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We propose a new class of probabilistic neural-symbolic models, that have symbolic functional programs as a latent, stochastic variable. Instantiated in the context of visual question answering, our probabilistic formulation offers two key…

We investigate the zonal polynomials, a family of symmetric polynomials that appear in many mathematical contexts, such as multivariate statistics, differential geometry, representation theory, and combinatorics. We present two computer…

Motivated by Stanley's generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a representation of the symmetric group, or more generally, of a symmetric…

Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion…

For an integrable Hamiltonian system we construct a representation of the phase space symmetry algebra over the space of functions on a Lagrangian manifold. The representation is a result of the canonical quantization of the integrable…

Two families (type $A$ and type $B$) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri type recurrence formulas for these families. In the…

We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using harmonic means and completely partitioned weighted geometric means. Our result…

This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a…

We introduce the notion of linearly representable games. Broadly speaking, these are TU games that can be described by as many parameters as the number of players, like weighted voting games, airport games, or bankruptcy games. We show that…

In the present article a new method of deriving integral representations of combinations and partitions in terms of harmonic products has been established. This method may be relevant to statistical mechanics and to number theory.

We consider the problem of symbolic-numeric integration of symbolic functions, focusing on rational functions. Using a hybrid method allows the stable yet efficient computation of symbolic antiderivatives while avoiding issues of…

In this work we provide a novel approach for computing the coefficients of the characteristic polynomial of a square matrix. We demonstrate that each coefficient can be efficiently represented by a set of circle graphs. Thus, one can employ…

The multivariate quantum $q$-Krawtchouk polynomials are shown to arise as matrix elements of "$q$-rotations" acting on the state vectors of many $q$-oscillators. The focus is put on the two-variable case. The algebraic interpretation is…

Multivariate Poisson random variables subject to linear integer constraints arise in several application areas, such as queuing and biomolecular networks. This note shows how to compute conditional statistics in this context, by employing…

Integral representations for continuous polynomial local functionals on convex functions are established in terms of a finite family of polynomials. This result is obtained by approximation from a classification of the dense subspace of…

A multivariate polynomial is {\em stable} if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra $\A_n$ that preserve stability. An important…

A functional representation of free L\'evy processes is established via an ensemble of unitarily invariant Hermitian matrix-valued L\'evy processes. This is accomplished by proving functional asymptotics of their empirical spectral…

In multicentric representation of piecewise holomorphic functions one combines Lagrange interpolation at roots of a polynomial $p$ with convergent power series of $p$ as the "coefficients" multiplying the Lagrange basis polynomials. When…

We derive a collection of identities for bivariate Fibonacci and Lucas polynomials using essentially a matrix approach as well as properties of such polynomials when the variables $x$ and $y$ are replaced by polynomials. A wealth of…

Trough the classical umbral calculus, we provide new, compact and easy to handle expressions of k-statistics, and more in general of U-statistics. In addition such a symbolic method can be naturally extended to multivariate case and to…