Equivariant quantum Schubert calculus

We study the T-equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients.     

On quantum stochastic differential equations

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.     

Further results on the existence,uniqueness and stability of strong solutions of quantum stochastic differential equations

Under a more general Lipschitz condition on the coefficients than our consideration in [E.O. Ayoola, Existence and stability results for strong solutions of quantum stochastic differential equations, Stochastic Anal. Appl. 20 (2) (2002) 263–281], we establish the existence, uniqueness and stability of strong solutions of quantum stochastic differential equations (QSDE). This enables us to exhibit a class of Lipschitzian QSDE whose coefficients are continuous on the locally convex space of solution.     

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.     

A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval

A method for construction of CF approximants in some cases of rational approximation of a rational function f on the unit disk and on the unit interval is presented. The inverted square root of the greatest positive eigenvalue and a corresponding eigenvector of an eigenvalue problem defined by the coefficients of f gives the solution.     

A Littlewood-Richardson rule for factorial Schur functions

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.

     

Scalar products in models with a GL(3) trigonometric R-matrix: Highest coefficient

We study quantum integrable models with a GL(3) trigonometric R-matrix solvable by the nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of bilinear combinations of the highest coefficients. We show that there exist two different highest coefficients in the models with a GL(3) trigonometric R-matrix. We obtain various representations for the highest coefficients in terms of sums over partitions. We also prove several important properties of the highest coefficients, which are necessary for evaluating the scalar products.     

Graded quiver varieties,quantum cluster algebras and dual canonical basis

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier–Sato–Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra.     

Clebsch-Gordan coefficients for the rank one quantum superalgebra. I.

We give general definitions of quantum Lie superalgebras using defining relations and the Cartan matrix, as well as the R-matrix. We present a realization of quantum superalgebra generators in terms of the q-oscillators and calculate the Clebsch—Gordan coefficients for the quantum superalgebra osp_q(1|2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 76–88, 1990.     

Tangential Quantum Cohomology of Arbitrary Order

Kock has previously defined a tangency quantum product on formal power series with coefficients in the cohomology ring of any smooth projective variety, and thus a ring that generalizes the quantum cohomology ring. We further generalize Kock's construction by defining a dth-order contact product and establishing its associativity.     

Quasi-binomial coefficients stemming from Nakayama algebras

It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.     

Inequalities for the Extremal Coefficients of Multivariate Extreme Value Distributions

The extremal coefficients are the natural dependence measures for multivariate extreme value distributions. For an m-variate distribution 2^m distinct extremal coefficients of different orders exist; they are closely linked and therefore a complete set of 2^m coefficients cannot take any arbitrary values. We give a full characterization of all the sets of extremal coefficients. To this end, we introduce a simple class of extreme value distributions that allows for a 1-1 mapping to the complete sets of extremal coefficients. We construct bounds that higher order extremal coefficients need to satisfy to be consistent with lower order extremal coefficients. These bounds are useful as lower order extremal coefficients are the most easily inferred from data.     

On the estimation of solutions for some linear systems of differential equations

In this paper we estimate the solutions of homogeneous linear system of differential equations with unbounded coefficients on the real lineR. We also give a necessary and sufficient condition in order that the linear differential operator with unbounded coefficients has a bounded inverse in the scalar case.     

Normalizing and variance stabilizing transformations of multivariate statistics under an elliptical population

Summary Normalizing and variance stabilizing transformations of a sample correlation, multiple correlation and canonical correlation coefficients are obtained under an elliptical population. It is shown that the Fisher'sz-transformation is efficient for these statistics. A normalizing transformation is also studied for a latent root of a sample covariance matrix in an elliptical sample.     

New information-entropic relations for Clebsch–Gordan coefficients

Using properties of the Shannon and Tsallis entropies, we obtain new inequalities for the Clebsch–Gordan coefficients of the group SU(2). For this, we use squares of the Clebsch–Gordan coefficients as probability distributions. The obtained relations are new characteristics of correlations in a quantum system of two spins. We also find new inequalities for Hahn polynomials and the hypergeometric functions ₃F₂.     

On quantum stochastic differential equations with unbounded coefficients

Summary We prove an existence, uniqueness and unitarity theorem for quantum stochastic differential equations with unbounded coefficients which satisfy an analyticity condition on a common dense invariant domain. This result, applied to the quantum harmonic oscillator, gives a rigorous meaning to a large class of stochastic differential equations that have been considered formally in quantum probability.     

Time reversal for modified oscillators

We consider a new completely integrable case of the time-dependent Schrödinger equation in ®ⁿ with variable coefficients for a modified oscillator that is dual (with respect to time reversal) to a model of the quantum oscillator. We find a second pair of dual Hamiltonians in the momentum representation. The examples considered show that in mathematical physics and quantum mechanics, a change in the time direction may require a total change of the system dynamics to return the system to its original quantum state. We obtain particular solutions of the corresponding nonlinear Schrödinger equations. We also consider a Hamiltonian structure of the classical integrable problem and its quantization.     

Remark on the theory of Sergeev frequencies of zeros,signs, and roots for solutions of linear differential equations: II

The theorem that claims that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order > 2 with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended numerical line is inverted for the spectra of upper frequencies of third-order equations under the assumption that the spectra contain zero. In addition, we construct examples of third-order equations with continuous coefficients whose Lebesgue sets of the upper Sergeev frequency of signs belong to the exact first Borel class, and the Lebesgue sets of upper Sergeev frequencies of zeros and roots belong to the exact second Borel class.     

Perturbation theory in large order

For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by−″ + (x²/4+εx⁴/4−E)y=0, y(±∞)=0,the perturbation coefficients A_n in the expansion for the ground-state energysimplify dramatically as n → ∞:.We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λ⁴ model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron.     

A quantum stochastic Lie-Trotter product formula

A Trotter product formula is established for unitary quantum stochastic processes governed by quantum stochastic differential equations with constant bounded coefficients.