Division of differential operators, intertwine relations and darboux transformations

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x 1,x 2,? and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N?? of renormalized Macdonald operators acting on symmetric polynomials in the variables x 1,?,x N . They are differential operators in terms of the power sum variables $p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots$ and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall---Littlewood symmetric functions of the variables x 1,x 2,?. Our result also yields elementary step operators for the Macdonald symmetric functions.

For a semisimple Lie group G satisfying the equal-rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In our work here we study some of the branching laws for discrete series when restricted to a subgroup H of the same type by combining classical results with recent work of Kobayashi; in particular, we prove discrete decomposability under Harish-Chandra’s condition of cusp form on the reproducing kernel. We show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators.

Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

For classes of periodic functions defined by constraints imposed on the L1-norm of the result of action of differential operators with constant coefficients and real spectrum on these functions, we determine the exact values of the best L1-approximations by generalized splines from the classes considered.

For the numerical solution of linear variational problems involving elliptic partial differential operators in n ≥ 2 space dimensions, iterative solution schemes are indispensable on account of their problem size. Our guiding principle is to devise iterative solvers which are optimal in the number of arithmetic operations, i.e., which are of linear complexity in the total number of unknowns. For these algorithms, asymptotically optimal preconditioners are required. The class of preconditioners for which this can be shown are of multilevel type, requiring nested approximation spaces to approximate the solution of the system on a fine user-specified grid. For smooth solutions of standard second and fourth order elliptic PDEs (partial differential equations) in variational form, approximations based on tensor products of higher-order B-splines yield high accuracy.For such problem classes, this survey collects the main ingredients for multilevel preconditioners in terms of higher order B-splines. There are three types of multilevel preconditioners for which asymptotic optimality can be shown. Two of them, the so-called additive preconditioners, are specified for isogeometric analysis involving linear elliptic partial differential operators in terms of variants of the BPX (Bramble-Pasciak-Xu) preconditioner and wavelet preconditioners. The third class are the so-called multiplicative preconditioners, specifically, multigrid methods.An essential ingredient for all these multilevel preconditioners are intergrid operators which transform vectors or matrices between grids of different grid spacing. For higher order B-splines, these intergrid operators can be derived from their refinement relations. In addition to a presentation of the theoretical ingredients, the performance of the different preconditioners will be demonstrated by some numerical examples.KeywordsVariational InequalityRiesz BasisBiorthogonal WaveletElliptic PDEsSpline SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier–Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space {varPhi }_h subset H^2({varOmega }) and prove that the triad {{varPhi }_h, {varvec{V}}_h, Q_h} (with {varvec{V}}_h and Q_h denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.

We explore the connections between singular Weyl–Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the results to spherical Schrödinger operators (also known as Bessel operators). We also investigate the connections with the generalized Bäcklund–Darboux transformation.

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.

We compute numerically the dimensions and OPE coefficients of several operators in the 3d Ising CFT, and then try to reverse-engineer the solution to crossing symmetry analytically. Our key tool is a set of new techniques for computing infinite sums of SL(2, mathrm{mathbb{R}} ) conformal blocks. Using these techniques, we solve the lightcone bootstrap to all orders in an asymptotic expansion in large spin, and suggest a strategy for going beyond the large spin limit. We carry out the first steps of this strategy for the 3d Ising CFT, deriving analytic approximations for the dimensions and OPE coefficients of several infinite families of operators in terms of the initial data {Δσ, Δϵ, fσσϵ, fϵϵϵ, cT}. The analytic results agree with numerics to high precision for about 100 low-twist operators (correctly accounting for O(1) mixing effects between large-spin families). Plugging these results back into the crossing equations, we obtain approximate analytic constraints on the initial data.

We analyze the dependence of the effective action and the entanglement entropy in the Maxwell theory on the gauge fixing parameter a in d dimensions. For a generic value of a the corresponding vector operator is nonminimal. The operator can be diagonalized in terms of the transverse and longitudinal modes. Using this factorization we obtain an expression for the heat kernel coefficients of the nonminimal operator in terms of the coefficients of two minimal Beltrami-Laplace operators acting on 0- and 1-forms. This expression agrees with an earlier result by Gilkey et al. Working in a regularization scheme with the dimensionful UV regulators we introduce three different regulators: for transverse, longitudinal and ghost modes, respectively. We then show that the effective action and the entanglement entropy do not depend on the gauge fixing parameter a provided the certain (a-dependent) relations are imposed on the regulators. Comparing the entanglement entropy with the black hole entropy expressed in terms of the induced Newton’s constant we conclude that their difference, the so-called Kabat’s contact term, does not depend on the gauge fixing parameter a. We consider this as an indication of gauge invariance of the contact term.

We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} } \) and \( \partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }^\dag \) and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.

We examine the remarkable connection, first discovered by Beukers, Kolk and Calabi, between ζ(2n), the value of the Riemann Zeta-function at an even positive integer, and the volume of some 2n-dimensional polytope. It can be shown that this volume is equal to the trace of a compact self-adjoint operator. We provide an explicit expression for the kernel of this operator in terms of Euler polynomials. This explicit expression makes it easy to calculate the volume of the polytope and hence ζ(2n). In the case of odd positive integers, we rediscover an integral representation for ζ(2n + 1), obtained by a different method by Cvijovic and Klinowski. Finally, we indicate that the origin of the miraculous Beukers–Kolk–Calabi change of variables in the multidimensional integral, which is at the heart of this circle of ideas, can be traced to the amoeba associated with the certain Laurent polynomial. The article is dedicated to the memory of Vladimir Arnold (1937–2010).

Parametric uncertainty assessment in hydrological modeling using the generalized polynomial chaos expansion

Bell polynomials of arbitrary (fractional) orders+

In this paper, we generalize classical Bell polynomials into super version, which are found to be effective in systematically constructing super bilinear representation, bilinear Bäcklund transformation, Lax pair, and infinite conservation laws of supersymmetric equations. We take \documentclass[12pt]{minimal}\begin{document}$\mathcal{N}=1$\end{document}N=1 supersymmetric KdV equation and \documentclass[12pt]{minimal}\begin{document}$\mathcal{N}=2$\end{document}N=2 supersymmetric sine-Gordon equation to illustrate this procedure.