scholarly journals Multivariate Rankin–Selberg Integrals on GL4 and GU(2, 2)

2018 ◽  
Vol 61 (4) ◽  
pp. 822-835
Author(s):  
Aaron Pollack ◽  
Shrenik Shah
Keyword(s):  
Integral Representation ◽  
Integral Representations ◽  
Cusp Forms ◽  
Selberg Integral ◽  
Selberg Integrals ◽  
Exterior Square

AbstractInspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on GSp4 for the product of the standard and spin L-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin–Selberg integral for cusp forms on PGL4 representing the product of the L-functions attached to the three fundamental representations of the Langlands L-group SL4(C). The second integral, which is closely related, is a two-variable Rankin–Selberg integral for cusp forms on PGU(2, 2) representing the product of the degree 8 standard L-function and the degree 8 exterior square L-function.

Langevin field equations: Properties and renormalization

2021 ◽  
pp. 857-874
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Integral Representation ◽  
Langevin Equation ◽  
Gauge Theories ◽  
Homogeneous Spaces ◽  
Brst Symmetry ◽  
Integral Representations ◽  
Constraint Equations ◽  
Dynamic Action ◽  
Geometric Origin ◽  
Field Integral

Langevin equations for fields have been proposed to describe the dynamics of critical phenomena, or as an alternative method of quantization, which could be useful in cases where ordinary methods lead to difficulties, like in gauge theories. Some of their general properties will be described here. For a number of problems, in particular related to perturbation theory, it is more convenient to work with an action and a field integral than with the equation directly, because standard methods of quantum field theory (QFT) then become available. For this purpose, one can associate a field integral representation, involving a dynamic action to the Langevin equation. The dynamic action can be interpreted as generated by the Langevin equation, considered as a constraint equation. Quite generally, the integral representation of constraint equations, including stochastic equations, leads to an action that has a Slavnov–Taylor (ST) symmetry and, in a different form, has an anticommuting type Becchi–Rouet–Stora–Tyutin (BRST) symmetry, a symmetry that involves anticommuting parameters. This symmetry has no geometric origin, but is merely a consequence of associating a specific form of integral representations to the constraint equations. This symmetry is used in a number of different examples to prove the renormalizability of non-Abelian gauge theories, or to prove the geometric stability of models defined on homogeneous spaces under renormalization. In this chapter, it is used to prove Ward-Takahashi (WT) identities, and to determine how the Langevin equation renormalizes.

Positive Forms on Nuclear *-Algebras and Their Integral Representations

1990 ◽  
Vol 42 (3) ◽  
pp. 410-469 ◽  
Author(s):  
Alain Bélanger ◽  
Erik G. F. Thomas
Keyword(s):  
Integral Representation ◽  
Positive Cone ◽  
Approximate Identity ◽  
Integral Representations ◽  
Regularity Conditions ◽  
Direct Integral ◽  
Positive Form ◽  
Representations Of Algebras ◽  
Almost All ◽  
Order Intervals

Abstract.The main result of this paper establishes the existence and uniqueness of integral representations of KMS functionals on nuclear *- algebras. Our first result is about representations of *-algebras by means of operators having a common dense domain in a Hilbert space. We show, under certain regularity conditions, that (Powers) self-adjoint representations of a nuclear *-algebra, which admit a direct integral decomposition, disintegrate into representations which are almost all self-adjoint. We then define and study the class of self-derivative algebras. All algebras with an identity are in this class and many other examples are given. We show that if is a self-derivative algebra with an equicontinuous approximate identity, the cone of all positive forms on is isomorphic to the cone of all positive invariant kernels on These in turn correspond bijectively to the invariant Hilbert subspaces of the dual space This shows that if is a nuclear -space, the positive cone of has bounded order intervals, which implies that each positive form on has an integral representation in terms of the extreme generators of the cone. Given a continuous exponentially bounded one-parameter group of *-automorphisms of we can define the subcone of all invariant positive forms satisfying the KMS condition. Central functionals can be viewed as KMS functionals with respect to a trivial group action. Assuming that is a self-derivative algebra with an equicontinuous approximate identity, we show that the face generated by a self-adjoint KMS functional is a lattice. If is moreover a nuclear *-algebra the previous results together imply that each self-adjoint KMS functional has a unique integral representation by means of extreme KMS functionals almost all of which are self-adjoint.

scholarly journals Integral representations of generalized Lauricella hypergeometric functions

1992 ◽  
Vol 15 (4) ◽  
pp. 653-657 ◽  
Author(s):  
Vu Kim Tuan ◽  
R. G. Buschman
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Appell Function

The generalized hypergeometric function was introduced by Srivastava and Daoust. In the present paper a new integral representation is derived. Similarly new integral representations of Lauricella and Appell function are obtained.

