On the Bogolyubov endomorphisms of the renormalized square of white noise algebra

We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.

Clausen’s Series 3F2(1) with Integral Parameter Differences

Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the open problem raised by Miller and Paris in 2012. Given a known value 3F2((a,b,x),(c,x+1),1), if f−x is an integer, then we construct an algorithm to obtain 3F2((a,b,f),(c,f+n),1) in an explicit closed form, where n is a positive integer and a,b,c and f are arbitrary complex numbers. We also extend our results to evaluate some specific forms of p+1Fp(1), for any positive integer p≥2.

Multi-Scale Pseudo-Bending Raytracing for Arbitrary Complex Media

Raytracing is a fast and effective numerical simulation method of the seismic wavefield. It plays an important role in field data acquisition design, wavefield analysis, identification, and tomography. In raytracing, pseudo-bending (PB) is a fast and efficient method, but it is unsuitable for complex media with sudden velocity changes. An improved pseudo-bending raytracing method is presented in this paper, which can be applied to any complex medium. The proposed method first decomposes complex medium into multi-scale velocity components and then applies the pseudo-bending approach to the velocity components of different scales. The numerical simulation of seismic wavefield from models shows that the improved multi-scale pseudo-bending (MSPB) method can be applied to a medium with continuous velocity variation and any complex medium with abrupt velocity change.

On the Spectrum of Two-Point Boundary Value Problems for the Dirac Operator

We consider the spectral problem for a Dirac operator with arbitrary two-point boundary conditions and an arbitrary complex-valued integrable potential. The existence of nontrivial boundary value problems of this type with an unbounded growth of the multiplicity of eigenvalues is established.

Toward the capacity limit of 2D planar Jones matrix with a single-layer metasurface

The Jones matrix is a useful tool to deal with polarization problems, and its number of degrees of freedom (DOFs) that can be manipulated represents its polarization-controlled capabilities. A metasurface is a planar structure that can control light in a desired manner, which, however, has a limited number of controlled DOFs (≤4) in the Jones matrix. Here, we propose a metasurface design strategy to construct a Jones matrix with six DOFs, approaching the upper-limit number of a 2D planar structure. We experimentally demonstrate several polarization functionalities that can only be achieved with high (five or six) DOFs of the Jones matrix, such as polarization elements with independent amplitude and phase tuning along its fast and slow axes, triple-channel complex-amplitude holography, and triple sets of printing-hologram integrations. Our work provides a platform to design arbitrary complex polarization elements, which paves the way to a broader exploitation of polarization optics.

SU(N) q-Toda equations from mass deformed ABJM theory

It is known that the partition functions of the U(N)k × U(N + M)−k ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painlevé III3 equation. In this paper we have suggested that a similar bilinear relation holds for the ABJM theory with $$ \mathcal{N} $$ N = 6 preserving mass deformation for an arbitrary complex value of mass parameter, to which we have provided several non-trivial checks by using the exact values of the partition function for various N, k, M and the mass parameter. For particular choices of the mass parameters labeled by integers ν, a as m1 = m2 = −πi(ν − 2a)/ν, the bilinear relation corresponds to the q-deformation of the affine SU(ν) Toda equation in τ-form.

Возрастающее объединение пространств Стейна с сингулярностями

We show that if $X$ is a Stein space and, if $\Omega\subset X$ is exhaustable by a sequence $\Omega_{1}\subset\Omega_{2}\subset\ldots\subset\Omega_{n}\subset\dots$ of open Stein subsets of $X$, then $\Omega$ is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for $X=\mathbb{C}^{n}$ and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When $X$ has dimension $2$, we prove that the same result follows if we assume only that $\Omega\subset\subset X$ is a domain of holomorphy in a Stein normal space. It is known, however, that if $X$ is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets $X_{1}\subset X_{2}\subset\dots\subset X_{n}\subset\dots$, it does not follow in general that $X$ is holomorphically-convex or~holomorphically-separate (even if $X$ has no singularities). One can even obtain $2$-dimensional complex manifolds on which all holomorphic functions are constant.