#### Sample records for bayess theorem

1. Bell's Theorem from Moore's Theorem

Fields, Chris

2012-01-01

It is shown that the restrictions of what can be inferred from classically-recorded observational outcomes that are imposed by the no-cloning theorem, the Kochen-Specker theorem and Bell's theorem also follow from restrictions on inferences from observations formulated within classical automata theory. Similarities between the assumptions underlying classical automata theory and those underlying universally-unitary quantum theory are discussed.

2. Waldhausen's Theorem

Schleimer, Saul

2009-01-01

This note is an exposition of Waldhausen's proof of Waldhausen's Theorem: the three-sphere has a single Heegaard splitting, up to isotopy, in every genus. As a necessary step we also give a sketch of the Reidemeister-Singer Theorem.

3. Frege's theorem

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

4. Noether's theorem

Noether's theorem attains its maximum simplicity and depth when formulated in curved space-time, gravitation being included. Extension to curved space-times is here made simple by the use of a formulation, for the flat case, due to Jackiw. The exposition purports to be pedagogical. (Author)

5. Probability Theories and the Justification of Theism

Portugal, Agnaldo Cuoco

2003-01-01

In the present paper I intend to analyse, criticise and suggest an alternative to Richard Swinburne"s use of Bayes"s theorem to justify the belief that there is a God. Swinburne"s contribution here lies in the scope of his project and the interpretation he adopts for Bayes"s formula, a very important theorem of the probability calculus.

6. A soft scalar theorem

We examine a soft scalar theorem which has proved useful in the evaluation of certain Feynman graphs. The use of this theorem is described in connection with the determination of the Λnphi coupling in a unified model of weak and electromagnetic interactions. (author)

7. The Parity Theorem Shuffle

Smith, Michael D.

2016-01-01

The Parity Theorem states that any permutation can be written as a product of transpositions, but no permutation can be written as a product of both an even number and an odd number of transpositions. Most proofs of the Parity Theorem take several pages of mathematical formalism to complete. This article presents an alternative but equivalent…

8. Godel's theorem is invalid

Saa, Diego

2005-01-01

Goedel's results have had a great impact in diverse fields such as philosophy, computer sciences and fundamentals of mathematics. The fact that the rule of mathematical induction is contradictory with the rest of clauses used by Goedel to prove his undecidability and incompleteness theorems is proved in this paper. This means that those theorems are invalid.

9. Algorithmic Meta-Theorems

Kreutzer, Stephan

2009-01-01

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a "logical" and a "structural" component, that is they are results of the form: every computational problem that can be formalised in a given logic L can be solved efficiently on every class C of structures satisfying certain conditions. This paper gives a survey of algorithmic meta-theorems obtained in recent years and the methods used to prove them. As many meta-theorems use results from graph minor theory, we give a brief introduction to the theory developed by Robertson and Seymour for their proof of the graph minor theorem and state the main algorithmic consequences of this theory as far as they are needed in the theory of algorithmic meta-theorems.

10. Intersection homology Kunneth theorems

Friedman, Greg

2008-01-01

Cohen, Goresky and Ji showed that there is a Kunneth theorem relating the intersection homology groups $I^{\\bar p}H_*(X\\times Y)$ to $I^{\\bar p}H_*(X)$ and $I^{\\bar p}H_*(Y)$, provided that the perversity $\\bar p$ satisfies rather strict conditions. We consider biperversities and prove that there is a K\\"unneth theorem relating $I^{\\bar p,\\bar q}H_*(X\\times Y)$ to $I^{\\bar p}H_*(X)$ and $I^{\\bar q}H_*(Y)$ for all choices of $\\bar p$ and $\\bar q$. Furthermore, we prove that the Kunneth theorem...

11. 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...

12. Virial Theorem and Hypervirial Theorem in a spherical geometry

Li, Yan; Zhang, Fu-Lin; Chen, Jing-Ling

2010-01-01

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 \\emph{perturbation theorem without wave functions}, corresponding to the Hypervirial-Hellmann-Feynman Theorem perturbation theorem of Euclidean geometry. The o...

13. Two preservation theorems

Zapletal, Jindrich

2005-01-01

I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.

14. The Fermi's Bayes Theorem

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.

15. Converse Barrier Certificate Theorem

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 by...

16. On Sylow's theorems

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....

17. Goldstone theorem revisited

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.

18. Formality theorem for gerbes

Bressler, Paul; Gorokhovsky, Alexander; Nest, Ryszard;

2015-01-01

The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an L∞L∞ deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe.......The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an L∞L∞ deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe....

19. 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)

20. Gödel's 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 of Logic and Structure. Comments are welcome.

1. Gödel's 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

2. An Improved Subadditive Ergodic Theorem

Liggett, Thomas M.

1985-01-01

A new version of Kingman's subadditive ergodic theorem is presented, in which the subadditivity and stationarity assumptions are relaxed without weakening the conclusions. This result applies to a number of situations that were not covered by Kingman's original theorem. The proof involves a rather simple reduction to the additive case, where Birkhoff's ergodic theorem can be applied.

3. How to Understand a Theorem?

Abramovitz, Buma; Berezina, Miryam; Berman, Abraham; Shvartsman, Ludmila

2009-01-01

In this article we describe the process of studying the assumptions and the conclusion of a theorem. We tried to provide the students with exercises and problems where we discuss the following questions: What are the assumptions of a theorem and what are the conclusions? What is the geometrical meaning of a theorem? What happens when one or more…

4. Virial theorem and hypervirial theorem in a spherical geometry

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)

5. Fixed points, intersection theorems, variational inequalities, and equilibrium theorems

Sehie Park

2000-07-01

Full Text Available From a fixed point theorem for compact acyclic maps defined on admissible convex sets in the sense of Klee, we first deduce collectively fixed point theorems, intersection theorems for sets with convex sections, and quasi-equilibrium theorems. These quasi-equilibrium theorems are applied to give simple and unified proofs of the known variational inequalities of the Hartman-Stampacchia-Browder type. Moreover, from our new fixed point theorem, we deduce new variational inequalities which can be used to obtain fixed point results for convex-valued maps. Finally, various general economic equilibrium theorems are deduced in the forms of the Nash type, the Tarafdar type, and the Yannelis-Prabhakar type. Our results are stated for not-necessarily locally convex topological vector spaces and for abstract economies with arbitrary number of commodities and agents. Our new results extend a lot of known works with much simpler proofs.

6. A tale of three theorems

Noether's theorem relates symmetries and conservation laws of Hamiltonians systems. Arnol'd's theorem uses those integrals of motion for the construction of sufficient stability conditions of hydrodynamical problems, which are Hamiltonian with a singular Poisson bracket. Finally, Andrews' theorem imposes restriction on the existence of Arnol'd stable solutions of symmetric systems. It is shown that denial of Andrews'theorem implies the divergence of the velocity component normal to the symmetric coordinate. This proof by reductio ad absurdum may be used to determine the strength of the symmetry breaking elements, necessary to overcome the limitations imposed by this theorem (Author)

7. The holographic F theorem

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.

8. Some Approximation Theorems

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.

9. The Recovery Theorem

Stephen A. Ross

2011-01-01

We can only estimate the distribution of stock returns but we observe the distribution of risk neutral state prices. Risk neutral state prices are the product of risk aversion - the pricing kernel - and the natural probability distribution. The Recovery Theorem enables us to separate these and to determine the market's forecast of returns and the market's risk aversion from state prices alone. Among other things, this allows us to determine the pricing kernel, the market risk premium, the pro...

10. Negishi's Theorem and Method

Vela Velupillai, K.

2011-01-01

Takashi Negishi's remarkable youthful contribution to welfare economics, general equilibrium theory and, with the benefit of hindsight, also to one strand of computable general equilibrium theory, all within the span of six pages in one article, has become one of the modern classics of general equilibrium theory and mathematical economics. Negishi's celebrated theorem and what has been called Negishi's Method have formed one foundation for computable general equilibrium theory. In this paper ...

11. Coevolution. Extending Prigogine Theorem

Leon, Antonio

2006-01-01

The formal consideration of the concept of interaction in thermodynamic analysis makes it possible to deduce, in the broadest terms, new results related to the coevolution of interacting systems, irrespective of their distance from thermodynamic equilibrium. In this paper I prove the existence of privileged coevolution trajectories characterized by the minimum joint production of internal entropy, a conclusion that extends Prigogine theorem to systems evolving far from thermodynamic equilibri...