Alternative Boundary-Integral Representations of Ship Waves

2002 ◽  
Author(s):  
Francis Noblesse ◽  
Chi Yang ◽  
Dane Hendrix ◽  
Rainald Lo¨hner
Keyword(s):  
Integral Representation ◽  
Fundamental Problem ◽  
Boundary Integral ◽  
Ship Hull ◽  
Integral Representations ◽  
Singular Boundary ◽  
Ship Waves ◽  
Classical Boundary ◽  
The Green Function ◽  
Boundary Integral Representation

The fundamental problem of determining the free-surface potential flow that corresponds to a given flow at a ship hull surface is reconsidered. Stokes’ theorem is used to transform the dipole distribution over the ship hull surface in the classical boundary-integral representation of the velocity potential. This Stokes’ transformation yields a weakly-singular boundary-integral representation that defines the potential in terms of the Green function G and related functions that are no more singular than G. Accordingly, the velocity representation only involves functions that are no more singular than ∇G.

scholarly journals Regularized Integral Representations of the Reciprocal Gamma Function

2019 ◽  
Vol 3 (1) ◽  
pp. 1 ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Integral Representation ◽  
Computer Algebra ◽  
Gamma Function ◽  
Real Line ◽  
Computer Algebra System ◽  
Integral Representations ◽  
Complex Representation ◽  
Hypersingular Integral ◽  
The Real ◽  
Two Parameter

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided.

scholarly journals ON INTEGRAL REPRESENTATIONS OF THE DRAZIN INVERSE IN BANACH ALGEBRAS

2002 ◽  
Vol 45 (2) ◽  
pp. 327-331 ◽  
Author(s):  
N. Castro González ◽  
J. J. Koliha ◽  
Yimin Wei
Keyword(s):  
Integral Representation ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Integral Representations ◽  
Subject Classification ◽  
Drazin Inverse ◽  
General Situation ◽  
Primary 46H05 ◽  
C Algebra

AbstractThe purpose of this paper is to derive an integral representation of the Drazin inverse of an element of a Banach algebra in a more general situation than previously obtained by the second author, and to give an application to the Moore–Penrose inverse in a $C^*$-algebra.AMS 2000 Mathematics subject classification:Primary 46H05; 46L05

A note on integral representations of the Skorokhod map

2016 ◽  
Vol 53 (1) ◽  
pp. 293-298 ◽  
Author(s):  
Patrick Buckingham ◽  
Brian Fralix ◽  
Offer Kella
Keyword(s):  
Continuous Function ◽  
Integral Representation ◽  
Bounded Variation ◽  
Integral Representations ◽  
The One ◽  
Skorokhod Reflection

Abstract We present a very short derivation of the integral representation of the two-sided Skorokhod reflection Z of a continuous function X of bounded variation, which is a generalization of the integral representation of the one-sided map featured in Anantharam and Konstantopoulos (2011) and Konstantopoulos et al. (1996). We also show that Z satisfies a simpler integral representation when additional conditions are imposed on X.

scholarly journals Periods of automorphic forms: the case of

2014 ◽  
Vol 151 (4) ◽  
pp. 665-712 ◽  
Author(s):  
Atsushi Ichino ◽  
Shunsuke Yamana
Keyword(s):  
Automorphic Form ◽  
Automorphic Forms ◽  
Selberg Integral ◽  
Selberg Integrals ◽  
Diagonal Subgroup ◽  
Periods Of Automorphic Forms

Following Jacquet, Lapid and Rogawski, we define a regularized period of an automorphic form on $\text{GL}_{n+1}\times \text{GL}_{n}$ along the diagonal subgroup $\text{GL}_{n}$ and express it in terms of the Rankin–Selberg integral of Jacquet, Piatetski-Shapiro and Shalika. This extends the theory of Rankin–Selberg integrals to all automorphic forms on $\text{GL}_{n+1}\times \text{GL}_{n}$.

scholarly journals On integral representation of the solutions of a model $\vec{2b}$-parabolic boundary value problem

2019 ◽  
Vol 11 (1) ◽  
pp. 193-203
Author(s):  
N.I. Turchyna ◽  
S.D. Ivasyshen
Keyword(s):  
Boundary Value Problem ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Integral Representations ◽  
Parabolic Boundary ◽  
Dirac Delta ◽  
Representation Of Solutions ◽  
Green’S Matrix ◽  
Complementing Condition

A general boundary value problem for Eidelman type $\overrightarrow{2b}$-parabolic system of equation without minor terms in the equations and boundary conditions, and with constant coefficients in the group of major terms is considered in the region $$\{(t,x_1,\dots,x_n)\in \mathbb{R}^{n+1}|t\in(0,T], x_j\in\mathbb{R}, j\in\{1,\dots,n-1\}, x_n>0\},$$ $T>0$, $n\ge 2$. It is assumed that the boundary conditions are connected with the system of equations by the complementing condition, which is analogous to the Lopatynsky complementing condition. Integral representations of the solutions for such a problem are derived. The kernels of the integrals from this representation form the Green's matrix of the problem. It is revealed that, in general, not all the elements of the Green's matrix are ordinary functions. Some of them contain terms that are linear combinations of Dirac delta functions and their derivatives. This occurs in cases when the boundary conditions include derivatives with respect to the variables $t$ and $x_n$ of orders that are equal or greater than the highest orders of derivatives with respect to these variables in the equations of the system. The obtained results are important, in particular, for the establishing of the correct solvability and integral representation of solutions for more general $\overrightarrow{2b}$-parabolic boundary value problems.

Integral Representation for U3 × GL2

2009 ◽  
Vol 61 (6) ◽  
pp. 1383-1406
Author(s):  
Eric Wambach
Keyword(s):  
Integral Representation ◽  
Unitary Group ◽  
Integral Representations ◽  
Automorphic Representations ◽  
Residual Spectrum ◽  
Cuspidal Automorphic Representations ◽  
Split Rank

Abstract Gelbart and Piatetskii-Shapiro constructed various integral representations of Rankin–Selberg type for groups G×GLn, where G is of split rank n. Here we show that their method can equally well be applied to the product U3 × GL2, where U3 denotes the quasisplit unitary group in three variables. As an application, we describe which cuspidal automorphic representations of U3 occur in the Siegel induced residual spectrum of the quasisplit U4.

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