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Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
It seems that the index theory for non-compact spaces has found its ultimate formulation in realm of coarse spaces and $K$-theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and…
In engineering practice one often encounters planar problems, where the corresponding vector space of forces, velocities or (infinitesimal) displacements is three dimensional. This paper shows how these spaces can be factorized, such that…
We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no…
In relativistic quantum mechanics, elementary particles are described by irreducible unitary representations of the Poincare group. The same applies to the center-of-mass kinematics of a multi-particle system that is not subject to external…
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from…
This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index…
For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.
The purpose of this paper is twofold. First, we axiomatize preference relations on a $\sigma$-algebra of a saturated measure space represented by a vector measure and furnish a utility representation in terms of a nonadditive measure…
Bernoulli-$p$ thinning has been well-studied for point processes. Here we consider three other cases: (1) sequences $(X_1,X_2,...)$; (2) gaps of such sequences $(X_{n+1}-X_1)_{n\in\mathbb{N}}$; (3) partition structures. For the first case…
We study a relation between brick $n$-tuples of subspaces of a finite dimensional linear space, and irreducible $n$-tuples of subspaces of a finite dimensional Hilbert (unitary) space such that a linear combination, with positive…
In this paper, we introduce a graph structure, called non-zero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the…
Motivated by the theory of locally definable groups, we study the theory of $K$-vector spaces with a predicate for the union $X$ of an infinite family of independent subspaces. We show that if $K$ is infinite then the theory is complete and…
A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.
Let $X$ be a topological vector space of complex-valued sequences and $Y$ be a subset of $X$. We provide conditions for $X \setminus Y \cup \{0\}$ to contain uncountably infinitely many linearly independent dense vector subspaces of $X$. We…
The state space of a homogeneous body is derived under two different assumptions: infinitesimal reducibility and irreducibility. The first assumption leads to a real vector space, used in classical mechanics, while the second one leads to a…
We study dynamical semigroups of positive, but not completely positive maps on finite-dimensional bipartite systems and analyze properties of their generators in relation to non-decomposability and bound-entanglement. An example of…
A relationally exchangeable structure is a random combinatorial structure whose law is invariant with respect to relabeling its relations, as opposed to its elements. Aside from exchangeable random partitions, examples include edge…
Relational quantum queries are sometimes capable to effectively decide between collections of mutually exclusive elementary cases without completely resolving and determining those individual instances. Thereby the set of mutually exclusive…