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The main concern of this paper is to study large-time behavior of solutions for an outflow problem to the reduced gravity two and a half layer model in a half space. We establish the existence of stationary solutions and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small. Moreover, we obtain the convergence rate of the solution toward the stationary solution in some weighted Sobolev spaces only for the supersonic case. The proof is based on the basic energy method. A key point for the proof of convergence rates is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity satisfy some additional conditions. [ABSTRACT FROM AUTHOR]

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley–Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron P λ , and we give a formula for the dominant weight λ . Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of (3 + 1) -free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the ζ map from diagonal harmonics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel series of second-order Backward Differentiation Formulas (BDFs) specifically designed to address phase-lag and its first derivative in the numerical resolution of Initial Value Problems (IVPs) with orbital solutions. Our methodology commences with an in-depth analysis of phase-lag phenomena associated with second-order BDFs. Following this, we construct a suite of equations by embedding algebraic functions into the operational framework of the 3-step second-order BDF (SOBDF) method. Additionally, we elaborate on equations that precisely describe the phase-lag and its derivatives, with a concentrated focus on the 3-step SOBDF method. The culmination of this work is the presentation of six distinct methods, each methodically crafted to negate both the real and imaginary elements of phase-lag and its derivatives in numerical computations. The study advances with a meticulous examination of the local truncation error and the stability regions pertinent to the six phase-fitted methods introduced. Furthermore, we scrutinize their computational performance by deploying these methods across a spectrum of initial value problems, offering valuable insights into their effectiveness in varying contexts. [ABSTRACT FROM AUTHOR]

Let |$A$| be the integral closure of the ring of polynomials |${{\mathbb {C}}}[t]$|⁠ , within the field of algebraic functions in one variable. We show that |$A$| interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a decidable theory in [ 12 ] and [ 4 ]. [ABSTRACT FROM AUTHOR]

Supercuspidal representations are usually infinite-dimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its L-parameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second result, using the first, is to verify that Kaletha's regular supercuspidal L-packets satisfy the conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we show that ∞-adic multiple zeta values associated to the function field of an algebraic curve of higher genus over a finite field are not zero, under certain assumption on the gap sequence associated to the rational point ∞ on the given curve. Using arguments and results of Sheats and Thakur for the case of the projective line, we calculate the absolute values of power sums in the series defining multiple zeta values, and show that the calculation implies the non-vanishing result. [ABSTRACT FROM AUTHOR]

Rotation symmetric Boolean functions are good candidates for stream ciphers because they have such advantages as simple structure, high operational speed and low implement cost. Recently, Mesnager et al. proposed for the first time an efficient method to construct balanced rotation symmetric Boolean functions with optimal algebraic immunity and good nonlinearity for an arbitrary even number of variables. However, the algebraic degree of their constructed n -variable (n > 4) function is always less than the maximum value n − 1. In this paper, by modifying the support of Boolean functions from Mesnager et al.'s construction, we present two new constructions of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity as well as higher algebraic degree and nonlinearity. The algebraic degree of Boolean functions in the first construction reaches the maximum value n − 1 if n 2 is odd and n 2 ≠ (2 k + 1) 2 or (2 k + 1) 2 + 2 for integer k , while that of the second construction reaches the maximum value for all n. Moreover, the nonlinearities of Boolean functions in both two constructions are higher than that of Mesnager et al.'s construction. [ABSTRACT FROM AUTHOR]

A classical problem due to Abel is to determine if a differential equation |$y^{\prime}=\eta y$| admits a non-trivial solution |$y$| algebraic over |$\mathbb C(x)$| when |$\eta $| is a given algebraic function over |$\mathbb C(x)$|⁠. Risch designed an algorithm that, given |$\eta $|⁠ , determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when |$\eta $| admits a Puiseux expansion with rational coefficients at some point in |$\mathbb C\cup \{\infty \}$|⁠ , which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of |$y^{\prime}=\eta y$| if and only if the coefficients of the Puiseux expansion of |$x\eta (x)$| at |$0$| satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations |$y^{\prime}=\eta y$| with an algebraic solution when |$x\eta (x)$| is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks. [ABSTRACT FROM AUTHOR]

