Equations In Terms Of Functions Research Articles

GENERALIZED ZETA INTEGRALS ON CERTAIN REAL PREHOMOGENEOUS VECTOR SPACES

AbstractLet X be a real prehomogeneous vector space under a reductive group G, such that X is an absolutely spherical G-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz–Bruhat functions on X against generalized matrix coefficients of admissible representations of $G(\mathbb {R})$ , twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation, as well as a functional equation in terms of abstract $\gamma $ -factors. This subsumes the archimedean zeta integrals of Godement–Jacquet, those of Sato–Shintani (in the spherical case), and the previous works of Bopp–Rubenthaler. The proof of functional equations is based on Knop’s results on Capelli operators.

Characterization of Probability Distributions via Functional Equations of Power-Mixture Type

We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line [0,∞). These equations arise when studying distributional equations of the type Z=dX+TZ, where the random variable T≥0 has known distribution, while the distribution of the random variable Z≥0 is a transformation of that of X≥0, and we want to find the distribution of X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results that are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.

Necessary and sufficient conditions for unique solution to functional equations of Poincaré type

Distributional equation is an important tool in the characterization theory because many characteristic properties of distributions can be transferred to such equations. Using a novel and natural approach, we retreat a remarkable distributional equation whose corresponding functional equation in terms of Laplace–Stieltjes transform is of the Poincaré type. The necessary and sufficient conditions for the equation to have a unique distributional solution with finite variance are provided. This complements the previous results which involve at most the mean of the distributional solution. Besides, more general distributional (or functional) equations are investigated as well.

ON THE HYPERSTABILITY OF THE DRYGAS FUNCTIONAL EQUATION ON A RESTRICTED DOMAIN

We prove hyperstability results for the Drygas functional equation on a restricted domain (a certain subset of a normed space). Our results are more general than the ones proposed by Aiemsomboon and Sintunavarat [‘Two new generalised hyperstability results for the Drygas functional equation’, Bull. Aust. Math. Soc.95 (2017), 269–280] and our proof does not rely on the fixed point theorem of Brzdęk as was the case there. A characterisation of the Drygas functional equation in terms of its asymptotic behaviour is given. Several examples are given to illustrate our generalisations. Finally, we point out a misleading statement in the proof of the second result in the paper by Aiemsomboon and Sintunavarat and propose its correction.

Sine and cosine type functional equations on hypergroups

AbstractLet{(X,*)}be a hypergroup and let{w_{0}}be a fixed measure onX. In this paper we study the two functional equations\langle\delta_{x}*\delta_{y}*\omega_{0},g\rangle+\langle\delta_{x}*\delta_{% \check{y}}*\omega_{0},g\rangle=2g(x)g(y),\quad x,y\in X,and\langle\delta_{x}*\delta_{\check{y}}*\omega_{0},f\rangle-\langle\delta_{x}*% \delta_{y}*\omega_{0},f\rangle=2f(x)f(y),\quad x,y\in X,where{g,f:X\to\mathbb{C}}are continuous and bounded functions to be determined. We express the solutions of the two functional equations in terms of multiplicative maps on{(X,*)}. As an application we give the solution of the two functional equations on polynomial and Sturm–Liouville hypergroups.

The Laurent Coefficients of the Hilbert Series of a Gorenstein Algebra

By a theorem of R. Stanley, a graded Cohen-Macaulay domain A is Gorenstein if and only if its Hilbert series satisfies the functional equation:where, d is the Krull dimension and a is the -invariant of A. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of at t = 1. The main idea consists of examining the graded algebra of formal power series in the variable x that fulfill the condition . As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of Gould and Carlitz on power sums of symmetric number triangles is established.

Structural equations of supermanifolds immersed in the superspace with a prescribed curvature

The aim of this paper is to construct the structural equations of supermanifolds immersed in Euclidean, hyperbolic and spherical superspaces parametrised with two bosonic and two fermionic variables. To perform this analysis, for each type of immersion, we split the supermanifold into its Grassmannian components and study separately each manifold generated. Even though we consider four variables in the Euclidean case, we obtain that the structural equations of each manifold are linked with the Gauss–Codazzi equations of a surface immersed in a Euclidean or spherical space. In the hyperbolic and spherical superspaces, we find that the body manifolds are linked with the classical Gauss–Codazzi equations for a surface immersed in hyperbolic and spherical spaces, respectively. For some soul manifolds, we show that the immersion of the manifolds must be in a hyperbolic space and that the structural equations split into two cases. In one case, the structural equations reduce to the Liouville equation, which can be completely solved. In the other case, we can express the geometric quantities solely in terms of the metric coefficients, which provide a geometric characterization of the structural equations in terms of functions linked with the Hopf differential, the mean curvature and a new function which does not appear in the characterization of a classical (not super) surface.

On the solution of functional equations of Wilson's type on monoids

espanolLet S be a monoid, C be the set of complex numbers, and let σ,τ ∈ Antihom(S,S) satisfy τ ○ τ =σ ○ σ= id. The aim of this paper is to describe the solution ⨍,g: S → C of the functional equation in terms of multiplicative and additive functions. EnglishLet S be a monoid, C be the set of complex numbers, and let σ,τ ∈ Antihom(S,S) satisfy τ ○ τ =σ ○ σ= id. The aim of this paper is to describe the solution ⨍,g: S → C of the functional equation ⨍(xσ(y)) + ⨍(τ(y)x) = 2f(x)g(y), x, y ∈ S, in terms of multiplicative and additive functions.

$$L$$ L -functions as distributions

We define an axiomatic class of L-functions extending the Selberg class. We show in particular that one can recast the traditional conditions of an Euler product, analytic continuation and functional equation in terms of distributional identities akin to Weil's explicit formula. The generality of our approach enables some new applications; for instance, we show that the L-function of any cuspidal automorphic representation of GL_3(A_Q) has infinitely many zeros of odd order.

