Note on the rank of quadratic twists of Mordell equations

Let E be the elliptic curve given by a Mordell equation y²=x³−A where A∈Z. Michael Stoll found a precise formula for the size of a Selmer group of E for certain values of A. For D∈Z, let ED denote the quadratic twistDy²=x³−A. We use Stoll's formula to show that for a positive square-free integer A≡1 or and for a nonnegative integer k, we can compute a lower bound for the proportion of square-free integers D up to X such that rankED(Q)?2k. We also compute an upper bound for a certain average rank of quadratic twists of E.     

Quaternary universal forms over \mathbf{Q}(\sqrt{13})

Let be a real quadratic field over Q with m a square-free positive rational integer and be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x ₁,…,x _n )=∑_1≤i,j≤n α _ij x _i x _j ( ) is called universal if f represents all totally positive integers in . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12. In this paper we prove that there are only two quaternary universal forms (up to equivalence) over . For the proof of universality we apply the theory of quadratic lattices.      

Five peculiar theorems on simultaneous representation of primes by quadratic forms

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It is a theorem of Kaplansky that a prime is representable by both or none of x²+32y² and x²+64y², whereas a prime is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states that a prime is representable by both or none of x²+20y² and x²+100y², whereas a prime is representable by exactly one of these forms. A heuristic argument is given why there are no other results of the same kind. This argument relies on the (plausible) conjecture that there are exactly 485 negative discriminants Δ such that the class group C(Δ) has exponent 4.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=l_yRq0oqKx4.     

Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions

We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions , where . We prove that if the final states

     

Fuzzy weighted equilibrium multi-job assignment problem and genetic algorithm

In this paper, the equilibrium optimization problem is proposed and the assignment problem is extended to the equilibrium multi-job assignment problem, equilibrium multi-job quadratic assignment problem and the minimum cost and equilibrium multi-job assignment problem. Furthermore, the mathematical models of the equilibrium multi-job assignment problem and the equilibrium multi-job quadratic assignment problem with fuzzy parameters are formulated. Finally, a genetic algorithm is designed for solving the proposed programming models and some numerical examples are given to verify the efficiency of the designed algorithm.     

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.     

Improved compact linearizations for the unconstrained quadratic 0-1 minimization problem

We present and compare three new compact linearizations for the quadratic 0-1 minimization problem, two of which achieve the same lower bound as does the “standard linearization”. Two of the linearizations require the same number of constraints with respect to Glover’s one, while the last one requires n additional constraints where n is the number of variables in the quadratic 0-1 problem. All three linearizations require the same number of additional variables as does Glover’s linearization. This is an improvement on the linearization of Adams, Forrester and Glover (2004) which requires n additional variables and 2n additional constraints to reach the same lower bound as does the standard linearization. Computational results show however that linearizations achieving a weaker lower bound at the root node have better global performances than stronger linearizations when solved by Cplex.     

Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.     

二次四元数系统正定解的迭代法

二次四元数系统XAX-BX=P是离散型Lyapunov方程正定解反问题的推广形式.本文在四元数体上讨论它的正定解存在性及迭代求解方法.利用等价二次方程的系数矩阵的极大极小特征值,获得其正定解的存在区间,并针对系数矩阵的不同情况构建出三种收敛的迭代格式.同时根据每种迭代的特点,给出了迭代初始矩阵的选取方法.最后通过四元数...     

基于乐观抱怨值和悲观抱怨值的合作博弈最优妥协值的求解模型

基于乐观抱怨值和悲观抱怨值,通过建立二次规划模型求解(Hou,et al.,2018)定义的平衡博弈的最优妥协值,二次规划模型及求解方法比(Hou,et al.,2018)提出的字典序方法简单易操作.此外,文章进一步给出了同时满足个体合理性和群体有效性的乐观最优妥协值的求解算法.最后,通过数值实例说明文章建立的模型和方...