The present work deals with the characteristic polynomial of Laplacian matrix for circulant graphs. We show that it can be decomposed into a finite product of algebraic function evaluated at the roots of a linear combination of Chebyshev polynomials. As an important consequence of this result, we get the periodicity of characteristic polynomials evaluated at the prescribed integer values. Moreover, we can show that the characteristic polynomials of circulant graphs are always perfect squares up to explicitly given linear factors. [ABSTRACT FROM AUTHOR]

This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form P / Q k and 1 / Q , are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple F 1 2 hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model. [ABSTRACT FROM AUTHOR]

We show—to our own surprise—that total electronic energies for a family of m × n rectangular graphene flakes can be very accurately represented by a simple function of the structural parameters m and n with errors not exceeding 1 kcal/mol. The energies of these flakes, usually referred to as multiple zigzag chains Z(m,n), are computed for m, n < 21 at their optimized geometries using the DFTB3 methodology. We have discovered that the structural parameters m and n (and their simple algebraic functions) provide a much better basis for the energy decomposition scheme than the various topological invariants usually used in this context. Most terms appearing in our energy decomposition scheme seem to have simple chemical interpretations. Our observation goes against the well-established knowledge stating that many-body energies are complicated functions of molecular parameters. Our observations might have far-reaching consequences for building accurate machine learning models. [ABSTRACT FROM AUTHOR]

Symmetry is often regarded as an integral aspect about aesthetics. This motivates the pursuit of interdisciplinary studies based on the use of subjects in mathematics concerned with symmetry in conjunction with aesthetics. What is referred to as a symmetric function in the field of algebraic combinatorics is an abstraction based on polynomials that exhibit a symmetric property, and this leads us to pursue an algebraic combinatorics-inspired exploration based on aesthetics. In particular, we use different bases and transitions between them to create aesthetically pleasing visualizations of symmetric functions. We see that these visualizations in turn raise new and interesting questions. [ABSTRACT FROM AUTHOR]

Here we exhibit multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or RN; N ∞ N, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We study also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the algebraic sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer. [ABSTRACT FROM AUTHOR]

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to C r and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K , but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K = F q (T) , and describe the denominators and support of the numerators. [ABSTRACT FROM AUTHOR]

A function that is analytic on a domain of C n is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein–Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the structure of certain sheaves of holonomic functions, such as the algebraic functions along a hypersurface, determining their direct sum decompositions into indecomposables, that further respect decompositions of Bernstein–Sato polynomials. When the space is endowed with the action of a linear algebraic group G, we study the class of G-finite analytic functions, i.e. functions that under the action of the Lie algebra of G generate a finite dimensional rational G-module. These are automatically algebraic functions on a variety with a dense orbit. When G is reductive, we give several representation-theoretic techniques toward the determination of Bernstein–Sato polynomials of G-finite functions. We classify the G-finite functions on all but one of the irreducible reduced prehomogeneous vector spaces, and compute the Bernstein–Sato polynomials for distinguished G-finite functions. The results can be used to construct explicitly equivariant D -modules. [ABSTRACT FROM AUTHOR]

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in Cheng and Hao [On the non-existence of common submanifolds of Kähler manifolds and complex space forms. Ann Glob Anal Geom. 2021;60(1):167–180] and Cheng X, Niu Y. Submanifolds of Cartan–Hartogs domains and complex Euclidean spaces. J Math Anal Appl 2017;452(2):1262–1268.] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic. [ABSTRACT FROM AUTHOR]

Symbolic 3-plithogenic sets as a generalization of classical concept of sets were applicable to algebraic structures. The symbolic 3-plithogenic rings and fields are good generalizations of classical corresponding systems. In this paper, we study the symbolic 3-plithogenic real functions with one variable by using a special algebraic function called AH-isometry. In addition, we discuss the symbolic 3-plithogenic simple differential equations and conic sections by using this isometry. Also, many examples will be presented to explain the novelty of this work. [ABSTRACT FROM AUTHOR]

Symbolic 2-plithogenic sets as a generalization of classical concept of sets were applicable to algebraic structures. The symbolic 2-plithogenic rings and fields are good generalizations of classical corresponding systems. In this paper, we study the symbolic 2-plithogenic real functions with one variable by using a special algebraic function called AH-isometry. In addition, we discuss the symbolic 2-plithogenic simple differential equations and conic sections by using this isometry. Also, many examples will be presented to explain the novelty of this work. [ABSTRACT FROM AUTHOR]

