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We introduce some classical concepts in the representation theory of compact groups, in order to use them for a new generalization of the Peter-Weyl Theorem. We mostly deal with functions on locally compact groups possessing large…

We consider lifting of mod p representations to mod p^2 representations in the setting of representations of (i) finite groups; (ii) absolute Galois groups of abstract fields; and (iii) absolute Galois groups of local and global fields.

We characterise the existence of balanced and pluriclosed metrics on compact quotients of real semisimple Lie groups equipped with regular complex structures, in terms of Vogan diagrams. Consequently, such complex manifolds cannot…

For a rational function of several variables with nonnegative imaginary part on the upper poly-half-plane, the matrix representations are obtained.

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…

Using geometric considerations, we provide a clear derivation of the integral representation for the error function, known as the Craig formula. We calculate the corresponding power series expansion and prove the convergence. The same…

We apply some recent developments of Baldoni-Beck-Cochet-Vergne on vector partition function, to Kostant's and Steinberg's formulae, for classical Lie algebras $A\_r$, $B\_r$, $C\_r$, $D\_r$. We therefore get efficient {\tt Maple} programs…

We consider the problem of the representation of real continuous functions by linear superpositions $\sum_{i=1}^{k}g_{i}\circ p_{i}$ with continuous $g_{i}$ and $p_{i}$. This problem was considered by many authors. But complete, and at the…

The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…

Motivated by asymptotic symmetry groups in general relativity, we consider projective unitary representations $\overline{\rho}$ of the Lie group $\mathrm{Diff}_c(M)$ of compactly supported diffeomorphisms of a smooth manifold $M$ that…

We study maximal representations of nonnegative sesquilinear forms in real or complex Hilbert spaces, that are not necessarily closed or even closable. We associate positive self-adjoint operators with such forms, in a sense similar to…

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the…

Hypercomplex numbers are unital algebras over the real numbers. We offer a short demonstration of the practical value of hypercomplex analytic functions in the field of partial differential equations.

Monte Carlo simulations are based on the manipulation of random numbers to evaluate probable outcomes, with applicability in a variety of different fields. By assigning probabilities, which can be determined a priori, to various events, it…

The efficacy of using complex numbers for understanding geometric questions related to polar equations and general cycloids is demonstrated.

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…

We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The goal of this paper is to investigate of this kind of operators. The main theorem is a…

In a recent work of Matteo Mio on compact quantitative equational theories (here compact means that all its consequences are derivable by means of finite proofs) convex algebras on the carrier set [0,1] whose operations are monotone and…

The description of irreducible finite dimensional representations of finite dimensional solvable Lie superalgebras over complex numbers given by V.~Kac is refined. In reality these representations are not just induced from a polarization…

Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…