Space Of Regular Functions Research Articles

Five-dimensional gauge theories and the local B-model

We propose an effective framework for computing the prepotential of the topological B-model on a class of local Calabi–Yau geometries related to the circle compactification of five-dimensional mathcal {N}=1 super Yang–Mills theory with simple gauge group. In the simply laced case, we construct Picard–Fuchs operators from the Dubrovin connection on the Frobenius manifolds associated with the extended affine Weyl groups of type mathrm {ADE}. In general, we propose a purely algebraic construction of Picard–Fuchs ideals from a canonical subring of the space of regular functions on the ramification locus of the Seiberg–Witten curve, encompassing non-simply laced cases as well. We offer several precision tests of our proposal for simply laced cases by comparing with the gauge theory prepotentials obtained from the K-theoretic blow-up equations, finding perfect agreement. Whenever there is more than one candidate Seiberg-Witten curve for non-simply laced gauge groups in the literature, we employ our framework to verify which one agrees with the K-theoretic blow-up equations.

Analytical Solution of the Cauchy Problem for a Nonstationary Three-dimensional Model of the Filtration Theory

This paper is devoted to the study of the Cauchy problem for a system of differential equations describing the unsteady flow of a compressible fluid in a homogeneous and inhomogeneous porous medium with a general nonlinear filtration law in three-dimensional space. In the work using the methods of four-dimensional mathematics, a special four-dimensional space of space and time coordinates was developed, as well as a functional space of regular functions, and analytical conditions were obtained on the general form of the nonlinear filtration law for which the Cauchy problem has a solution.

On the Bargmann-Fock-Fueter and Bergman-Fueter integral transforms

This paper deals with some special integral transforms of Bargmann-Fock type in the setting of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. The construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions, we can construct an Appell system of quaternionic regular polynomials. The ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic setting are obtained in both the Fock and Bergman cases.

Gleason’s problem, rational functions and spaces of left-regular functions: The split-quaternion setting

We study Gleason’s problem, rational functions and spaces of regular functions in the setting of split-quaternions. There are two natural symmetries in the algebra of split-quaternions. The first symmetry allows to define positive matrices with split-quaternionic entries, and also reproducing Hilbert spaces of regular functions. The second leads to reproducing kernel Krein spaces.

From Hankel Operators to Carleson Measures in a Quaternionic Variable

AbstractWe introduce and study Hankel operators defined on the Hardy space of regular functions of a quaternionic variable. Theorems analogous to those of Nehari and Fefferman are proved.

The Hall algebra of a curve

Let X be a smooth projective curve over a finite field. We describe H, the full Hall algebra of vector bundles on X, as a Feigin–Odesskii shuffle algebra. This shuffle algebra corresponds to the scheme S of all cusp eigenforms and to the rational function of two variables on S coming from the Rankin–Selberg L-functions. This means that the zeroes of these L-functions control all the relations in H. The scheme S is a disjoint union of countably many \({\mathbb G}_m\)-orbits. In the case when X has a theta-characteristic defined over the base field, we embed H into the space of regular functions on the symmetric powers of S.

Induced orbit data and induced unitary representations for complex groups

We use induced orbit covers to define induced orbit data. By studying the space of regular functions on orbit cover, we know that the induced representation has close connection with the induced orbit datum under the meaning of Vogan's conjecture. Therefore, when verifying Vogan's conjecture, many cases can be reduced to the case of rigid orbit data.

Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups

The extended weight semigroup of a homogeneous space of a connected semisimple algebraic group characterizes the spectra of the representations of on spaces of regular sections of homogeneous line bundles over , including the space of regular functions on . We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces , where is a simply connected non-simple semisimple complex algebraic group and is a connected closed subgroup of . In all cases we also find the highest-weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups.

Fokker–Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces

We consider a Kolmogorov operator L 0 in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drift F = F ( t , ⋅ ) : H → H , defined on a suitable space of regular functions. We show that L 0 is essentially m-dissipative in the space L p ( [ 0 , T ] × H ; ν ) , p ⩾ 1 , where ν ( d t , d x ) = ν t ( d x ) d t and the family ( ν t ) t ∈ [ 0 , T ] is a solution of the Fokker–Planck equation given by L 0 . As a consequence, the closure of L 0 generates a Markov C 0 -semigroup. We also prove uniqueness of solutions to the Fokker–Planck equation for singular drifts F. Applications to reaction–diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753–774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397–418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631–640] to infinite dimensions.