Birational properties of generically finite morphisms X→Y$X\rightarrow Y$ of algebraic varieties can be understood locally by a valuation of the function field of X. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation, there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields, it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing up. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in defect Artin–Schreier extensions which can have any prescribed distance. [ABSTRACT FROM AUTHOR]

At Eurocrypt 2016, Méaux et al. presented FLIP, a new family of stream ciphers that aimed to enhance the efficiency of homomorphic encryption frameworks. Motivated by FLIP, recent research has focused on the study of Boolean functions with good cryptographic properties when restricted to subsets of the space F 2 n. If an n -variable Boolean function has the property of balancedness when restricted to each set of vectors with fixed Hamming weight between 1 and n − 1 , it is a weightwise perfectly balanced (WPB) Boolean function. In the literature, a few algebraic constructions of WPB functions are known, in which there are some constructions that use iterative method based on functions with low degrees of 1, 2, or 4. In this paper, we generalize the iterative method and contribute a unified construction of WPB functions based on functions with algebraic degrees that can be any power of 2. For any given positive integer d not larger than m , we first provide a class of 2 m -variable Boolean functions with a degree of 2 d − 1. Utilizing these functions, we then present a construction of 2 m -variable WPB functions g m ; d. In particular, g m ; d includes four former classes of WPB functions as special cases when d = 1 , 2 , 3 , m. When d takes other integer values, g m ; d has never appeared before. In addition, we prove the algebraic degree of the constructed WPB functions and compare the weightwise nonlinearity of WPB functions known so far in 8 and 16 variables. [ABSTRACT FROM AUTHOR]

The main goals of this paper are to provide an introduction to the idea of interval-valued n -polynomial s -type convex functions and to investigate the algebraic properties of this type of function. This new generalization aims to show the existence of new Hermite–Hadamard inequalities for the recently presented class of interval-valued n -polynomials of s -type convex describing the φ -fractional integral operator. In the classical sense, some special cases are figured out, and the two examples are also given. There are some recently discovered inequalities for interval-valued functions that are regulated by fractional calculus applicable to interval-valued n -polynomial s -type convexity. The results obtained show that future research will be simple to implement, highly efficient, feasible, and extremely precise in its investigation. It could also help solve modeling problems, optimization problems, and fuzzy interval-valued functions that involve both discrete and continuous variables. [ABSTRACT FROM AUTHOR]

In this paper, we first give a relationship between generalized algebraic geometry codes (GAG codes) and algebraic geometry codes (AG codes). More precisely, we show that a GAG code is contained (up to isomorphism) in a suitable AG code. Next we recall the concept of an N1N2-automorphism group, a subgroup of the automorphism group of a GAG code. With the use of the relation we obtained between these two classes of codes, we show that the N1N2-automorphism group is a subgroup of the automorphism group of an AG code. [ABSTRACT FROM AUTHOR]

In this study, we introduce and study the concept of strongly hyperbolic type convexity functions and their some algebraic properties. We obtain Hermite-Hadamard type inequalities for the strongly hyperbolic type convex functions. After that, by using an identity, we get some inequalities for strongly hyperbolic type convex functions. In addition, we compare the results obtained with both Hölder, Hölder-İşcan inequalities and power-mean, improved power-mean integral inequalities. [ABSTRACT FROM AUTHOR]

We study the existence of target patterns in oscillatory media with weak local coupling and in the presence of an impurity, or defect. We model these systems using a viscous eikonal equation posed on the plane and represent the defect as a perturbation. In contrast to previous results, we consider large defects, which we describe using a function with slow algebraic decay, i.e., g ∼ O (1 / | x | m) for m ∈ (1 , 2 ] . We prove that these defects are able to generate target patterns and that, just as in the case of strongly localized impurities, their frequency is small beyond all orders of the small parameter describing their strength. Our analysis consists of finding two approximations to target pattern solutions, one which is valid at intermediate scales and a second one which is valid in the far field. This is done using weighted Sobolev spaces, which allow us to recover Fredholm properties of the relevant linear operators, as well as the implicit function theorem, which is then used to prove existence. By matching the intermediate and far-field approximations, we then determine the frequency of the pattern that is selected by the system. [ABSTRACT FROM AUTHOR]

