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Representation of Elliptic Ding-Iohara algebra

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Frontiers of Mathematics in China 2020年第1期15卷 155-166页

作者： Lifang WANG Ke WU Jie YANG Zifeng YANG School of Mathematics and Statistics Henan universityKaifeng 475004China School of Mathematical Sciences Capital Normal UniversityBeijing 100048China

We define a vector representation V(u)of Elliptic Ding-Iohara algebra u(q,t,p).Furthermore,we construct the tensor products of the vector representations and the Fock modulesF(u)by taking the inductive limit of certai... 详细信息

关键词： Elliptic Ding-Iohara algebra vector representation partition

Efficient construction of a substitution box based on a Mordell Elliptic curve over a finite field

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Frontiers of Information Technology & Electronic Engineering 2019年第10期20卷 1378-1389页

作者： Naveed Ahmed AZAM Umar HAYAT Ikram ULLAH Department of Applied Mathematics and Physics Graduate School of InformaticsKyoto UniversityKyoto 606-8501Japan Department of Mathematics Quaid-i-Azam UniversityIslamabad 44000Pakistan

Elliptic curve cryptography has been used in many security systems due to its small key size and high security compared with other cryptosystems. In many well-known security systems, a substitution box (S-box) is the only non-linear component. Recently, it has been shown that the security of a cryptosystem can be improved using dynamic S-boxes instead of a static S-box. This necessitates the construction of new secure S-boxes. We propose an efficient method to generate S-boxes that are based on a class of Mordell Elliptic curves over prime fields and achieved by defining different total orders. The proposed scheme is devel-oped in such a way that for each input it outputs an S-box in linear time and constant space. Due to this property, our method takes less time and space than the existing S-box construction methods over Elliptic curves. Computational results show that the pro-posed method is capable of generating cryptographically strong S-boxes with security comparable to some of the existing S-boxes constructed via different mathematical structures.

关键词： Substitution box Finite field Mordell Elliptic curve Total order Computational complexity

A Study on the Far Wake of Elliptic Cylinders

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Computer Modeling in Engineering & Sciences 2017年第1期113卷 35-55页

作者： Subhankar Sen Sanjay Mittal Department of Mechanical Engineering Indian Institute of Technology KanpurIndia Department of Aerospace Engineering Indian Institute of Technology KanpurIndia

The evolution of far wake of stationary Elliptic cylinders with angle of attack is investigated for the unsteady flow at Re=200 using a stabilized finite-element method.The Reynolds number,Re,is based on the major axis of cylinder and free-stream speed.Cylinder aspect ratios,AR considered are 0.2,0.5 and 0.8 while the angle of attack,αvaries from 0° to 90°.With increasingα,the Karman or primary shedding frequency of each cylinder decreases monotonically while the primary wavelength and time-averaged as well as r.***.s.drag increase monotonically.The time-averaged lift and moment coefficients initially increase withαand then decrease.The power spectrum analysis of unsteady transverse velocity signal at various stations along the wake centerline is used to study the associated frequencies.The Karman or primary shedding frequency exists for eachα.For the AR=0.2 cylinder,the secondary and subsequently,low magnitude tertiary frequencies appear in the wake asαcontinues to increase from 30°.In contrast to the decaying nature of primary frequency withα,secondary frequency exhibits a nonmonotonic variation.The secondary frequency in general is not a sub-harmonic of the primary.This suggests that the secondary structures do not form due to merging of primary vortices.With increasingα,the location of formation of secondary structures advances upstream.For a given orientation,the upstream advancement is more dominant with decreasing AR.Power spectrum analysis indicates that the wake state for confined flow is characterized by the primary frequency alone,i.***.the imposition of blockage suppresses the formation of secondary structures.

关键词： Elliptic cylinder stabilized finite-element far wake angle of attack blockage

A CASCADIC MULTIGRID METHOD FOR SEMILINEAR Elliptic EQUATIONS

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Journal of Computational Mathematics 2019年第1期37卷 112-129页

作者： Fei Xu Fusheng Luo Beijing Institute for Scientific and Engineering Computing Beijing University of Technology Beijing 100124China The Third Institute of Oceanography SOAXiamen 361005China

This paper introduce a cascadic multigrid method for solving semilinear Elliptic equations based on a multilevel correction method.Instead of the common costly way of directly solving semilinear Elliptic equation on a very fine space,the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear Elliptic equations on a very coarse space.To prove the efficiency of the new method,we derive two results,one of the optimal convergence rate by choosing the appro- priate sequence of finite element spaces and the number of smoothing steps,and the other of the optimal computational work by applying the parallel computing technique.Moreover,the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method.Some numerical experiments are presented to validate our theoretical analysis.

