Polya's Theorem Research Articles

Distribution of Base Pair Alternations in a Periodic DNA Chain: Application of Pólya Counting to a Physical System

In modeling DNA chains, the number of alternations between Adenine-Thymine (AT) and Guanine-Cytosine (GC) base pairs can be considered as a measure of the heterogeneity of the chain, which in turn could affect its dynamics. A probability distribution function of the number of these alternations is derived for circular or periodic DNA. Since there are several symmetries to account for in the periodic chain, necklace counting methods are used. In particular, Polya's Enumeration Theorem is extended for the case of a group action that preserves partitioned necklaces. This, along with the treatment of generating functions as formal power series, allows for the direct calculation of the number of possible necklaces with a given number of AT base pairs, GC base pairs and alternations. The theoretically obtained probability distribution functions of the number of alternations are accurately reproduced by Monte Carlo simulations and fitted by Gaussians. The effect of the number of base pairs on the characteristics of these distributions is also discussed, as well as the effect of the ratios of the numbers of AT and GC base pairs.

Structure and enumeration theorems for hereditary properties in finite relational languages

Given a finite relational language L, a hereditary L-property is a class of finite L-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary L-properties, where L is any finite relational language with maximum arity at least two. In particular, the goal of this paper is to generalize how extremal results and stability theorems can be combined with well-known techniques and tools to yield approximate enumeration and structure theorems. We accomplish this by generalizing the notions of extremal graphs, asymptotic density, and graph stability theorems using structures in an auxiliary language associated to a hereditary L-property. Given a hereditary L-property H, we prove an approximate asymptotic enumeration theorem for H in terms of its generalized asymptotic density. Further we prove an approximate structure theorem for H, under the assumption of that H has a stability theorem. The tools we use include a new application of the hypergraph containers theorem (Balogh–Morris–Samotij [16], Saxton–Thomason [38]) to the setting of L-structures, a general supersaturation theorem for hereditary L-properties (also new), and a general graph removal lemma for L-structures proved by Aroskar and Cummings in [5]. Similar results in the setting of multicolored graphs and hypergraphs were recently proved independently by Falgas-Ravry, O'Connel, Strömberg, and Uzzell [21].

On the Van Vleck theorem for limit-periodic continued fractions of general form

The boundary properties of functions representable as limit-periodic continued fractions of the form A 1(z)/(B 1(z) + A 2(z)/(B 2(z) +...)) are studied; here the sequence of polynomials {A n } =1 ∞ has periodic limits with zeros lying on a finite set E, and the sequence of polynomials {B n } =1 ∞ has periodic limits with zeros lying outside E. It is shown that the transfinite diameter of the boundary of the convergence domain of such a continued fraction in the external field associated with the fraction coincides with the upper limit of the averaged generalized Hankel determinants of the function defined by the fraction. As a consequence of this result combined with the generalized Polya theorem, it is shown that the functions defined by the continued fractions under consideration do not have a single-valued meromorphic continuation to any neighborhood of any nonisolated point of the boundary of the convergence set.

Effective spin physics in two-dimensional cavity QED arrays

We investigate a strongly correlated system of light and matter in two-dimensional cavity arrays. We formulate a multimode Tavis–Cummings (TC) Hamiltonian for two-level atoms coupled to cavity modes and driven by an external laser field which reduces to an effective spin Hamiltonian in the dispersive regime. In one-dimension we provide an exact analytical solution. In two-dimensions, we perform mean-field study and large scale quantum Monte Carlo simulations of both the TC and the effective spin models. We discuss the phase diagram and the parameter regime which gives rise to frustrated interactions between the spins. We provide a quantitative description of the phase transitions and correlation properties featured by the system and we discuss graph-theoretical properties of the ground states in terms of graph colourings using Pólya’s enumeration theorem.

Analisis Kemampuan Mahasiswa dalam Pemecahkan Masalah Matematis menurut Teori Polya

This research aimed to describe students’ ability in solving mathematics problems according to Polya theory. According to writer’s experience in the subject of basic mathematics, the students had low ability in solving mathemathics problems. Therefore, the writer used the descriptive qualitative method to analyze problem-solving of mathematics according to Polya theory of semester two of IAIN students. Test and interview were used in collecting data. Based on analysis of simple statistic, it was found that students, whose high prior knowledge thought algorithmically in solving the question of problem-solving of mathematics, could understand the problems correctly and fluently. Students whose moderate prior knowledge thought algorithmically and not completely solved the questions of problem-solving. Furthermore, students whose low prior knowledge thought heuristic in solving the questions of problem-solving of mathematics.