The present research explores the Stokesian flow of an axisymmetric, incompressible couple stress fluid through a porous material enclosing a solid sphere in the presence of a uniform magnetic field. Utilizing the separation of variables technique, the problem is solved analytically. Expressions for the stream function, velocity components, vorticity component, fluid pressure, stress tensors, and couple stresses are obtained, which involved modified Bessel functions, trigonometric functions, and algebraic functions. Impenetrability, no-slip, and nil-vorticity conditions have been applied at the interface. A novel result for the drag force experienced by a solid sphere enclosed in porous material under the applied uniform magnetic field is reported. The evidence already in place leads to the identification and validation of our results for flow past the impermeable sphere. Tangential velocity and normal velocity, drag force, and streamline patterns are displayed for different values of flow parameters, which incorporate the Hartmann number, permeability parameter, and couple stress parameter. [ABSTRACT FROM AUTHOR]

We develop the theory of central ideals on commutative domains. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties. We provide a construction of the central seminormalization by a decomposition theorem in elementary central gluings. The existence of a central seminormalization is established in the affine case and for real schemes. [ABSTRACT FROM AUTHOR]

Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function being either analytic on and within an ellipse and |$\lambda \leqslant 0$| or differentiable and |$\lambda \leqslant 1$|⁠ , where |$\lambda $| is the parameter in Gegenbauer projections. If the underlying function is analytic and |$\lambda>0$| or differentiable and |$\lambda>1$|⁠ , then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of |$n^{\lambda }$| and |$n^{\lambda -1}$|⁠ , respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all |$\lambda>-1/2$|⁠. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral methods. [ABSTRACT FROM AUTHOR]

We study polynomials given by the discrete Rodrigues formula, which generalizes a similar formula for Meixner polynomials. Such polynomials are associated with the theory of Diophantine approximations. The saddle point method is used to find the limit distribution of zeros of scaled polynomials. An answer is received in terms of a meromorphic function on a compact Riemann surface and is interpreted using the vector equilibrium problem of the logarithmic potential theory. [ABSTRACT FROM AUTHOR]

The Lichtenbaum–Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni–Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the n-th Greek letter family is detected by a commutative ring spectrum R, then we conjecture that the n+1-st Greek letter family will be detected by the algebraic K-theory of R. We prove this in the case n=1 for R=\mathrm {K}(\mathbb {F}_q) modulo (p,v_1) where p\ge 5 and q=\ell ^k is a prime power generator of the units in \mathbb {Z}/p^2\mathbb {Z}. In particular, we prove that the commutative ring spectrum \mathrm {K}(\mathrm {K}(\mathbb {F}_q)) detects the part of the p-primary \beta-family that survives mod (p,v_1). The method of proof also implies that these \beta elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to integral modular forms satisfying certain congruences. [ABSTRACT FROM AUTHOR]

Analytical solutions are obtained for two problems for a discrete analogue of the Helmholtz equation on a triangular lattice: (1) the problem of radiation from a point source on a plane; (2) the problem of diffraction on a half-line with Dirichlet boundary conditions. It is shown that in these problems, the complete field can be represented as an integral of an algebraic function over a family of contours located on some complex manifold. The solution to the first problem is found as an integral of some differential form over this manifold, and the asymptotics of the far field for this solution is obtained. The second problem is solved using an analogue of the Sommerfeld integral. It is checked that the obtained solution coincides with the solution of this problem by the Wiener–Hopf method. [ABSTRACT FROM AUTHOR]

A sampling error‐based finite‐set predictive current control (FS‐PCC) is proposed in this article for the open‐end winding induction motor (OEWIM) drive. The proposed scheme controls the zero‐sequence current (ZSC) alongside the stator currents. In a model‐free approach, this method predicts the future of ZSC and stator current components by the stator current and voltage sampling errors. In this way, the parameters of the OEWIM are not utilised in the prediction algorithm of the FS‐PCC. So, the proposed method is robust against the variation of the parameter. Moreover, this article presents a simple vector selection technique for the FS‐PCC of the OEWIM. The proposed technique has two cost functions and a simple algebraic equation to put the voltage vectors (VVs) in the prediction algorithm. The first cost function uses VVs that do not have the zero‐sequence voltage component. Then, the algebraic equation determines VVs that must be utilised in the second cost function. Finally, the optimum VV is selected by the second cost function. In the proposed scheme, the prediction algorithm is iterated 14 times instead of 27 iterations of the conventional predictive algorithm. So, besides establishing a novel model‐free prediction algorithm, the proposed method has almost 50% fewer calculations. The validity of the proposed sampling error‐based FS‐PCC and the simplified vector selection technique has been verified through experimental tests. [ABSTRACT FROM AUTHOR]