12. Bourgain's discretization theorem

Giladi, Ohad; Schechtman, Gideon

2011-01-01

Bourgain's discretization theorem asserts that there exists a universal constant $C\\in (0,\\infty)$ with the following property. Let $X,Y$ be Banach spaces with $\\dim X=n$. Fix $D\\in (1,\\infty)$ and set $\\d= e^{-n^{Cn}}$. Assume that $\\mathcal N$ is a $\\d$-net in the unit ball of $X$ and that $\\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem. We also obtain an improvement of Bourgain's theorem in the important case when $Y=L_p$ for some $p\\in [1,\\infty)$: in this case it suffices to take $\\delta= C^{-1}n^{-5/2}$ for the same conclusion to hold true. The case $p=1$ of this improved discretization result has the following consequence. For arbitrarily large $n\\in \\N$ there exists a family $\\mathscr Y$ of $n$-point subsets of ${1,...,n}^2\\subseteq \\R^2$ such that if we write $|\\mathscr ... 13. The Non-Signalling theorem in generalizations of Bell's theorem Walleczek, Jan; Groessing, Gerhard 2014-01-01 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 even nonlocal hidden... 14. Arrow's Theorem in Judgement Aggregation Franz Dietrich; Christian List 2005-01-01 In response to recent work on the aggregation of individual judgements on logically connected propositions into collective judgements, it is often asked whether judgement aggregation is a special case of Arrowian preference aggregation. We argue the opposite. After proving a general impossibility theorem, we construct an embedding of preference aggregation into judgement aggregation and prove Arrow's theorem as a corollary of our result. Although we provide a new proof of Arrow's theorem, our... 15. Perspectives on the CAP Theorem Gilbert, Seth; Lynch, Nancy Ann 2012-01-01 Almost twelve years ago, in 2000, Eric Brewer introduced the idea that there is a fundamental trade-off between consistency, availability, and partition tolerance. This trade-off, which has become known as the CAP Theorem, has been widely discussed ever since. In this paper, we review the CAP Theorem and situate it within the broader context of distributed computing theory. We then discuss the practical implications of the CAP Theorem, and explore some general techniques for coping with the i... 16. A theorem in relativistic electronics Yongjian, Yu 1990-04-01 This paper presents a theorem that connects the dispersion relation of the Electron Cyclotron Maser' and the oscillation equation of the Gyromonotron. This theorem gives us a simple way of obtaining the osscillating characteristics of the Gyromonotron provided that dispersion relation of the ECRM is given. Though the theorem is proved only with the case of ECRM and Gyromonotron, it holds for other kinds of Electron Masers, FEL4etc. and corresponding osscillators. 17. Goedel's theorem and leapfrog We in the nuclear power industry consider ourselves to be at the forefront of civilised progress. Yet, all too often, even we ourselves don't believe our public relations statements about nuclear power. Why is this? Let us approach the question by considering Godel's Theorem. Godel's Theorem is extremely complicated mathematically, but for our purposes can be simplified to the maxim that one cannot validate a system from within that system. Scientists, especially those in the fields of astronomy and nuclear physics, have long realised the implications of Godel's Theorem. The people to whom we must communicate look to us, who officially know everything about our industry, to comfort and reassure them. And we forget that we can only comfort them by addressing their emotional needs, not by demonstrating our chilling objectivity. Let us try something completely new in communication. Instead of looking for incremental rules which will help us marginally differentiate the way we communicate about minor or major incidents, let us leapfrog across 'objectivity' to meaning and relevance. If we truly believe that nuclear energy is a good thing, this leap should not be difficult. Finally, if we as communicators are not prepared to be meaningful and relevant - not prepared to leapfrog beyond weasel terms like 'minor incident' - what does that say about the kinds of people we believe the nuclear community to be? Are nuclear people a group apart, divisible from the rest of the human race by their evil? In fact the nuclear community is a living, laughing, normal part of a whole society; and is moreover a good contributor to the technological progress that society demands. When we ourselves recognise this, we will start to communicate nuclear issues in the same language as the rest of society. We will start to speak plainly and convincingly, and our conviction will leapfrog our audience into being able to believe us 18. Cobham's theorem for substitutions Durand, Fabien 2010-01-01 The seminal theorem of Cobham has given rise during the last 40 years to a lot of works around non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let$\\alpha$and$\\beta$be two multiplicatively independent Perron numbers. Then, a sequence$x\\in A^\\mathbb{N}$, where$A$is a finite alphabet, is both$\\alpha$-substitutive and$\\beta$-substitutive if and only if$x$is ultimately periodic. 19. Fluctuation theorem in spintronics Microscopic reversibility is a key in deriving the Onsager relation. It even leads a new exact relationship that would be valid far from equilibrium, called fluctuation theorem (FT). The FT provides a precise statement for the second law of thermodynamics; and remarkably, reproduces the linear response theory. We consider the FT in the spin-dependent transport and derive universal relations among nonlinear spin and charge transport coefficients. We apply the relations to a quantum dot embedded in a two-terminal Aharonov-Bohm interferometer and check that the relations are satisfied. 20. The Clique Density Theorem Reiher, Christian 2012-01-01 Tur\\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer$r \\geq 2$every graph on$n$vertices with more than${\\tfrac{r-2}{2(r-1)}\\cdot n^2}$edges contains a clique of size$r$, i.e.,$r$mutually adjacent vertices. The corresponding extremal graphs are balanced$(r-1)$-partite graphs. The question as to how many such$r$-cliques appear at least in any$n$-vertex graph with$\\gamma n^2$edges has been intensively studied in the literature. In particula... 1. Abelian theorems for Whittaker transforms Richard D. Carmichael 1987-01-01 Full Text Available Initial and final value Abelian theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches 0 or ∞ in absolute value inside a wedge region in the right half plane. 2. Geometry of the Adiabatic Theorem Lobo, Augusto Cesar; Ribeiro, Rafael Antunes; Ribeiro, Clyffe de Assis; Dieguez, Pedro Ruas 2012-01-01 We present a simple and pedagogical derivation of the quantum adiabatic theorem for two-level systems (a single qubit) based on geometrical structures of quantum mechanics developed by Anandan and Aharonov, among others. We have chosen to use only the minimum geometric structure needed for the understanding of the adiabatic theorem for this case.… 3. Andreev's Theorem on hyperbolic polyhedra Roeder, R K W; Dunbar, W D; Roeder, Roland K. W.; Hubbard, John H.; Dunbar, William D. 2004-01-01 In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron,$C$, Andreev's Theorem provides five classes of linear inequalities, depending on$C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing$C$with the assigned dihedral angles. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev's Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev's Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev's proof of existence and also provide a readable new proof of the other parts of the proof of And... 4. Some Theorems on Generalized Basic Hypergeometric Series 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. 5. Herbrand's Fundamental Theorem - an encyclopedia article Wirth, Claus-Peter 2015-01-01 Herbrand's Fundamental Theorem provides a constructive characterization of derivability in first-order predicate logic by means of sentential logic. Sometimes it is simply called "Herbrand's Theorem", but the longer name is preferable as there are other important "Herbrand theorems" and Herbrand himself called it "Th\\'eor\\eme fondamental". It was ranked by Bernays [1957] as follows: "In its proof-theoretic form, Herbrand's Theorem can be seen as the central theorem of predicate logic. It exp... 6. 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. 7. 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 8. -Dimensional Fractional Lagrange's Inversion Theorem 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. 9. Opechowski's theorem and commutator groups It is shown that the conditions of application of Opechowski's theorem for double groups of subgroups of O(3) are directly associated to the structure of their commutator groups. Some characteristics of the structure of classes are also discussed. (Author) 10. KAM Theorem and Renormalization Group E. Simone; Kupiainen, A. 2007-01-01 We give an elementary proof of the analytic KAM theorem by reducing it to a Picard iteration of a PDE with quadratic nonlinearity, the so called Polchinski renormalization group equation studied in quantum field theory. 11. The Kramer sampling theorem revisited García García, Antonio; Hernandez Medina, Miguel Angel; Muñoz Bouto, María José 2013-01-01 The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the pa... 12. Complex extension of Wigner's theorem Brody, Dorje C 2013-01-01 Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorphically projective (complex geodesic-curves preserving) transformation on a quantum state space must be generated by a Hamiltonian that is not necessarily Hermitian. 13. Kazhdan's Theorem on Arithmetic Varieties Milne, J S 2001-01-01 Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again arithmetic. This article simplifies Kazhdan's proof. In particular, it avoids recourse to the classification theorems. It was originally completed on March 28, 1984, and distribu... 14. 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... 15. Acceptable Complexity Measures of Theorems Grenet, Bruno 2009-01-01 In 1931, G\\"odel presented in K\\"onigsberg his famous Incompleteness Theorem, stating that some true mathematical statements are unprovable. Yet, this result gives us no idea about those independent (that is, true and unprovable) statements, about their frequency, the reason they are unprovable, and so on. Calude and J\\"urgensen proved in 2005 Chaitin's "heuristic principle" for an appropriate measure: the theorems of a finitely-specified theory cannot be significantly more complex than the t... 16. Goedel's Incompleteness Theorems hold vacuously Anand, Bhupinder Singh 2002-01-01 In an earlier paper, "Omega-inconsistency in Goedel's formal system: a constructive proof of the Entscheidungsproblem" (math/0206302), I argued that a constructive interpretation of Goedel's reasoning establishes any formal system of Arithmetic as omega-inconsistent. It follows from this that Goedel's Theorem VI holds vacuously. In this paper I show that Goedel's Theorem XI essentially states that, if we assume there is a P-formula [Con(P)] whose standard interpretation is equivalent to the a... 17. 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. PMID:21863837 18. Soft theorems from anomalous symmetries Huang, Yu-tin 2015-01-01 We discuss constraints imposed by soft limits for effective field theories arising from symmetry breaking. In particular, we consider those associated with anomalous conformal symmetry as well as duality symmetries in supergravity. We verify these soft theorems for the dilaton effective action relevant for the a-theorem, as well as the one-loop effective action for N=4 supergravity. Using the universality of leading transcendental coefficients in the alpha' expansion of string theory amplitudes, we study the matrix elements of operator R^4 with half maximal supersymmetry. We construct the non-linear completion of R^4 that satisfies both single and double soft theorems up to seven points. This supports the existence of duality invariant completion of R^4. 19. 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. 20. Nonperturbative Adler-Bardeen theorem The Adler-Bardeen theorem has been proven only as a statement valid at all orders in perturbation theory, without any control on the convergence of the series. In this paper we prove a nonperturbative version of the Adler-Bardeen theorem in d=2 by using recently developed technical tools in the theory of Grassmann integration. The proof is based on the assumption that the boson propagator decays fast enough for large momenta. If the boson propagator does not decay, as for Thirring contact interactions, the anomaly in the WI (Ward Identities) is renormalized by higher order contributions 1. Two extensions of Ramsey's theorem Conlon, David; Fox, Jacob; Sudakov, Benny 2011-01-01 Ramsey’s theorem, in the version of Erdős and Szekeres, states that every$2$-coloring of the edges of the complete graph on$\\{1,2,\\ldots,n\\}$contains a monochromatic clique of order$({1}/{2})\\log n$. In this article, we consider two well-studied extensions of Ramsey’s theorem. Improving a result of Rödl, we show that there is a constant$c\\gt 0$such that every$2$-coloring of the edges of the complete graph on$\\{2,3,\\ldots,n\\}$contains a monochromatic clique$S$for which the sum of... 2. 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. 3. On the Putnam-Fuglede theorem Yin Chen 2004-01-01 Full Text Available We extend the Putnam-Fuglede theorem and the second-degree Putnam-Fuglede theorem to the nonnormal operators and to an elementary operator under perturbation by quasinilpotents. Some asymptotic results are also given. 4. 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.) 