemented framework is composed of two different stages: the first one is the inhibition stage, which turns the problem of estimating simultaneous DOAs (directions of arri...
Bell's theorem without inequalities and without alignments
Cabello, A
2003-01-01
A proof of Bell's theorem without inequalities is presented which exhibits three remarkable properties: (a) reduced local states are immune to collective decoherence, (b) local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups, and (c) local measurements require only individual measurements on the qubits. Indeed, it is shown that this proof is essentially the only one which fulfils (a), (b), and (c).
On the equivalence theorem for integrable systems
Melikyan, A; Rivelles, V O
2014-01-01
We investigate the equivalence theorem for integrable systems using two formulations of the Alday-Arutyunov-Frolov model. We show that the S-matrix is invariant under the field transformation which reduces the non-linear Dirac brackets of one formulation into the standard commutation relations in the second formulation. We also explain how to perform the direct diagonalization of the transformed Hamiltonian by constructing the states corresponding to self-adjoint extensions.
Resource Adaptive Agents in Interactive Theorem Proving
Benzmueller, Christoph
2009-01-01
We introduce a resource adaptive agent mechanism which supports the user in interactive theorem proving. The mechanism uses a two layered architecture of agent societies to suggest appropriate commands together with possible command argument instantiations. Experiments with this approach show that its effectiveness can be further improved by introducing a resource concept. In this paper we provide an abstract view on the overall mechanism, motivate the necessity of an appropriate resource concept and discuss its realization within the agent architecture.
Reciprocity Theorems for Ab Initio Force Calculations
Wei, C; Mele, E J; Rappe, A M; Lewis, Steven P.; Rappe, Andrew M.
1996-01-01
We present a method for calculating ab initio interatomic forces which scales quadratically with the size of the system and provides a physically transparent representation of the force in terms of the spatial variation of the electronic charge density. The method is based on a reciprocity theorem for evaluating an effective potential acting on a charged ion in the core of each atom. We illustrate the method with calculations for diatomic molecules.
Theorems for asymptotic safety of gauge theories
Energy Technology Data Exchange (ETDEWEB)
Bond, Andrew D.; Litim, Daniel F. [University of Sussex, Department of Physics and Astronomy, Brighton (United Kingdom)
2017-06-15
We classify the weakly interacting fixed points of general gauge theories coupled to matter and explain how the competition between gauge and matter fluctuations gives rise to a rich spectrum of high- and low-energy fixed points. The pivotal role played by Yukawa couplings is emphasised. Necessary and sufficient conditions for asymptotic safety of gauge theories are also derived, in conjunction with strict no go theorems. Implications for phase diagrams of gauge theories and physics beyond the Standard Model are indicated. (orig.)
A Central Limit Theorem for Repeating Patterns
Abrams, Aaron; Landau, Henry; Landau, Zeph; Pommersheim, James
2012-01-01
This note gives a central limit theorem for the length of the longest subsequence of a random permutation which follows some repeating pattern. This includes the case of any fixed pattern of ups and downs which has at least one of each, such as the alternating case considered by Stanley in [2] and Widom in [3]. In every case considered the convergence in the limit of long permutations is to normal with mean and variance linear in the length of the permutations.
Haag's Theorem and Parameterized Quantum Field Theory
Seidewitz, Edwin
2017-01-01
``Haag's theorem is very inconvenient; it means that the interaction picture exists only if there is no interaction''. In traditional quantum field theory (QFT), Haag's theorem states that any field unitarily equivalent to a free field must itself be a free field. But the derivation of the Dyson series perturbation expansion relies on the use of the interaction picture, in which the interacting field is unitarily equivalent to the free field, but which must still account for interactions. So, the usual derivation of the scattering matrix in QFT is mathematically ill defined. Nevertheless, perturbative QFT is currently the only practical approach for addressing realistic scattering, and it has been very successful in making empirical predictions. This success can be understood through an alternative derivation of the Dyson series in a covariant formulation of QFT using an invariant, fifth path parameter in addition to the usual four position parameters. The parameterization provides an additional degree of freedom that allows Haag's Theorem to be avoided, permitting the consistent use of a form of interaction picture in deriving the Dyson expansion. The extra symmetry so introduced is then broken by the choice of an interacting vacuum.
The "Nernst Theorem" and Black Hole Thermodynamics
Wald, R M
1997-01-01
The Nernst formulation of the third law of ordinary thermodynamics (often referred to as the ``Nernst theorem'') asserts that the entropy, $S$, of a system must go to zero (or a ``universal constant'') as its temperature, $T$, goes to zero. This assertion is commonly considered to be a fundamental law of thermodynamics. As such, it seems to spoil the otherwise perfect analogy between the ordinary laws of thermodynamics and the laws of black hole mechanics, since rotating black holes in general relativity do not satisfy the analog of the ``Nernst theorem''. The main purpose of this paper is to attempt to lay to rest the ``Nernst theorem'' as a law of thermodynamics. We consider a boson (or fermion) ideal gas with its total angular momentum, $J$, as an additional state parameter, and we analyze the conditions on the single particle density of states, $g(\\epsilon,j)$, needed for the Nernst formulation of the third law to hold. (Here, $\\epsilon$ and $j$ denote the single particle energy and angular momentum.) Alt...
Four theorems on the psychometric function.
Directory of Open Access Journals (Sweden)
Keith A May
Full Text Available In a 2-alternative forced-choice (2AFC discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, Δx. This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull "slope" parameter, β, can be approximated by β(Noise x β(Transducer, where β(Noise is the β of the Weibull function that fits best to the cumulative noise distribution, and β(Transducer depends on the transducer. We derive general expressions for β(Noise and β(Transducer, from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when d' ∝ (Δx(b, β ≈ β(Noise x b. We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4-0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull β reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of β for contrast discrimination suggests that, if internal noise is
Optical applications of von Neumann's alternating-projection theorem.
Montgomery, W D
1982-01-01
The previous work of Gerchberg [Opt. Acta 21, 709 (1974)], Papoulis [IEEE Trans. Circuits Syst. CAS-22, 735 (1975)], Youla [IEEE Trans. Circuits Syst. CAS-25, 694 (1978)], and Stark et al. [J. Opt. Soc. Am. 71, 735 (1981)] on image restoration is presented in a generalized setting. New examples using three projections are given. The method is used to solve the general problem of diffraction by a perfectly conducting plane screen of arbitrary configuration.
The implicit function theorem history, theory, and applications
Krantz, Steven G
2003-01-01
The implicit function theorem is part of the bedrock of mathematics analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for Ck functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth function, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash-Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume. The history of the implicit function theorem is a lively and complex store, and intimately bound up with the development of fundamental ideas in a...
Fluctuation theorem for out-of-time-ordered correlator
Halpern, Nicole Yunger
2016-01-01
The out-of-time-ordered correlator (OTOC) diagnoses quantum chaos and the scrambling of quantum information via the spread of entanglement. The OTOC encodes forward and reverse evolutions and has deep connections with the flow of time. So do fluctuation theorems such as Jarzynski's Equality, derived in nonequilibrium statistical mechanics. I unite these two powerful, seemingly disparate tools by deriving a fluctuation theorem for the OTOC. The fluctuation theorem is analogous to Jarzynski's Equality. The theorem's left-hand side equals the OTOC. The right-hand side implies a platform-nonspecific protocol for experimentally measuring the OTOC in an indirect manner fundamentally different from existing proposals. Time evolution need not be reversed in any trial. The theorem opens holography, condensed matter, and quantum information to new insights from fluctuation theorems and vice versa.
On the linearization theorem for proper Lie groupoids
Crainic, Marius
2011-01-01
We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual interpretation: as a manifestation of Morita invariance. We also clarify the precise conditions needed for the theorem to hold (which often have been misstated in the literature).
Logic for computer science foundations of automatic theorem proving
Gallier, Jean H
2015-01-01
This advanced text for undergraduate and graduate students introduces mathematical logic with an emphasis on proof theory and procedures for algorithmic construction of formal proofs. The self-contained treatment is also useful for computer scientists and mathematically inclined readers interested in the formalization of proofs and basics of automatic theorem proving. Topics include propositional logic and its resolution, first-order logic, Gentzen's cut elimination theorem and applications, and Gentzen's sharpened Hauptsatz and Herbrand's theorem. Additional subjects include resolution in fir
Inconsistency of Carnot's theorem's proof by R. Clausius
Ihnatovych, V
2013-01-01
R. Clausius proved Carnot's theorem basing on postulate "Heat cannot, of itself, pass from a colder to a hotter body". Alexander Gukhman demonstrated that Carnot's theorem can be proved based on the postulate "Heat cannot, of itself, pass from a hotter to a colder body". He concluded that Carnot's theorem does not follow from Clausius' postulate. The following paper gives a detailed justification of Gukhman's derivation.
Virial Theorem for a Class of Quantum Nonlinear Harmonic Oscillators
Institute of Scientific and Technical Information of China (English)
王雪红; 郭军义; 李艳
2012-01-01
In this paper,the Virial Theorem based on a class of quantum nonlinear harmonic oscillators is presented.This relationship has to do with parameter λ and ?/?λ,where the λ is a real number.When λ=0,the nonlinear harmonic oscillator naturally reduces to the usual quantum linear harmonic oscillator,and the Virial Theorem also reduces to the usual Virial Theorem.
A Converse of the Mean Value Theorem Made Easy
Mortici, Cristinel
2011-01-01
The aim of this article is to discuss some results about the converse mean value theorem stated by Tong and Braza [J. Tong and P. Braza, "A converse of the mean value theorem", Amer. Math. Monthly 104(10), (1997), pp. 939-942] and Almeida [R. Almeida, "An elementary proof of a converse mean-value theorem", Internat. J. Math. Ed. Sci. Tech. 39(8)…
Goedel incompleteness theorems and the limits of their applicability. I
Energy Technology Data Exchange (ETDEWEB)
Beklemishev, Lev D [Steklov Mathematical Institute, Russian Academy of Sciences, Moscow (Russian Federation)
2011-01-25
This is a survey of results related to the Goedel incompleteness theorems and the limits of their applicability. The first part of the paper discusses Goedel's own formulations along with modern strengthenings of the first incompleteness theorem. Various forms and proofs of this theorem are compared. Incompleteness results related to algorithmic problems and mathematically natural examples of unprovable statements are discussed. Bibliography: 68 titles.
Stochastic thermodynamics, fluctuation theorems and molecular machines.
Seifert, Udo
2012-12-01
Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.
Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem
Directory of Open Access Journals (Sweden)
Juliana Bueno-Soler
2016-09-01
Full Text Available This paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs. We show that LFIs encode very naturally an extension of the notion of probability able to express sophisticated probabilistic reasoning under contradictions employing appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes’ theorem for conditionalization. We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Some critical historical and conceptual points about probability theory are also reviewed.
Number systems and the Chinese Remainder Theorem
van de Woestijne, Christiaan E
2011-01-01
A well-known generalisation of positional numeration systems is the case where the base is the residue class of $x$ modulo a given polynomial $f(x)$ with coefficients in (for example) the integers, and where we try to construct finite expansions for all residue classes modulo $f(x)$, using a suitably chosen digit set. We give precise conditions under which direct or fibred products of two such polynomial number systems are again of the same form. The main tool is a general form of the Chinese Remainder Theorem. We give applications to simultaneous number systems in the integers.
APPROXIMATE SAMPLING THEOREM FOR BIVARIATE CONTINUOUS FUNCTION
Institute of Scientific and Technical Information of China (English)
杨守志; 程正兴; 唐远炎
2003-01-01
An approximate solution of the refinement equation was given by its mask, and the approximate sampling theorem for bivariate continuous function was proved by applying the approximate solution. The approximate sampling function defined uniquely by the mask of the refinement equation is the approximate solution of the equation, a piece-wise linear function, and posseses an explicit computation formula. Therefore the mask of the refinement equation is selected according to one' s requirement, so that one may controll the decay speed of the approximate sampling function.
Fixed point theorems in spaces and -trees
Directory of Open Access Journals (Sweden)
Kirk WA
2004-01-01
Full Text Available We show that if is a bounded open set in a complete space , and if is nonexpansive, then always has a fixed point if there exists such that for all . It is also shown that if is a geodesically bounded closed convex subset of a complete -tree with , and if is a continuous mapping for which for some and all , then has a fixed point. It is also noted that a geodesically bounded complete -tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory.
A singularity theorem based on spatial averages
Indian Academy of Sciences (India)
J M M Senovilla
2007-07-01
Inspired by Raychaudhuri's work, and using the equation named after him as a basic ingredient, a new singularity theorem is proved. Open non-rotating Universes, expanding everywhere with a non-vanishing spatial average of the matter variables, show severe geodesic incompletness in the past. Another way of stating the result is that, under the same conditions, any singularity-free model must have a vanishing spatial average of the energy density (and other physical variables). This is very satisfactory and provides a clear decisive difference between singular and non-singular cosmologies.
Adiabatic theorems for generators of contracting evolutions
Avron, J E; Graf, G M; Grech, P
2011-01-01
We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics in the manifold of instantaneous stationary states and transversal to it have distinct characteristics: The former is irreversible and the latter is transient in a sense that we explain. Both the gapped and gapless cases are considered. Some applications are discussed.
Nekhoroshev theorem for the periodic Toda lattice.
Henrici, Andreas; Kappeler, Thomas
2009-09-01
The periodic Toda lattice with N sites is globally symplectomorphic to a two parameter family of N-1 coupled harmonic oscillators. The action variables fill out the whole positive quadrant of R(N-1). We prove that in the interior of the positive quadrant as well as in a neighborhood of the origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev's theorem applies on (almost) all parts of phase space (2000 Mathematics Subject Classification: 37J35, 37J40, 70H06).
Central limit theorems under special relativity.
McKeague, Ian W
2015-04-01
Several relativistic extensions of the Maxwell-Boltzmann distribution have been proposed, but they do not explain observed lognormal tail-behavior in the flux distribution of various astrophysical sources. Motivated by this question, extensions of classical central limit theorems are developed under the conditions of special relativity. The results are related to CLTs on locally compact Lie groups developed by Wehn, Stroock and Varadhan, but in this special case the asymptotic distribution has an explicit form that is readily seen to exhibit lognormal tail behavior.
Gerlach, Moritz
2011-01-01
We prove that every bounded, positive, irreducible, stochastically continuous semigroup on the space of bounded, measurable functions which is strong Feller, consists of kernel operators and possesses an invariant measure converges pointwise. This differs from Doob's theorem in that we do not require the semigroup to be Markovian and request a fairly weak kind of irreducibility. In addition, we elaborate on the various notions of kernel operators in this context, show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domains. The proofs are based on the theory of positive semigroups and do not use probability theory.
On Clifford's theorem for singular curves
Franciosi, Marco
2011-01-01
Let C be a 2-connected Gorenstein curve either reduced or contained in a smooth algebraic surface and let S be a subcanonical cluster (i.e. a 0-dim scheme such that the space H^0(C, I_S K_C) contains a generically invertible section). Under some general assumptions on S or C we show that h^0(C, I_S K_C) <= p_a(C) - deg (S)/2 and if equality holds then either S is trivial, or C is honestly hyperelliptic or 3-disconnected. As a corollary we give a generalization of Clifford's theorem for reduced curves.
Godel's Incompleteness Theorems and Platonic Metaphysics
Mikovic, Aleksandar
2015-01-01
We argue by using Godel's incompletness theorems in logic that platonism is the best metaphysics for science. This is based on the fact that a natural law in a platonic metaphysics represents a timeless order in the motion of matter, while a natural law in a materialistic metaphysics can be only defined as a temporary order which appears at random in the chaotic motion of matter. Although a logical possibility, one can argue that this type of metaphysics is highly implausible. Given that mathematics fits naturally within platonism, we conclude that a platonic metaphysics is more preferable than a materialistic metaphysics.