Tropical Newton-Puiseux polynomials, defined as piece-wise linear functions with rational coefficients of the variables, play a role as tropical algebraic functions. We provide explicit formulas for tropical Newton-Puiseux polynomials being the tropical zeroes of a univariate tropical polynomial with parametric coefficients. [ABSTRACT FROM AUTHOR]

In this discussion, we introduced the Lambert-W relation, how to solve equations that are solvable via this relation and not solvable otherwise, discussed how to numerically calculate values, how to generate the real and imaginary parts of W for which have sufficiently negative or complex arguments. We then gave a brief insight into one of the physical applications of the Lambert-W relation that is used in fluid mechanics. Although one could introduce several new functions into the algebraic toolbox for those exploring mathematics, we have demonstrated that the Lambert-W relation is one that can easily be adapted into the toolbox of elementary-level mathematics students. [ABSTRACT FROM AUTHOR]

The analysis of the derivation of the Riemann Analytic Continuation Formula from Euler's Quadratic Equation is presented in this paper. The connections between the roots of Euler's quadratic equation and the Analytic Continuation Formula of the Riemann Zeta equation are also considered. The method of partial summation is applied twice on the resulting series, thus leading to the Riemann Analytic Continuation Formula. A polynomial approach is anticipated to prove the Riemann hypothesis; thus, a general equation for the zeros of the Analytic Continuation Formula of the Riemann Zeta equation based on a polynomial function is also obtained. An expression in Terms of Prime numbers and their products is considered and obtained. A quadratic function, G(tn), that is required for Euler's quadratic equation (EQE) to give the Analytic Continuation Formula of the Riemann Zeta equation (ACF) is presented. This function thus allows a new way of defining the Analytic Continuation Formula of the Riemann Zeta equation (ACF) via this equivalent equation. By and large, the Riemann Zeta function is shown to be a type of L function whose solutions are connected to some algebraic functions. These algebraic functions are shown and presented to be connected to some polynomials. These Polynomials are also shown to be some of the algebraic functions' solutions. Conclusively, &(z) is redefined as the product of a new function which is called H(tn; z) and this new function is shown to be dependent on the polynomial function, G(tn). [ABSTRACT FROM AUTHOR]

Functions that are integrals of products of generalized hypergeometric functions of some kind are considered. The conditions for the representability of these functions as a polynomial in hypergeometric ones are found. The necessary and sufficient conditions for the algebraic independence of such functions are established. [ABSTRACT FROM AUTHOR]

Here we exhibit multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝN, N ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We study also the case of approximation by iterated operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type in- equalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the algebraic sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer. [ABSTRACT FROM AUTHOR]

We obtain the Boomerang Connectivity Table of power permutations F (x) = x 2 m − 1 of F 2 n with m ∈ { 3 , n − 1 2 , n + 1 2 , n − 2 }. In particular, we obtain the Boomerang uniformity and the Boomerang uniformity set of F (x) at b ∈ F 2 n ∖ F 2. Moreover we determine the complete Boomerang distribution spectrum of F(x) using the number of rational points of certain concrete algebraic curves over F 2 n . We also determine the distribution spectra of Boomerang uniformities explicitly. [ABSTRACT FROM AUTHOR]

Given an algebraic function field F|K$F|K$ and a place ℘ on K, we prove that the places that are composite with extensions of ℘ to finite extensions of K lie dense in the space of all places of F, in a strong sense. We apply the result to the case of K=R$K=R$ any real closed field and the fixed place on R being its natural (finest) real place. This leads to a new description of the real holomorphy ring of F, which can be seen as an analog to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space M(F)$M(F)$ of all R$\mathbb {R}$‐places of F (places with residue field contained in R$\mathbb {R}$), its subspace of all R$\mathbb {R}$‐places of F that are composite with the natural R$\mathbb {R}$‐place of R, and the topological space of all R‐rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper, the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space M(F)$M(F)$ has only finitely many topological components. [ABSTRACT FROM AUTHOR]

Schur's inequality for the sum of products of the differences of real numbers states that for x, y, z, t ≥ 0, xt(x − y)(x − z) + yt(y − z)(y − x) + zt(z − x)(z − y) ≥ 0. In this paper, we study a generalization of this inequality to more terms, more general functions of the variables and algebraic structures such as vectors and Hermitian matrices. [ABSTRACT FROM AUTHOR]

