scholarly journals Asymptotic formulas for generalized gcd-sum and lcm-sum functions over $ r $-regular integers $ (\bmod\ n^{r}) $

2021 ◽  
Vol 6 (12) ◽  
pp. 13157-13169
Author(s):  
Zhengjin Bu ◽  
◽  
Zhefeng Xu
Keyword(s):  
Asymptotic Formulas ◽  
Euler Product ◽  
Dirichlet Product

<abstract><p>In this paper we perform a further investigation for $ r $-gcd-sum function over $ r $-regular integers $ (\bmod\ n^{r}) $, and we derive two kinds of asymptotic formulas by making use of Dirichlet product, Euler product and some techniques. Moreover, we also establish estimates for the generalized $ r $-lcm-sum function over $ r $-regular integers $ (\bmod\ n) $.</p></abstract>

Asymptotics for rank moments of overpartitions

2014 ◽  
Vol 10 (08) ◽  
pp. 2011-2036 ◽  
Author(s):  
Renrong Mao
Keyword(s):  
Asymptotic Formulas ◽  
Error Terms ◽  
Cranks Of Partitions

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.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

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.

scholarly journals Numerical finite-key analysis of quantum key distribution

2020 ◽  
Vol 6 (1) ◽  
Author(s):  
Darius Bunandar ◽  
Luke C. G. Govia ◽  
Hari Krovi ◽  
Dirk Englund
Keyword(s):  
Finite Number ◽  
Quantum Key Distribution ◽  
Key Distribution ◽  
Numerical Approach ◽  
Asymptotic Formulas ◽  
Quantum Computers ◽  
Secure Communications ◽  
Quantum Relative Entropy ◽  
Asymptotic Equipartition Property ◽  
Optimization Solvers

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.

scholarly journals On Monotonic Pattern in Periodic Boundary Solutions of Cylindrical and Spherical Kortweg–De Vries–Burgers Equations

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 220
Author(s):  
Alexey Samokhin
Keyword(s):  
Periodic Boundary ◽  
Asymptotic Formulas ◽  
Burgers Equations ◽  
Spherical Waves ◽  
Constant Height ◽  
De Vries ◽  
Spatial Dimensions ◽  
Integral Characteristics ◽  
Boundary Solutions ◽  
Shock Fronts

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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