Jackson's Theorem on Bounded Symmetric Domains
Institute of Scientific and Technical Information of China (English)
Ming Zhi WANG; Guang Bin REN
2007-01-01
Polynomial approximation is studied on bounded symmetric domain Ω in C n for holo-morphic function spaces X ,such as Bloch-type spaces,Bergman-type spaces,Hardy spaces,Ω algebra and Lipschitz space.We extend the classical Jackson ’s theorem to several complex variables:E k f,X ) ω (1 /k,f,X ),where E k f,X )is the deviation of the best approximation of f ∈X by polynomials of degree at mostk with respect to the X -metric and ω (1/k,f,X )is the corresponding modulus of continuity.
Stone's representation theorem in fuzzy topology
Institute of Scientific and Technical Information of China (English)
LIU; Yingming(刘应明); ZHANG; Dexue(张德学)
2003-01-01
In this paper, a complete solution to the problem of Stone's repesentation theorem in fuzzy topology is given for a class of completely distributive lattices. Precisely, it is proved that if L is a frame such that 0 ∈ L is a prime or 1 ∈ L is a coprime, then the category of distributive lattices is dually equivalent to the category of coherent L-locales and that if L is moreover completely distributive, then the category of distributive lattices is dually equivalent to the category of coherent stratified L-topological spaces.
Lampart, Jonas; Lewin, Mathieu
2015-12-01
We prove a generalized version of the RAGE theorem for N-body quantum systems. The result states that only bound states of systems with {0 ≤slant n ≤slant N} particles persist in the long time average. The limit is formulated by means of an appropriate weak topology for many-body systems, which was introduced by the second author in a previous work, and is based on reduced density matrices. This topology is connected to the weak-* topology of states on the algebras of canonical commutation or anti-commutation relations, and we give a formulation of our main result in this setting.
Lifespan theorem for constrained surface diffusion flows
McCoy, James; Williams, Graham; 10.1007/s00209-010-0720-7
2012-01-01
We consider closed immersed hypersurfaces in $\\R^{3}$ and $\\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for which a smooth solution exists, and a small upper bound on a power of the total curvature during this time. By phrasing the theorem in terms of the concentration of curvature in the initial surface, our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows.
Theorem proving support in programming language semantics
Bertot, Yves
2007-01-01
We describe several views of the semantics of a simple programming language as formal documents in the calculus of inductive constructions that can be verified by the Coq proof system. Covered aspects are natural semantics, denotational semantics, axiomatic semantics, and abstract interpretation. Descriptions as recursive functions are also provided whenever suitable, thus yielding a a verification condition generator and a static analyser that can be run inside the theorem prover for use in reflective proofs. Extraction of an interpreter from the denotational semantics is also described. All different aspects are formally proved sound with respect to the natural semantics specification.
Generalizations of the Nash Equilibrium Theorem in the KKM Theory
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Park Sehie
2010-01-01
Full Text Available The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases for -convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.
Generalizations of the Nash Equilibrium Theorem in the KKM Theory
Directory of Open Access Journals (Sweden)
Sehie Park
2010-01-01
Full Text Available The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases for G-convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.
A Formal Proof Of The Riesz Representation Theorem
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Anthony Narkawicz
2011-01-01
Full Text Available This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation. This result follows from the equivalence of the Riemann Stieltjes and Darboux Stieltjes integrals, which would have been a lengthy result to prove in PVS, so a simpler lemma was proved that captures the underlying concept of this integral equivalence. In order to prove the Riesz theorem, the Hahn Banach theorem was proved in the case where the normed linear spaces are the continuous and bounded functions on a closed interval. The proof of the Riesz theorem follows the proof in Haaser and Sullivan's book Real Analysis. The formal proof of this result in PVS revealed an error in textbook's proof. Indeed, the proof of the Riesz representation theorem is constructive, and the function constructed in the textbook does not satisfy a key property. This error illustrates the ability of formal verification to find logical errors. A specific counterexample is given to the proof in the textbook. Finally, a corrected proof of the Riesz representation theorem is presented.
Fluctuation-dissipation theorem and quantum tunneling with dissipation
Fujikawa, K
1998-01-01
We suggest to take the fluctuation-dissipation theorem of Callen and Welton as a basis to study quantum dissipative phenomena (such as macroscopic quantum tunneling) in a manner analogous to the Nambu-Goldstone theorem for spontaneous symmetry breakdown. It is shown that the essential physical contents of the Caldeira-Leggett model such as the suppression of quantum coherence by Ohmic dissipation are derived from general principles only, namely, the fluctuation-dissipation theorem and unitarity and causality (i.e., dispersion relations), without referring to an explicit form of the Lagrangian. An interesting connection between quantum tunneling with Ohmic dissipation and the Anderson's orthogonality theorem is also noted.
Fluctuation theorem in dynamical systems with quenched disorder
Drocco, Jeffrey; Olson Reichhardt, Cynthia; Reichhardt, Charles
2010-03-01
We demonstrate that the fluctuation theorem of Gallavotti and Cohen can be used to characterize far from equilibrium dynamical nonthermal systems in the presence of quenched disorder where strong fluctuations or crackling noise occur. By observing the frequency of entropy-destroying trajectories, we show that the theorem holds in specific dynamical regimes near the threshold for motion, indicating that these systems might be ideal candidates for understanding what types of nonthermal fluctuations could be used in constructing generalized fluctuation theorems. We also discuss how the theorem could be tested with global or local probes in systems such as superconducting vortices, magnetic domain walls, stripe phases, Coulomb glasses and earthquake models.
New theorem relating two-mode entangled tomography to two-mode Fresnel operator
Institute of Scientific and Technical Information of China (English)
Xie Chuan-Mei; Fan Hong-Yi
2012-01-01
Based on the Fan-Hu's formalism,i.e.,the tomogram of two-mode quantum states can be considered as the module square of the states' wave function in the intermediate representation,which is just the eigenvector of the Fresnel quadrature phase,we derive a new theorem for calculating the quantum tomogram of two-mode density operators,i.e.,the tomogram of a two-mode density operator is equal to the marginal integration of the classical Weyl correspondence function of F+2pF2,where F2 is the two-mode Fresnel operator. An application of the theorem in evaluating the tomogram of an optical chaotic field is also presented.
Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes
Woolgar, Eric; Wylie, William
2016-02-01
We study Lorentzian manifolds with a weight function such that the N-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the "pure Bakry-Émery" N = ∞ case with f uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite N-values N ∈ (n, ∞) and N ∈ (-∞, 1]. In the N ∈ (n, ∞) case, no bound on f is required, while for N ∈ (-∞, 1] and N = ∞, we are able to replace the boundedness of f by a weaker condition on the integral of f along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when N = 1, the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in a prior work on the N-Bakry-Émery curvature in Riemannian signature when N = 1 and appears to be a general feature.
Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes
Energy Technology Data Exchange (ETDEWEB)
Woolgar, Eric, E-mail: ewoolgar@ualberta.ca [Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1 (Canada); Wylie, William, E-mail: wwylie@syr.edu [215 Carnegie Building, Department of Mathematics, Syracuse University, Syracuse, New York 13244 (United States)
2016-02-15
We study Lorentzian manifolds with a weight function such that the N-Bakry-Émery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the “pure Bakry-Émery” N = ∞ case with f uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite N-values N ∈ (n, ∞) and N ∈ (−∞, 1]. In the N ∈ (n, ∞) case, no bound on f is required, while for N ∈ (−∞, 1] and N = ∞, we are able to replace the boundedness of f by a weaker condition on the integral of f along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when N = 1, the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in a prior work on the N-Bakry-Émery curvature in Riemannian signature when N = 1 and appears to be a general feature.
Directory of Open Access Journals (Sweden)
Ronghua He
2012-04-01
Full Text Available Let $I$ be any index set. By using some existence theorems of maximal elements for a family of set-valued mappings involving a better admissible set-valued mapping under noncompact setting of $FC$-spaces, we first present some nonempty intersection theorems for a family $\\{G_{i}\\}_{i\\in I}$ in a product space of $FC$-spaces. Next we give a coincidence theorem and a Fan-Browder type fixed point theorem. Finally, as applications, some equilibrium existence theorems for a system of generalized vector equilibrium problems are proved in product $FC$-space, some existence theorems of solutions for a system of Ky Fan type minimax inequalities involving a family of $G_{\\cal B}$-majorized mappings defined on the product space of $FC$-space are also obtained. Our results improve and generalize some recent results.
On the inversion of Fueter's theorem
Dong, Baohua; Kou, Kit Ian; Qian, Tao; Sabadini, Irene
2016-10-01
The well known Fueter theorem allows to construct quaternionic regular functions or monogenic functions with values in a Clifford algebra defined on open sets of Euclidean space R n + 1, starting from a holomorphic function in one complex variable or, more in general, from a slice hyperholomorphic function. Recently, the inversion of this theorem has been obtained for odd values of the dimension n. The present work extends the result to all dimensions n by using the Fourier multiplier method. More precisely, we show that for any axially monogenic function f defined in a suitable open set in R n + 1, where n is a positive integer, we can find a slice hyperholomorphic function f → such that f =Δ (n - 1) / 2 f →. Both the even and the odd dimensions are treated with the same, viz., the Fourier multiplier, method. For the odd dimensional cases the result obtained by the Fourier multiplier method coincides with the existing result obtained through the pointwise differential method.
Limit theorems for functions of marginal quantiles
Babu, G Jogesh; Choi, Kwok Pui; Mangalam, Vasudevan; 10.3150/10-BEJ287
2011-01-01
Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur's representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that \\[\\sqrt{n}\\Biggl(\\frac{1}{n}\\sum_{i=1}^n\\phi\\bigl(X_{n:i}^{(1)},...,X_{n:i}^{(d)}\\bigr)-\\bar{\\gamma}\\Biggr)=\\frac{1}{\\sqrt{n}}\\sum_{i=1}^nZ_{n,i}+\\mathrm{o}_P(1)\\] as $n\\rightarrow\\infty$, where $\\bar{\\gamma}$ is a constant and $Z_{n,i}$ are i.i.d. random variables for each $n$. This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.
Birth of a theorem a mathematical adventure
Villani, Cédric
2015-01-01
This man could plainly do for mathematics what Brian Cox has done for physics" (Sunday Times). What goes on inside the mind of a rock-star mathematician? Where does inspiration come from? With a storyteller's gift, Cedric Villani takes us on a mesmerising journey as he wrestles with a new theorem that will win him the most coveted prize in mathematics. Along the way he encounters obstacles and setbacks, losses of faith and even brushes with madness. His story is one of courage and partnership, doubt and anxiety, elation and despair. We discover how it feels to be obsessed by a theorem during your child's cello practise and throughout your dreams, why appreciating maths is a bit like watching an episode of Columbo, and how sometimes inspiration only comes from locking yourself away in a dark room to think. Blending science with history, biography with myth, Villani conjures up an inimitable cast of characters including the omnipresent Einstein, mad genius Kurt Godel, and Villani's personal hero, John Nash. Bir...
De Finetti Theorem on the CAR Algebra
Crismale, Vitonofrio; Fidaleo, Francesco
2012-10-01
The symmetric states on a quasi local C*-algebra on the infinite set of indices J are those invariant under the action of the group of the permutations moving only a finite, but arbitrary, number of elements of J. The celebrated De Finetti Theorem describes the structure of the symmetric states (i.e. exchangeable probability measures) in classical probability. In the present paper we extend the De Finetti Theorem to the case of the CAR algebra, that is for physical systems describing Fermions. Namely, after showing that a symmetric state is automatically even under the natural action of the parity automorphism, we prove that the compact convex set of such states is a Choquet simplex, whose extremal (i.e. ergodic w.r.t. the action of the group of permutations previously described) are precisely the product states in the sense of Araki-Moriya. In order to do that, we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self-containing interest.
De Finetti theorem on the CAR algebra
Crismale, Vito
2012-01-01
The symmetric states on a quasi local C*-algebra on the infinite set of indices J are those invariant under the action of the group of the permutations moving only a finite, but arbitrary, number of elements of J. The celebrated De Finetti Theorem describes the structure of the symmetric states (i.e. exchangeable probability measures) in classical probability. In the present paper we extend De Finetti Theorem to the case of the CAR algebra, that is for physical systems describing Fermions. Namely, after showing that a symmetric state is automatically even under the natural action of the parity automorphism, we prove that the compact convex set of such states is a Choquet simplex, whose extremal (i.e. ergodic w.r.t. the action of the group of permutations previously described) are precisely the product states in the sense of Araki-Moriya. In order to do that, we also prove some ergodic properties naturally enjoyed by the symmetric states which have a self--containing interest.
Bell's theorem, inference, and quantum transactions
Garrett, A. J. M.
1990-04-01
Bell's theorem is expounded as an analysis in Bayesian inference. Assuming the result of a spin measurement on a particle is governed by a causal variable internal (hidden, “local”) to the particle, one learns about it by making a spin measurement; thence about the internal variable of a second particle correlated with the first; and from there predicts the probabilistic result of spin measurements on the second particle. Such predictions are violated by experiment: locality/causality fails. The statistical nature of the observations rules out signalling; acausal, superluminal, or otherwise. Quantum mechanics is irrelevant to this reasoning, although its correct predictions of experiment imply that it has a nonlocal/acausal interpretation. Cramer's new transactional interpretation, which incorporates this feature by adapting the Wheeler-Feynman idea of advanced and retarded processes to the quantum laws, is advocated. It leads to an invaluable way of envisaging quantum processes. The usual paradoxes melt before this, and one, the “delayed choice” experiment, is chosen for detailed inspection. Nonlocality implies practical difficulties in influencing hidden variables, which provides a very plausible explanation for why they have not yet been found; from this standpoint, Bell's theorem reinforces arguments in favor of hidden variables.
Randomized central limit theorems: A unified theory
Eliazar, Iddo; Klafter, Joseph
2010-08-01
The central limit theorems (CLTs) characterize the macroscopic statistical behavior of large ensembles of independent and identically distributed random variables. The CLTs assert that the universal probability laws governing ensembles’ aggregate statistics are either Gaussian or Lévy, and that the universal probability laws governing ensembles’ extreme statistics are Fréchet, Weibull, or Gumbel. The scaling schemes underlying the CLTs are deterministic—scaling all ensemble components by a common deterministic scale. However, there are “random environment” settings in which the underlying scaling schemes are stochastic—scaling the ensemble components by different random scales. Examples of such settings include Holtsmark’s law for gravitational fields and the Stretched Exponential law for relaxation times. In this paper we establish a unified theory of randomized central limit theorems (RCLTs)—in which the deterministic CLT scaling schemes are replaced with stochastic scaling schemes—and present “randomized counterparts” to the classic CLTs. The RCLT scaling schemes are shown to be governed by Poisson processes with power-law statistics, and the RCLTs are shown to universally yield the Lévy, Fréchet, and Weibull probability laws.
A vector bundle proof of Poncelet theorem
Vallès, Jean
2012-01-01
In the town of Saratov where he was prisonner, Poncelet, continuing the work of Euler and Steiner on polygons simultaneously inscribed in a circle and circumscribed around an other circle, proved the following generalization : "Let C and D be two smooth conics in the projective complex plane. If D passes through the n(n-1)/2 vertices of a complete polygon with n sides tangent to C then D passes through the vertices of infinitely many such polygons." According to Marcel Berger this theorem is the nicest result about the geometry of conics. Even if it is, there are few proofs of it. To my knowledge there are only three. The first proof, published in 1822 and based on infinitesimal deformations, is due to Poncelet. Later, Jacobi proposed a new proof based on finite order points on elliptic curves; his proof, certainly the most famous, is explained in a modern way and in detail by Griffiths and Harris. In 1870 Weyr proved a Poncelet theorem in space (more precisely for two quadrics) that implies the one above whe...