This paper provides several illustrations of the numerous remarkable properties of the lambda extensions of the two-point correlation functions of the Ising model, shedding some light on the non-linear ODEs of the Painlevé type they satisfy. We first show that this concept also exists for the factors of the two-point correlation functions focusing, for pedagogical reasons, on two examples, namely C (0 , 5) and C (2 , 5) at ν = − k . We then display, in a learn-by-example approach, some of the puzzling properties and structures of these lambda extensions: for an infinite set of (algebraic) values of λ these power series become algebraic functions, and for a finite set of (rational) values of lambda they become D-finite functions, more precisely polynomials (of different degrees) in the complete elliptic integrals of the first and second kind K and E . For generic values of λ these power series are not D-finite, they are differentially algebraic. For an infinite number of other (rational) values of λ these power series are globally bounded series, thus providing an example of an infinite number of globally bounded differentially algebraic series. Finally, taking the example of a product of two diagonal two-point correlation functions, we suggest that many more families of non-linear ODEs of the Painlevé type remain to be discovered on the two-dimensional Ising model, as well as their structures, and in particular their associated lambda extensions. The question of their possible reduction, after complicated transformations, to Okamoto sigma forms of Painlevé VI remains an extremely difficult challenge. [ABSTRACT FROM AUTHOR]

We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then obtained by making corrections at singular points. We report on an implementation and provide some examples in MAGMA which demonstrate an improvement over Tuitman's algorithm. [ABSTRACT FROM AUTHOR]

The theoretical foundation has been developed for establishing whether cluster perturbation (CP) series for the energy, molecular properties, and excitation energies are convergent or divergent and for using a two-state model to describe the convergence rate and convergence patterns of the higher-order terms in the CP series. To establish whether the perturbation series are convergent or divergent, a fictitious system is introduced, for which the perturbation is multiplied by a complex scaling parameter z. The requirement for convergent perturbation series becomes that the energy or molecular property, including an excitation energy, for the fictitious system is an analytic, algebraic function of z that has no singularities when the norm |z| is smaller than one. Examples of CP series for the energy and molecular properties, including excitation energies, are also presented, and the two-state model is used for the interpretation of the convergence rate and the convergence patterns of the higher-order terms in these series. The calculations show that the perturbation series effectively become a two-state model at higher orders. [ABSTRACT FROM AUTHOR]

For an algebraic number α$\alpha$ and γ∈R$\gamma \in \mathbb {R}$, let be the house, h(α)$h(\alpha)$ be the (logarithmic) Weil height, and hγ(α)=(degα)γh(α)$h_\gamma (\alpha)=(\deg \alpha)^\gamma h(\alpha)$ be the γ$\gamma$‐weighted (logarithmic) Weil height of α$\alpha$. Let f:Q¯→[0,∞)$f:\overline{\mathbb {Q}}\rightarrow [0,\infty)$ be a function on the algebraic numbers Q¯$\overline{\mathbb {Q}}$, and let S⊂Q¯$S\subset \overline{\mathbb {Q}}$. The Northcott number Nf(S)$\mathcal {N}_f(S)$ of S$S$, with respect to f$f$, is the infimum of all X⩾0$X\geqslant 0$ such that {α∈S;f(α)

Arrays are commonly used in a variety of software to store and process data in loops. Automatically proving safety properties of such programs that manipulate arrays is challenging. We present a novel verification technique, called full-program induction, for proving (a sub-class of) quantified as well as quantifier-free properties of programs manipulating arrays of parametric size N. Instead of inducting over individual loops, our technique inducts over the entire program (possibly containing multiple loops) directly via the program parameter N. The technique performs non-trivial transformations of the given program and pre-conditions during the inductive step. The transformations assist in effectively reducing the assertion checking problem by transforming a program with multiple loops to a program which has fewer and simpler loops or is loop free. Significantly, full-program induction does not require generation or use of loop-specific invariants. To assess the efficacy of our technique, we have developed a prototype tool called Vajra. We demonstrate the performance of Vajra vis-a-vis several state-of-the-art tools on a large set of array manipulating benchmarks from the international software verification competition (SV-COMP) and on several programs inspired by algebraic functions that perform polynomial computations. [ABSTRACT FROM AUTHOR]