Теорема Пойа і міграція та захоплення квантової частинки

Теорема Пойа і міграція та захоплення квантової частинки

Numerical Algorithm for Pólya Enumeration Theorem

Although the Pólya enumeration theorem has been used extensively for decades, an optimized, purely numerical algorithm for calculating its coefficients is not readily available. We present such an algorithm for finding the number of unique colorings of a finite set under the action of a finite group.

基于有限结合代数的 MOR 公钥密码安全性分析

The Shor algorithm is effective for public-key cryptosystems based on an abelian group. At CRYPTO 2001, Paeng (2001) presented a MOR cryptosystem using a non-abelian group, which can be considered as a candidate scheme for post-quantum attack. This paper analyses the security of a MOR cryptosystem based on a finite associative algebra using a quantum algorithm. Specifically, let L be a finite associative algebra over a finite field F. Consider a homomorphism φ: Aut(L) → Aut(H)×Aut(I), where I is an ideal of L and H ≅ L/I. We compute dim Im(φ) and dim Ker(φ), and combine them by dim Aut(L) = dim Im(φ)+dim Ker(φ). We prove that Im(φ) = StabComp(H,I)(μ + B 2(H, I)) and Ker(φ) ≅ Z 1(H, I). Thus, we can obtain dim Im(φ), since the algorithm for the stabilizer is a standard algorithm among abelian hidden subgroup algorithms. In addition, Z 1(H, I) is equivalent to the solution space of the linear equation group over the Galois fields GF(p), and it is possible to obtain dim Ker(φ) by the enumeration theorem. Furthermore, we can obtain the dimension of the automorphism group Aut(L). When the map ϕ ∈ Aut(L), it is possible to effectively compute the cyclic group 〈ϕ〉 and recover the private key a. Therefore, the MOR scheme is insecure when based on a finite associative algebra in quantum computation.

General Glivenko–Cantelli theorems

A Glivenko–Cantelli theorem is a fundamental result in statistics. It says that an empirical distribution function uniformly approximates the true distribution function for a sufficiently large sample size. We prove general Glivenko–Cantelli theorems for three types of sequences of random variables: delayed regenerative, delayed stationary and delayed exchangeable. In particular, our results hold for irreducible Harris recurrent Markov chains that admit a stationary probability distribution but are not necessarily in the stationary state. We also do not assume any mixing conditions on the Markov chain. This is useful in the application of Markov chain Monte Carlo methods. A key tool used is a generalized version of Polya's theorem on the convergence of distribution functions. Copyright © 2016 John Wiley & Sons, Ltd.

Affine equivalence of monomial rotation symmetric Boolean functions: A Pólya’s theorem approach

AbstractTwo Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

An analogue of Polya's theorem for piecewise holomorphic functions

A well-known result due to Polya for a function given by its holomorphic germ at is extended to the case of a piecewise holomorphic function on an arbitrary compact set in . This result is applied to the problem of the existence of compact sets that have the minimum transfinite diameter in the external field of the logarithmic potential of a negative unit charge among all compact sets such that a certain multivalued analytic function is single-valued and piecewise holomorphic on their complement. Bibliography: 13 titles.

Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of $H_\infty$ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

Stability Analysis of T–S Fuzzy Control Systems by Using Set Theory

This paper is concerned with the stability analysis for Takagi–Sugeno (T–S) fuzzy control systems. By exploiting the property of the structure of a fuzzy inference engine, an equivalence relation on the index set of the product of fuzzy rule weights is defined. Furthermore, a new stability criterion is proposed by using the equivalence relation and formulated into progressively less-conservative sets of linear matrix inequalities. By using an extension of Polya's theorem, the new criterion is proved to be with no conservatism for quadratic stability analysis of T–S fuzzy control systems with a product inference engine and any possible fuzzy membership functions. A numerical example is given to illustrate the effectiveness of the proposed method.

Multi-base representations of integers: Asymptotic enumeration and central limit theorems

In a multi-base representation, in contrast to the common b-ary representation, the base is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, meaning that each integer can have multiple different digit expansions. We provide a general asymptotic formula for the number of multi-base representations of a positive integer. Moreover, we prove central limit theorems for different statistics associated to a multi-base representation.