A note on the weighted Khintchine-Groshev Theorem
DEFF Research Database (Denmark)
Hussain, Mumtaz; Yusupova, Tatiana
Let W(m,n;ψ−−) denote the set of ψ1,…,ψn-approximable points in Rmn. The classical Khintchine-Groshev theorem assumes a monotonicity condition on the approximating functions ψ−−. Removing monotonicity from the Khintchine-Groshev theorem is attributed to different authors for different cases of m...
Caristi Fixed Point Theorem in Metric Spaces with a Graph
Directory of Open Access Journals (Sweden)
M. R. Alfuraidan
2014-01-01
Full Text Available We discuss Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi’s fixed point theorem. It extends some recent works on the extension of Banach contraction principle to metric spaces with graph.
Linear Strategy for Boolean Ring Based Theorem Proving
Institute of Scientific and Technical Information of China (English)
WU Jinzhao; LIU Zhuojun
2000-01-01
Two inference rules are discussed in boolean ring based theorem proving, and linear strategy is developed. It is shown that both of them are complete for linear strategy. Moreover, by introducing a partial ordering on atoms, pseudo O-linear and O-linear strategies are presented. The former is complete, the latter, however, is complete for clausal theorem proving.
The Boundary Crossing Theorem and the Maximal Stability Interval
Directory of Open Access Journals (Sweden)
Jorge-Antonio López-Renteria
2011-01-01
useful tools in the study of the stability of family of polynomials. Although both of these theorem seem intuitively obvious, they can be used for proving important results. In this paper, we give generalizations of these two theorems and we apply such generalizations for finding the maximal stability interval.
On Euler's Theorem for Homogeneous Functions and Proofs Thereof.
Tykodi, R. J.
1982-01-01
Euler's theorem for homogenous functions is useful when developing thermodynamic distinction between extensive and intensive variables of state and when deriving the Gibbs-Duhem relation. Discusses Euler's theorem and thermodynamic applications. Includes six-step instructional strategy for introducing the material to students. (Author/JN)
COMPARISON THEOREM OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper is devoted to deriving a comparison theorem of solutions to backward doubly stochastic differential equations driven by Brownian motion and backward It-Kunita integral. By the application of this theorem, we give an existence result of the solutions to these equations with continuous coefficients.
Mann Iteration of Weak Convergence Theorems in Banach Space
Institute of Scientific and Technical Information of China (English)
Liang-gen Hu; Jin-ping Wang
2009-01-01
In this paper, by using Mann's iteration process we will establish several weak convergence theorems for approximating a fixed point of κ-strictly pseudocontractive mappings with respect to p in p-uniformly convex Banach spaces. Our results answer partially the open question proposed by Marino and Xu, and extend Reich's theorem from nonexpausive mappings to κ-strict pseudocontractive mappings.
The Non-countable Summation Type Hahn-Schur Theorems
Institute of Scientific and Technical Information of China (English)
LUETong-fu; HUJian-hua; CHOMin-hyung
2005-01-01
The classical countable summation type Hahn-Schur theorem is a famous result in summation theory and measure theory. An interesting problem is whether the theorem can be generalized to non-countable summation case? In this paper, we show that the answer is true.
AN EXISTENCE THEOREM FOR MAXIMAL ELEMENTS AND ITS APPLICATIONS
Institute of Scientific and Technical Information of China (English)
Hou Jicheng; He Wei
2007-01-01
An existence theorem of maximal elements for an L*-majorized correspondence de-fined on a non-paracompact H-space is established. As applications of the result, an equilibrium existence theorem for a non-paracompact generalized game in H-spaces with infinitely many players and with L*-majorized correspondences is given.
A parametrised version of Moser's modifying terms theorem
Wagener, F.
2009-01-01
A sharpened version of Moser's `modifying terms' KAM theorem is derived, and it is shown how this theorem can be used to investigate the persistence of invariant tori in general situations, including those where some of the Floquet exponents of the invariant torus may vanish. The result is `structur
A parametrised version of Moser's modifying terms theorem
Wagener, F.
2010-01-01
A sharpened version of Moser's 'modifying terms' KAM theorem is derived, and it is shown how this theorem can be used to investigate the persistence of invariant tori in general situations, including those where some of the Floquet exponents of the invariant torus may vanish. The result is 'structur
Noether's theorem of a rotational relativistic variable mass system
Institute of Scientific and Technical Information of China (English)
方建会; 赵嵩卿
2002-01-01
Noether's theory of a rotational relativistic variable mass system is studied. Firstly, Jourdain's principle of therotational relativistic variable mass system is given. Secondly, on the basis of the invariance of the Jourdain's principleunder the infinitesimal transformations of groups, Noether's theorem and its inverse theorem of the rotational relativisticvariable mass system are presented. Finally, an example is given to illustrate the application of the result.
N(o)ther-type theorem of piecewise algebraic curves
Institute of Scientific and Technical Information of China (English)
WANG Renhong; ZHU Chungang
2004-01-01
The piecewise algebraic curve is a generalization of the classical algebraic curve.This paper describes the improvement of the Nother-type theorem of piecewise algebraic curves on the star region.Moreover,the Nother-type theorem of piecewise algebraic curves on the cross-cut partition is discussed.
Sufficient Condition for Validity of Quantum Adiabatic Theorem
Institute of Scientific and Technical Information of China (English)
TAO Yong
2012-01-01
In this paper, we attempt to give a sufficient condition of guaranteeing the validity of the proof of the quantum adiabatic theorem. The new sufficient condition can clearly remove the inconsistency and the counterexample of the quantum adiabatic theorem pointed out by Marzlin and Sanders [Phys. Rev. Lett. 93 （2004） 160408].
Discovering Theorems in Abstract Algebra Using the Software "GAP"
Blyth, Russell D.; Rainbolt, Julianne G.
2010-01-01
A traditional abstract algebra course typically consists of the professor stating and then proving a sequence of theorems. As an alternative to this classical structure, the students could be expected to discover some of the theorems even before they are motivated by classroom examples. This can be done by using a software system to explore a…
A simple proof of Renner's exponential de Finetti theorem
Vidick, Thomas; Yuen, Henry
2016-01-01
We give a simple proof of the exponential de Finetti theorem due to Renner. Like Renner's proof, ours combines the post-selection de Finetti theorem, the Gentle Measurement lemma, and the Chernoff bound, but avoids virtually all calculations, including any use of the theory of types.
A q-analogue of the four functions theorem
Christofides, Demetres
2009-01-01
In this article we give a proof of a q-analogue of the celebrated four functions theorem. This analogue was conjectured by Bjorner and includes as special cases both the four functions theorem and also Bjorner's q-analogue of the FKG inequality.
Generalizations of Karp's theorem to elastic scattering theory
Tuong, Ha-Duong
Karp's theorem states that if the far field pattern corresponding to the scattering of a time-harmonic acoustic plane wave by a sound-soft obstacle in R2 is invariant under the group of rotations, then the scatterer is a circle. The theorem is generalized to the elastic scattering problems and the axisymmetric scatterers in R3.
Systematic Approaches to Experimentation: The Case of Pick's Theorem
Papadopoulos, Ioannis; Iatridou, Maria
2010-01-01
In this paper two 10th graders having an accumulated experience on problem-solving ancillary to the concept of area confronted the task to find Pick's formula for a lattice polygon's area. The formula was omitted from the theorem in order for the students to read the theorem as a problem to be solved. Their working is examined and emphasis is…
Yan Theorem in L∞ with Applications to Asset Pricing
Institute of Scientific and Technical Information of China (English)
2007-01-01
We prove an L∞ version of the Yan theorem and deduce from it a necessary condition for the absence of free lunches in a model of financial markets, in which asset prices are a continuous Rd valued process and only simple investment strategies are admissible. Our proof is based on a new separation theorem for convex sets of finitely additive measures.
Leaning on Socrates to Derive the Pythagorean Theorem
Percy, Andrew; Carr, Alistair
2010-01-01
The one theorem just about every student remembers from school is the theorem about the side lengths of a right angled triangle which Euclid attributed to Pythagoras when writing Proposition 47 of "The Elements". Usually first met in middle school, the student will be continually exposed throughout their mathematical education to the formula b2 +…
An ideal topology type convergent theorem on scale effect algebras
Institute of Scientific and Technical Information of China (English)
WU JunDe; ZHOU XuanChang; Minhyung CHO
2007-01-01
The famous Antosik-Mikusinski convergent theorem on the Abel topological groups has very extensive applications in measure theory, summation theory and other analysis fields. In this paper, we establish the theorem on a class of effect algebras equipped with the ideal topology. This paper shows also that the ideal topology of effect algebras is a useful topology in studying the quantum logic theory.
FIXED POINTS THEOREMS IN MULTI-METRIC SPACES
Directory of Open Access Journals (Sweden)
Laurentiu I. Calmutchi
2011-07-01
Full Text Available The aim of the present article is to give some general methods inthe fixed point theory for mappings of general topological spaces. Using the notions of the multi-metric space and of the E-metric space, we proved the analogous of several classical theorems: Banach fixed point principle, Theorems of Edelstein, Meyers, Janos etc.
The Structure Theorem for Complete Intersections of Grade 4
Institute of Scientific and Technical Information of China (English)
Oh-Jin Kang; Hyoung J. Ko
2005-01-01
Serre showed that a Gorenstein ideal of grade 2 is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for Gorenstein ideals of grade 3. It is found that a certain complete matrix defines a perfect ideal K3(f).As an application,we present a structure theorem for complete intersections of grade 4.
THE KILLING FORMS AND DECOMPOSITION THEOREMS OF LIE SUPERTRIPLE SYSTEMS
Institute of Scientific and Technical Information of China (English)
Zhang Zhixue; Jia Peipei
2009-01-01
In this article, the Killing form of a Lie supertriple system (LSTS) and that of its imbedding Lie superalgebra (LSA) are investigated, and a unique decomposition theorem for a quasiclassical LSTS with trivial center is established by means of the parallel decomposition theorem for a quasiclassical LSA.
Computer Algebra Systems and Theorems on Real Roots of Polynomials
Aidoo, Anthony Y.; Manthey, Joseph L.; Ward, Kim Y.
2010-01-01
A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems. (Contains 1 figure.)
A Hamiltonian KAM theorem for bundles of Lagrangean tori
Broer, HW; Cushman, RH; Fasso, F; Dumortier, F; Broer, H; Mawhin, J; Vanderbauwhede, A; Lunel, SV
2005-01-01
The classical KAM theorem deals with Lagrangean invariant tori in nearly integrable Hamiltonian systems. The stability formulation of the KAM theorem states that, when restricting to a large measure Diophantine "Cantor set" of such tori, the integrable approximation is smoothly conjugate to the near
The Anosov theorem for flat generalized Hantzsche-Wendt manifolds
Dekimpe, K.; De Rock, B.; Malfait, W.
2004-10-01
In this paper we prove that N( f)=| L( f)| for any continuous map f on a given orientable flat generalized Hantzsche-Wendt manifold. This is the analogue of a theorem of Anosov for continuous maps on nilmanifolds. We also show that the theorem always fails in the non-orientable case.
Generalization of Bombieri’s theorem and its applications
Institute of Scientific and Technical Information of China (English)
林家发; 展涛
1995-01-01
By using the so-called Hooley-Huxley Contour and zero density estimates for Dirichlet L-function, Bombieri’s theorem is established for a class of arithmetic functions whose generating functions satisfy certain analytic conditions. As applications of our theorem, the mean value estimates of L-functions and the distribution of integers representable as sums of two squares are discussed.
An Algebraic Proof of Quillen's Resolution Theorem for K_1
Whale, Ben
2009-01-01
In his 1973 paper Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of a paper by Nenashev, we are able to give an algebraic proof of Quillen's Resolution Theorem for K_1 of an exact category.
A note on the homomorphism theorem for hemirings
Directory of Open Access Journals (Sweden)
D. M. Olson
1978-01-01
Full Text Available The fundamental homomorphism theorem for rings is not generally applicable in hemiring theory. In this paper, we show that for the class of N-homomorphism of hemirings the fundamental theorem is valid. In addition, the concept of N-homomorphism is used to prove that every hereditarily semisubtractive hemiring is of type (K.
Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems
Directory of Open Access Journals (Sweden)
Du Wei-Shih
2010-01-01
Full Text Available We establish some new coupled fixed point theorems for various types of nonlinear contractive maps in the setting of quasiordered cone metric spaces which not only obtain several coupled fixed point theorems announced by many authors but also generalize them under weaker assumptions.
An integral Riemann-Roch theorem for surface bundles
DEFF Research Database (Denmark)
Madsen, Ib Henning
2010-01-01
This paper is a response to a conjecture by T. Akita about an integral Riemann–Roch theorem for surface bundles.......This paper is a response to a conjecture by T. Akita about an integral Riemann–Roch theorem for surface bundles....
Note on two theorems in nonequilibrium statistical mechanics
Energy Technology Data Exchange (ETDEWEB)
Cohen, E.G.D.; Gallavotti, G.
1999-09-01
An attempt is made to clarify the difference between a theorem derived by Evans and Searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of fluctuations on phase space trajectory segments in a nonequilibrium stationary state.
Level reduction and the quantum threshold theorem
Aliferis, Panagiotis (Panos)
Computers have led society to the information age revolutionizing central aspects of our lives from production and communication to education and entertainment. There exist, however, important problems which are intractable with the computers available today and, experience teaches us, will remain so even with the more advanced computers we can envision for tomorrow.Quantum computers promise speedups to some of these important but classically intractable problems. Simulating physical systems, a problem of interest in a diverse range of areas from testing physical theories to understanding chemical reactions, and solving number factoring, a problem at the basis of cryptographic protocols that are used widely today on the internet, are examples of applications for which quantum computers, when built, will offer a great advantage over what is possible with classical computer technology.The construction of a quantum computer of sufficient scale to solve interesting problems is, however, especially challenging. The reason for this is that, by its very nature, operating a quantum computer will require the coherent control of the quantum state of a very large number of particles. Fortunately, the theory of quantum error correction and fault-tolerant quantum computation gives us confidence that such quantum states can be created, can be stored in memory and can also be manipulated provided the quantum computer can be isolated to a sufficient degree from sources of noise.One of the central results in the theory of fault-tolerant quantum computation, the quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the
A most compendious and facile quantum de Finetti theorem
König, R; Koenig, Robert; Mitchison, Graeme
2007-01-01
In its most basic form, the finite quantum de Finetti theorem states that the reduced k-partite density operator of an n-partite symmetric state can be approximated by a convex combination of k-fold product states. Variations of this result include Renner's "exponential" approximation by "almost-product" states, a theorem which deals with certain triples of representations of the unitary group, and D'Cruz et al.'s result for infinite-dimensional systems. We show how these theorems follow from a single, general de Finetti theorem for representations of symmetry groups, each instance corresponding to a particular choice of symmetry group and representation of that group. This gives some insight into the nature of the set of approximating states, and leads to some new results, including an exponential theorem for infinite-dimensional systems.
Soft Photon and Graviton Theorems in Effective Field Theory
Elvang, Henriette; Naculich, Stephen G
2016-01-01
Extensions of the photon and graviton soft theorems are derived in 4d local effective field theories with massless particles of arbitrary spin. We prove that effective operators can result in new terms in the soft theorems at subleading order for photons and subsubleading order for gravitons. The new soft terms are unique and we provide a complete classification of all local operators responsible for such modifications. We show that no local operators can modify the subleading soft graviton theorem. The soft limits are taken in a manifestly on-locus manner using a complex double deformation of the external momenta. In addition to the new soft theorems, the resulting master formula yields consistency conditions such as the conservation of electric charge, the Einstein equivalence principle, supergravity Ward identities, and the Weinberg-Witten theorem.
