On the Carey–Helton–Howe–Pincus trace formula

In this article, we give a new proof of the Carey–Helton–Howe–Pincus trace formula using Kato's theory of “relatively-smooth” operators and Krein's trace formula.     

Koplienko Trace Formula

Koplienko (Sib Math J 25(5): 735–743, 1984) gave a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class ${\mathcal{B}_2(\mathcal{H})}$ . Recently Gesztesy et?al. (Basics Z Mat Fiz Anal Geom 4(1):63–107, 2008) gave an alternative proof of the trace formula when the operators involved are bounded. In this article, we give a still another proof and extend the formula for unbounded case by reducing the problem to a finite dimensional one as in the proof of Krein trace formula by Voiculescu (On a Trace Formula of M. G. Krein. Operator Theory: Advances and Applications, vol. 24, pp. 329–332. Birkhauser, Basel, 1987), Sinha and Mohapatra (Proc Indian Acad Sci (Math Sci) 104(4):819–853, 1994).     

A “direct” method to prove the generalized Itô–Venttsel’ formula for a generalized stochastic differential equation

For the first time we present a complete proof (from the standpoint of stochastic analysis) of the generalized Itô–Venttsel’ formula whose ideas were adduced in [8]. The proposed proof is an approach to construct the generalized Itô–Venttsel’ formula based on the direct application of the generalized Itô formula and the theory of stochastic approximation in contrast to the proof presented in [17] and based on the method of the integral invariants of a stochastic differential equation.     

A combinatorial way of counting unicellular maps and constellations

Our work is devoted to the bijective enumeration of the set of factorizations of a permutation into m factors with a given number of cycles. Previously, this major problem in combinatorics and its various specializations were considered mainly from the character theoretic or algebraic geometry point of view. Let us specially mention here the works of Harer and Zagier or Kontsevich. In 1988, Jackson reported a very general formula solving this problem. However, to the author’s own admission this result left little room for combinatorial interpretation and no bijective proof of it was known yet. In 2001, Lass found a combinatorial proof of the celebrated special case of Jackson’s formula known as the Harer–Zagier formula. This work was followed by Goulden and Nica, who presented in 2004 another combinatorial proof involving a direct bijection. In the past two years, we have introduced new sets of objects called partitioned maps and partitioned cacti, the enumeration of which allowed us to construct bijective proofs for more general cases of Jackson’s formula.     

Touchard–Riordan formulas, T-fractions, and Jacobi’s triple product identity

Touchard–Riordan-like formulas are certain expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove such formulas, related to integer partitions. This gives a new perspective on the original result of Touchard and Riordan. But the main goal is to give a combinatorial proof of a Touchard–Riordan-like formula for q-secant numbers discovered by the first author. An interesting limit case of these objects can be directly interpreted in terms of partitions, so that we obtain a connection between the formula for q-secant numbers, and a particular case of Jacobi’s triple product identity. Building on this particular case, we obtain a “finite version” of the triple product identity. It is in the form of a finite sum which is given a combinatorial meaning, so that the triple product identity can be obtained by taking the limit. Here the proof is non-combinatorial and relies on a functional equation satisfied by a T-fraction. Then from this result on the triple product identity, we derive a whole new family of Touchard–Riordan-like formulas whose combinatorics is not yet understood. Eventually, we prove a Touchard–Riordan-like formula for a q-analog of Genocchi numbers, which is related with Jacobi’s identity for (q;q)³ rather than the triple product identity.     

From Tanaka's formula to Ito's formula: The fundamental theorem of stochastic calculus

In this article we give a new proof of Ito's formula inR ⁿ starting from the one-dimensional Tanaka formula. The proof is algebraic and does not use any limiting procedure. It uses the integration by parts formula, Fubini's theorem for stochastic integrals and essential properties of local times.     

Quantum monodromy and semi-classical trace formulæ

We consider an h-pseudodifferential operator whose symbol has a closed Hamiltonian trajectory. There exists a Fourier integral operator which quantizes in a natural way the Poincaré map. With the help of this monodromy operator, we give a trace formula which leads to a new proof of the trace formula of Duistermaat–Guillemin and Gutzwiller.     

Entropy Bumps and another sufficient condition for the two-weight boundedness of sparse operators

In this short note, we give a very efficient proof of a recent result of Treil–Volberg and Lacey–Spencer giving sufficient conditions for the two-weight boundedness of a sparse operator. We also give a new sufficient condition for the two-weight boundedness of a sparse operator. We make critical use of a formula of Hytönen in [6].     

