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We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials $\BR[\underline{x}]$ shall be described in terms of the generators of the Weyl-Heisenberg algebra. The…

In this paper we consider the numerical approximation of systems of Boussinesq-type to model surface wave propagation. Some theoretical properties of these systems (multi-symplectic and Hamiltonian formulations, well-posedness and existence…

We introduce and study certain asymptotic invariants associated with fusion algebras (equipped with a dimension function), which arise naturally in the representation theory of compact quantum groups. Our invariants generalise the analogous…

The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and…

We first introduce the notion of Hamiltonian structure for a partial difference equation. Then we construct some infinite quivers, and realize the discrete KdV equation, the Hirota-Miwa equation and its various reductions as the mutation…

In this paper we prove a useful formula for the graded commutator of the Hodge codifferential with the left wedge multiplication by a fixed $p$-form acting on the de Rham algebra of a Riemannian manifold. Our formula generalizes a formula…

Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite…

We recapture Kuperberg's numerical invariant of 3-manifolds associated to a semisimple and cosemisimple Hopf algebra through a `planar algebra construction'. A result of possibly independent interest, used during the proof, which relates…

In this note we consider compact homomorphisms and endomorphisms between various Dales-Davie algebras. In particular, we obtain fairly complete results when the underlying set is the disc or the unit circle. Comparable results when the…

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be…

We obtain a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ in the weight aspect. We show that correlations of masses coming from off-diagonal terms dissipate…

In this paper, based on the classical K. Yano's formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction $J\vec{H}$ and…

The homotopy type and homotopy groups of some spectra TAF of topological automorphic forms associated to a unitary similitude group GU of type (1,1) are explicitly described in quasi-split cases. The spectrum TAF is shown to be closely…

We provide a finite element discretization of $\ell$-form-valued $k$-forms on triangulation in $\mathbb{R}^{n}$ for general $k$, $\ell$ and $n$ and any polynomial degree. The construction generalizes finite element Whitney forms for the…

In this paper, deformations of $L_\infty$-algebras are defined in such a way that the bases of deformations are $L_\infty$-algebras, as well. A universal and a semiuniversal deformation is constructed for $L_\infty$-algebras, whose…

For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov…

Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups -- coordinate functions, differential…

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field…

We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of non-uniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses…

$G_\infty$-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a $B_\infty$-algebra by an associative algebra. Actions of $B_\infty$-algebras on…