The virial theorem for the polarizable continuum model
Energy Technology Data Exchange (ETDEWEB)
Cammi, R., E-mail: roberto.cammi@unipr.it [Dipartimento di Chimica, Università di Parma, Parco Area delle Scienze 17/A, I-43100 Parma (Italy)
2014-02-28
The electronic virial theorem is extended to molecular systems within the framework of the Polarizable Continuum Model (PCM) to describe solvation effects. The theorem is given in the form of a relation involving the components of the energy (kinetic and potential) of a molecular solute and its electrostatic properties (potential and field) at the boundary of the cavity in the continuum medium. The virial theorem is also derived in the presence of the Pauli repulsion component of the solute-solvent interaction. Furthermore, it is shown that these forms of the PCM virial theorem may be related to the virial theorem of more simple systems as a molecule in the presence of fixed point charges, and as an atom in a spherical box with confining potential.
The virial theorem for the Polarizable Continuum Model.
Cammi, R
2014-02-28
The electronic virial theorem is extended to molecular systems within the framework of the Polarizable Continuum Model (PCM) to describe solvation effects. The theorem is given in the form of a relation involving the components of the energy (kinetic and potential) of a molecular solute and its electrostatic properties (potential and field) at the boundary of the cavity in the continuum medium. The virial theorem is also derived in the presence of the Pauli repulsion component of the solute-solvent interaction. Furthermore, it is shown that these forms of the PCM virial theorem may be related to the virial theorem of more simple systems as a molecule in the presence of fixed point charges, and as an atom in a spherical box with confining potential.
Primitive Near-rings-Some Structure Theorems
Institute of Scientific and Technical Information of China (English)
Gerhard Wendt
2007-01-01
We show that any zero symmetric 1-primitive near-ring with descending chain condition on left ideals can be described as a centralizer near-ring in which the multipli-cation is not the function composition but sandwich multiplication.This result follows from a more general structure theorem on 1-primitive near-rings with multiplicative right identity,not necessarily having a chain condition on left ideals.We then use our results to investigate more closely the multiplicative semigroup of a 1-primitive near-ring.In par-ticular,we show that the set of regular elements forms a right ideal of the multiplicative semigroup of the near-ring.
Some Generalizations of Fedorchuk Duality Theorem -- II
Dimov, Georgi Dobromirov
2007-01-01
As it was shown in the first part of this paper, there exists a duality between the category DSkeLC (introduced there) and the category SkeLC of locally compact Hausdorff spaces and continuous skeletal maps. We describe here the subcategories of the category DSkeLC which are dually equivalent to the following eight categories: all of them have as objects the locally compact Hausdorff spaces and their morphisms are, respectively, the injective (respectively, surjective) continuous skeletal maps, the injective (resp., surjective) open maps, the injective (resp., surjective) skeletal perfect maps, the injective (resp., surjective) open perfect maps. The particular cases of these theorems for the full subcategories of the last four categories having as objects all compact Hausdorff spaces are formulated and proved. The DSkeLC-morphisms which are LCA-embeddings and the dense homeomorphic embeddings are characterized through their dual morphisms. For any locally compact space X, a description of the frame of all op...
On a curvature-statistics theorem
Energy Technology Data Exchange (ETDEWEB)
Calixto, M [Departamento de Matematica Aplicada y Estadistica, Universidad Politecnica de Cartagena, Paseo Alfonso XIII 56, 30203 Cartagena (Spain); Aldaya, V [Instituto de Astrofisica de Andalucia, Apartado Postal 3004, 18080 Granada (Spain)], E-mail: Manuel.Calixto@upct.es
2008-08-15
The spin-statistics theorem in quantum field theory relates the spin of a particle to the statistics obeyed by that particle. Here we investigate an interesting correspondence or connection between curvature ({kappa} = {+-}1) and quantum statistics (Fermi-Dirac and Bose-Einstein, respectively). The interrelation between both concepts is established through vacuum coherent configurations of zero modes in quantum field theory on the compact O(3) and noncompact O(2; 1) (spatial) isometry subgroups of de Sitter and Anti de Sitter spaces, respectively. The high frequency limit, is retrieved as a (zero curvature) group contraction to the Newton-Hooke (harmonic oscillator) group. We also make some comments on the physical significance of the vacuum energy density and the cosmological constant problem.
Elementary Theorems Regarding Blue Isocurvature Perturbations
Chung, Daniel J H
2015-01-01
Blue CDM-photon isocurvature perturbations are attractive in terms of observability and may be typical from the perspective of generic mass relations in supergravity. We present and apply three theorems useful for blue isocurvature perturbations arising from linear spectator scalar fields. In the process, we give a more precise formula for the blue spectrum associated with the work of 0904.3800, which can in a parametric corner give a factor of O(10) correction. We explain how a conserved current associated with Peccei-Quinn symmetry plays a crucial role and explicitly plot several example spectra including the breaks in the spectra. We also resolve a little puzzle arising from a naive multiplication of isocurvature expression that sheds light on the gravitational imprint of the adiabatic perturbations on the fields responsible for blue isocurvature fluctuations.
Oscillation and the mean ergodic theorem
Avigad, Jeremy
2012-01-01
Let B be a uniformly convex Banach space, let T be a nonexpansive linear operator, and let A_n x denote the ergodic average (1/n) sum_{i 0, the sequence has only finitely many fluctuations greater than epsilon. Drawing on calculations by Kohlenbach and Leustean, we provide a uniform bound on the number of fluctuations that depends only on rho := || x || / epsilon and a modulus, eta, of uniform convexity for B. Specifically, we show that the sequence of averages (A_n x) has O(rho^2 log rho * eta(1/(8 rho))^{-1})-many epsilon-fluctuations, and if B is a Hilbert space, the sequence has O(rho^3 log rho)-many epsilon-fluctuations. The proof is fully explicit, providing a remarkably uniform, quantitative, and constructive formulation of the mean ergodic theorem.
Weak Convergence Theorems for Nonself Mappings
Institute of Scientific and Technical Information of China (English)
Liu Yong-quan; Guo Wei-ping; Ji You-qing
2015-01-01
Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retrac-tion. Let T1, T2 : K → E be two weakly inward nonself asymptotically nonexpan-sive mappings with respect to P with a sequence {k(i)n } ⊂ [1,∞) (i = 1, 2), and F := F (T1)∩F (T2) = ∅. An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Fr´echet differentiable norm or its dual E∗ has Kadec-Klee property, then weak convergence theorems are obtained.
The Birkhoff theorem and string clouds
Bronnikov, K A; Skvortsova, M V
2016-01-01
We consider spherically symmetric space-times in GR under the unconventional assumptions that the spherical radius $r$ is either a constant or has a null gradient in the $(t,x)$ subspace orthogonal to the symmetry spheres (i.e., $(\\partial r)^2 = 0$). It is shown that solutions to the Einstein equations with $r = \\rm const$ contain an extra (fourth) spatial or temporal Killing vector and thus satisfy the Birkhoff theorem under an additional physically motivated condition that the lateral pressure is functionally related to the energy density. This leads to solutions that directly generalize the Bertotti-Robinson, Nariai and Plebanski-Hacyan solutions. Under similar conditions, solutions with $(\\partial r)^2 = 0$ but $r\
What price the spin–statistics theorem?
Indian Academy of Sciences (India)
E C G Sudarshan; I M Duck
2003-10-01
We examine a number of recent proofs of the spin–statistics theorem. All, of course, get the target result of Bose–Einstein statistics for identical integral spin particles and Fermi–Dirac statistics for identical half-integral spin particles. It is pointed out that these proofs, distinguished by their purported simple and intuitive kinematic character, require assumptions that are outside the realm of standard quantum mechanics. We construct a counterexample to these non-dynamical kinematic ‘proofs’ to emphasize the necessity of a dynamical proof as distinct from a kinematic proof. Sudarshan’s simple non-relativistic dynamical proof is brieﬂy described. Finally, we make clear the price paid for any kinematic ‘proof’.
Steinitz theorems for simple orthogonal polyhedra
Directory of Open Access Journals (Sweden)
David Eppstein
2014-09-01
Full Text Available We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex.By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.
Courcelle's Theorem - A Game-Theoretic Approach
Kneis, Joachim; Rossmanith, Peter
2011-01-01
Courcelle's Theorem states that every problem definable in Monadic Second-Order logic can be solved in linear time on structures of bounded treewidth, for example, by constructing a tree automaton that recognizes or rejects a tree decomposition of the structure. Existing, optimized software like the MONA tool can be used to build the corresponding tree automata, which for bounded treewidth are of constant size. Unfortunately, the constants involved can become extremely large - every quantifier alternation requires a power set construction for the automaton. Here, the required space can become a problem in practical applications. In this paper, we present a novel, direct approach based on model checking games, which avoids the expensive power set construction. Experiments with an implementation are promising, and we can solve problems on graphs where the automata-theoretic approach fails in practice.
A Theorem on Grid Access Control
Institute of Scientific and Technical Information of China (English)
XU ZhiWei(徐志伟); BU GuanYing(卜冠英)
2003-01-01
The current grid security research is mainly focused on the authentication of grid systems. A problem to be solved by grid systems is to ensure consistent access control. This problem is complicated because the hosts in a grid computing environment usually span multiple autonomous administrative domains. This paper presents a grid access control model, based on asynchronous automata theory and the classic Bell-LaPadula model. This model is useful to formally study the confidentiality and integrity problems in a grid computing environment. A theorem is proved, which gives the necessary and sufficient conditions to a grid to maintain confidentiality.These conditions are the formalized descriptions of local (node) relations or relationship between grid subjects and node subjects.
An entropic view of Pickands' theorem
Bercher, J -F
2008-01-01
It is shown that distributions arising in Renyi-Tsallis maximum entropy setting are related to the Generalized Pareto Distributions (GPD) that are widely used for modeling the tails of distributions. The relevance of such modelization, as well as the ubiquity of GPD in practical situations follows from Balkema-De Haan-Pickands theorem on the distribution of excesses (over a high threshold). We provide an entropic view of this result, by showing that the distribution of a suitably normalized excess variable converges to the solution of a maximum Tsallis entropy, which is the GPD. This highlights the relevance of the so-called Tsallis distributions in many applications as well as some relevance to the use of the corresponding entropy.
Virial Theorem in Nonlocal Newtonian Gravity
Directory of Open Access Journals (Sweden)
Bahram Mashhoon
2016-05-01
Full Text Available Nonlocal gravity is the recent classical nonlocal generalization of Einstein’s theory of gravitation in which the past history of the gravitational field is taken into account. In this theory, nonlocality appears to simulate dark matter. The virial theorem for the Newtonian regime of nonlocal gravity theory is derived and its consequences for “isolated” astronomical systems in virial equilibrium at the present epoch are investigated. In particular, for a sufficiently isolated nearby galaxy in virial equilibrium, the galaxy’s baryonic diameter D 0 —namely, the diameter of the smallest sphere that completely surrounds the baryonic system at the present time—is predicted to be larger than the effective dark matter fraction f D M times a universal length that is the basic nonlocality length scale λ 0 ≈ 3 ± 2 kpc.
Virial Theorem in Nonlocal Newtonian Gravity
Mashhoon, B
2015-01-01
Nonlocal gravity is the recent classical nonlocal generalization of Einstein's theory of gravitation in which the past history of the gravitational field is taken into account. In this theory, nonlocality appears to simulate dark matter. The virial theorem for the Newtonian regime of nonlocal gravity theory is derived and its consequences for "isolated" astronomical systems in virial equilibrium at the present epoch are investigated. In particular, for a sufficiently isolated nearby galaxy in virial equilibrium, the galaxy's baryonic diameter---namely, the diameter of the smallest sphere that completely surrounds the baryonic system at the present time---is predicted to be larger than the effective dark matter fraction times a universal length that is the basic nonlocality length scale of about 3 kpc.
Virial Theorem in Nonlocal Newtonian Gravity
Mashhoon, Bahram
2016-05-01
Nonlocal gravity is the recent classical nonlocal generalization of Einstein's theory of gravitation in which the past history of the gravitational field is taken into account. In this theory, nonlocality appears to simulate dark matter. The virial theorem for the Newtonian regime of nonlocal gravity theory is derived and its consequences for "isolated" astronomical systems in virial equilibrium at the present epoch are investigated. In particular, for a sufficiently isolated nearby galaxy in virial equilibrium, the galaxy's baryonic diameter---namely, the diameter of the smallest sphere that completely surrounds the baryonic system at the present time---is predicted to be larger than the effective dark matter fraction times a universal length that is the basic nonlocality length scale of about 3 kpc.
The Recursion Theorem and Infinite Sequences
Miller, Arnold W
2008-01-01
In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove that there exists an increasing sequence such that W_{e_n}={e_{n+1},e_{n+2},...} for every n. We call a nonempty computably enumerable set A self-constructing if W_e=A for every e in A. We show that every nonempty computable enumerable set which is disjoint from an infinite computable set is one-one equivalent to a self-constructing set
Theorem Proving in Intel Hardware Design
O'Leary, John
2009-01-01
For the past decade, a framework combining model checking (symbolic trajectory evaluation) and higher-order logic theorem proving has been in production use at Intel. Our tools and methodology have been used to formally verify execution cluster functionality (including floating-point operations) for a number of Intel products, including the Pentium(Registered TradeMark)4 and Core(TradeMark)i7 processors. Hardware verification in 2009 is much more challenging than it was in 1999 - today s CPU chip designs contain many processor cores and significant firmware content. This talk will attempt to distill the lessons learned over the past ten years, discuss how they apply to today s problems, outline some future directions.
Asset management using an extended Markowitz theorem
Directory of Open Access Journals (Sweden)
Paria Karimi
2014-06-01
Full Text Available Markowitz theorem is one of the most popular techniques for asset management. The method has been widely used to solve many applications, successfully. In this paper, we present a multi objective Markowitz model to determine asset allocation by considering cardinality constraints. The resulted model is an NP-Hard problem and the proposed study uses two metaheuristics, namely genetic algorithm (GA and particle swarm optimization (PSO to find efficient solutions. The proposed study has been applied on some data collected from Tehran Stock Exchange over the period 2009-2011. The study considers four objectives including cash return, 12-month return, 36-month return and Lower Partial Moment (LPM. The results indicate that there was no statistical difference between the implementation of PSO and GA methods.
An Extension of Gregus Fixed Point Theorem
Directory of Open Access Journals (Sweden)
J. O. Olaleru
2007-03-01
Full Text Available Let C be a closed convex subset of a complete metrizable topological vector space (X,d and T:CÃ¢Â†Â’C a mapping that satisfies d(Tx,TyÃ¢Â‰Â¤ad(x,y+bd(x,Tx+cd(y,Ty+ed(y,Tx+fd(x,Ty for all x,yÃ¢ÂˆÂˆC, where 0theorem, which is a generalization and an extension of the results of several authors, is proved in this paper. In addition, we use the Mann iteration to approximate the fixed point of T.
No-Hair Theorem for Weak Pulsar
Gruzinov, Andrei
2015-01-01
It is proposed that there exists a class of pulsars, called weak pulsars, for which the large-scale magnetosphere, and hence the gamma-ray emission, are independent of the detailed pattern of plasma production. The weak pulsar magnetosphere and its gamma-ray emission are uniquely determined by just three parameters: spin, dipole, and the spin-dipole angle. We calculate this supposedly unique pulsar magnetosphere in the axisymmetric case. The magnetosphere is found to be very close to (although interestingly not fully identical with) the magnetosphere we have previously calculated, explaining the phenomenological success of the old calculation. We offer only a highly tentative proof of this "Pulsar No-Hair Theorem". Our analytics, while convincing in its non-triviality, is incomplete, and counts only as a plausibility argument. Our numerics, while complete, is dubious. The plasma flow in the weak pulsar magnetosphere turns out to be even more intricate than what we have previously proposed: some particles, aft...