5. Global Lie-Tresse theorem Kruglikov, Boris 2011-01-01 We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of polynomial-rational differential invariants and invariant derivations. 6. Birkhoff Theorems in General Relativity Torre, Charles G. 2014-01-01 In the following Maple worksheet I demonstrate three versions of Birkhoff's theorem, which is a characterization of spherically symmetric solutions of the Einstein equations. The three versions considered here correspond to taking the "Einstein equations" to be: (1) the vacuum Einstein equations; (2) the Einstein equations with a cosmological constant (3) the Einstein-Maxwell equations. 7. Microwave electronics Slater's perturbation theorem Slater's perturbation theorem is one of the most useful for both experiments and theories of microwave electronics. In particular, this is applied to measurements of the field strengths in standing-wave systems. Since a traveling wave can be represented by a linear combination of two standing waves, the field measurement is also possible in a traveling-wave system. The theorem tells us the amount of the shift in a resonant frequency arising from a metallic body. Since the amount is dependent upon the square of the electric and magnetic field strengths at the metallic body, one can obtain the field strengths at the metallic body from the measured frequency shift. First the theorem is derived in Sec. 2. We then discuss the implications of the theorem by deriving it intuitively in Sec. 3. The perturbation of the field due to a metallic body is described in Sec. 4, where the frequency shift is actually related to the field strengths. In Sec. 5, we describe how to determine the impedance by using the data thus measured. Examples of field measurement are shown in Sec. 6 together with the impedance measurement. (author) 8. JACKSON'S THEOREM FOR COMPACT GROUPS 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. 9. Illustrating the Central Limit Theorem Corcoran, Mimi 2016-01-01 Statistics is enjoying some well-deserved limelight across mathematics curricula of late. Some statistical concepts, however, are not especially intuitive, and students struggle to comprehend and apply them. As an AP Statistics teacher, the author appreciates the central limit theorem as a foundational concept that plays a crucial role in… 10. Discovering the Inscribed Angle Theorem Roscoe, Matt B. 2012-01-01 Learning to play tennis is difficult. It takes practice, but it also helps to have a coach--someone who gives tips and pointers but allows the freedom to play the game on one's own. Learning to act like a mathematician is a similar process. Students report that the process of proving the inscribed angle theorem is challenging and, at times,… 11. Almost Subadditive Extensions of Kingman's Ergodic Theorem Schurger, Klaus 1991-01-01 Based on two notions of almost subadditivity which were introduced by Derriennic and Schurger, two a.s. limit theorems are proved which both generalize Kingman's subadditive ergodic theorem. These results, being valid under weak moment conditions, are obtained by short proofs. One of these proofs is completely elementary and does not even make use of Birkhoff's ergodic theorem which, instead, is obtained as a by-product. Finally, an improvement of Liggett's a.s. limit theorem is given. 12. 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 13. Pythagorean Theorem Proofs: Connecting Interactive Websites Lin, Cheng-Yao 2007-01-01 There are over 400 proofs of the Pythagorean Theorem. Some are visual proofs, others are algebraic. This paper features several proofs of the Pythagorean Theorem in different cultures--Greek, Chinese, Hindu and American. Several interactive websites are introduced to explore ways to prove this beautiful theorem. (Contains 8 figures.) 14. An Algebraic Identity Leading to Wilson Theorem Ruiz, Sebastian Martin 2004-01-01 In most text books on number theory Wilson Theorem is proved by applying Lagrange theorem concerning polynomial congruences.Hardy and Wright also give a proof using cuadratic residues. In this article Wilson theorem is derived as a corollary to an algebraic identity. 15. A generalized no-broadcasting theorem Barnum, H.; Barrett, J; Leifer, M.; Wilce, A. 2007-01-01 We prove a generalized version of the no-broadcasting theorem, applicable to essentially \\emph{any} nonclassical finite-dimensional probabilistic model satisfying a no-signaling criterion, including ones with super-quantum'' 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. 16. Tight closure and vanishing theorems Tight closure has become a thriving branch of commutative algebra since it was first introduced by Mel Hochster and Craig Huneke in 1986. Over the past few years, it has become increasingly clear that tight closure has deep connections with complex algebraic geometry as well, especially with those areas of algebraic geometry where vanishing theorems play a starring role. The purpose of these lectures is to introduce tight closure and to explain some of these connections with algebraic geometry. Tight closure is basically a technique for harnessing the power of the Frobenius map. The use of the Frobenius map to prove theorems about complex algebraic varieties is a familiar technique in algebraic geometry, so it should perhaps come as no surprise that tight closure is applicable to algebraic geometry. On the other hand, it seems that so far we are only seeing the tip of a large and very beautiful iceberg in terms of tight closure's interpretation and applications to algebraic geometry. Interestingly, although tight closure is a 'characteristic p' tool, many of the problems where tight closure has proved useful have also yielded to analytic (L2) techniques. Despite some striking parallels, there had been no specific result directly linking tight closure and L∼ techniques. Recently, however, the equivalence of an ideal central to the theory of tight closure was shown to be equivalent to a certain 'multiplier ideal' first defined using L2 methods. Presumably, deeper connections will continue to emerge. There are two main types of problems for which tight closure has been helpful: in identifying nice structure and in establishing uniform behavior. The original algebraic applications of tight closure include, for example, a quick proof of the Hochster-Roberts theorem on the Cohen-Macaulayness of rings of invariants, and also a refined version of the Brianqon-Skoda theorem on the uniform behaviour of integral closures of powers of ideals. More recent, geometric 17. The Classical Version of Stokes' Theorem Revisited Markvorsen, Steen 2005-01-01 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......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...... general version of Stokes' theorem for differential forms on manifolds. The main points in the present paper, however, is firstly that this latter fact usually does not get within reach for students in first year calculus courses and secondly that calculus textbooks in general only just hint at the... 18. 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. 19. 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 20. A Randomized Central Limit Theorem Eliazar, Iddo; Klafter, Joseph 2010-05-01 The Central Limit Theorem (CLT), one of the most elemental pillars of Probability Theory and Statistical Physics, asserts that: the universal probability law of large aggregates of independent and identically distributed random summands with zero mean and finite variance, scaled by the square root of the aggregate-size (√{n}), is Gaussian. The scaling scheme of the CLT is deterministic and uniform - scaling all aggregate-summands by the common and deterministic factor √{n}. This Letter considers scaling schemes which are stochastic and non-uniform, and presents a "Randomized Central Limit Theorem" (RCLT): we establish a class of random scaling schemes which yields universal probability laws of large aggregates of independent and identically distributed random summands. The RCLT universal probability laws, in turn, are the one-sided and the symmetric Lévy laws. 1. A Randomized Central Limit Theorem The Central Limit Theorem (CLT), one of the most elemental pillars of Probability Theory and Statistical Physics, asserts that: the universal probability law of large aggregates of independent and identically distributed random summands with zero mean and finite variance, scaled by the square root of the aggregate-size (√(n)), is Gaussian. The scaling scheme of the CLT is deterministic and uniform - scaling all aggregate-summands by the common and deterministic factor √(n). This Letter considers scaling schemes which are stochastic and non-uniform, and presents a 'Randomized Central Limit Theorem' (RCLT): we establish a class of random scaling schemes which yields universal probability laws of large aggregates of independent and identically distributed random summands. The RCLT universal probability laws, in turn, are the one-sided and the symmetric Levy laws. 2. Bell's theorem, accountability and nonlocality Bell's theorem is a fundamental theorem in physics concerning the incompatibility between some correlations predicted by quantum theory and a large class of physical theories. In this paper, we introduce the hypothesis of accountability, which demands that it is possible to explain the correlations of the data collected in many runs of a Bell experiment in terms of what happens in each single run. Under this assumption, and making use of a recent result by Colbeck and Renner (2011 Nature Commun. 2 411), we then show that any nontrivial account of these correlations in the form of an extension of quantum theory must violate parameter independence. Moreover, we analyze the violation of outcome independence of quantum mechanics and show that it is also a manifestation of nonlocality. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell's theorem’. (paper) 3. Lectures on Fermat's last theorem The report presents the main ideas involved in the approach towards the so-called Fermat's last theorem (FLT). The discussion leads to the point where recent work of A. Wiles starts and his work is not discussed. After a short history of the FLT and of the present approach, are discussed the elliptic curves and the modular forms with their relations, the Taniyama-Shimura-Well conjecture and the FLT 4. 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... 5. Dynamic Newton-Puiseux Theorem Mannaa, Bassel; Coquand, Thierry 2013-01-01 A constructive version of Newton-Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field need not be algebraically closed and a factorization algorithm of polynomials over the base field is not needed. The extensions obtained are a type of regular algebras over the base field and the expansions are given as formal power series over these algebras. 6. A Polar de Rham Theorem Khesin, B.; Rosly, A.; Thomas, R. P. 2003-01-01 We prove an analogue of the de Rham theorem for polar homology; that the polar homology$HP_q(X)$of a smooth projective variety$X$is isomorphic to its$H^{n,n-q}$Dolbeault cohomology group. This analogue can be regarded as a geometric complexification where arbitrary (sub)manifolds are replaced by complex (sub)manifolds and de Rham's operator$d$is replaced by Dolbeault's$\\bar\\partial$. 7. Notes on the area theorem Hawking's area theorem can be understood from a quasi-stationary process in which a black hole accretes positive energy matter, independent of the details of the gravity action. I use this process to study the dynamics of the inner as well as the outer horizons for various black holes which include the recently discovered exotic black holes and three-dimensional black holes in higher derivative gravities as well as the usual BTZ black hole and the Kerr black hole in four dimensions. I find that the area for the inner horizon 'can decrease', rather than increase, with the quasi-stationary process. However, I find that the area for the outer horizon 'never decreases' such that the usual area theorem still works in our examples, though this is quite non-trivial in general. There exists an instability problem of the inner horizons but it seems that the instability is not important in my analysis. I also find a generalized area theorem by combining those of the outer and inner horizons 8. On Harnack's theorem and extensions Costa, Antonio F.; Parlier, Hugo Harnack's theorem states that the fixed points of an orientation reversing involution of a compact orientable surface of genus g are a set of k disjoint simple closed geodesic where 0≤ k≤ g+1 . The first goal of this article is to give a purely geometric, complete and self-contained proof of this fact. In the case where the fixed curves of the involution do not separate the surface, we prove an extension of this theorem, by exhibiting the existence of auxiliary invariant curves with interesting properties. Although this type of extension is well known (see, for instance, Comment. Math. Helv. 57(4): 603-626 (1982) and Transl. Math. Monogr., vol. 225, Amer. Math. Soc., Providence, RI, 2004), our method also extends the theorem in the case where the surface has boundary. As a byproduct, we obtain a geometric method on how to obtain these auxiliary curves. As a consequence of these constructions, we obtain results concerning presentations of Non-Euclidean crystallographic groups and a new proof of a result on the set of points corresponding to real algebraic curves in the compactification of the Moduli space of complex curves of genus g , overline{M_{g}} . More concretely, we establish that given two real curves there is a path in overline{M_{g}} which passes through at most two singular curves, a result of M. Seppaelae (Ann. Sci. Ecole Norm. Sup. (4), 24(5), 519-544 (1991)). 