关键词： Sernilinear Elliptic equation Parallel computing Cascadic multigrid Multilevel correction Finite element method

Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the L^p Dirichlet Problem

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Acta Mathematica Sinica,English Series 2019年第6期35卷 749-770页

作者： Martin DINDO? Jill PIPHER School of Mathematics The University of Edinburgh and Maxwell Institute of Mathematical Sciences Department of Mathematics Brown University

We establish a Dahlberg-type perturbation theorem for second order divergence form Elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L_0 = div A^0(x)? + B^0(x) · ? is a p-Elliptic operator satisfying the assumptions of Theorem 1.1 then the LpDirichlet problem for the operator L_0 is solvable in the upper half-space Rn+. In this paper we prove that the Lpsolvability is stable under small perturbations of L_0. That is if L_1 is another divergence form Elliptic operator with complex coefficients and the coefficients of the operators L_0 and L_1 are sufficiently close in the sense of Carleson measures, then the LpDirichlet problem for the operator L_1 is solvable for the same value of p. As a corollary we obtain a new result on Lpsolvability of the Dirichlet problem for operators of the form L = div A(x)? + B(x) · ? where the matrix A satisfies weaker Carleson condition(expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiable and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoˇs,Petermichl and Pipher.

关键词： Complex coefficients Elliptic PDEs perturbation theory boundary value problems

A New Elliptic Measure on Lower Dimensional Sets

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Acta Mathematica Sinica,English Series 2019年第6期35卷 876-902页

作者： Guy DAVID Joseph FENEUIL Svitlana MAYBORODA Université Paris-Sud Laboratoire de Mathématiques Department of Mathematics Temple University School of Mathematics University of Minnesota

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n-1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate Elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging Elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.

关键词： Elliptic measure in higher codimension degenerate Elliptic operators absolute continuity Dahlberg’s theorem Dirichlet solvability

Symmetrization for Fractional Elliptic and Parabolic Equations and an Isoperimetric Application(Dedicated to Haim Brezis,great master of analysis,on his 70th birthday)

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Chinese Annals of Mathematics,Series B 2017年第2期38卷 661-686页

作者： Yannick SIRE Juan Luis VAZQUEZ Bruno VOLZONE johns hopkins university Krieger Hall3401 N.Charles StBaltimoreM D21218USA. departamento de matemticas universidad autnoma de madrid28049 M adridSpain dipartimento di ingegneria universit degli studi di napoli"parthenope"80143 Italia.

This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both Elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R^N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.

关键词： Symmetrization, Fractional Laplacian, Nonlocal Elliptic and parabolicequations, Faber-Krahn inequality

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

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Acta Mathematica Sinica,English Series 2018年第4期34卷 612-628页

作者： Jun GENG School of Mathematics and Statistics Lanzhou University

We consider a family of second-order Elliptic operators {LE } in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in Rn. We are able to show that the uniform WI＇p estimate of second order Elliptic systems holds for 2n/n＋1-δ〈p〈2n/n-1＋δ where δ 〉 0 is independent of ~ and the ranges are sharp for n = 2, 3. And for Elliptic equations in Lipschitz domains,the W1,p estimate is true for 3/2 -δ 〈 p〈 3 ＋ δ ifn ≥ 4, similar estimate was extended to domains, the WI＇p estimate is true for nconvex domains for 1 〈 p 〈 ∞.

关键词： Homogenization Elliptic non-smooth

A Combined Finite Element Scheme for Second Elliptic Problems Posted in Domains with Rough Boundaries

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Journal of Donghua University(English Edition) 2020年第2期37卷 130-136页

作者：周迦南黄磊徐世鹏 School of Mathematics and Computer Science Jiangxi Science and Technology Normal UniversityNanchang 330038China

A finite element method(FEM) is proposed to solve second Elliptic problems posted in domains with rough boundaries. We also present an FEM for Elliptic problems as an extension of the original method. The optimal error estimates in the energy norm are also proved. Numerical results for second Elliptic equations in regular and rough domains are presented to illustrate the efficiency of our method, respectively, which is consistent with theoretical analysis.

关键词： finite element multiscale problem Elliptic problem rough domain penalty technique