Regge trajectories for the (2, 0) theories

We investigate the structure of conformal Regge trajectories for the maximally supersymmetric (2, 0) theories in six dimensions. The different conformal multiplets in a single superconformal multiplet must all have similarly-shaped Regge trajectories. We show that these super-descendant trajectories interact in interesting ways, leading to new constraints on their shape. For the four-point function of the stress tensor multiplet supersymmetry also softens the Regge behavior in some channels, and consequently we observe that ‘analyticity in spin’ holds for all spins greater than −3. All the physical operators in this correlator therefore lie on Regge trajectories and we describe an iterative scheme where the Lorentzian inversion formula can be used to bootstrap the four-point function. Some numerical experiments yield promising results, with OPE data approaching the numerical bootstrap results for all theories with rank greater than one.

From dS to AdS and back

We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.

A note on the axisymmetric diffusion equation

We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for \(u(r,t)\) as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series. doi:10.1017/S1446181121000110

The wave packet transform in the framework of linear canonical transform

In this paper, we introduce the concept of linear canonical wave packet transform (LCWPT) based on the idea of linear canonical transform (LCT) and wave packet transform (WPT). Parseval’s identity and some properties of LCWPT are discussed. The inversion formula of LCWPT is formulated. Moreover, the composition of LCWPTs is defined and some properties are studied related to it. The LCWPTs of Mexican hat wavelet function are obtained.

A PARAMETRIZATION OF δ-SPHERICAL FUNCTIONS ON COMMUTATIVE TRIPLES ASSOCIATED WITH NILPOTENT LIE GROUPS

Let $N$ be a connected and simply connected nilpotent Lie group, $K$ be a compact subgroup of $Aut(N)$, the group of automorphisms of $N$ and $\delta$ be a class of unitary irreducible representations of $K$. The triple $(N,K,\delta)$ is a commutative triple if the convolution algebra $\mathfrak{U}_{\delta}^{1}(N)$ of $\delta$-radial integrable functions is commutative. In this paper, we obtain first a parametrization of $\delta$ spherical functions by means of the unitary dual $\widehat{N}$ and then an inversion formula for the spherical transform of $F\in \mathfrak{U}_{\delta}^{1}(N)$.

A new aspect of generalized integral operator and an estimation in a generalized function theory

In this paper we investigate certain integral operator involving Jacobi–Dunkl functions in a class of generalized functions. We utilize convolution products, approximating identities, and several axioms to allocate the desired spaces of generalized functions. The existing theory of the Jacobi–Dunkl integral operator (Ben Salem and Ahmed Salem in Ramanujan J. 12(3):359–378, 2006) is extended and applied to a new addressed set of Boehmians. Various embeddings and characteristics of the extended Jacobi–Dunkl operator are discussed. An inversion formula and certain convergence with respect to δ and Δ convergences are also introduced.

A NOTE ON THE AXISYMMETRIC DIFFUSION EQUATION

We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for $u(r,t)$ as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as a rapidly converging series.

A new structure of an integral operator associated with trigonometric Dunkl settings

In this paper, we discuss a generalization to the Cherednik–Opdam integral operator to an abstract space of Boehmians. We introduce sets of Boehmians and establish delta sequences and certain class of convolution products. Then we prove that the extended Cherednik–Opdam integral operator is linear, bijective and continuous with respect to the convergence of the generalized spaces of Boehmians. Moreover, we derive embeddings and discuss properties of the generalized theory. Moreover, we obtain an inversion formula and provide several results.