PBR theorem and Einstein's quantum hole argument
Weinstein, Galina
2013-01-01
This note discusses the latest hot topic: Quantum states: ontic or epistemic? and the PBR theorem. Upon reading Einstein's views on quantum incompleteness in publications or in his correspondence after 1935 (the EPR paradox), one gets a very intense feeling of deja-vu. Einstein presents a quantum hole argument, which somewhat reminds of the hole argument in his 1914 "Entwurf" general theory of relativity. In their paper, PBR write the following: "an important step towards the derivation of our result is the idea that the quantum state is physical if distinct quantum states correspond to non-overlapping distributions for [the set of possible physical states that a system can be in]", and they then refer to Einstein's argument and views.
If 1+1=2 then the Pythagorean theorem holds, or one more proof of the oldest theorem of mathematics
Directory of Open Access Journals (Sweden)
Alexandru HORVÁTH
2013-06-01
Full Text Available The Pythagorean theorem is one of the oldest theorems of mathematics. It gained during the time a central position and even today it continues to be a source of inspiration. In this note we try to give a proof which is based on a hopefully new approach. Our treatment will be as intuitive as it can be.
Generalizations of some Zero Sum Theorems
Indian Academy of Sciences (India)
M N Chintamani; B K Moriya
2012-02-01
Given an abelian group of order , and a finite non-empty subset of integers, the Davenport constant of with weight , denoted by $D_A(G)$, is defined to be the least positive integer such that, for every sequence $(x_1,\\ldots,x_t)$ with $x_i\\in G$, there exists a non-empty subsequence $(x_{j_1},\\ldots,x_{j_l})$ and $a_i\\in A$ such that $\\sum^l_{i=1}a_ix_{j_i}=0$. Similarly, for an abelian group of order $n,E_A(G)$ is defined to be the least positive integer such that every sequence over of length contains a subsequence $(x_{j_1},\\ldots,x_{j_n})$ such that $\\sum^n_{i=1}a_ix_{j_i}=0$, for some $a_i\\in A$. When is of order , one considers to be a non-empty subset of $\\{1,\\ldots,n-1\\}$. If is the cyclic group $\\mathbb{Z}/n\\mathbb{Z}$, we denote $E_A(G)$ and $D_A(G)$ by $E_A(n)$ and $D_A(n)$ respectively. In this note, we extend some results of Adhikari et al(Integers 8(2008) Article A52) and determine bounds for $D_{R_n}(n)$ and $E_{R_n}(n)$, where $R_n=\\{x^2:x\\in(\\mathbb{Z}/n\\mathbb{Z})^∗\\}$. We follow some lines of argument from Adhikari et al(Integers 8 (2008) Article A52) and use a recent result of Yuan and Zeng (European J. Combinatorics 31 (2010) 677–680), a theorem due to Chowla (Proc. Indian Acad. Sci. (Math. Sci.) 2 (1935) 242–243) and Kneser’s theorem (Math. Z.58(1953) 459–484;66(1956) 88–110;61(1955) 429–434).
Cosmological singularity theorems and splitting theorems for N-Bakry-Emery spacetimes
Woolgar, Eric
2015-01-01
We study Lorentzian manifolds with a weight function such that the $N$-Bakry-\\'Emery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the "pure Bakry-\\'Emery" $N= \\infty$ case with $f$ uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite $N$-values $N\\in (n,\\infty)$ and $N\\in (-\\infty,1]$. In the $N\\in (n,\\infty)$ case, no bound on $f$ is required, while for $N\\in (-\\infty,1]$ and $N= \\infty$, we are able to replace the boundedness of $f$ by a weaker condition on the integral of $f$ along future-inextendible timel...
Traves, Will
2011-01-01
Using a new point of view inspired by hyperplane arrangements, we generalize the converse to Pascal's Theorem, sometimes called the Braikenridge-Maclaurin Theorem. In particular, we show that if 2k lines meet a given line, colored green, in k triple points and if we color the remaining lines so that each triple point lies on a red and blue line then the points of intersection of the red and blue lines lying off the green line lie on a unique curve of degree k-1. We also use these ideas to extend a second generalization of the Braikenridge-Maclaurin Theorem, due to M\\"obius. Finally we use Terracini's Lemma and secant varieties to show that this process constructs a dense set of curves in the space of plane curves of degree d, for degrees d <= 5. The process cannot produce a dense set of curves in higher degrees. The exposition is embellished with several exercises designed to amuse the reader.
Institute of Scientific and Technical Information of China (English)
同小军; 同登科; 陈绵云
2001-01-01
For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation has been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.
Mode-matching for Optical Antennas
Feichtner, Thorsten; Christiansen, Silke; Hecht, Bert
2016-01-01
The emission rate of a point dipole can be strongly increased in presence of a well-designed optical antenna. Yet, optical antenna design is largely based on radio-frequency rules, ignoring e.g.~ohmic losses and non-negligible field penetration in metals at optical frequencies. Here we combine reciprocity and Poynting's theorem to derive a set of optical-frequency antenna design rules for benchmarking and optimizing the performance of optical antennas driven by single quantum emitters. Based ...
Direct and converse theorems the elements of symbolic logic
Gradshtein, I S; Stark, M; Ulam, S
1963-01-01
Direct and Converse Theorems: The Elements of Symbolic Logic, Third Edition explains the logical relations between direct, converse, inverse, and inverse converse theorems, as well as the concept of necessary and sufficient conditions. This book consists of two chapters. The first chapter is devoted to the question of negation. Connected with the question of the negation of a proposition are interrelations of the direct and converse and also of the direct and inverse theorems; the interrelations of necessary and sufficient conditions; and the definition of the locus of a point. The second chap
The Differential Virial Theorem with Gradient Formulas for the Operators
Finley, James P
2016-01-01
A gradient dependent formula is derived for the spinless one-particle density-matrix operator z from the differential virial theorem. A gradient dependent formula is also derived for a spinless one-particle density-matrix operator that can replace the two operators of the differential virial theorem that arise from the kinetic energy operator. Other operators are also derived that can replace the operators mentioned above in the differential virial theorem; these operators depend on the real part of spinless one-particle density-matrix.
Fluctuation theorem for Hamiltonian Systems: Le Chatelier's principle
Evans, Denis J.; Searles, Debra J.; Mittag, Emil
2001-05-01
For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields.
Bottom-Left Placement Theorem for Rectangle Packing
Huang, Wenqi; Chen, Duanbing
2011-01-01
This paper proves a bottom-left placement theorem for the rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given rectangles into a rectangular container without overlapping, then we can achieve a feasible packing by successively placing a rectangle onto a bottom-left corner in the container. This theorem shows that even for the real-parameter rectangle packing problem, we can solve it after finite times of bottom-left placement actions. Based on this theorem, we might develop efficient heuristic algorithms for solving the rectangle packing problem.
Convergence theorems for lattice group-valued measures
Boccuto, Antonio
2015-01-01
Convergence Theorems for Lattice Group-valued Measures explains limit and boundedness theorems for measures taking values in abstract structures. The eBook begins with a historical survey about these topics since the beginning of the last century, moving on to basic notions and preliminaries on filters/ideals, lattice groups, measures and tools which are featured in the rest of this text. Readers will also find a survey on recent classical results about limit, boundedness and extension theorems for lattice group-valued measures followed by information about recent developments on these kinds o
Noether Theorem for Nonholonomic Systems with Time Delay
Directory of Open Access Journals (Sweden)
Shi-Xin Jin
2015-01-01
Full Text Available The paper focuses on studying the Noether theorem for nonholonomic systems with time delay. Firstly, the differential equations of motion for nonholonomic systems with time delay are established, which is based on the Hamilton principle with time delay and the Lagrange multiplier rules. Secondly, based upon the generalized quasi-symmetric transformations for nonconservative systems with time delay, the Noether theorem for corresponding holonomic systems is given. Finally, we obtain the Noether theorem for the nonholonomic nonconservative systems with time delay. At the end of the paper, an example is given to illustrate the application of the results.
Theorems on Positive Data: On the Uniqueness of NMF
Directory of Open Access Journals (Sweden)
Hans Laurberg
2008-01-01
Full Text Available We investigate the conditions for which nonnegative matrix factorization (NMF is unique and introduce several theorems which can determine whether the decomposition is in fact unique or not. The theorems are illustrated by several examples showing the use of the theorems and their limitations. We have shown that corruption of a unique NMF matrix by additive noise leads to a noisy estimation of the noise-free unique solution. Finally, we use a stochastic view of NMF to analyze which characterization of the underlying model will result in an NMF with small estimation errors.
ON THE PRIMARY DECOMPOSITION THEOREM OF MODULAR LIE SUPERALGEBRAS
Institute of Scientific and Technical Information of China (English)
CHEN LIANGYUN; MENG DAOJI
2005-01-01
This gives some identities of associative Lie superalgebras and some properties of modular Lie superalgebras. Furthermore, the primry decomposition theorem of modular Lie superalgebras is shown. It is well known that the primary decomposition theorem of modular Lie algebras has played an important role in the classification of the finite-dimensional simple modular Lie algebras (see [5, 6]). Analogously, the primary decomposition theorem of modular Lie superalgebras may play an important role in the open classification of the finite dimensional simple modular Lie superalgebras.
Inconsistency of Carnot's theorem's proof by William Thomson
Ihnatovych, V
2013-01-01
William Thomson proved Carnot's theorem basing on postulate: "It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects". The present paper demonstrates that Carnot's theorem can be proved based on the contrary Thomson's postulate: "It is impossible to use the mechanical effect to the heating the coldest of surrounding objects". A conclusion that Carnot's theorem does not follow from the Thomson's postulate has been drawn.
A global Torelli theorem for hyperkaehler manifolds (after Verbitsky)
Huybrechts, Daniel
2011-01-01
Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in general not determined by its natural weight-two Hodge structure. The text gives an account of a recent theorem of M. Verbitsky, which can be regarded as a weaker version of the Global Torelli theorem phrased in terms of the injectivity of the period map on the connected components of the moduli space of marked manifolds.
On Pythagoras Theorem for Products of Spectral Triples
D'Andrea, Francesco; Martinetti, Pierre
2013-05-01
We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K-homology.
ON REGULARITY THEOREMS FOR LINEARLY INVARIANT FAMILIES OF HARMONIC FUNCTIONS
Directory of Open Access Journals (Sweden)
E. G. Ganenkova
2015-05-01
Full Text Available The classical theorem of growth regularity in the class S of analytic and univalent in the unit disc ∆ functions ƒ describes the growth character of different functionals of ƒ Є S and z Є ∆ as z tends to δ∆. Earlier the authors proved the theorems of growth and decrease regularity for harmonic and sense-preserving in ∆ functions which generalized the classical result for the class S. In the presented paper we establish new properties of harmonic sense-preserving functions, connected with the regularity theorems. The effects both common for analytic and harmonic case and specific for harmonic functions are displayed.
Asymmetric de Finetti Theorem for Infinite-dimensional Quantum Systems
Niu, Murphy Yuezhen
2016-01-01
The de Finetti representation theorem for continuous variable quantum system is first developed to approximate an N-partite continuous variable quantum state with a convex combination of independent and identical subsystems, which requires the original state to obey permutation symmetry conditioned on successful experimental verification on k of N subsystems. We generalize the de Finetti theorem to include asymmetric bounds on the variance of canonical observables and biased basis selection during the verification step. Our result thereby enables application of infinite-dimensional de Finetti theorem to situations where two conjugate measurements obey different statistics, such as the security analysis of quantum key distribution protocols based on squeezed state against coherent attack.
A reconstruction theorem for almost-commutative spectral triples
Ćaćić, Branimir
2011-01-01
We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric, and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of the reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in Connes's reconstruction theorem for commutative spectral triples, and, following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectral triples under suitable perturbation of the Dirac operator.
Noncommutative topology and the world's simplest index theorem.
van Erp, Erik
2010-05-11
In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoelliptic operators on contact manifolds. Finally, we discuss the noncommutative topology that is employed in the proof of this theorem. The article is intended to illustrate that noncommutative topology can be a powerful tool for proving results in classical analysis and geometry.
The direct Flow parametric Proof of Gauss' Divergence Theorem revisited
DEFF Research Database (Denmark)
Markvorsen, Steen
The standard proof of the divergence theorem in undergraduate calculus courses covers the theorem for static domains between two graph surfaces. We show that within first year undergraduate curriculum, the flow proof of the dynamic version of the divergence theorem - which is usually considered...... only much later in more advanced math courses - is comprehensible with only a little extension of the first year curriculum. Moreover, it is more intuitive than the static proof. We support this intuition further by unfolding and visualizing a few examples with increasing complexity. In these examples...
Convolution theorems: partitioning the space of integral transforms
Lindsey, Alan R.; Suter, Bruce W.
1999-03-01
Investigating a number of different integral transforms uncovers distinct patterns in the type of translation convolution theorems afforded by each. It is shown that transforms based on separable kernels (aka Fourier, Laplace and their relatives) have a form of the convolution theorem providing for a transform domain product of the convolved functions. However, transforms based on kernels not separable in the function and transform variables mandate a convolution theorem of a different type; namely in the transform domain the convolution becomes another convolution--one function with the transform of the other.
The Basic Existence Theorem of Riemann-Stieltjes Integral
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Nakasho Kazuhisa
2016-12-01
Full Text Available In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].
Institute of Scientific and Technical Information of China (English)
Li Shan LIU
2001-01-01
In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234-240) is true for1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with intK ≠φ.This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractivemaps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-self-maps are proved under various well-known boundary conditions. Our results are generalizations andimprovements of the recent results obtained by many authors.
The Michaelis-Menten-Stueckelberg Theorem
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Alexander N. Gorban
2011-05-01
Full Text Available We study chemical reactions with complex mechanisms under two assumptions: (i intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS and (ii they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE. Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the “rarefied gas” limit or the “molecular chaos” hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density (low density of elementary events. This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann H-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.
The Ewald-Oseen Extinction Theorem
Mansuripur, Masud
2015-01-01
When a beam of light enters a material medium, it sets in motion the resident electrons, whether these electrons are free or bound. The electronic oscillations in turn give rise to electromagnetic radiation which, in the case of linear media, possess the frequency of the exciting beam. Because Maxwell's equations are linear, one expects the total field at any point in space to be the sum of the original (exciting) field and the radiation produced by all the oscillating electrons. However, in practice the original beam appears to be absent within the medium, as though it had been replaced by a different beam, one having a shorter wavelength and propagating in a different direction. The Ewald-Oseen theorem resolves this paradox by showing how the oscillating electrons conspire to produce a field that exactly cancels out the original beam everywhere inside the medium. The net field is indeed the sum of the incident beam and the radiated field of the oscillating electrons, but the latter field completely masks th...
Maximum entropy production and the fluctuation theorem
Energy Technology Data Exchange (ETDEWEB)
Dewar, R C [Unite EPHYSE, INRA Centre de Bordeaux-Aquitaine, BP 81, 33883 Villenave d' Ornon Cedex (France)
2005-05-27
Recently the author used an information theoretical formulation of non-equilibrium statistical mechanics (MaxEnt) to derive the fluctuation theorem (FT) concerning the probability of second law violating phase-space paths. A less rigorous argument leading to the variational principle of maximum entropy production (MEP) was also given. Here a more rigorous and general mathematical derivation of MEP from MaxEnt is presented, and the relationship between MEP and the FT is thereby clarified. Specifically, it is shown that the FT allows a general orthogonality property of maximum information entropy to be extended to entropy production itself, from which MEP then follows. The new derivation highlights MEP and the FT as generic properties of MaxEnt probability distributions involving anti-symmetric constraints, independently of any physical interpretation. Physically, MEP applies to the entropy production of those macroscopic fluxes that are free to vary under the imposed constraints, and corresponds to selection of the most probable macroscopic flux configuration. In special cases MaxEnt also leads to various upper bound transport principles. The relationship between MaxEnt and previous theories of irreversible processes due to Onsager, Prigogine and Ziegler is also clarified in the light of these results. (letter to the editor)
A stem cell niche dominance theorem
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Shibata Darryl K
2011-01-01
Full Text Available Abstract Background Multilevelness is a defining characteristic of complex systems. For example, in the intestinal tissue the epithelial lining is organized into crypts that are maintained by a niche of stem cells. The behavior of the system 'as a whole' is considered to emerge from the functioning and interactions of its parts. What we are seeking here is a conceptual framework to demonstrate how the "fate" of intestinal crypts is an emergent property that inherently arises from the complex yet robust underlying biology of stem cells. Results We establish a conceptual framework in which to formalize cross-level principles in the context of tissue organization. To this end we provide a definition for stemness, which is the propensity of a cell lineage to contribute to a tissue fate. We do not consider stemness a property of a cell but link it to the process in which a cell lineage contributes towards tissue (malfunction. We furthermore show that the only logically feasible relationship between the stemness of cell lineages and the emergent fate of their tissue, which satisfies the given criteria, is one of dominance from a particular lineage. Conclusions The dominance theorem, conceived and proven in this paper, provides support for the concepts of niche succession and monoclonal conversion in intestinal crypts as bottom-up relations, while crypt fission is postulated to be a top-down principle.
