Related papers: Open Questions for operators related to Rectangula…

In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive…

The history of the quadratic stochastic operators can be traced back to work of S.Bernshtein (1924). During more than 80 years this theory developed and many papers were published. In recent years it has again become of interest in…

The goal of this paper is to develop the theory of Schur complementation in the context of operators acting on anti-dual pairs. As a byproduct, we obtain a natural generalization of the parallel sum and parallel difference, as well as the…

We prove some Schur positivity results for the chromatic symmetric function $X_G$ of a (hyper)graph $G$, using connections to the group algebra of the symmetric group. The first such connection works for (hyper)forests $F$: we describe the…

The aim of this note is to introduce a compound basis for the space of symmetric functions. Our basis consists of products of Schur functions and $Q$-functions. The basis elements are indexed by the partitions. It is well known that the…

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum…

It is shown that the squared operation of the Dirac equation which is widely applied may create new solutions and moreover may change the inner nature of original equation. Some illustrating examples are considered as well.

This paper is devoted to conditions defined in terms of the generalized shift operator for a rational number to be representable by certain positive generalizations of $q$-ary expansions.

We consider the Krall-Sheffer class of admissible, partial differential operators in the plane. We concentrate on algebraic structures, such as the role of commuting operators and symmetries. For the polynomial eigenfunctions, we give…

We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies…

We establish an eigenfunctional theorem for positive operators, evocative of the Krein--Rutman theorem. A more general version gives a joint eigenfunctional for commuting operators.

In this paper necessary conditions and sufficient conditions are given for a linear operator to be a positive operators of an Extended Lorentz cone. Similarities and differences with the positive operators of Lorentz cones are investigated.

We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…

Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half-line Schrodinger operator with square integrable potential.

There exist a number of well known multiplicative generating functions for series of Schur functions. Amongst these are some related to the dual Cauchy identity whose expansion coefficients are rather simple, and in some cases periodic in…

We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse…

We give an explicit combinatorial formula for the Schur expansion of Macdonald polynomials indexed by partitions with second part at most two. This gives a uniform formula for both hook and two column partitions. The proof comes as a…

This paper deals with some simple results about spherical functions of type $\delta$, namely new integral formulas, new results about behavior at infinity and some facts about the related $C_\sigma$ functions.

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity…

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying…