Josef Meixner: His life and his orthogonal polynomials

This paper starts with a biographical sketch of the life of Josef Meixner. Then his motivations to work on orthogonal polynomials and special functions are reviewed. Meixner’s 1934 paper introducing the Meixner and Meixner–Pollaczek polynomials is discussed in detail. Truksa’s forgotten 1931 paper, which already contains the Meixner polynomials, is mentioned. The paper ends with a survey of the reception of Meixner’s 1934 paper.     

Three sphere inequality for second order elliptic equations with coefficients with jump discontinuity

This is a short note to complete the paper appeared in Francini et al. (2016) [4], where a rough version of the classical well known Hadamard three-circle theorem for solution of an elliptic PDE in divergence form has been proved. Precisely, instead of circles, the authors obtain a similar inequality in a more complicated geometry. In this paper we clean the geometry and obtain a generalized version of the three-circle inequality for elliptic equation with coefficients with discontinuity of jump type.     

On weighted q-Daehee polynomials with their applications

In this paper, we first consider a generalization of Kim’s $p$-adic $q$-integral on ${\mathbb{Z}}_p$ including parameters $\alpha$ and $\beta$. By using this integral, we introduce the $q$-Daehee polynomials and numbers with weight $\left(\alpha ,\beta \right)$. Then, we obtain some interesting relationships and identities for these numbers and polynomials. We also derive some correlations among $q$-Daehee polynomials with weight $\left(\alpha ,\beta \right)$, $q$-Bernoulli polynomials with weight $\left(\alpha ,\beta \right)$ and Stirling numbers of second kind.     

Sheaves and duality

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf representations of universal algebras over stably compact spaces and frame homomorphisms from the dual frames of such spaces into subframes of pairwise commuting congruences of the congruence lattices of the universal algebras. For distributive-lattice-ordered algebras this allows us to dualize such sheaf representations.     

Operator-stable and operator-self-similar random fields

Two classes of multivariate random fields with operator-stable marginals are constructed. The random fields $\mathbb{X}=\{X\left(t\right):t\in {\mathbb{R}}^d\}$ with values in ${\mathbb{R}}^m$ are invariant in law under operator-scaling in both the time-domain and the state-space. The construction is based on operator-stable random measures utilizing certain homogeneous functions.     

Integrality and gauge dependence of Hennings TQFTs

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbb{k}$, starting from a finite Hopf algebra over $\mathbb{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds.We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotomic integers for surfaces with one boundary component.We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element.These results are combined and applied to show that the Hennings invariant associated to quantum-${\mathfrak{sl}}_2$ takes values in the cyclotomic integers. Using prior results of Chen et al. we infer integrality also of the Witten–Reshetikhin–Turaev $SO(3)$ invariant for rational homology spheres.As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras.     

Derived recollements and generalised AR formulas

The Defect Recollement, Restriction Recollement, Auslander–Gruson–Jensen Recollement, and others, are shown to be instances of a general construction using zeroth derived functors and methods from stable module theory. The right derived functors ${\mathsf{W}}_k:=R_k{(\underset{\_}{})}^{?}$ are computed and it is shown that the functor ${\mathsf{W}}_2:=R_2{(\underset{\_}{})}^{?}$ is right exact and restricts to a duality $\mathsf{W}$ of the defect zero functors. The duality $\mathsf{W}$ satisfies two identities which we call the Generalised Auslander–Reiten formulas. We show that $\mathsf{W}$ induces the generalised Auslander–Bridger transpose and show that the Generalised Auslander–Reiten formulas reduce to the well-known Auslander–Reiten formulas.     

Flow-contractible configurations and group connectivity of signed graphs

Jaeger, Linial, Payan and Tarsi (JCTB, 1992) introduced the concept of group connectivity as a generalization of nowhere-zero flow for graphs. In this paper, we introduce group connectivity for signed graphs and establish some fundamental properties. For a finite abelian group $A$, it is proved that an $A$-connected signed graph is a contractible configuration for $A$-flow problem of signed graphs. In addition, we give sufficient edge connectivity conditions for signed graphs to be $A$-connected and study the group connectivity of some families of signed graphs.     

Nilpotence and generation in the stable module category

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.     

Discretisation and duality of optimal Skorokhod embedding problems

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem.The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (2017), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.     

The enhanced Sanov theorem and propagation of chaos

We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the ($k$-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean–Vlasov type limit, as shown in two corollaries.     

Dichotomies for generalized ordinary differential equations and applications

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.     

AF inverse monoids and the structure of countable MV-algebras

This paper is a further contribution to the developing theory of Boolean inverse monoids. These monoids should be regarded as non-commutative generalizations of Boolean algebras; indeed, classical Stone duality can be generalized to this non-commutative setting to yield a duality between Boolean inverse monoids and a class of étale topological groupoids. MV-algebras are also generalizations of Boolean algebras which arise from many-valued logics. It is the goal of this paper to show how these two generalizations are connected. To do this, we define a special class of Boolean inverse monoids having the property that their lattices of principal ideals naturally form an MV-algebra. We say that an arbitrary MV-algebra can be co-ordinatized if it is isomorphic to an MV-algebra arising in this way. Our main theorem is that every countable MV-algebra can be so co-ordinatized. The particular Boolean inverse monoids needed to establish this result are examples of what we term AF inverse monoids and are the inverse monoid analogues of AF $C^{?}$-algebras. In particular, they are constructed from Bratteli diagrams as direct limits of finite direct products of finite symmetric inverse monoids.     

Monochromatic tree covers and Ramsey numbers for set-coloured graphs

We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a set of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions.Our results for tree covers in complete graphs imply that a stronger version of Ryser’s conjecture holds for $k$-intersecting $r$-partite $r$-uniform hypergraphs: they have a transversal of size at most $r?k$. (Similar results have been obtained by Király et al., see below.) However, we also show that the bound $r?k$ is not best possible in general.     

Universal algebra of a Hom-Lie algebra and group-like elements

We construct the universal enveloping algebra of a Hom-Lie algebra and endow it with a Hom-Hopf algebra structure. We discuss group-like elements that we see as a Hom-group integrating the initial Hom-Lie algebra.     

slN-web categories and categorified skew Howe duality

In this paper we show how the colored Khovanov–Rozansky ${\mathfrak{sl}}_N$-matrix factorizations, due to Wu [45] and Y.Y. [46], [47], can be used to categorify the type A quantum skew Howe duality defined by Cautis, Kamnitzer and Morrison in [14]. In particular, we define ${\mathfrak{sl}}_N$-web categories and 2-representations of Khovanov and Lauda's categorical quantum ${\mathfrak{sl}}_m$ on them. We also show that this implies that each such web category is equivalent to the category of finite-dimensional graded projective modules over a certain type A cyclotomic KLR-algebra.     

The Soergel category and the redotted Webster algebra

We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.