Some Representation Theorems for Recovering Contraction Relations
Institute of Scientific and Technical Information of China (English)
Ping Hou
2005-01-01
One of the important topics in the study of contraction inference relations is to establish the representation theorems for them. Various methods have been employed for giving representation of a broad class of contraction operations.However, there was not any canonical approach to dealing with the representation results for the contraction relations in the literature. Recently, in order to obtain the representation result for recovering contraction inference relations satisfying the condition weak conjunctive inclusion (wci), a notion of an image structure associated with the canonical epistemic state has been introduced. Based on the image structure, this paper establishes three representation results for recovering contraction inference relations which satisfy the conditions CL, CR1 and DR* respectively by the standard epistemic AGM states. A unique technique and uniform proofs to represent these contraction relations are adopted, which could overcome the core objection in previous description of contraction relations. The paper shows as well that the image structure and canonical epistemic states can be used not only to get the representation result for wci-recovering contraction relation, but also to provide semantic characterizations for a wide range of recovering contraction relations.
Isotropy theorem for cosmological vector fields
Cembranos, J A R; Maroto, A L; Jareño, S J Núñez
2012-01-01
We consider homogeneous abelian vector fields in an expanding universe. We find a mechanical analogy in which the system behaves as a particle moving in three dimensions under the action of a central potential. In the case of bounded and rapid evolution compared to the rate of expansion, we show by making use of the virial theorem that for arbitrary potential and polarization pattern, the average energy-momentum tensor is always diagonal and isotropic despite the intrinsic anisotropic evolution of the vector field. For simple power-law potentials of the form V=\\lambda (A^\\mu A_\\mu)^n, the average equation of state is found to be w=(n-1)/(n+1). This implies that vector coherent oscillations could act as natural dark matter or dark energy candidates. Finally, we show that under very general conditions, the average energy-momentum tensor of a rapidly evolving bounded vector field in any background geometry is always isotropic and has the perfect fluid form for any locally inertial observer.
Gödel’s Incompleteness Theorems and Physics
Directory of Open Access Journals (Sweden)
Newton C. A. da Costa
2011-12-01
Full Text Available This paper is a summary of a lecture in which I presented some remarks on Gödel’s incompleteness theorems and their meaning for the foundations of physics. The entire lecture will appear elsewhere.
SEVERAL UNIQUENESS THEOREMS OF ALGEBROID FUNCTIONS ON ANNULI
Institute of Scientific and Technical Information of China (English)
Yang TAN
2016-01-01
In this paper, we discuss the uniqueness problem of algebroid functions on an-nuli, we get several uniqueness theorems of algebroid functions on annuli, which extend the Nevanlinna value distribution theory for algebroid functions on annuli.
A Generalized Carpenter's Rule Theorem for Self-Touching Linkages
Abbott, Timothy G; Gassend, Blaise
2009-01-01
The Carpenter's Rule Theorem states that any chain linkage in the plane can be folded continuously between any two configurations while preserving the bar lengths and without the bars crossing. However, this theorem applies only to strictly simple configurations, where bars intersect only at their common endpoints. We generalize the theorem to self-touching configurations, where bars can touch but not properly cross. At the heart of our proof is a new definition of self-touching configurations of planar linkages, based on an annotated configuration space and limits of nontouching configurations. We show that this definition is equivalent to the previously proposed definition of self-touching configurations, which is based on a combinatorial description of overlapping features. Using our new definition, we prove the generalized Carpenter's Rule Theorem using a topological argument. We believe that our topological methodology provides a powerful tool for manipulating many kinds of self-touching objects, such as...
An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem
Ephremidze, Lasha
2010-01-01
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
Wigner-Araki-Yanase theorem beyond conservation laws
Tukiainen, Mikko
2017-01-01
The ability to measure every quantum observable is ensured by a fundamental result in quantum measurement theory. Nevertheless, additive conservation laws associated with physical symmetries, such as the angular momentum conservation, may lead to restrictions on the measurability of the observables. Such limitations are imposed by the theorem of Wigner, Araki, and Yanase (WAY). In this paper a formulation of the WAY theorem is presented rephrasing the measurability limitations in terms of quantum incompatibility. This broader mathematical basis enables us to both capture and generalize the WAY theorem by allowing us to drop the assumptions of additivity and even conservation of the involved quantities. Moreover, we extend the WAY theorem to the general level of positive operator-valued measures.
Forest Carbon Uptake and the Fundamental Theorem of Calculus
Zobitz, John
2013-01-01
Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.
POISSON LIMIT THEOREM FOR COUNTABLE MARKOV CHAINS IN MARKOVIAN ENVIRONMENTS
Institute of Scientific and Technical Information of China (English)
方大凡; 王汉兴; 唐矛宁
2003-01-01
A countable Markov chain in a Markovian environment is considered. A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved. In order to prove this theorem, the entropy function h is introduced and the Shannon-McMillan-Breiman theorem for the Markov chain in a Markovian environment is shown. It' s well-known that a Markov process in a Markovian environment is generally not a standard Markov chain, so an example of Poisson approximation for a process which is not a Markov process is given. On the other hand, when the environmental process degenerates to a constant sequence, a Poisson limit theorem for countable Markov chains, which is the generalization of Pitskel's result for finite Markov chains is obtained.
Comparison theorems for neutral stochastic functional differential equations
Bai, Xiaoming; Jiang, Jifa
2016-05-01
The comparison theorems under Wu and Freedman's order are proved for neutral stochastic functional differential equations with finite or infinite delay whose drift terms satisfy the quasimonotone condition and diffusion term is the same.
Generalized Chung-Feller Theorems for Lattice Paths (Thesis)
Huq, Aminul
2009-01-01
In this thesis we develop generalized versions of the Chung-Feller theorem for lattice paths constrained in the half plane. The beautiful cycle method which was developed by Devoretzky and Motzkin as a means to prove the ballot problem is modified and applied to generalize the classical Chung-Feller theorem. We use Lagrange inversion to derive the generalized formulas. For the generating function proof we study various ways of decomposing lattice paths. We also show some results related to equidistribution properties in terms of Narayana and Catalan generating functions. We then develop generalized Chung-Feller theorems for Motzkin and Schroeder paths. Finally we study generalized paths and the analogue of the Chung-Feller theorem for them.
Forest Carbon Uptake and the Fundamental Theorem of Calculus
Zobitz, John
2013-01-01
Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.
Limit theorems for solutions of stochastic differential equation problems
Directory of Open Access Journals (Sweden)
J. Vom Scheidt
1980-01-01
Full Text Available In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.
On some fixed point theorems in Banach spaces
Directory of Open Access Journals (Sweden)
D. V. Pai
1982-01-01
Full Text Available In this paper, some fixed point theorems are proved for multi-mappings as well as a pair of mappings. These extend certain known results due to Kirk, Browder, Kanna, Ćirić and Rhoades.
A tropical proof of the Brill-Noether Theorem
Cools, Filip; Draisma, Jan; Payne, Sam; Robeva, Elina
2010-01-01
We produce Brill-Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill-Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.
Modified Noether theorem and arrow of time in quantum mechanics
Energy Technology Data Exchange (ETDEWEB)
Asadov, V V; Kechkin, O V, E-mail: asadov@neurok.co, E-mail: kechkin@gmail.co [Neur OK-III, Lomonosov Moscow State University, Vorob' jovy Gory, 119899 Moscow (Russian Federation)
2010-06-01
Relativistic quantum mechanics is presented with modified Noether theorem. It was shown that Noether charges are related with thermodynamic potentials in such scheme. Broken symmetries generated by thermodynamic mode lead to gravity appearance as effective quantum field.
Hyperbolic positive mass theorem under modified energy condition
Institute of Scientific and Technical Information of China (English)
XIE NaQing
2008-01-01
We provide two new positive mass theorems under respective modified energy conditions allowing Too negative on some compact set for certain modified asymptotically hyperbolic manifolds. This work is analogous to Zhang's previous result for modified asymptotically fiat initial data sets.
Product Integral Formalism and Non-Abelian Stokes Theorem
Karp, R L; Rno, J S; Karp, Robert L.; Mansouri, Freydoon; Rno, Jung S.
1999-01-01
We make use of the properties of product integrals to obtain a surface product integral representation for the Wilson loop operator. The result can be interpreted as the non-abelian version of Stokes' theorem.
Quantum nonlocality and reality 50 years of Bell's theorem
Gao, Shan
2016-01-01
Description Contents Resources Courses About the Authors Combining twenty-six original essays written by an impressive line-up of distinguished physicists and philosophers of physics, this anthology reflects some of the latest thoughts by leading experts on the influence of Bell's theorem on quantum physics. Essays progress from John Bell's character and background, through studies of his main work, and on to more speculative ideas, addressing the controversies surrounding the theorem, and investigating the theorem's meaning and its deep implications for the nature of physical reality. Combined, they present a powerful comment on the undeniable significance of Bell's theorem for the development of ideas in quantum physics over the past 50 years. Questions surrounding the assumptions and significance of Bell's work still inspire discussion in the field of quantum physics. Adding to this with a theoretical and philosophical perspective, this balanced anthology is an indispensable volume for students and researc...
Duality Theorems on Multi-objective Programming of Generalized Functions
Institute of Scientific and Technical Information of China (English)
Li-ping Pang; Wei Wang; Zun-quan Xia
2006-01-01
The form of a dual problem of Mond-Weir type for multi-objective programming problems of generalized functions is defined and theorems of the weak duality, direct duality and inverse duality are proven.
Representation Theorems for Fuzzy Random Sets and Fuzzy Stochastic Processes
Institute of Scientific and Technical Information of China (English)
无
1999-01-01
The fuzzy static and dynamic random phenomena in an abstract separable Banach space is discussed in this paper. The representation theorems for fuzzy set-valued random sets, fuzzy random elements and fuzzy set-valued stochastic processes are obtained.
Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces
Institute of Scientific and Technical Information of China (English)
H.M.Abu-Donia; A.A.Nasef
2008-01-01
The purpose of this paper is to introduce some types of compatibility of maps, and we prove some common fixed point theorems for mappings satisfying some conditions on intuitionistic fuzzy metric spaces which defined by J.H.Park.
A compactness theorem for surfaces with Bounded Integral Curvature
Debin, Clément
2016-01-01
We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation of singularities is allowed.
Generalized Browder's and Weyl's theorems for Banach space operators
Curto, Raúl E.; Han, Young Min
2007-12-01
We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem. We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of [sigma](T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f[set membership, variant]H((T)), the space of functions analytic on an open neighborhood of [sigma](T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f[set membership, variant]H([sigma](T)).
Koopmans' theorem in statistical Hartree-Fock theory
Pain, Jean-Christophe
2011-01-01
In this short paper, the validity of Koopmans' theorem in the Hartree-Fock theory at non-zero temperature (Hartree-Fock statistical theory) is investigated. It is shown that Koopmans' theorem does not apply in the grand-canonical ensemble, due to a missing contribution to the energy proportional to the interaction between two electrons belonging to the same orbital. Hartree-Fock statistical theory has also been applied in the canonical ensemble [Blenski et al., Phys. Rev. E 55, R4889 (1997)] for the purpose of photo-absorption calculations. In that case, the Hartree-Fock self-consistent-field equations are derived in the super-configuration approximation. It is shown that Koopmans' theorem does not hold in the canonical ensemble, but that a restricted version of the theorem can be obtained, by assuming that a particular quantity multiplying the interaction matrix element in the expression of the energy does not change during the removal of an electron.
A Computer Science Version of Goedel’s Theorem.
1983-08-01
The author presents a simplified proof of Godel’s theorem by appealing to well-known programming concepts. The significance of Goedel’s result to computer science , mathematics and logic is discussed. (Author)
Generalized -Bernstein-Schurer Operators and Some Approximation Theorems
Directory of Open Access Journals (Sweden)
M. Mursaleen
2013-01-01
Full Text Available We study statistical approximation properties of -Bernstein-Shurer operators and establish some direct theorems. Furthermore, we compute error estimation and show graphically the convergence for a function by operators and give its algorithm.
Bell's theorem without inequalities and without unspeakable information
Cabello, A
2004-01-01
A proof of Bell's theorem without inequalities is presented in which distant local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups.
Critical Point Theorems and Applications to Differential Equations
Institute of Scientific and Technical Information of China (English)
A. R. EL AMROUSS
2005-01-01
This paper contains a generalization of the well-known Palais-Smale and Cerami compactness conditions. The compactness condition introduced is used to prove some general existence theorems for critical points. Some applications are given to differential equations.
T1 theorem for Besov and Triebel-Lizorkin spaces
Institute of Scientific and Technical Information of China (English)
DENG Donggao; HAN Yongsheng
2005-01-01
Using the discrete Calderon type reproducing formula and the PlancherelPolya characterization for the Besov and Triebel-Lizorkin spaces, the T1 theorem for the Besov and Triebel-Lizorkin spaces was proved.
The Penrose singularity theorem in regularity $C^{1,1}$
Kunzinger, Michael; Vickers, James A
2015-01-01
We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity $C^{1,1}$. The proof is based on regularisation techniques, combined with recent results in low regularity causality theory.
On the generalized virial theorem for systems with variable mass
Ganghoffer, Jean-François; Rahouadj, Rachid
2016-03-01
We presently extend the virial theorem for both discrete and continuous systems of material points with variable mass, relying on developments presented in Ganghoffer (Int J Solids Struct 47:1209-1220, 2010). The developed framework is applicable to describe physical systems at very different scales, from the evolution of a population of biological cells accounting for growth to mass ejection phenomena occurring within a collection of gravitating objects at the very large astrophysical scales. As a starting basis, the field equations in continuum mechanics are written to account for a mass source and a mass flux, leading to a formulation of the virial theorem accounting for non-constant mass within the considered system. The scalar and tensorial forms of the virial theorem are then written successively in both Lagrangian and Eulerian formats, incorporating the mass flux. As an illustration, the averaged stress tensor in accreting gravitating solid bodies is evaluated based on the generalized virial theorem.
Analogy to Derive an Extended Pythagorean Theorem to ''N'' Dimensions
Directory of Open Access Journals (Sweden)
Acosta-Robledo J.U.
2012-01-01
Full Text Available This article demonstrates that it is possible to extend the Pythagorean Theorem to ''N'' dimensions. This demonstration is mainly done based on linear algebra, especially in the vector product of ''N'' dimensions.
HEAT KERNEL AND HARDY'S THEOREM FOR JACOBI TRANSFORM
Institute of Scientific and Technical Information of China (English)
T. KAWAZOE; LIU JIANMING(刘建明)
2003-01-01
In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associatedwith Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform byusing the sharp estimate of the heat kernel.