Some asymptotics of Topological quantum field theory via skein theory

For each oriented surface Σ of genus g, we study a limit of quantum representations of the mapping class group arising in topological quantum field theory (TQFT) derived from the Kauffman bracket. We determine that these representations converge in the Fell topology to the representation of the mapping class group on H(Σ), the space of regular functions on the SL(2,C)-representation variety with its Hermitian structure coming from the symplectic structure of the SU(2)-representation variety. As a corollary, we give a new proof of the asymptotic faithfulness of quantum representations

Higher-Dimensional Generalizations of Affine Kac-Moody and Virasoro Conformal Lie Algebras

We discuss the generalizations of the notion of Conformal Algebra and Local Distribution Lie algebras for multi-dimensional bases. We replace the algebra of Laurent polynomials on Open image in new window by an infinite-dimensional representation (with some additional structures) of a simple finite-dimensional Lie algebra Open image in new window in the space of regular functions on the corresponding Grassmann variety Open image in new window that can be described as a ``right'' higher-dimensional generalization of Open image in new window from the point of view of a corresponding group action. For Open image in new window it gives us the usual Vertex Algebra notion. We construct the higher dimensional generalizations of the Virasoro and the Affine Kac-Moody Conformal Lie algebras explicitly and in terms of the Operator Product Expansion.

A surjectivity theorem for differential operators on spaces of regular functions

In this article we show that it is possible to construct a Koszul-type complex for maps given by suitable pairwise commuting matrices of polynomials. This result has applications to surjectivity theorems for constant coefficients differential operators of finite and infinite order. In particular, we construct a large class of constant coefficients differential operators which are surjective on the space of regular (or monogenic) functions on open convex sets.

Uniform estimates in two periodic homogenization problems

We analyze two partial differential equations that are posed on perforated domains. We provide a priori estimates that do not depend on the size of the perforation: A sequence of solutions is uniformly bounded in a Sobolev space of regular functions. The first homogenization problem concerns the Laplace and the mean-curvature operator with Neumann boundary conditions. We derive uniform Lipschitz estimates for the solutions. The result is used in the analysis of a free boundary system of fluid mechanics. A contractive iteration yields the existence of solutions and uniform estimates. The key is the use of function spaces that are different from the usual Lp-spaces. © 2000 John Wiley & Sons, Inc.

RANDOM PROCESSES GENERATED BY A HYPERBOLIC SEQUENCE OF MAPPINGS. II

This paper is a direct continuation of [1]. It contains proofs of the theorems announced there on the existence of invariant classes of functional elements and functional deformations for a hyperbolic sequence of mappings, as well as theorems on the existence of invariant cones in spaces of regular functions defined on a set of functional elements.

ON THE INTEGRABILITY OF INVARIANT HAMILTONIAN SYSTEMS WITH HOMOGENEOUS CONFIGURATION SPACES

All homogeneous spaces ( is a semisimple complex (compact) Lie group, a reductive subgroup) are enumerated for which arbitrary Hamiltonian flows on with -invariant Hamiltonians are integrable in the class of Noether integrals. It is proved that only for these spaces does the quasiregular representation of in the space of regular functions of the algebraic variety have a simple spectrum.Bibliography: 21 titles.

Poisson integrals of regular functions

Tangential convergence of Poisson integrals is proved for certain spaces of regular functions which contain the spaces of Bessel potentials of L p {L^p} functions, 1 > p > ∞ 1 > p > \infty , and of functions in the local Hardy space h 1 {h^1} , and the corresponding tangential maximal functions are shown to be of strong p p type, p ⩾ 1 p \geqslant 1 .

Realizability Theory of Continuous Linear Operators on Groups

Let G and H be a certain type of locally compact (abelian) group. $D(G)$ denotes the space of regular functions with compact support on G and $D'(G)$ is the corresponding space of distributions. Linear mappings from $D(G)$ into $D'(G)(D'(H))$ are the subject of our investigations. We have carried out some systems-theoretic investigation in a distributional setting where distributions are defined on groups. Such investigations in (Schwartz’s) distributional setting have been carried out by several authors. We have chosen the distribution theory on groups as developed by F. Bruhat, K. Maurin and G. I. Kac. This choice is motivated by the existence of Bruhat’s kernel theorem and the nuclearity of the space $D(G)$. Properties such as continuity, regularity, translation-invariance, causality, semipassivity and passivity (a certain positivity property) are imposed on the linear mappings and their effects are studied. Representations of continuous linear mappings from $D(G)$ into $D'(H)$ which are regular into $C(H)$, of continuous linear and translation-invariant mappings from $D(G)$ into $D'(G)$ and of linear and scatter-semipassive mappings from $D(G)$ into $D'(G)$ are obtained. We establish one-to-one correspondence between contractions in $L_2 (G)$ and linear and scatter-semipassive mappings from $D(G)$ into $D'(G)$. We also show that causality and semipassivity imply passivity in the scattering formalism and passivity implies causality in the immittance formalism. A characterization of linear, scatter-semipassive, and real mappings in terms of a positivity criterion is established.