scholarly journals Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q,t)-Gelfand-Tsetlin Graph

Author(s):  
Cesar Cuenca ◽  
2014 ◽  
Vol 10 (08) ◽  
pp. 2011-2036 ◽  
Author(s):  
Renrong Mao

Bringmann, Mahlburg and Rhoades proved asymptotic formulas for all the even moments of the ranks and cranks of partitions with polynomial error terms. In this paper, motivated by their work, we apply the same method and obtain asymptotics for the two rank moments of overpartitions.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.


Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 779
Author(s):  
Charles F. Dunkl

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.


2020 ◽  
Vol 6 (1) ◽  
Author(s):  
Darius Bunandar ◽  
Luke C. G. Govia ◽  
Hari Krovi ◽  
Dirk Englund

AbstractQuantum key distribution (QKD) allows for secure communications safe against attacks by quantum computers. QKD protocols are performed by sending a sizeable, but finite, number of quantum signals between the distant parties involved. Many QKD experiments, however, predict their achievable key rates using asymptotic formulas, which assume the transmission of an infinite number of signals, partly because QKD proofs with finite transmissions (and finite-key lengths) can be difficult. Here we develop a robust numerical approach for calculating the key rates for QKD protocols in the finite-key regime in terms of two semi-definite programs (SDPs). The first uses the relation between conditional smooth min-entropy and quantum relative entropy through the quantum asymptotic equipartition property, and the second uses the relation between the smooth min-entropy and quantum fidelity. The numerical programs are formulated under the assumption of collective attacks from the eavesdropper and can be promoted to withstand coherent attacks using the postselection technique. We then solve these SDPs using convex optimization solvers and obtain numerical calculations of finite-key rates for several protocols difficult to analyze analytically, such as BB84 with unequal detector efficiencies, B92, and twin-field QKD. Our numerical approach democratizes the composable security proofs for QKD protocols where the derived keys can be used as an input to another cryptosystem.


Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 220
Author(s):  
Alexey Samokhin

We studied, for the Kortweg–de Vries–Burgers equations on cylindrical and spherical waves, the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary. The regular profile at the vicinity of perturbation looks like a periodical chain of shock fronts with decreasing amplitudes. Further on, shock fronts become decaying smooth quasi-periodic oscillations. After the oscillations cease, the wave develops as a monotonic convex wave, terminated by a head shock of a constant height and equal velocity. This velocity depends on integral characteristics of a boundary condition and on spatial dimensions. In this paper the explicit asymptotic formulas for the monotonic part, the head shock and a median of the oscillating part are found.


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