THREE SOLUTIONS THEOREMS FOR NONLINEAR OPERATOR EQUATIONS AND APPLICATIONS
Institute of Scientific and Technical Information of China (English)
SUN Jingxian; XU Xi'an
2005-01-01
In this paper, some three solutions theorems about a class of operators which are said to be limit-increasing are obtained. Some applications to the second order differential equations boundary value problems are given.
Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem
Energy Technology Data Exchange (ETDEWEB)
Gallavotti, G. [Rutgers Univ., New Brunswick, NJ (United States)
1996-09-01
It is shown that the chaoticity hypothesis recently introduced in statistical mechanics, which is analogous to Ruelle`s principle for turbulence, implies the Onsager reciprocity and the fluctuation-dissipation theorem in various reversible models for coexisting transport phenomena.
Chaotic hypothesis Onsager reciprocity and fluctuation-dissipation theorem
Gallavotti, G
1996-01-01
It is shown that the "chaoticity hypothesis", analogous to Ruelle's principle for turbulence and recently introduced in statistical mechanics, implies the Onsager reciprocity and the fluctuation dissipation theorem in various models for coexisting transport phenomena.
Fluctuation theorem in driven nonthermal systems with quenched disorder
Energy Technology Data Exchange (ETDEWEB)
Reichhardt, Charles [Los Alamos National Laboratory; Reichhardt, C J [Los Alamos National Laboratory; Drocco, J A [PRINCETON UNIV.
2009-01-01
We demonstrate that the fluctuation theorem of Evans and Searles can be used to characterize the class of dynamics that arises in nonthermal systems of collectively interacting particles driven over random quenched disorder. By observing the frequency of entropy-destroying trajectories, we show that there are specific dynamical regimes near depinning in which this theorem holds. Hence the fluctuation theorem can be used to characterize a significantly wider class of non-equilibrium systems than previously considered. We discuss how the fluctuation theorem could be tested in specific systems where noisy dynamics appear at the transition from a pinned to a moving phase such as in vortices in type-II superconductors, magnetic domain walls, and dislocation dynamics.
The Decoration Theorem for Mandelbrot and Multibrot Sets
Dudko, Dzmitry
2010-01-01
We prove the decoration theorem for the Mandelbrot set (and Multibrot sets) which says that when a "little Mandelbrot set" is removed from the Mandelbrot set, then most of the resulting connected components have small diameters.
Noncommutative topology and the world's simplest index theorem
van Erp, Erik
2010-01-01
This is an expository article. It discusses an approach to hypoelliptic Fredholm index theory based on noncommutative methods (groupoids, C*-algebras, K-theory). The paper starts with an explicit index theorem for scalar second order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how this theorem is a special case of a much more general index theorem for subelliptic operators on contact manifolds. Finally we discuss the noncommutative topology that is employed in the proof of this theorem. We present these results as an instance in which noncommutative topology is fruitful in proving a very explicit (analytic/geometric) classical result.
An application of Pappus' Involution Theorem in euclidean and non-euclidean geometry
Vigara, Ruben
2014-01-01
Pappus' Involution Theorem is a powerful tool for proving theorems about non-euclidean triangles and generalized triangles in Cayley-Klein models. Its power is illustrated by proving with it some theorems about euclidean and non-euclidean polygons of different types. A $n$-dimensional euclidean version of these theorems is stated too.
Proof of the equivalence theorem in the chiral lagrangian formalism
He, H J; Li, X; Hong-Jian He; He, Hong-Jian; Kuang, Yu-Ping; Li, Xiaoyuan; Xiaoyuan Li
1994-01-01
A general proof of the equivalence theorem in electroweak theories with the symmetry breaking sector described by the chiral Lagrangian is given in the $R_{\\xi}$ gauge by means of the Ward-Takahashi identities. The precise form of the theorem contains a modification factor $C_{mod}$ associated with each external Goldstone boson similar to that in the standard model. $C_{mod}$ is exactly unity in our previously proposed renormalization scheme, {\\it Scheme-II}.
Liouville`s theorem and phase-space cooling
Energy Technology Data Exchange (ETDEWEB)
Mills, R.L. [Ohio State Univ., Columbus, OH (United States); Sessler, A.M. [Lawrence Berkeley Lab., CA (United States)
1993-09-28
A discussion is presented of Liouville`s theorem and its consequences for conservative dynamical systems. A formal proof of Liouville`s theorem is given. The Boltzmann equation is derived, and the collisionless Boltzmann equation is shown to be rigorously true for a continuous medium. The Fokker-Planck equation is derived. Discussion is given as to when the various equations are applicable and, in particular, under what circumstances phase space cooling may occur.
The convolution theorem for two-dimensional continuous wavelet transform
Institute of Scientific and Technical Information of China (English)
ZHANG CHI
2013-01-01
In this paper , application of two -dimensional continuous wavelet transform to image processes is studied. We first show that the convolution and correlation of two continuous wavelets satisfy the required admissibility and regularity conditions ,and then we derive the convolution and correlation theorem for two-dimensional continuous wavelet transform. Finally, we present numerical example showing the usefulness of applying the convolution theorem for two -dimensional continuous wavelet transform to perform image restoration in the presence of additive noise.
Exchangeability and Realizability: De Finetti Theorems on Graphs
Jayram, T. S.; Vondrák, Jan
2014-01-01
A classic result in probability theory known as de Finetti's theorem states that exchangeable random variables are equivalent to a mixture of distributions where each distribution is determined by an i.i.d. sequence of random variables (an "i.i.d. mix"). Motivated by a recent application and more generally by the relationship of local vs. global correlation in randomized rounding, we study weaker notions of exchangeability that still imply the conclusion of de Finetti's theorem. We say that a...
Coupled Best Proximity Point Theorem in Metric Spaces
Directory of Open Access Journals (Sweden)
Animesh Gupta
2013-11-01
Full Text Available The purpose of this article is to generalized the result of W. Sintunavarat and P. Kumam [29]. We also give an example in support of our theorem for which result of W. Sintunavarat and P. Kumam [29] is not true. Moreover we establish the existence and convergence theorems of coupled best proximity points in metric spaces, we apply this results in a uniformly convex Banach space.
Integrating factors and conservation theorems of constrained Birkhoffian systems
Institute of Scientific and Technical Information of China (English)
Qiao Yong-Fen; Zhao Shu-Hong; Li Ren-Jie
2006-01-01
In this paper the conservation theorems of the constrained Birkhoffian systems are studied by using the method of integrating factors. The differential equations of motion of the system are written. The definition of integrating factors is given for the system. The necessary conditions for the existence of the conserved quantity for the system are studied.The conservation theorem and its inverse for the system are established. Finally, an example is given to illustrate the application of the results.
Conformal frames and the validity of Birkhoff's theorem
Capozziello, Salvatore
2012-01-01
Birkhoff's theorem is one of the most important statements of Einstein's general relativity, which generally can not be extended to modified theories of gravity. Here we study the validity of the theorem in scalar-tensor theories using a perturbative approach, and compare the results in the so-called Einstein and Jordan frames. The implications of the results question the physical equivalence between both frames, at least in perturbations.
Some Limit Theorems for Negatively Associated Random Variables
Indian Academy of Sciences (India)
Yu Miao; Wenfei Xu; Shanshan Chen; Andre Adler
2014-08-01
Let $\\{X_n,n≥ 1\\}$ be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the $L^p$-convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.
A strong completeness theorem in intuitionistic quantified modal logic
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Based on the intuitionistic first order predicate calculus H given by Thomason with the modal machinery of MIPC put forward by Prior this paper obtains the intuitionistic quantified modal logic system MIPC*, gives it a semantic interpretation and proves its strong (thus also weak) completeness theorem and soundness theorem with respect to that semantic. Since Zorn lemma plays a decisive role in our discussion, methodologically, it was even farther from the intuitionistic point of view than Thomason's result.
A strong completeness theorem in intuitionistic quantified modal logic
Institute of Scientific and Technical Information of China (English)
高恒珊
2000-01-01
Based on the intuitionistic first order predicate calculus H given by Thomason with the modal machinery of MIPC put forward by Prior this paper obtains the intuitionistic quantified modal logic system MIPC* , gives it a semantic interpretation and proves its strong (thus also weak) completeness theorem and soundness theorem with respect to that semantic. Since Zom lemma plays a decisive role in our discussion, methodologically, it was even farther from the intuitionistic point of view than Thomason’s result.
A short list color proof of Grotzsch's theorem
DEFF Research Database (Denmark)
Thomassen, Carsten
2000-01-01
We give a short proof of the result that every planar graph of girth $5$is $3$-choosable and hence also of Gr\\"{o}tzsch's theorem saying that everyplanar triangle-free graph is $3$-colorable.......We give a short proof of the result that every planar graph of girth $5$is $3$-choosable and hence also of Gr\\"{o}tzsch's theorem saying that everyplanar triangle-free graph is $3$-colorable....
Some functional limit theorems for compound Cox processes
Korolev, Victor Yu.; Chertok, A. V.; Korchagin, A. Yu.; Kossova, E. V.; Zeifman, Alexander I.
2016-06-01
An improved version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with variance-mean mixed normal distributions, in particular, to stable Lévy processes.
Numerical cubature from Archimedes' hat-box theorem
Kuperberg, Greg
2004-01-01
Archimedes' hat-box theorem states that uniform measure on a sphere projects to uniform measure on an interval. This fact can be used to derive Simpson's rule. We present various constructions of, and lower bounds for, numerical cubature formulas using moment maps as a generalization of Archimedes' theorem. We realize some well-known cubature formulas on simplices as projections of spherical designs. We combine cubature formulas on simplices and tori to make new formulas on spheres. In partic...
An $\\alpha$-stable limit theorem under sublinear expectation
Erhan Bayraktar; Alexander Munk
2014-01-01
For $\\alpha\\in (1,2)$, we present a generalized central limit theorem for $\\alpha$-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, provided a mild but natural additional condition holds. Our approach contrasts with previous arguments for the result in the linear setting which have typically relied ...
Conformal Killing vector fields and a virial theorem
Cariñena, José F; Martínez, Eduardo; Santos, Patrícia
2014-01-01
The virial theorem is formulated both intrinsically and in local coordinates for a Lagrangian system of mechanical type on a Riemann manifold. An import case studied in this paper is that of an affine virial function associated to a vector field on the configuration manifold. The special cases of a virial function associated to a Killing, a homothetic and a conformal Killing vector field are considered and the corresponding virial theorems are established for this type of functions.
Generalization of the hypervirial and Feynman-Hellman theorems
Nadareishvili, Teimuraz
2013-01-01
Using well-known methods we generalize (hyper)virial theorems to case of singular potential. Discussion is carried on for most general second order differential equation, which involves all physically interesting cases, such as Schr\\"odinger and two-body Klein-Gordon equations with singular potentials. Some physical consequences are discussed. The connection with Feynman-Hellmann like theorems are also considered and some relevant differences are underlined.
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The classical Hardy theorem asserts that f and its Fourier transform (f) can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU(1, 1) there are infinitely many "good"functions in the sense that f and its spherical Fourier transform (f) both have good decay.In this paper, we shall characterize such functions on SU(1,1).
Generalized Panofsky-Wenzel theorem and hybrid coupling
Smirnov, A. V.
2001-08-01
The Panofsky-Wenzel theorem is reformulated for the case in which phase slippage between the wave and beam is not negligible. The extended theorem can be applied in analysis of detuned waveguides, RF injectors, bunchers, some tapered waveguides or high-power sources and multi-cell cavities for dipole and higher order modes. As an example, the relative contribution of the Lorentz' component of the deflecting force is calculated for a conventional circular disk-loaded waveguide.
Generalized Panofsky-Wenzel theorem and hybrid coupling
Smirnov, A V
2001-01-01
The Panofsky-Wenzel theorem is reformulated for the case in which phase slippage between the wave and beam is not negligible. The extended theorem can be applied in analysis of detuned waveguides, RF injectors, bunchers, some tapered waveguides or high-power sources and multi-cell cavities for dipole and higher order modes. As an example, the relative contribution of the Lorentz' component of the deflecting force is calculated for a conventional circular disk-loaded waveguide.
Hojman's conservation theorems for generalized Raitzin canonical equations of motion
Institute of Scientific and Technical Information of China (English)
Qiao Yong-Fen; Li Ren-Jie; Zhao Shu-Hong
2005-01-01
Using the Lie symmetry under infinitesimal transformations in which the time is not variable, Hojman's conservation theorems for Raitzin's canonical equations of motion in generalized classical mechanics are studied. The generalized Raitzin's canonical equations of motion are established. The determining equations of Lie symmetry under infinitesimal transformations are given. The Hojman conservation theorems of the system are established. Finally, an example is also presented to illustrate the application of the result.
No-broadcasting theorem and its classical counterpart.
Kalev, Amir; Hen, Itay
2008-05-30
Although it is widely accepted that "no-broadcasting"-the nonclonability of quantum information-is a fundamental principle of quantum mechanics, an impossibility theorem for the broadcasting of general density matrices has not yet been formulated. In this Letter, we present a general proof for the no-broadcasting theorem, which applies to arbitrary density matrices. The proof relies on entropic considerations, and as such can also be directly linked to its classical counterpart, which applies to probabilistic distributions of statistical ensembles.
On Kummer's second theorem involving product of generalized hypergeometric series
Directory of Open Access Journals (Sweden)
Arjun K. Rathie
1995-11-01
Full Text Available The aim of this paper is to obtain single series expression for e-x/2 1F1(α;2 α+i;x for i=1 and i=-1. For i=0, we have the well known Kummer's second theorem. The results are derived with the help of generalized Gauss's second summation theorem recently obtained by Lavoie, Grondin and Rathie.
A Z_m Ham Sandwich Theorem for Complex Measures
Simon, Steven
2010-01-01
A "ham sandwich" theorem is derived for n complex measures on C^n. For each integer m >= 2, a complex hyperplane H and m corresponding "1/m" sectors are found, satisfying the condition that the "rotational average" of the measures of these sectors is zero. The result can be seen as a kind of rotational equipartition of each measure by a regular "m-fan". The proof of the theorem is topological, based on a Z_m version of the Borsuk-Ulam theorem for punctured (2n - 1)-spheres in C^n. Any finite real measure on R^{2n} can be seen as a type of complex measure on C^n, so the theorem can be applied to n finite measures on R^{2n}. In the case that m = 3, this shows the existence of a regular 3-fan which trisects each of the n measures. Similarly, any pair of finite real measures on R^{2n} can be naturally identified with a complex measure on C^n, so the theorem applies to 2n finite measures on R^{2n}. This yields a rotational condition on pairs of finite measures, and the original ham sandwich theorem for R^{2n} is r...
The strength of Ramsey Theorem for coloring relatively large sets
Carlucci, Lorenzo
2012-01-01
We characterize the computational content and the proof-theoretic strength of a Ramsey-type theorem for bi-colorings of so-called {\\em exactly large} sets. An {\\it exactly large} set is a set $X\\subset\\Nat$ such that $\\card(X)=\\min(X)+1$. The theorem we analyze is as follows. For every infinite subset $M$ of $\\Nat$, for every coloring $C$ of the exactly large subsets of $M$ in two colors, there exists and infinite subset $L$ of $M$ such that $C$ is constant on all exactly large subsets of $L$. This theorem is essentially due to Pudl\\`ak and R\\"odl and independently to Farmaki. We prove that --- over Computable Mathematics --- this theorem is equivalent to closure under the $\\omega$ Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey...
New Generalized L-KKM Type Theorems in Topological Spaces with Applications
Institute of Scientific and Technical Information of China (English)
Min FANG
2013-01-01
In this paper,some new generalized L-KKM type theorems with finitely open values and with finitely closed values are established without any convexity structure in topological spaces.As applications,some new matching theorem,fixed point theorem and existence theorem of equilibrium problem with lower and upper bounds are also given under some suitable conditions.These theorems presented in this paper unify and generalize some corresponding known results in recent literatures.
[Health care systems and impossibility theorems].
Penchas, Shmuel
2004-02-01
results are Kurt Godel's seminal paper in 1931: "Ueber formal unentscheidbare Saetze der Principia Mathematica and verwandter System I" and Arrow's Nobel Prize winning "Impossibility Theorem" (Social Choice and Individual Values, 1951). Godel showed, unequivocally, that there is an enormous gap between what is being perceived as truth and what in fact can be proven as such. Arrow showed that the translation of individual preferences into a social order is impossible--except in a dictatorship. The unsolved controversies concerning the desirable or ideal structure of health care systems are impinged upon by these findings generally, and, in the case of the impossibility theorem, also directly. There is the impossibility of aggregating preferences and, at a deeper level, the impossibility of defining certain fundamental values, coupled with the problematic use of certain words, the absence of the possibility of creating, on a logically defined base, a complex system, complete and comprehensive in its own right. This is added to the fact that according to the elaboration by Stephen Wolfram in "A New Kind of Science", it is not easy to reduce complicated systems to simple components and to predict the continuation of their development even from simple basic laws without complicated calculations. All of these factors impede the construction of satisfying health care systems and leave obvious problems which overshadow the structure and the operation of health care systems.
KKM Type Theorems and Coincidence Theorems in Topological Spaces%拓扑空间中的KKM型定理和重合点定理
Institute of Scientific and Technical Information of China (English)
郑莲; 丁协平
2008-01-01
A class of finitely continuous topological spaces(in short,FC-spaces)is introduced.Some new KKM type theorems and coincidence theorems involving admissible set-valued mappings and the set-valued mapping with compactly local intersection property are proved in FCspaces. As applications,some new fixed point theorems are obtained in FC-spaces.These theorems improve and generalize many known results in recent literature.
Gleason's Theorem for Rectangular JBW-Triples
Edwards, C. Martin; Rüttimann, Gottfried T.
bounded sesquilinear functionals φm on pAp×qAq with the property that the action of the centroid Z(B) of B commutes with the adjoint operation. When B is a complex Hilbert space of dimension greater than two, this result reduces to Gleason's Theorem.
Entanglement in Classical Optics
Ghose, Partha
2013-01-01
The emerging field of entanglement or nonseparability in classical optics is reviewed, and its similarities with and differences from quantum entanglement clearly pointed out through a recapitulation of Hilbert spaces in general, the special restrictions on Hilbert spaces imposed in quantum mechanics and the role of Hilbert spaces in classical polarization optics. The production of Bell-like states in classical polarization optics is discussed, and new theorems are proved to discriminate between separable and nonseparable states in classical wave optics where no discreteness is involved. The influence of the Pancharatnam phase on a classical Bell-like state is deived. Finally, to what extent classical polarization optics can be used to simulate quantum information processing tasks is also discussed. This should be of great practical importance because coherence and entanglement are robust in classical optics but not in quantum systems.
A homological selection theorem implying a division theorem for Q-manifolds
2015-01-01
We prove that a space ▫$M$▫ with the Disjoint Disk Property is a ▫$Q$▫-manifold if and only if ▫$M times X$▫ is a ▫$Q$▫-manifold for some ▫$C$▫-space ▫$X$▫. This implies that the product ▫$M times I^2$▫ of a space $M$ with the disk is a ▫$Q$▫-manifold if and only if ▫$M times X$▫ is a ▫$Q$▫-manifold for some ▫$C$▫-space ▫$X$▫. The proof of these theorems exploits the homological characterization of ▫$Q$▫-manifolds due to R. J. Daverman and J. J. Walsh, combined with the existence of ▫$G$▫-sta...
Simulation of biological evolution and the NFL theorems.
Meester, Ronald
2009-09-01
William Dembski (No free lunch: why specified complexity cannot be purchased without intelligence, 2002) claimed that the NFL theorems from optimization theory render darwinian biological evolution impossible. Häggström (Biology and Philosophy 22:217-230, 2007) argued that the NFL theorems are not relevant for biological evolution at all, since the assumptions of the NFL theorems are not met. Although I agree with Häggström (Biology and Philosophy 22:217-230, 2007), in this article I argue that the NFL theorems should be interpreted as dealing with an extreme case within a much broader context. This broader context is in fact relevant for scientific research of certain evolutionary processes; not in the sense that the theorems can be used to draw conclusions about any intelligent design inference, but in the sense that it helps us to interpret computer simulations of evolutionary processes. As a result of this discussion, I will argue that from simulations, we do not learn much about how complexity arises in the universe. This position is in contrast with certain claims in the literature that I will discuss.
Formalization of the Integral Calculus in the PVS Theorem Prover
Directory of Open Access Journals (Sweden)
Ricky Wayne Butler
2009-04-01
Full Text Available The PVS Theorem prover is a widely used formal verification tool used for the analysis of safetycritical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht’s classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.
Experimental Test of Quantum No-Hiding Theorem
Samal, Jharana Rani; Kumar, Anil
2010-01-01
Linearity and unitarity are two fundamental tenets of quantum theory. Any consequence that follows from these must be respected in the quantum world. The no-cloning theorem and the no-deleting theorem are the consequences of the linearity and the unitarity. Together with the stronger no-cloning theorem they provide permanence to quantum information, thus, suggesting that in the quantum world information can neither be created nor be destroyed. In this sense quantum information is robust, but at the same time it is also fragile because any interaction with the environment may lead to loss of information. Recently, another fundamental theorem was proved, namely, the no-hiding theorem that addresses precisely the issue of information loss. It says that if any physical process leads to bleaching of quantum information from the original system, then it must reside in the rest of the universe with no information being hidden in the correlation between these two subsystems. This has applications in quantum teleporta...
On local-hidden-variable no-go theorems
Methot, A. A.
2006-06-01
The strongest attack against quantum mechanics came in 1935 in the form of a paper by Einstein, Podolsky, and Rosen. It was argued that the theory of quantum mechanics could not be called a complete theory of Nature, for every element of reality is not represented in the formalism as such. The authors then put forth a proposition: we must search for a theory where, upon knowing everything about the system, including possible hidden variables, one could make precise predictions concerning elements of reality. This project was ultimately doomed in 1964 with the work of Bell, who showed that the most general local hidden variable theory could not reproduce correlations that arise in quantum mechanics. There exist mainly three forms of no-go theorems for local hidden variable theories. Although almost every physicist knows the consequences of these no-go theorems, not every physicist is aware of the distinctions between the three or even their exact definitions. Thus, we will discuss here the three principal forms of no-go theorems for local hidden variable theories of Nature. We will define Bell theorems, Bell theorems without inequalities, and pseudo-telepathy. A discussion of the similarities and differences will follow.
Generalized Fourier slice theorem for cone-beam image reconstruction.
Zhao, Shuang-Ren; Jiang, Dazong; Yang, Kevin; Yang, Kang
2015-01-01
The cone-beam reconstruction theory has been proposed by Kirillov in 1961, Tuy in 1983, Feldkamp in 1984, Smith in 1985, Pierre Grangeat in 1990. The Fourier slice theorem is proposed by Bracewell 1956, which leads to the Fourier image reconstruction method for parallel-beam geometry. The Fourier slice theorem is extended to fan-beam geometry by Zhao in 1993 and 1995. By combining the above mentioned cone-beam image reconstruction theory and the above mentioned Fourier slice theory of fan-beam geometry, the Fourier slice theorem in cone-beam geometry is proposed by Zhao 1995 in short conference publication. This article offers the details of the derivation and implementation of this Fourier slice theorem for cone-beam geometry. Especially the problem of the reconstruction from Fourier domain has been overcome, which is that the value of in the origin of Fourier space is 0/0. The 0/0 type of limit is proper handled. As examples, the implementation results for the single circle and two perpendicular circle source orbits are shown. In the cone-beam reconstruction if a interpolation process is considered, the number of the calculations for the generalized Fourier slice theorem algorithm is O(N^4), which is close to the filtered back-projection method, here N is the image size of 1-dimension. However the interpolation process can be avoid, in that case the number of the calculations is O(N5).
The Goldstone equivalence theorem and AdS/CFT
Energy Technology Data Exchange (ETDEWEB)
Anand, Nikhil; Cantrell, Sean [Department of Physics & Astronomy, Johns Hopkins University,Baltimore, MD 21218 (United States)
2015-08-03
The Goldstone equivalence theorem allows one to relate scattering amplitudes of massive gauge fields to those of scalar fields in the limit of large scattering energies. We generalize this theorem under the framework of the AdS/CFT correspondence. First, we obtain an expression of the equivalence theorem in terms of correlation functions of creation and annihilation operators by using an AdS wave function approach to the AdS/CFT dictionary. It is shown that the divergence of the non-conserved conformal current dual to the bulk gauge field is approximately primary when computing correlators for theories in which the masses of all the exchanged particles are sufficiently large. The results are then generalized to higher spin fields. We then go on to generalize the theorem using conformal blocks in two and four-dimensional CFTs. We show that when the scaling dimensions of the exchanged operators are large compared to both their spins and the dimension of the current, the conformal blocks satisfy an equivalence theorem.
Formal analysis of electromagnetic optics
Khan-Afshar, Sanaz; Hasan, Osman; Tahar, Sofiène
2014-09-01
Optical systems are increasingly being used in safety-critical applications. Due to the complexity and sensitivity of optical systems, their verification raises many challenges for engineers. Traditionally, the analysis of such systems has been carried out by paper-and-pencil based proofs and numerical computations. However, these techniques cannot provide accurate results due to the risk of human error and inherent approximations of numerical algorithms. In order to overcome these limitations, we propose to use theorem proving (i.e., a computer-based technique that allows to express mathematical expressions and reason about their correctness by taking into account all the details of mathematical reasoning) as a complementary approach to improve optical system analysis. This paper provides a higher-order logic (a language used to express mathematical theories) formalization of electromagnetic optics in the HOL Light theorem prover. In order to demonstrate the practical effectiveness of our approach, we present the analysis of resonant cavity enhanced photonic devices.
A novel sampling theorem on the rotation group
McEwen, J D; Leistedt, B; Peiris, H V; Wiaux, Y
2015-01-01
We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by associating the rotation group with the three-torus through a periodic extension. Our sampling theorem requires $4L^3$ samples to capture all of the information content of a signal band-limited at $L$, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as $O(L^4)$, compared to the naive scaling of $O(L^6)$. For the common case of a low directional band-limit $N$, complexity is reduced to $O(N L^3)$. Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.
The Measure-theoretic Identity Underlying Transient Fluctuation Theorems
Shargel, Benjamin Hertz
2010-01-01
We prove a measure-theoretic identity that underlies all transient fluctuation theorems (TFTs) for entropy production and dissipated work in inhomogeneous deterministic and stochastic processes, including those of Evans and Searles, Crooks, and Seifert. The identity is used to deduce a tautological physical interpretation of TFTs in terms of the arrow of time, and its generality reveals that the self-inverse nature of the various trajectory and process transformations historically relied upon to prove TFTs, while necessary for these theorems from a physical standpoint, is not necessary from a mathematical one. The moment generating functions of thermodynamic variables appearing in the identity are shown to converge in general only in a vertical strip in the complex plane, with the consequence that a TFT that holds over arbitrary timescales may fail to give rise to an asymptotic fluctuation theorem for any possible speed of the corresponding large deviation principle. The case of strongly biased birth-death ch...
Bell on Bell's theorem: The changing face of nonlocality
Brown, Harvey R
2015-01-01
Between 1964 and 1990, the notion of nonlocality in Bell's papers underwent a profound change as his nonlocality theorem gradually became detached from quantum mechanics, and referred to wider probabilistic theories involving correlations between separated beables. The proposition that standard quantum mechanics is itself nonlocal (more precisely, that it violates `local causality') became divorced from the Bell theorem per se from 1976 on, although this important point is widely overlooked in the literature. In 1990, the year of his death, Bell would express serious misgivings about the mathematical form of the local causality condition, and leave ill-defined the issue of the consistency between special relativity and violation of the Bell-type inequality. In our view, the significance of the Bell theorem, both in its deterministic and stochastic forms, can only be fully understood by taking into account the fact that a fully Lorentz-covariant version of quantum theory, free of action-at-a-distance, can be a...
Note on Identities Inspired by New Soft Theorems
Rao, Junjie
2016-01-01
The new soft theorems, for both gravity and gauge amplitudes, have inspired a number of works, including the discovery of new identities related to amplitudes. In this note, we present the proof and discussion for two sets of identities. The first set includes an identity involving the half-soft function which had been used in the soft theorem for one-loop rational gravity amplitudes, and another simpler identity as its byproduct. The second set includes two identities involving the KLT momentum kernel, as the consistency conditions of the KLT relation plus soft theorems for both gravity and gauge amplitudes. We use the CHY formulation to prove the first identity, and transform the second one into a convenient form for future discussion.
Quantum de Finetti theorem in phase-space representation
Leverrier, Anthony; Cerf, Nicolas J.
2009-07-01
The quantum versions of de Finetti’s theorem derived so far express the convergence of n -partite symmetric states, i.e., states that are invariant under permutations of their n parties, toward probabilistic mixtures of independent and identically distributed (IID) states of the form σ⊗n . Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads to a quantum de Finetti theorem particularly relevant to continuous-variable systems. Specifically, an n -mode bosonic state that is invariant with respect to this continuous symmetry in phase space is proven to converge toward a probabilistic mixture of IID Gaussian states (actually, n identical thermal states).
Asymptotic symmetries of QED and Weinberg's soft photon theorem
Campiglia, Miguel
2015-01-01
Various equivalences between so-called soft theorems which constrain scattering amplitudes and Ward identities related to asymptotic symmetries have recently been established in gauge theories and gravity. So far these equivalences have been restricted to the case of massless matter fields, the reason being that the asymptotic symmetries are defined at null infinity. The restriction is however unnatural from the perspective of soft theorems which are insensitive to the masses of the external particles. In this work we remove the aforementioned restriction in the context of scalar QED. Inspired by the radiative phase space description of massless fields at null infinity, we introduce a manifold description of time-like infinity on which the asymptotic phase space for massive fields can be defined. The "angle dependent" large gauge transformations are shown to have a well defined action on this phase space, and the resulting Ward identities are found to be equivalent to Weinberg's soft photon theorem.
Limit theorems for multi-indexed sums of random variables
Klesov, Oleg
2014-01-01
Presenting the first unified treatment of limit theorems for multiple sums of independent random variables, this volume fills an important gap in the field. Several new results are introduced, even in the classical setting, as well as some new approaches that are simpler than those already established in the literature. In particular, new proofs of the strong law of large numbers and the Hajek-Renyi inequality are detailed. Applications of the described theory include Gibbs fields, spin glasses, polymer models, image analysis and random shapes. Limit theorems form the backbone of probability theory and statistical theory alike. The theory of multiple sums of random variables is a direct generalization of the classical study of limit theorems, whose importance and wide application in science is unquestionable. However, to date, the subject of multiple sums has only been treated in journals. The results described in this book will be of interest to advanced undergraduates, graduate students and researchers who ...
A Macro for Reusing Abstract Functions and Theorems
Directory of Open Access Journals (Sweden)
Sebastiaan J. C. Joosten
2013-04-01
Full Text Available Even though the ACL2 logic is first order, the ACL2 system offers several mechanisms providing users with some operations akin to higher order logic ones. In this paper, we propose a macro, named instance-of-defspec, to ease the reuse of abstract functions and facts proven about them. Defspec is an ACL2 book allowing users to define constrained functions and their associated properties. It contains macros facilitating the definition of such abstract specifications and instances thereof. Currently, lemmas and theorems derived from these abstract functions are not automatically instantiated. This is exactly the purpose of our new macro. instance-of-defspec will not only instantiate functions and theorems within a specification but also many more functions and theorems built on top of the specification. As a working example, we describe various fold functions over monoids, which we gradually built from arbitrary functions.