9. Carnot's theorem as Noether's theorem for thermoacoustic engines Onset in thermoacoustic engines, the transition to spontaneous self-generation of oscillations, is studied here as both a dynamical critical transition and a limiting heat engine behavior. The critical transition is interesting because it occurs for both dissipative and conservative systems, with common scaling properties. When conservative, the stable oscillations above the critical point also implement a reversible engine cycle satisfying Carnot's theorem, a universal conservation law for entropy flux. While criticality in equilibrium systems is naturally associated with symmetries and universal conservation laws, these are usually exploited with global minimization principles, which dynamical critical systems may not have if dissipation is essential to their criticality. Acoustic heat engines furnish an example connecting equilibrium methods with dynamical and possibly even dissipative critical transitions: A reversible engine is shown to map, by a change of variables, to an equivalent system in apparent thermal equilibrium; a Noether symmetry in the equilibrium field theory implies Carnot's theorem for the engine. Under the same association, onset is shown to be a process of spontaneous symmetry breaking and the scaling of the quality factor predicted for both the reversible and irreversible engines is shown to arise from the Ginzburg-Landau description of the broken phase. copyright 1998 The American Physical Society 10. 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. 11. Bringing Theorem Proving to the (sonic) Masses Gallego Arias, Emilio Jesús; Pin, Benoît; Jouvelot, Pierre, 2015-01-01 We explore the intersection of interactive theorem proving and digital signal processing through the use of web-based, rich interfaces. Traditionally, the barrier to entry to interactive theorem proving has been high.Provers are complex systems using obscure programming languages, and libraries may be underdocumented and use formalisms and notations far from the standard domain-specific practice. Thus, it doesn't come at a surprise that interactive theorem proving has seldom been explored in ... 12. 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. H... 13. The Equivalence Theorem and Effective Lagrangians Grosse-Knetter, Carsten; Kuss, Ingolf 1994-01-01 We point out that the equivalence theorem, which relates the amplitude for a process with external longitudinally polarized vector bosons to the amplitude in which the longitudinal vector bosons are replaced by the corresponding pseudo-Goldstone bosons, is not valid for effective Lagrangians. However, a more general formulation of this theorem also holds for effective interactions. The generalized theorem can be utilized to determine the high-energy behaviour of scattering processes just by p... 14. The exchange fluctuation theorem in quantum mechanics Akagawa, Shiho; Hatano, Naomichi 2009-01-01 We study the heat transfer between two finite quantum systems initially at different temperatures. We find that a recently proposed fluctuation theorem for heat exchange, namely the exchange fluctuation theorem [C. Jarzynski and D. K. Wojcik, Phys. Rev. Lett. 92, 230602 (2004)], does not generally hold in the presence of a finite heat transfer as in the original form proved for weak coupling. As the coupling is weakened, the deviation from the theorem and the heat transfer vanish in the same ... 15. Expanding the Interaction Equivalency Theorem 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. 16. Shell theorem for spontaneous emission Kristensen, Philip Trøst; Mortensen, Jakob Egeberg; Lodahl, Peter; Stobbe, Søren 2013-01-01 We investigate spontaneous emission from excitons beyond the point source dipole approximation and show how the symmetry of the exciton wave function plays a crucial role. We find that for spherically symmetric wave functions, the Purcell effect is independent of the wave function size 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.... 17. Herbrand Theorems for Substructural Logics Cintula, Petr; Metcalfe, G. Berlin: Springer, 2013 - (McMillan, K.; Middeldorp, A.; Voronkov, A.), s. 584-600. (Lecture Notes in Computer Science. Advanced Research in Computing and Software Science. 8312). ISBN 978-3-642-45221-5. ISSN 0302-9743. [LPAR-19. International Conference /19./. Stellenbosch (ZA), 14.12.2013-19.12.2013] R&D Projects: GA ČR GAP202/10/1826 Institutional support: RVO:67985807 Keywords : substructural logics * residuated lattices * Herbrand theorem * Skolemization * predicate logics Subject RIV: BA - General Mathematics 18. The inverse Fueter mapping theorem Colombo, Fabrizio; Sabadini, Irene; Sommen, Franciscus 2011-01-01 In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function f of the form f = alpha + (omega) under bar beta (where alpha, beta satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function f = A + (omega) under barB (where A, B satisfy the Vekua's system) given by f(x) = Delta n-1/2 f (x) where Delta is t... 19. A generalized preimage theorem in global analysis 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. 20. Bit-Blasting ACL2 Theorems 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. 1. Cosmological Perturbations and the Weinberg Theorem Akhshik, Mohammad; Jazayeri, Sadra 2015-01-01 The celebrated Weinberg theorem in cosmological perturbation theory states that there always exist two adiabatic scalar modes in which the comoving curvature perturbation is conserved on super-horizon scales. In particular, when the perturbations are generated from a single source, such as in single field models of inflation, both of the two allowed independent solutions are adiabatic and conserved on super-horizon scales. There are few known examples in literature which violate this theorem. We revisit the theorem and specify the loopholes in some technical assumptions which violate the theorem in models of non-attractor inflation, fluid inflation, solid inflation and in the model of pseudo conformal universe. 2. Constructive and Computable Hahn-Banach Theorems for the (Second) Fundamental Theorem of Welfare Economics Vela Velupillai, K. 2014-01-01 The Hahn-Banach Theorem plays a crucial role in the second fundamental theorem of welfare economics. To date, all mathematical economics and advanced general equilibrium textbooks concentrate on using nonconstructive or incomputable versions of this celebrated theorem. In this paper we argue for the introduction of constructive or computable Hahn-Banach theorems in mathematical economics and advanced general equilibrium theory. The suggested modification would make applied and policy-oriented... 3. 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. 4. Two extensions of Ramsey's theorem Conlon, David; Sudakov, Benny 2011-01-01 Ramsey's theorem, in the version of Erd\\H{o}s and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,...,n} contains a monochromatic clique of order 1/2\\log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of R\\"odl, we show that there is a constant$c>0$such that every 2-coloring of the edges of the complete graph on \\{2, 3,...,n\\} contains a monochromatic clique S for which the sum of 1/\\log i over all vertices i \\in S is at least c\\log\\log\\log n. This is tight up to the constant factor c and answers a question of Erd\\H{o}s from 1981. Motivated by a problem in model theory, V\\"a\\"an\\"anen asked whether for every k there is an n such that the following holds. For every permutation \\pi of 1,...,k-1, every 2-coloring of the edges of the complete graph on {1, 2, ..., n} contains a monochromatic clique a_1a_{\\pi(2)+1}-a_{\\pi(2)}>...>a_{\\pi(k-1)+1}-a_{\\pi(k-1)}. That is, not only do we want a monochromatic clique, but the difference... 5. Abel's Theorem in the Noncommutative Case Leitenberger, Frank 2005-01-01 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. 6. The classical version of Stokes' Theorem revisited 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... 7. Anisotropic weak Hardy spaces and interpolation theorems 2008-01-01 In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces. 8. Convergence theorems for intermediate problems. II Beattie, C. A.; Greenlee, W. M. 2002-01-01 Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in previous literature. 9. Interpolation theorems on weighted Lorentz martingale spaces 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. 10. AN ABSTRACT ORLICS: PETTIS THEOREM AND APPLICATIONS LI RONGLU 2008-08-01 Full Text Available In this paper we establish two abstract versions of the classical Orlicz-Pettis Theorem for multiplier convergent series. We show that these abstract results yield known versions of the Orlicz-Pettis Theorem for locally convex spaces as well as versions for operator valued series. We also give applications to vector valued measures and spaces of continuous functions. 11. The Ahlfors lemma and Picard's theorems Simonič, Aleksander 2015-01-01 The article introduces Ahlfors' generalization of the Schwarz lemma. With this powerful geometric tool of complex functions in one variable, we are able to prove some theorems concerning the size of images under holomorphic mappings, including the celebrated Picard's theorems. The article concludes with a brief insight into the theory of Kobayashi hyperbolic complex manifolds. 12. Generalized Fibonacci Numbers and Blackwell's Renewal Theorem Christensen, Sören 2010-01-01 We investigate a connection between generalized Fibonacci numbers and renewal theory for stochastic processes. Using Blackwell's renewal theorem we find an approximation to the generalized Fibonacci numbers. With the help of error estimates in the renewal theorem we figure out an explicit representation. 13. Szemeredi's theorem and problems on arithmetic progressions Szemeredi's famous theorem on arithmetic progressions asserts that every subset of integers of positive asymptotic density contains arithmetic progressions of arbitrary length. His remarkable theorem has been developed into a major new area of combinatorial number theory. This is the topic of the present survey. 14. No-cloning theorem in thermofield dynamics Prudencio, Thiago 2011-01-01 Here we apply the no-cloning theorem from quantum information in the thermofield dynamics (TFD) scenario, relating the doubling procedure of TFD to a cloning machine process. As a consequence we use the no-cloning theorem to demonstrate that the thermal vaccuum state defined in TFD is necessarilly a mixed state. 15. No-cloning theorem in thermofield dynamics Prudencio, Thiago 2011-01-01 We discuss the relation between the no-cloning theorem from quantum information and the doubling procedure used in the formalism of thermofield dynamics (TFD). We also discuss how to apply the no-cloning theorem in the context of thermofield states defined in TFD. Consequences associated to mixed states, von Neumann entropy and thermofield vacuum are also addressed. 16. A New Fixed Point Theorem and Applications 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. 17. Non perturbative Adler-Bardeen Theorem Mastropietro, Vieri 2006-01-01 The Adler-Bardeen theorem has been proved only as a statement valid at all orders in perturbation theory, without any control on the convergence of the series. In this paper we prove a nonperturbative version of the Adler-Bardeen theorem ind=2$by using recently developed technical tools in the theory of Grassmann integration. 18. The Classical Version of Stokes' Theorem Revisited 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 R[superscript 3] follows from an application of Gauss' divergence theorem to a suitable modification of the vector field in a tubular shell around the… 19. A Metrized Duality Theorem for Markov Processes Kozen, Dexter; Mardare, Radu Iulian; Panangaden, Prakash 2014-01-01 We extend our previous duality theorem for Markov processes by equipping the processes with a pseudometric and the algebras with a notion of metric diameter. We are able to show that the isomorphisms of our previous duality theorem become isometries in this quantitative setting. This opens the wa... 20. A Generalization of the Prime Number Theorem Bruckman, Paul S. 2008-01-01 In this article, the author begins with the prime number theorem (PNT), and then develops this into a more general theorem, of which many well-known number theoretic results are special cases, including PNT. He arrives at an asymptotic relation that allows the replacement of certain discrete sums involving primes into corresponding differentiable… 1. Visualizing the Central Limit Theorem through Simulation Ruggieri, Eric 2016-01-01 The Central Limit Theorem is one of the most important concepts taught in an introductory statistics course, however, it may be the least understood by students. Sure, students can plug numbers into a formula and solve problems, but conceptually, do they really understand what the Central Limit Theorem is saying? This paper describes a simulation… 2. Boundary contributions to the hypervirial theorem Esteve, J. G.; Falceto, F.; Giri, Pulak Ranjan 2012-01-01 It is shown that under certain boundary conditions the virial theorem has to be modified. We analyze the origin of the extra term and compute it in particular examples. The Coulomb and harmonic oscillator with point interaction have been studied in the light of this generalization of the virial theorem. 3. A Simple Vector Proof of Feuerbach's Theorem Scheer, Michael 2011-01-01 The celebrated theorem of Feuerbach states that the nine-point circle of a nonequilateral triangle is tangent to both its incircle and its three excircles. In this note, we give a simple proof of Feuerbach's Theorem using straightforward vector computations. All required preliminaries are proven here for the sake of completeness. 