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This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of the…
We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function $f : \mathbb{N} \to \mathbb{N}$ such that the $n$-ary part of $C$ is equal to the $n$-ary part of…
In a number of papers, Y. Sternfeld investigated the problems of representation of continuous and bounded functions by linear superpositions. In particular, he proved that if such representation holds for continuous functions, then it holds…
We introduce a new class of unitary transformations based on the su(1,1) Lie algebra that generalizes, for certain particular representations of its generators, well-known squeezing transformations in quantum optics. To illustrate our…
We study reflexivity and structure properties of operator algebras generated by representations of the discrete Heisenberg semi-group. We show that the left regular representation of this semi-group gives rise to a semi-simple reflexive…
In [M. R\"osler and M. Voit. Integral Representation and Uniform Limits for Some Heckman-Opdam Hypergeometric Functions of type BC, Transactions of the American Mathematical Society, Vol. 368, No. 8, 6005-6032, 2016.], R\"osler and Voit…
The norm closure of the algebra generated by the set $\{n\mapsto {\lambda}^{n^k}:$ $\lambda\in{\mathbb {T}}$ and $k\in{\mathbb{N}}\}$ of functions on $({\mathbb {Z}}, +)$ was studied in \cite{S} (and was named as the Weyl algebra). In this…
We formulate some problems and conjectures about semigroups of rational functions under composition. The considered problems arise in different contexts, but most of them are united by a certain relationship to the concept of amenability.
We defined (n+1)-ary derivations of $n$-ary algebras. We described (n+1)-derivations of simple and semisimple finite-dimensional Filippov algebras over algebraically closed field zero characteristic. We constructed new examples of…
In a recent paper, Cohl and Costas-Santos derived a number of interesting multi-derivative and multi-integral relations for associated Legendre and Ferrers functions in which the orders of those functions are changed in integral steps.…
Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan-Ginzburg algebra of…
We study plane partitions satisfying condition $a_{n+1,m+1}=0$ (this condition is called "pit") and asymptotic conditions along three coordinate axes. We find the formulas for generating function of such plane partitions. Such plane…
The purpose of this paper is to study some results of constructions on Hom-Poisson superal-gebras we use the representations and Rota-Baxter operators. We introduce the structures ofn-ary Hom-Nambu Poisson superalgebras and their…
We generalize two integral representation formulae of Nevanlinna to functions of several variables. We show that for a large class of analytic functions that have non-negative imaginary part on the upper polyhalfplane there are…
By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…
A consistent description of images on the disk and of their transformations is given as elements of a vector space and of an operators algebra. The vector space of images on the disk $\mathbb{D}$ is the Hilbert space $L^2(\mathbb{D})$ that…
In our previous paper q-alg/9605011 we proposed several algebraic methods for constructing new solutions to the bispectral problem. In the present note the corresponding eigenfunctions are explicitly constructed as multiple Laplace…
Let E be an operator algebra on a Hilbert space with finite-dimensional generated C*-algebra. A classification is given of the locally finite algebras and the operator algebras obtained as limits of direct sums of matrix algebras over E…
It is known that local operators in quantum field theory transform in representations of ordinary global symmetry groups. The purpose of this paper is to generalise this statement to extended operators such as line and surface defects. We…
We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of $Q \in \C$, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models…