A q-series expansion formula and the Askey–Wilson polynomials

Previously, we proved a q-series expansion formula which allows us to recover many important classical results for q-series. Based on this formula, we derive a new q-formula in this paper, which clearly includes infinitely many q-identities. This new formula is used to give a new proof of the orthogonality relation for the Askey–Wilson polynomials. A curious q-transformation formula is proved, and many applications of this transformation to Hecke type series are given. Some Lambert series identities are also derived.     

Combinatorial proof of the inversion formula on the Kazhdan–Lusztig R-polynomials

In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\) -polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equidistribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma’s equidistribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element.     

Specializations of nonsymmetric Macdonald–Koornwinder polynomials

The purpose of this article is to work out the details of the Ram–Yip formula for nonsymmetric Macdonald–Koornwinder polynomials for the double affine Hecke algebras of not-necessarily reduced affine root systems. It is shown that the \(t\rightarrow 0\) equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular, our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type, the proof requires the Ram–Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given at \(t\rightarrow \infty \). As a consequence, we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at \(q\rightarrow \infty \), which yields the p-adic Iwahori–Whittaker functions of Brubaker, Bump, and Licata.     

A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof

The celebrated Frame-Robinson-Thrall (Canad. J. Math. 6 (1954) 316–324) hook-lengths formula, counting the Young tableaux of a specified shape, is given a short bijective proof. This proof was obtained by translating the elegant Greene-Nijenhuis-Wilf proof (Adv. in Math. 31 (1979) 104–109) into bijective language.     

The symbolical and cancellation-free formulae for Schur elements

In this paper we give the symbolical formula and cancellation-free formula for the Schur elements associated to the simple modules of the degenerate cyclotomic Hecke algebras. As some applications, we show that the Schur elements are symmetric polynomials with rational integer coefficients and give a different proof of Ariki–Mathas–Rui’s criterion on the semisimplicity of the degenerate cyclotomic Hecke algebras.     

Hecke–Rogers type series representations for infinite products

The Ramanujan Journal - In this paper, we give a new proof of Liu’s extension of the non-terminating $$_6\phi _5$$ summation formula. Based on this formula, some Hecke–Rogers type...     

A bijective enumeration of tiered trees

Tiered trees were introduced by Dugan–Glennon–Gunnells–Steingrímsson as a generalization of intransitive trees that were introduced by Postnikov, the latter of which have exactly two tiers. Tiered trees arise naturally in counting the absolutely indecomposable representations of certain quivers, and enumerating torus orbits on certain homogeneous varieties over finite fields. By employing generating function arguments and geometric results, Dugan et al. derived an elegant formula concerning the enumeration of tiered trees, which is a generalization of Postnikov’s formula for intransitive trees. In this paper, we provide a bijective proof of this formula by establishing a bijection between tiered trees and certain rooted labeled trees. As an application, our bijection also enables us to derive a refinement of the enumeration of tiered trees with respect to level of the node 1.     

On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a \({{\mathrm{{{\mathrm{Spin}}}^c}}}\)-Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).     

A new proof of Suzuki's formula

We give a different proof of a formula of Suzuki and its strengthening by Zaidenberg for the topological Euler characteristic of an affine surface fibered over a curve. We deduce this formula using the ideas for the proof of an analogous formula for a proper morphism.     

The polynomial bounds of proof complexity in Frege systems

The concept of a determinative set of variables for a propositional formula was introduced by one of the authors, which made it possible to distinguish the set of hard-determinable formulas. The proof complexity of a formula of this sort has exponential lower bounds in some proof systems of classical propositional calculus (cut-free sequent system, resolution system, analytic tableaux, cutting planes, and bounded Frege systems). In this paper we prove that the property of hard-determinability is insufficient for obtaining a superpolynomial lower bound of proof lines (sizes) in Frege systems: an example of a sequence of hard-determinable formulas is given whose proof complexities are polynomially bounded in every Frege system.     

On an inequality of Brendle in the hyperbolic space

We give a spinorial proof of a Heintze–Karcher-type inequality in the hyperbolic space proved by Brendle [4]. The proof relies on a generalized Reilly formula on spinors recently obtained in [7].