4. A density Corradi-Hajnal theorem Allen, P.; Böttcher, J.; Hladký, Jan; Piguet, D. 2015-01-01 Roč. 67, č. 4 (2015), s. 721-758. ISSN 0008-414X Institutional support: RVO:67985840 Keywords : extremal graph theory * Mantel's theorem * Corradi-Hajnal theorem Subject RIV: BA - General Mathematics Impact factor: 0.765, year: 2014 http://cms.math.ca/10.4153/CJM-2014-030-6 5. New proofs of basic theorems in calculus Reem, Daniel 2007-01-01 In this note we present new proofs of three basic theorems in calculus. Although these theorems are well-known, in each proof we obtain something which seems to be unknown. We start with the Heine-Cantor theorem about uniform continuity and obtain explicitly the optimal delta for the given epsilon. We then proceed with the Weierstrass extreme value theorem and present two proofs of it: the envelope proof'' in which the largest possible maximal point is found using an envelope function, and the programmer proof'', which does not use the costume argument of proving boundedness first, and in which an explicit sequence is shown to converge monotonically to the maximal value. We finish with the intermediate value theorem, which is generalized to a class of discontinuous functions and in which the meaning of the intermediate value property is re-examined. In the end we discuss in which sense the proofs are constructive. 6. 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... 7. Dirac's theorem for random graphs Lee, Choongbum 2011-01-01 A classical theorem of Dirac from 1952 asserts that every graph on$n$vertices with minimum degree at least$\\lceil n/2 \\rceil$is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of random graph properties we prove that if$p \\gg \\log n /n$, then a.a.s. every subgraph of$G(n,p)$with minimum degree at least$(1/2+o(1))np$is Hamiltonian. Our result improves on previously known bounds, and answers an open problem of Sudakov and Vu. Both, the range of edge probability$p$and the value of the constant 1/2 are asymptotically best possible. 8. Ehrenfest Theorem in Precanonical Quantization Kanatchikov, I V 2015-01-01 We discuss the precanonical quantization of fields, which is based on the De Donder-Weyl (DW) Hamiltonian formulation and does not distinguish between the space and time variables. Classical field equations in DW Hamiltonian form are derived as the equations on the expectation values of the corresponding precanonical quantum operators. This field-theoretic generalization of the quantum mechanical Ehrenfest theorem demonstrates the consistency of three aspects of precanonical field quantization: the precanonical representation of operators in terms of the Clifford (Dirac) algebra valued partial differential operators, the Dirac-like precanonical generalization of the Schr\\"odinger equation without the distinguished time dimension, and the prescription of calculating the expectation values of operators using the Clifford-valued precanonical wave functions. 9. Singlet and triplet instability theorems Yamada, Tomonori; Hirata, So, E-mail: sohirata@illinois.edu [Department of Chemistry, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Urbana, Illinois 61801 (United States); CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012 (Japan) 2015-09-21 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. 10. Magnetohydrodynamic stability comparison theorems revisited Magnetohydrodynamic (MHD) stability comparison theorems are presented for several different plasma models, each one corresponding to a different level of collisionality: a collisional fluid model (ideal MHD), a collisionless kinetic model (kinetic MHD), and two intermediate collisionality hybrid models (Vlasov-fluid and kinetic MHD-fluid). Of particular interest is the re-examination of the often quoted statement that ideal MHD makes the most conservative predictions with respect to stability boundaries for ideal modes. Some of the models have already been investigated in the literature and we clarify and generalize these results. Other models are essentially new and for them we derive new comparison theorems. Three main conclusions can be drawn: (1) it is crucial to distinguish between ergodic and closed field line systems; (2) in the case of ergodic systems, ideal MHD does indeed make conservative predictions compared to the other models; (3) in closed line systems undergoing perturbations that maintain the closed line symmetry this is no longer true. Specifically, when the ions are collisionless and their gyroradius is finite, as in the Vlasov-fluid model, there is no compressibility stabilization. The Vlasov-fluid model is more unstable than ideal MHD. The reason for this is related to the wave-particle resonance associated with the perpendicular precession drift motion of the particles (i.e., the ExB drift and magnetic drifts), combined with the absence of any truly toroidally trapped particles. The overall conclusion is that to determine macroscopic stability boundaries for ideal modes for any magnetic geometry using a simple conservative approach, one should analyze the ideal MHD energy principle for incompressible displacements. 11. Singlet and triplet instability theorems 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 12. OTTER, Resolution Style Theorem Prover 1 - Description of program or function: OTTER (Other Techniques for Theorem-proving and Effective Research) is a resolution-style theorem-proving program for first-order logic with equality. OTTER includes the inference rules binary resolution, hyper-resolution, UR-resolution, and binary para-modulation. These inference rules take as small set of clauses and infer a clause. If the inferred clause is new and useful, it is stored and may become available for subsequent inferences. Other capabilities are conversion from first-order formulas to clauses, forward and back subsumption, factoring, weighting, answer literals, term ordering, forward and back demodulation, and evaluable functions and predicates. 2 - Method of solution: For its inference process OTTER uses the given-clause algorithm, which can be viewed as a simple implementation of the set of support strategy. OTTER maintains three lists of clauses: axioms, sos (set of support), and demodulators. OTTER is not automatic. Even after the user has encoded a problem into first-order logic or into clauses, the user must choose inference rules, set options to control the processing of inferred clauses, and decide which input formulae or clauses are to be in the initial set of support and which, if any, equalities are to be demodulators. If OTTER fails to find a proof, the user may try again different initial conditions. 3 - Restrictions on the complexity of the problem - Maxima of: 5000 characters in an input string, 64 distinct variables in a clause, 51 characters in any symbol. The maxima can be changed by finding the appropriate definition in the header.h file, increasing the limit, and recompiling OTTER. There are a few constraints on the order of commands 13. The pointwise Hellmann-Feynman theorem 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. 14. An algebraic spin and statistics theorem Guido, I D 1994-01-01 Abstract. A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption. 15. A Generalization of Chaplygin's Reducibility Theorem Fernandez, O E; Bloch, A M 2009-01-01 In this paper we study Chaplygin's Reducibility Theorem and extend its applicability to nonholonomic systems with symmetry described by the Hamilton-Poincare-d'Alembert equations in arbitrary degrees of freedom. As special cases we extract the extension of the Theorem to nonholonomic Chaplygin systems with nonabelian symmetry groups as well as Euler-Poincare-Suslov systems in arbitrary degrees of freedom. In the latter case, we also extend the Hamiltonization Theorem to nonholonomic systems which do not possess an invariant measure. Lastly, we extend previous work on conditionally variational systems using the results above. We illustrate the results through various examples of well-known nonholonomic systems. 16. 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 17. Limit theorems for 2D invasion percolation Damron, Michael 2010-01-01 We prove limit theorems and variance estimates for quantities related to ponds and outlets for 2D invasion percolation. We first exhibit several properties of a sequence (O(n)) of outlet variables, the n-th of which gives the number of outlets in the box centered at the origin of side length 2^n. The most important of these properties describe the sequence's renewal structure and exponentially fast mixing behavior. We use these to prove a central limit theorem and strong law of large numbers for (O(n)). We then show consequences of these limit theorems for the pond radii and outlet weights. 18. Quadratic Goldreich-Levin Theorems Tulsiani, Madhur 2011-01-01 Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich-Levin algorithm can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients. In the study of "quadratic Fourier analysis", higher-degree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part correspondingly weaker. For example, it has previously been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the$U^3$norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition. A key part of the algorithm is a local self-correction procedure for Re... 19. Lorentz violating kinematics: threshold theorems Baccetti, Valentina; Tate, Kyle; Visser, Matt 2012-03-01 Recent tentative experimental indications, and the subsequent theoretical speculations, regarding possible violations of Lorentz invariance have attracted a vast amount of attention. An important technical issue that considerably complicates detailed calculations in any such scenario, is that once one violates Lorentz invariance the analysis of thresholds in both scattering and decay processes becomes extremely subtle, with many new and naively unexpected effects. In the current article we develop several extremely general threshold theorems that depend only on the existence of some energy momentum relation E(p), eschewing even assumptions of isotropy or monotonicity. We shall argue that there are physically interesting situations where such a level of generality is called for, and that existing (partial) results in the literature make unnecessary technical assumptions. Even in this most general of settings, we show that at threshold all final state particles move with the same 3-velocity, while initial state particles must have 3-velocities parallel/anti-parallel to the final state particles. In contrast the various 3-momenta can behave in a complicatedand counter-intuitive manner. 20. Security Theorems via Model Theory 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. 1. TRANSVERSAL SPACES AND FIXED POINT THEOREMS Sinia N. Ješić; Milan R. Tasković; Nataša Babačev 2007-01-01 In this paper we define Transversal functional probabilistic spaces (upper and lower) as a natural extension of Metric spaces, Probabilistic metric spaces and Fuzzy metric spaces. Also, we formulate and prove some fixed and common fixed point theorems. 2. Remarks on the Cayley-Hamilton Theorem Gatto, Letterio; Scherbak, Inna 2015-01-01 We revisit the classical theorem by Cayley and Hamilton, "{\\em each endomorphism is a root of its own characteristic polynomial}", from the point of view of {\\em Hasse--Schmidt derivations on an exterior algebra} 3. Yet another proof of Szemeredi's theorem Green, Ben 2010-01-01 Using the density-increment strategy of Roth and Gowers, we derive Szemeredi's theorem on arithmetic progressions from the inverse conjectures GI(s) for the Gowers norms, recently established by the authors and Ziegler. 4. Limit Theorems in Free Probability Theory I Chistyakov, G. P.; Götze, F. 2006-01-01 Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory. 5. 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(... 6. A generalized preimage theorem in global analysis 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. 7. Interval logic. Proof theory and theorem proving Rasmussen, Thomas Marthedal 2002-01-01 Real-time systems are computer systems which have to meet real-time constraints. To increase the confidence in such systems, formal methods and formal verification are utilized. The class of logics known as interval logics can be used for expressing properties and requirements of real-time systems...... labelled natural deduction system. We conduct theoretical investigations of the systems with respect to subformula properties, proof search, etc. The generic theorem proving system Isabelle is used as a framework for encoding both proof theoretical systems. We consider a number of examples/small case....... By theorem proving we understand the activity of proving theorems of a logic with the assistance of a computer. The goal of this thesis is to improve theorem proving support for interval logics such that larger and more realistic case-studies of real-time systems can be conducted using these... 8. Lambda-mu-calculus and Bohm's theorem David, René; Py, Walter 2001-01-01 The lambda mu-calculus is an extension of the lambda-calculus that has been introduced by M. Parigot to give an algorithmic content to classical proofs. We show that Bohm's theorem fails in this calculus. 9. 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. 10. Lie Algebras and the Four Color Theorem Bar-Natan, Dror 1996-01-01 We present a reasonable'' statement about Lie algebras that is equivalent to the Four Color Theorem. The notions appearing in the statement also appear in the theory of finite-type invariants of knots (Vassiliev invariants) and 3-manifolds. 11. On the failure of Bell's theorem Bene, Gyula 1997-01-01 Using a new approach to quantum mechanics we revisit Hardy's proof for Bell's theorem and point out a loophole in it. We also demonstrate on this example that quantum mechanics is a local realistic theory. 