Asymptotic Statistics of Cycles in Surrogate-Spatial Permutations

We propose an extension of the Ewens measure on permutations by choosing the cycle weights to be asymptotically proportional to the degree of the symmetric group. This model is primarily motivated by a natural approximation to the so-called spatial random permutations recently studied by V. Betz and D. Ueltschi (hence the name "surrogate-spatial"), but it is of substantial interest in its own right. We show that under the suitable (thermodynamic) limit both measures have the similar critical behaviour of the cycle statistics characterized by the emergence of infinitely long cycles. Moreover, using a greater analytic tractability of the surrogate-spatial model, we obtain a number of new results about the asymptotic distribution of the cycle lengths (both small and large) in the full range of subcritical, critical and supercritical domains. In particular, in the supercritical regime there is a parametric "phase transition" from the Poisson-Dirichlet limiting distribution of ordered cycles to the occurrence of a single giant cycle. Our techniques are based on the asymptotic analysis of the corresponding generating functions using Polya's Enumeration Theorem and complex variable methods.

Order Polynomials and Pólya's Enumeration Theorem

Pólya’s enumeration theorem states that the number of labelings of a finite set up to symmetry is given by a polynomial in the number of labels. We give a new perspective on this theorem by generalizing it to partially ordered sets and order preserving maps. Further we prove a reciprocity statement in terms of strictly order preserving maps generalizing a classical result by Stanley (1970). We apply our results to counting graph colorings up to symmetry.

Functions of exponential type and the cardinal series

In this paper, various growth rates and oscillation conditions for entire functions of exponential type π are given which ensure validity of the classical cardinal series. Among other applications, a theorem of Plancherel and Polya is extended to show that any function of exponential type π which decays to 0 along the real axis is a convergent cardinal series. Examples of functions which show the limitations of this extension are also presented.

Enumerating stereo-isomers of tree-like polyinositols

Enumeration of molecules is one of the fundamental problems in bioinformatics and chemoinformatics which is also important from a practical viewpoint. We consider the problem of enumerating the stereo-isomers of tree-like polyinositol molecules (with chemical formula $$\hbox {C}_{6n}\hbox {O}_{5n+6}\hbox {H}_{4n+2}$$ where $$n$$ is the number of hexagonal oinositol rings) and monosubstituted tree-like polyinositols (with chemical formula $$\hbox {C}_{6n}\hbox {O}_{5n+6}\hbox {H}_{4n+1}\hbox {Z}$$ ). We establish recursion counting formulas for the numbers of the stereo-isomers for these two classes of molecules, in which chirality is also taken into account. In our study, the generating function, Polya enumeration theory and ‘Dissimilarity Characteristic Theorem’ play important roles. Compared to some known computer programs such as ISOMERS, MOLGEN, exhaustive construction and Dynamic Programming etc., our method is more efficient to our enumeration problem with larger number of inositol rings. Further more, based on the obtained recursion formulas, we derive the asymptotic values for the numbers of these two stereo-isomers from which we conclude that almost all tree-like and monosubstituted tree-like polyinositols are chiral.

Counting rotation symmetric functions using Polya’s theorem

Homogeneous rotation symmetric (invariant under cyclic permutation of the variables) Boolean functions have been extensively studied in recent years due to their applications in cryptography. In this paper we give an explicit formula for the number of homogeneous rotation symmetric functions over the finite field GF(pm) using Polya’s enumeration theorem, which completely solves the open problem proposed by Yuan Li in 2008. This result simplifies the proof and the nonexplicit counting formula given by Shaojing Fu et al. over the field GF(p). This paper also gives an explicit count for n-variable balanced rotation symmetric Boolean functions with n=pq, where p and q are distinct primes. Previous work only gave an explicit count for the case where n is prime and lower bounds for the case where n is a prime power.

Generalization of the Harary–Palmer power group theorem to all irreducible representations of object and color groups- color combinatorial group theory

The Harary–Palmer classic power group enumeration theorem applies to a group G acting on a set D of objects such as vertices, edges, faces and simultaneously with a group H acting on another set R of colors such that the power group \(\hbox {H}^\mathrm{G}\) acts on the set \(\hbox {R}^\mathrm{D}\) of all functions from D to R. In this paper we show for the first time that the power group enumeration can be generalized to all irreducible representations of the object group G of D and also all irreducible representations of the color group H of R. We have also provided interpretation of various power group generating functions for different irreducible representations in the context of color symmetry group theory. Special cases of the power group enumeration with all irreducible representations of G keeping the color group representation fixed to the totally symmetric representation are shown to have important enumerative combinatorial applications in a number of problems of chemistry, physics and biology that involve color symmetry (color inversions) such as magnetism, neutron imaging, NMR, ESR spectroscopy, catalytic functions of non-rigid disordered proteins, and quantum chromodynamics of strong interactions of fundamental particles.