Classification of integrable Vlasov-type equations

The classification of integrable Vlasov-type equations reduces to a functional equation for a generating function. We find a general solution of this functional equation in terms of hypergeometric functions.

The stability of the equilibrium of two-phase elastic solids

The distinctive features of the loss of stability of elastic solids which undergo phase transitions are investigated for the case of small deformations. The non-uniqueness of the solution of the boundary-value problem for the describing of the thermodynamic equilibrium of a two-phase body is caused by the non-linearity associated with the unknown interface. The solution can be chosen by comparing the potential energies of the body in the two-phase and single phase states and by analysing of the local stability of the two-phase states. A linearized boundary-value problem is formulated which describes infinitesimal small perturbations of an initial two-phase state which is in thermodynamic equilibrium. Analysis of the stability of the two-phase state reduces to an investigation of the bifurcation points and the behaviour of the small solutions of the system of integrodifferential equations in terms of functions describing the perturbations of the interface. The problem of the non-uniqueness and loss of stability of centrisymmetric equilibrium two-phase deformations is investigated as an example. A theorem concerning the number of centrisymmetric solutions is proved. The energy changes accompanying the formation and development of two-phase states and the stability of the solutions obtained are investigated. The concept of topological instability as a bifurcation is introduced, as a result of which the type of geometry of a solution of the boundary-value problem changes and surfaces of separation of the phases actually appear and disappear. Macrodiagrams of the deformational are constructed which demonstrate the effect of deformation softening in the path of a phase transition.

Universal amplitude ratios and Coxeter geometry in the dilute AL model

The leading excitations of the dilute A L model in regime 2 are considered using analytic arguments. The model can be identified with the integrable φ 1,2 perturbation of the unitary minimal series M L, L+1 . It is demonstrated that the excitation spectrum of the transfer matrix satisfies the same functional equations in terms of elliptic functions as the exact S-matrices of the φ 1,2 perturbation do in terms of trigonometric functions. In particular, the bootstrap equation corresponding to a self-fusing process is recovered. For the special cases L=3,4,6 corresponding to the Ising model in a magnetic field, and the leading thermal perturbations of the tricritical Ising and three-state Potts model, as well as for the unrestricted model, L=∞, we relate the structure of the Bethe roots to the Lie algebras E 8,7,6 and D 4 using Coxeter geometry. In these cases Coxeter geometry also allows for a single formula in generic Lie algebraic terms describing all four cases. For general L we calculate the spectral gaps associated with the leading excitation which allows us to compute universal amplitude ratios characteristic of the universality class. The ratios are of field theoretic importance as they enter the bulk vacuum expectation value of the energy momentum tensor associated with the corresponding integrable quantum field theories.

THERMODYNAMIC BETHE ANSATZ AND THREE-FOLD TRIANGULATIONS

In the thermodynamic Bethe ansatz approach to 2D integrable, ADE-related quantum field theories, one derives a set of algebraic functional equations (a Y system) which play a prominent role. This set of equations is mapped onto the problem of finding finite triangulations of certain 3D manifolds. This mapping allows us to find a general explanation of the periodicity of the Y system. For the AN related theories, and more generally for the various restrictions of the fractionally supersymmetric sine—Gordon models, we find an explicit, surprisingly simple solution of such functional equations in terms of a single unknown function of the rapidity. The recently found dilogarithm functional equations associated to the Y system simply express the invariance of the volume of a manifold for deformations of its triangulations.

Solution of the ‘Cube’ Functional Equation in Terms of ‘Trilinear Coefficients’

We consider the following three functional equations123where f:R3→R.Considering their geometric meaning, equations (1) and (2) are known as ‘Cube’ and ‘Octahedron’ functional equations, respectively. Under the assumption of continuity, Haruki [2] has proved that (1) and (2) are equivalent. Etigson [3], has proved the equivalence of (1) and (2) under no regularity assumption. We will give here another proof. Also, under the assumption of continuity, Haruki has solved the ‘Cube’ functional equation. He gave the solution as a certain polynomial of fifth degree in x, y, z individually whose terms are the partial derivatives of a given polynomial.

Generating functionals and the Ising model

The Ising model is studied by the generating functional approach in order to provide a better understanding of that method. It is shown how to derive a general solution of a functional equation in terms of infinite-dimensional integrals. This solution is not unique; the different possibilities are characterized by different paths of integration. Further, the saddle point approximation is used for the integrals in order to obtain second-order correlation functions. It is shown that besides the “normal” solution, one obtains several anomalous ones, which correspond directly to the nonphysical solutions of the transfer matrix method for treating the partition function. It is also shown that only the correct solution can give a realistic behavior of the correlation function at large distances. The relevance of the saddle point methods for describing phase transitions is also discussed.

On the Space-Time Functional Formalism for Turbulence

The functional formalism for turbulence mechanics originated with Hopf is generalized in such a way that it can embrace a space-time correlation tensor of turbulent fields, i.e. correlation tensor between non-simultaneous fields, and furthermore any external random forcing field can be taken into account in the formalism. The first work in this direction was recently done by Lewis and Kraichnan, but here the theory is more compactly and generally developed together with the deduction of a general solution of the full basic functional equations in terms of functional integration .