12. Stable convergence and stable limit theorems Häusler, Erich 2015-01-01 The authors present a concise but complete exposition of the mathematical theory of stable convergence and give various applications in different areas of probability theory and mathematical statistics to illustrate the usefulness of this concept. Stable convergence holds in many limit theorems of probability theory and statistics – such as the classical central limit theorem – which are usually formulated in terms of convergence in distribution. Originated by Alfred Rényi, the notion of stable convergence is stronger than the classical weak convergence of probability measures. A variety of methods is described which can be used to establish this stronger stable convergence in many limit theorems which were originally formulated only in terms of weak convergence. Naturally, these stronger limit theorems have new and stronger consequences which should not be missed by neglecting the notion of stable convergence. The presentation will be accessible to researchers and advanced students at the master's level... 13. Slowly changing potential problems in Quantum Mechanics: Adiabatic theorems, ergodic theorems, and scattering Fishman, S.; Soffer, A. 2016-07-01 We employ the recently developed multi-time scale averaging method to study the large time behavior of slowly changing (in time) Hamiltonians. We treat some known cases in a new way, such as the Zener problem, and we give another proof of the adiabatic theorem in the gapless case. We prove a new uniform ergodic theorem for slowly changing unitary operators. This theorem is then used to derive the adiabatic theorem, do the scattering theory for such Hamiltonians, and prove some classical propagation estimates and asymptotic completeness. 14. The matrix Euler-Fermat theorem We prove many congruences for binomial and multinomial coefficients as well as for the coefficients of the Girard-Newton formula in the theory of symmetric functions. These congruences also imply congruences (modulo powers of primes) for the traces of various powers of matrices with integer elements. We thus have an extension of the matrix Fermat theorem similar to Euler's extension of the numerical little Fermat theorem 15. A Theorem on Combinatorial Group Theory 何伯和 2000-01-01 Let F= F(X) be a free group of rand n, A be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of A if x is in the normal closure of A in F(X). Finally, an application of the theorem in Heegaard splitting of 3manifolds is given. 16. On Newton’s shell theorem In the present letter, Newton’s theorem for the gravitational field outside a uniform spherical shell is considered. In particular, a purely geometric proof of proposition LXXI/theorem XXXI of Newton’s Principia, which is suitable for undergraduates and even skilled high-school students, is proposed. Minimal knowledge of elementary calculus and three-dimensional Euclidean geometry are required. (letters and comments) 17. Fluctuation theorems for a molecular refrigerator. Kim, Kyung Hyuk; Qian, Hong 2007-02-01 We extend fluctuation theorems to a molecular refrigeration system that consists of Brownian particles in a heat bath under feedback control of their velocities. Such control can actively remove heat from the bath due to an entropy-pumping mechanism [Phys. Rev. Lett. 93, 120602 (2004)]. The presence of entropy pumping in an underdamped Brownian system modifies both the Jarzynski equality and the fluctuation theorems. We discover that the entropy pumping has a dual role of work and heat. PMID:17358382 18. Levi-Civita's Theorem for Noncommutative Tori Jonathan Rosenberg 2013-11-01 Full Text Available We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas. 19. 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. 20. Levi-Civita's Theorem for Noncommutative Tori Jonathan Rosenberg 2013-01-01 We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas. 1. The two Bell's theorems of John Bell Many of the heated arguments about the meaning of ‘Bell's theorem’ arise because this phrase 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 dual assumptions of locality and determinism. His 1976 theorem is the incompatibility of quantum phenomena with the unitary property of local causality. This is contrary to Bell's own later assertions, that his 1964 theorem began with that single, and indivisible, assumption of local causality (even if not by that name). While there are other forms of Bell's theorems—which I present to explain the relation between Jarrett-completeness, ‘fragile locality’, and EPR-completeness—I maintain that Bell's two versions are the essential ones. Although the two Bell's theorems are logically equivalent, their assumptions are not, and the different versions of the theorem suggest quite different conclusions, which are embraced by different communities. For realists, the notion of local causality, ruled out by Bell's 1976 theorem, is motivated implicitly by Reichenbach's principle of common cause and explicitly by the principle of relativistic causality, and it is the latter which must be forgone. Operationalists pay no heed to Reichenbach's principle, but wish to keep the principle of relativistic causality, which, bolstered by an implicit ‘principle of agent-causation’, implies their notion of locality. Thus for operationalists, Bell's theorem is the 1964 one, and implies that it is determinism that must be forgone. I discuss why the two ‘camps’ are drawn to these different conclusions, and what can be done to increase mutual understanding. (review article) 2. A new proof of Goodstein's Theorem Perez, Juan A. 2009-01-01 Goodstein sequences are numerical sequences in which a natural number m, expressed as the complete normal form to a given base a, is modified by increasing the value of the base a by one unit and subtracting one unit from the resulting expression. As initially defined, the first term of the Goodstein sequence is the complete normal form of m to base 2. Goodstein's Theorem states that, for all natural numbers, the Goodstein sequence eventually terminates at zero. Goodstein's Theorem was origin... 3. Epistemological Consequences of the Incompleteness Theorems Raguní, Giuseppe 2016-01-01 After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main epistemological consequences of the first incompleteness Theorem for the two fundamental arithmetical theories are shown: the non-mechanizability for the truths of the first-order arithmetic and the peculiarities for the model of the second-order arithmetic. Finally, the common epistemological interpretation of the second incompleteness Theorem is... 4. Virial theorems for trapped cold atoms Werner, Félix 2008-01-01 A few small corrections We present a general virial theorem for quantum particles with arbitrary zero-range or finite-range interactions in an arbitrary external potential. We deduce virial theorems for several situations relevant to trapped cold atoms: zero-range interactions with and without Efimov effect, hard spheres, narrow Feshbach resonances, and finite-range interactions. If the scattering length$a$is varied adiabatically in the BEC-BCS crossover, we find that the trapping potent... 5. Virial theorem for radiating accretion discs Mach, Patryk 2011-01-01 A continuum version of the virial theorem is derived for a radiating self-gravitating accretion disc around a compact object. The central object is point-like, but we can avoid the regularization of its gravitational potential. This is achieved by applying a modified Pohozaev-Rellich identity to the gravitational potential of the disk only. The theorem holds for general stationary configurations, including discontinuous flows (shock waves, contact discontinuities). It is used to test numerica... 6. Shafranov's virial theorem and magnetic plasma confinement Faddeev, Ludvig; Freyhult, Lisa; Niemi, Antti J.; Rajan, Peter 2000-01-01 Shafranov's virial theorem implies that nontrivial magnetohydrodynamical equilibrium configurations must be supported by externally supplied currents. Here we extend the virial theorem to field theory, where it relates to Derrick's scaling argument on soliton stability. We then employ virial arguments to investigate a realistic field theory model of a two-component plasma, and conclude that stable localized solitons can exist in the bulk of a finite density plasma. These solitons entail a non... 7. q-Deformed Dynamics and Virial Theorem Zhang, Jian-Zu 2002-01-01 In the framework of the q-deformed Heisenberg algebra the investigation of$q$-deformation of Virial theorem explores that q-deformed quantum mechanics possesses better dynamical property. It is clarified that in the case of the zero potential the theoretical framework for the q-deformed Virial theorem is self-consistent. In the selfadjoint states the q-deformed uncertainty relation essentially deviates from the Heisenberg one. 8. The Fundamental Theorem of Vassiliev Invariants Bar-Natan, Dror; STOIMENOW, Alexander 1997-01-01 The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M. Hutchings, a geometrical approach following Kontsevich, an algebraic approach following Drinfel'd's theory of associators, and a physical approach coming from the Chern-Simons quantum field theory. Each of these approaches is unsatisfactory in one way or anothe... 9. Has the Goldstone theorem been revisited? Guerrieri, A 2014-01-01 A recent paper (arXiv:1404.5619) claimed the presence of a loophole in the current-algebra proof of Goldstone Theorem. The enforcing of manifest covariance would lead to contradictory results also in scalar theory. We show that the argument proposed is not in contradiction with covariance, thus not invalidating the theorem. Moreover, the counterexample proposed of a scalar operator with a non-zero vacuum expectation value in an unbroken theory is ill-defined. 10. Positive energy theorems in General Relativity Dain, Sergio 2013-01-01 The aim of this chapter is to present an introduction and also an overview of some of the most relevant results concerning positivity energy theorems in General Relativity. These theorems provide the answer to a long standing problem that has been proved remarkably difficult to solve. They constitute one of the major results in classical General Relativity and they uncover a deep self-consistence of the theory. 11. Integral fluctuation theorem for the housekeeping heat The housekeeping heat Qhk is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states. By analysing the evolution of its probability distribution, we prove an integral fluctuation theorem (exp[-βQhk]) = 1 valid for arbitrary-driven transitions between steady states. We discuss Gaussian limiting cases and the difference between the new theorem and both the Hatano-Sasa and the Jarzynski relation. (letter to the editor) 12. Integral fluctuation theorem for the housekeeping heat Speck, T.; Seifert, U. 2005-01-01 The housekeeping heat$Q\\hk$is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states. By analyzing the evolution of its probability distribution, we prove an integral fluctuation theorem$\\mean{\\exp[-\\beta Q\\hk]}=1$valid for arbitrary driven transitions between steady states. We discuss Gaussian limiting cases and the difference between the new theorem and both the Hatano-Sasa and the Jarzynski relation. 13. A Converse of Fermat's Little Theorem Bruckman, P. S. 2007-01-01 As the name of the paper implies, a converse of Fermat's Little Theorem (FLT) is stated and proved. FLT states the following: if p is any prime, and x any integer, then x[superscript p] [equivalent to] x (mod p). There is already a well-known converse of FLT, known as Lehmer's Theorem, which is as follows: if x is an integer coprime with m, such… 14. Sperner and KKM-type theorems on trees and cycles Niedermaier, Andrew; Su, Francis Edward 2009-01-01 In this paper we prove a new combinatorial theorem for labellings of trees, and show that it is equivalent to a KKM-type theorem for finite covers of trees and to discrete and continuous fixed point theorems on finite trees. This is in analogy with the equivalence of the classical Sperner's lemma, KKM lemma, and the Brouwer fixed point theorem on simplices. Furthermore, we use these ideas to develop new KKM and fixed point theorems for infinite covers and infinite trees. Finally, we extend the KKM theorem on trees to an entirely new KKM theorem for cycles, and discuss interesting social consequences, including an application in voting theory. 15. Combinatorial theorems in sparse random sets Conlon, D 2010-01-01 We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\\'an's theorem, Szemer\\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For instance, we extend Tur\\'an's theorem to the random setting by showing that for every epsilon > 0 and every positive integer t >= 3 there exists a constant C such that, if G is a random graph on n vertices where each edge is chosen independently with probability at least C n^{-2/(t+1)}, then, with probability tending to 1 as n tends to infinity, every subgraph of G with at least (1 - \\frac{1}{t-1} + epsilon) e(G) edges contains a copy of K_t. This is sharp up to the constant C. We also show how to prove sparse analogues of structural results, giving two main applications, a stability version of the random Tur\\'an theorem stated above and a sparse hypergraph removal lemma. Many similar results have recently been obtained independently in a different way by Schacht and by Friedgut... 16. Optical theorem detectors for active scatterers Marengo, Edwin A.; Tu, Jing 2015-10-01 We develop a new theory of the optical theorem for scalar fields in nonhomogeneous media which can be bounded or unbounded. It applies to arbitrary lossless backgrounds and quite general probing fields. The derived formulation holds for arbitrary passive scatterers, which can be dissipative, as well as for the more general class of active scatterers which are composed of a (passive) scatterer component and an active, radiating (antenna) component. The generalization of the optical theorem to active scatterers is relevant to many applications such as surveillance of active targets including certain cloaks and invisible scatterers and wireless communications. The derived theoretical framework includes the familiar real power optical theorem describing power extinction due to both dissipation and scattering as well as a novel reactive optical theorem related to the reactive power changes. The developed approach naturally leads to three optical theorem indicators or statistics which can be used to detect changes or targets in unknown complex media. The paper includes numerical simulation results that illustrate the application of the derived optical theorem results to change detection in complex and random media. 17. 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... 18. Mental Constructions for The Group Isomorphism Theorem 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. 19. 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... 20. Coincidence Theorems with Applications to Minimax Inequalities, Section Theorem, Best Approximation and Multiobjective Games in Topological Spaces 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. 1. A Unifying Impossibility Theorem for Compact Metricsocial Alternatives Space Priscilla Man; Shino Takayama 2013-01-01 In Man and Takayama (2013) (henceforth MT) we show that many classical impossibility theorems follow from three simple and intuitive axioms on the social choice correspondence when the set of social alternatives is finite. This note extends the main theorem (Theorem 1) in MT to the case where the set of social alternatives is a compact metric space. We also qualify how versions of Arrow's Impossibility Theorem and the Muller-Satterthwaite Theorem (Muller and Satterthwaite, 1977) can be obtain... 2. Stability theorems for multidimensional linear systems with variable parameters Shrivastava, S. K. 1981-01-01 A Liapunov-type approach is used to derive two equivalent theorems which govern the stability of coupled linear systems with varying multiple parameters. The theorems generalize some of the existing theorems applicable to systems with constant parameters and the Sonin-Polya theorem applicable to a single-degree-of-freedom system with variable coefficients. As an illustration, the proposed theorems are applied to mechanical systems with varying inertia, stiffness, gyroscopic, and damping terms, and velocity and position-dependent forces. 3. Theorem on magnet fringe field Transverse particle motion in particle accelerators is governed almost totally by non-solenoidal magnets for which the body magnetic field can be expressed as a series expansion of the normal (bn) and skew (an) multipoles, By + iBx = summation(bn + ian)(x + iy)n, where x, y, and z denote horizontal, vertical, and longitudinal (along the magnet) coordinates. Since the magnet length L is necessarily finite, deflections are actually proportional to ''field integrals'' such as bar BL ≡ ∫ B(x,y,z)dz where the integration range starts well before the magnet and ends well after it. For bar an, bar bn, bar Bx, and bar By defined this way, the same expansion Eq. 1 is valid and the ''standard'' approximation is to neglect any deflections not described by this expansion, in spite of the fact that Maxwell's equations demand the presence of longitudinal field components at the magnet ends. The purpose of this note is to provide a semi-quantitative estimate of the importance of |Δp∝|, the transverse deflection produced by the ion-gitudinal component of the fringe field at one magnet end relative to |Δp0|, the total deflection produced by passage through the whole magnet. To emphasize the generality and simplicity of the result it is given in the form of a theorem. The essence of the proof is an evaluation of the contribution of the longitudinal field Bx from the vicinity of one magnet end since, along a path parallel to the magnet axis such as path BC 4. Kharitonov's theorem: Generalizations and algorithms Rublein, George 1989-01-01 In 1978, the Russian mathematician V. Kharitonov published a remarkably simple necessary and sufficient condition in order that a rectangular parallelpiped of polynomials be a stable set. Here, stable is taken to mean that the polynomials have no roots in the closed right-half of the complex plane. The possibility of generalizing this result was studied by numerous authors. A set, Q, of polynomials is given and a necessary and sufficient condition that the set be stable is sought. Perhaps the most general result is due to Barmish who takes for Q a polytope and proceeds to construct a complicated nonlinear function, H, of the points in Q. With the notion of stability which was adopted, Barmish asks that the boundary of the closed right-half plane be swept, that the set G is considered = to (j(omega)(bar) - infinity is less than omega is less than infinity) and for each j(omega)(sigma)G, require H(delta) is greater than 0. Barmish's scheme has the merit that it describes a true generalization of Kharitonov's theorem. On the other hand, even when Q is a polyhedron, the definition of H requires that one do an optimization over the entire set of vertices, and then a subsequent optimization over an auxiliary parameter. In the present work, only the case where Q is a polyhedron is considered and the standard definition of stability described, is used. There are straightforward generalizations of the method to the case of discrete stability or to cases where certain root positions are deemed desirable. The cases where Q is non-polyhedral are less certain as candidates for the method. Essentially, a method of geometric programming was applied to the problem of finding maximum and minimum angular displacements of points in the Nyquist locus (Q(j x omega)(bar) - infinity is less than omega is less than infinity). There is an obvious connection with the boundary sweeping requirement of Barmish. 5. The Hellmann–Feynman theorem, the comparison theorem, and the envelope theory Claude Semay 2015-01-01 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. 6. 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. PMID:25691697 7. 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. 8. The Non-Signalling theorem in generalizations of Bell's theorem 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 9. Equilibrium fluctuation theorems compatible with anomalous response Velazquez, L.; Curilef, S. 2010-12-01 Previously, we have derived a generalization of the canonical fluctuation relation between heat capacity and energy fluctuations C = β2langδU2rang, which is able to describe the existence of macrostates with negative heat capacities C < 0. In this work, we extend our previous results for an equilibrium situation with several control parameters to account for the existence of states with anomalous values in other response functions. Our analysis leads to the derivation of three different equilibrium fluctuation theorems: the fundamental and the complementary fluctuation theorems, which represent the generalization of two fluctuation identities already obtained in previous works, and the associated fluctuation theorem, a result that has no counterpart in the framework of Boltzmann-Gibbs distributions. These results are applied to study the anomalous susceptibility of a ferromagnetic system, in particular, the case of the 2D Ising model. 10. On Bayes' theorem for improper mixtures McCullagh, Peter; 10.1214/11-AOS892 2011-01-01 Although Bayes's theorem demands a prior that is a probability distribution on the parameter space, the calculus associated with Bayes's theorem sometimes generates sensible procedures from improper priors, Pitman's estimator being a good example. However, improper priors may also lead to Bayes procedures that are paradoxical or otherwise unsatisfactory, prompting some authors to insist that all priors be proper. This paper begins with the observation that an improper measure on Theta satisfying Kingman's countability condition is in fact a probability distribution on the power set. We show how to extend a model in such a way that the extended parameter space is the power set. Under an additional finiteness condition, which is needed for the existence of a sampling region, the conditions for Bayes's theorem are satisfied by the extension. Lack of interference ensures that the posterior distribution in the extended space is compatible with the original parameter space. Provided that the key finiteness conditio... 11. Bayes' theorem: scientific assessment of experience 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. 12. 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. 13. Asymptotic symmetries and subleading soft graviton theorem Campiglia, Miguel; Laddha, Alok 2014-12-01 Motivated by the equivalence between the soft graviton theorem and Ward identities for the supertranslation symmetries belonging to the Bondi, van der Burg, Metzner and Sachs (BMS) group, we propose a new extension (different from the so-called extended BMS) of the BMS group that is a semidirect product of supertranslations and Diff(S2) . We propose a definition for the canonical generators associated with the smooth diffeomorphisms and show that the resulting Ward identities are equivalent to the subleading soft graviton theorem of Cachazo and Strominger. 14. 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 15. 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. 16. 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. 17. GENERALIZATIONS OF THE ORLICZ-PETTIS THEOREM CHRISTOPHER STUART 2005-05-01 Full Text Available The Orlicz-Pettis Theorem for locally convex spaces asserts that a series in the space which is subseries convergent in the weak topology is actually subseries convergent in the original topology of the space. A subseries convergent series can be viewed as a multiplier convergent series where the terms of the series are multiplied by elements of the scalar sequence space m0 of sequences with finite range. In this paper we show that the conclusion of the Orlicz-Pettis Theorem holds (and can be strengthened if the multiplier space m0 is replaced by a sequence space with the signed weak gliding hump property 18. Two extensions of Ramsey’s theorem Conlon, David; Fox, Jacob; Sudakov, Benny 2013-01-01 Ramsey’s theorem, in the version of Erdos and Szekeres, states that every 2-coloring of the edges of the complete graph on {1,2,…,n} contains a monochromatic clique of order (1/2)logn. In this article, we consider two well-studied extensions of Ramsey’s theorem. Improving a result of Rodl, we show that there is a constant c > 0 such that every 2-coloring of the edges of the complete graph on {2,3,…,n} contains a monochromatic clique S for which the sum of 1/logi over all vertices i ∈ S is at ... 19. 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. 20. 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. 1. Jarzynski's theorem for lattice gauge theory Caselle, Michele; Costagliola, Gianluca; Nada, Alessandro; Panero, Marco; Toniato, Arianna 2016-08-01 Jarzynski's theorem is a well-known equality in statistical mechanics, which relates fluctuations in the work performed during a nonequilibrium transformation of a system, to the free-energy difference between two equilibrium ensembles. In this article, we apply Jarzynski's theorem in lattice gauge theory, for two examples of 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 Schrödinger functional and for simulations at finite density using reweighting techniques. 2. Limit Theorems for Dispersing Billiards with Cusps Bálint, P.; Chernov, N.; Dolgopyat, D. 2011-12-01 Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting "intermittent" behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. We prove that a non-classical central limit theorem holds, with a scaling factor of {sqrt{nlog n}} replacing the standard {sqrt{n}} . We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds. 3. The Goldstone boson equivalence theorem with fermions Durand, Loyal; Riesselmann, Kurt 1995-01-01 The calculation of the leading electroweak corrections to physical transition matrix elements in powers of$M_H^2/v^2$can be greatly simplified in the limit$M_H^2\\gg M_W^2,\\, M_Z^2$through the use of the Goldstone boson equivalence theorem. This theorem allows the vector bosons$W^\\pm$and$Z$to be replaced by the associated scalar Goldstone bosons$w^\\pm$,$z$which appear in the symmetry breaking sector of the Standard Model in the limit of vanishing gauge couplings. In the present pape... 4. Adiabatic Theorems and Reversible Isothermal Processes Abou-Salem, W K 2005-01-01 Reversible isothermal processes of a finitely extended, driven quantum system in contact with an infinite heat bath are studied from the point of view of quantum statistical mechanics. Notions like heat flux, work and entropy are defined for trajectories of states close to, but distinct from states of joint thermal equilibrium. A theorem characterizing reversible isothermal processes as quasi-static processes ("isothermal theorem") is described. Corollaries concerning the changes of entropy and free energy in reversible isothermal processes and on the 0th law of thermodynamics are outlined. 5. A Note on a Broken-Cycle Theorem for Hypergraphs 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 6. An existence theorem for Volterra integrodifferential equations with infinite delay 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. 7. Hindman's Theorem: An Ultrafilter Argument in Second Order Arithmetic Towsner, Henry 2009-01-01 Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic. 8. A multivariate central limit theorem for continuous local martingales Zanten, van, M. 1998-01-01 A theorem on the weak convergence of a properly normalized multivariate continuous local martingale is proved. The time-change theorem used for this purpose allows for short and transparent arguments. 9. 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... 10. Some Generalizations of Jungck's Fixed Point Theorem 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. 11. 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. 12. On the Non-Abelian Stokes Theorem Diakonov, Dmitri; Petrov, Victor 2000-01-01 We present the non-Abelian Stokes theorem for the Wilson loop in various forms and discuss its meaning. Its validity has been recently questioned by Faber, Ivanov, Troitskaya and Zach. We demonstrate that all points of their criticism are based on mistakes in mathematics. Finally, we derive a variant of our formula for the Wilson loop in lattice regularization. 13. A coupling approach to Doob's theorem Kulik, Alexei; Scheutzow, Michael 2014-01-01 We provide a coupling proof of Doob's theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure$\\mu$converge to$\\mu$in the total variation distance. In addition we show that non-singularity (rather than equivalence) of the transition probabilities suffices to ensure convergence of the transition probabilities for$\\mu$-almost all initial conditions. 14. Gap theorems for Ricci-harmonic solitons Tadano, Homare 2015-01-01 In the present paper, by using estimates for the generalized Ricci curvature, we shall give some gap theorems for Ricci-harmonic solitons showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Our results may be regarded as a generalization of recent works by H. Li, and M. Fernandez-Lopez and E. Garcia-Rio. 15. A strictly-positive mass theorem We show that the ADM 4-momentum of an isolated gravitational system (spatially asymptotically flat spacetime) satisfying the dominant energy condition cannot be null-like unless it is flat. Together with the positive mass theorem, this implies that the ADM 4-momentum of an isolated gravitational system must be strictly time-like. (orig.) 16. Multiplier theorems for special Hermite expansions on 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. 17. Sandwich reactor lattices and Bloch's theorem The study of the neutron flux distribution in repetitive sandwiches of reactor material leads to results analogous to the 1-dimensional form of Bloch's theorem for the electronic structure in crystals. This principle makes it possible to perform analytically accurate homogenisations of sandwich lattices The method has been extended to cover multi group diffusion and transport theory. (author) 18. Non-Archimedean Big Picard Theorems Cherry, William 2002-01-01 A non-Archimedean analog of the classical Big Picard Theorem, which says that a holomorphic map from the punctured disc to a Riemann surface of hyperbolic type extends accross the puncture, is proven using Berkovich's theory of non-Archimedean analytic spaces. 19. INTERPOLATION THEOREMS FOR SELF-ADJOINT OPERATORS Shijun Zheng 2009-01-01 We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator L, without assuming the gra-dient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schr(o)dinger operator with electro-magnetic potential. 20. Donsker-Type Theorem for BSDEs Briand, Philippe; Delyon, Bernard; Mémin, Jean 2001-01-01 This paper is devoted to the proof of Donsker's theorem for backward stochastic differential equations (BSDEs for short). The main objective is to give a simple method to discretize in time a BSDE. Our approach is based upon the notion of convergence of filtrations'' and covers the case of a$(y,z)$-dependent generator. 1. A Fixed Point Theorem for Discontinuous Functions Herings, Jean-Jacques; Laan, Gerard van der; Talman, Dolf; Yang, Zaifu 2004-01-01 Any function from a non-empty polytope into itself that is locally gross direction preserving is shown to have the fixed point property. Brouwer's fixed point theorem for continuous functions is a special case. We discuss the application of the result in the area of non-cooperative game theory. 2. Green's Theorem for Generalized Fractional Derivatives Odzijewicz, Tatiana; Malinowska, Agnieszka B.; Delfim F. M. Torres 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. 3. Central Limit Theorem for Coloured Hard Dimers 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. 4. Generalizations of the Lax-Milgram Theorem Dimosthenis Drivaliaris 2007-05-01 Full Text Available We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular, we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations. 5. Generalizations of the Lax-Milgram Theorem Yannakakis Nikos 2007-01-01 Full Text Available We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular, we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations. 6. Deduction Theorems in Weakly Implicative Logics Cintula, Petr Barcelona: Universitat de Barcelona, 2005. s. 19-20. [Algebraic and Topological Methods in Non-Classical Logics /2./. 15.06.2005-18.06.2005, Barcelona] Institutional research plan: CEZ:AV0Z10300504 Keywords : deduction theorem * substructural logic * BCI logic * weakly implicative logic Subject RIV: BA - General Mathematics 7. Random fixed point theorems on product spaces Ismat Beg; Naseer Shahzad 1993-01-01 The existence of random fixed point of a locally contractive random operator in first variable on product spaces is proved. The concept continuous random height-selection is discussed. Some random fixed point theorems for nonexpansive self and nonself maps are also obtained in product spaces. 8. 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. 9. On Noethers theorem in quantum field theory Extending an earlier construction of local generators of symmetries in (S. Doplicher, 1982) to space-time and supersymmetries, we establish a weak form of Noethers theorem in quantum field theory. We also comment on the physical significance of the 'split property', underlying our analysis, and discuss some local aspects of superselection rules following from our results. (orig./HSI) 10. 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. 11. 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 Schrodinger equation is a difference equation. 12. 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. 13. Lagrange’s Four-Square Theorem 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]. 14. JACKSON‘S THEOREM FOR COMPACT GROUPS 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. 15. Extended Kelvin theorem in relativistic magnetohydrodynamics Bekenstein, Jacob D.; Oron, Asaf 2000-01-01 We prove the existence of a generalization of Kelvin's circulation theorem in general relativity which is applicable to perfect isentropic magnetohydrodynamic flow. The argument is based on a new version of the Lagrangian for perfect magnetohydrodynamics. We illustrate the new conserved circulation with the example of a relativistic magnetohydrodynamic flow possessing three symmetries. 16. The virial theorem and planetary atmospheres Toth, Viktor T. 2010-01-01 We derive a version of the virial theorem that is applicable to diatomic planetary atmospheres that are in approximate thermal equilibrium at moderate temperatures and pressures and are sufficiently thin such that the gravitational acceleration can be considered constant. We contrast a pedagogically inclined theoretical presentation with the actual measured properties of air. 17. The virial theorem for nonlinear problems Amore, Paolo; Fernández, Francisco M. 2009-01-01 We show that the virial theorem provides a useful simple tool for approximating nonlinear problems. In particular we consider conservative nonlinear oscillators and a bifurcation problem. In the former case we obtain the same main result derived earlier from the expansion in Chebyshev polynomials. 18. 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. 19. A Bijective Proof For Forest Reciprocity Theorem Huang, ShinnYih 2009-01-01 In this paper, we study the graph polynomial that counts spanning rooted forests f_g of a given graph. This polynomial has a remarkable reciprocity property. We give a new bijective proof for this theorem which has Prufer coding as a special case. 20. Limit theorems for Markov random fields Markov Random Fields (MRF's) have been extensively applied in Statistical Mechanics as well as in Bayesian Image Analysis. MRF's are a special class of dependent random variables located at the vertices of a graph whose joint distribution includes a parameter called the temperature. When the number of vertices of the graph tends to infinity, the normalized distribution of statistics based on these random variables converge in distribution. It can happen that for certain values of the temperature, that the rate of growth of these normalizing constants change drastically. This feature is generally used to explain the phenomenon of phase transition as understood by physicist. In this dissertation the author will show that this drastic change in normalizing constants occurs even in the relatively smooth case when all the random variables are Gaussian. Hence any image analytic MRF ought to be checked for such discontinuous behavior before any analysis is performed. Mixed limit theorems in Bayesian Image Analysis seek to replace intensive simulations of MRF's with limit theorems that approximate the distribution of the MRF's as the number of sites increases. The problem of deriving mixed limit theorems for MRF's on a one dimensional lattice graph with an acceptor function that has a second moment has been studied by Chow. A mixed limit theorem for the integer lattice graph is derived when the acceptor function does not have a second moment as for instance when the acceptor function is a symmetric stable density of index 0 < α < 2 1. Kelvin's Canonical Circulation Theorem in Hall Magnetohydrodynamics Shivamoggi, B K 2016-01-01 The purpose of this paper is to show that, thanks to the restoration of the legitimate connection between the current density and the plasma flow velocity in Hall magnetohydrodynamics (MHD), Kelvin's Circulation Theorem becomes valid in Hall MHD. The ion-flow velocity in the usual circulation integral is now replaced by the canonical ion-flow velocity. 2. Scaling Identities for Solitons beyond Derrick's Theorem Manton, Nicholas S. 2008-01-01 New integral identities satisfied by topological solitons in a range of classical field theories are presented. They are derived by considering independent length rescalings in orthogonal directions, or equivalently, from the conservation of the stress tensor. These identities are refinements of Derrick's theorem. 3. Pauli and The Spin-Statistics Theorem 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 'everyone knows the spin-statistics theorem, but no one understands it'. This book simplifies and clarifies the formal statements of the theorem, and also corrects the invariably flawed intuitive explanations which are frequently put forward. The book will be of interest to many practising physicists in all fields who have long been frustrated by the impenetrable discussions on the subject which have been available until now.It will also be accessible to students at an advanced undergraduate level as an introduction to modern physics based directly on the classical writings of the founders, including Pauli, Dirac, Heisenberg, Einstein and many others 4. Fundamental theorems of extensional untyped$\\lambda$-calculus revisited Alexandre Lyaletsky 2015-10-01 Full Text Available This paper presents new proofs of three following fundamental theorems of the untyped extensional$\\lambda$-calculus: the$\\eta$-Postpo-nement theorem, the$\\beta\\eta$-Normal form theorem, and the Norma-lization theorem for$\\beta\\eta$-reduction. These proofs do not involve any special extensions of the standard language of$\\lambda$-terms but nevertheless are shorter and much more comprehensive than their known analogues. 5. 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. 6. Theorems of Tarski's Undefinability and Godel's Second Incompleteness - Computationally Salehi, Saeed 2015-01-01 We show that the existence of a finitely axiomatized theory which can prove all the true$\\Sigma_1\$ sentences may imply Godel's Second Incompleteness Theorem, by incorporating some bi-theoretic version of the derivability conditions (first discussed by Detlefsen~2001). We also argue that Tarski's theorem on the undefinability of truth is Godel's first incompleteness theorem relativized to definable oracles; here a unification of these two theorems is shown.

7. 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.

8. A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem

Tian, Guoqiang

2012-01-01

This paper provides necessary and sufficient conditions for fixed-point theorems, minimax inequalities and some related theorems defined on arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which fully characterizes the existence of equilibrium of minimax inequality without imposing any kind of convexity nor any restriction on topological space. The result then is employed to ful...

9. ON GÖDEL'S INCOMPLETENESS THEOREM(S), ARTIFICIAL INTELLIGENCE/LIFE, AND HUMAN MIND

CHRISTIANTO, V.; FLORENTIN SMARANDACHE

2015-01-01

In the present paper we have discussed concerning Gödel’s incompleteness theorem(s) and plausible implications to artificial intelligence/life and human mind. Perhaps we should agree with Sullins III, that the value of this finding is not to discourage certain types of research in AL, but rather to help move us in a direction where we can more clearly define the results of that research.

10. The direct Flow parametric Proof of Gauss' Divergence Theorem revisited

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...