Minkowski Normals for Polycircles

In this paper we will discuss the average number of normals through a point in a convex body in a Minkowski plane where the unit disk is a polygon.     

Central Limit Theorems for Exchangeable Random Variables When Limits Are Scale Mixtures of Normals

Central limit theorems for exchangeable random variables are studied when limits are scale mixtures of normals. First, necessary and sufficient conditions are given under the asymptotic tail probability condition for the mixands:

Numerical model of a pine in a wind

Steady-state nonlinear differential equations govering the stem curve of a wind-loaded pine are derived and solved numerically. Comparison is made between the results computed and the data from photographs of a pine stem during strong wind. The pine breaking is solved at the end.     

Normals in a Minkowski Plane

In this paper we will establish bounds on the average number of normals through a point in a convex body in a Minkowski plane for certain classes of convex bodies. Also, a related Euler relation is discussed.     

Proximal normal analysis approach to optimal control problems in infinite-dimensional spaces

In this paper, we extend the Pontryagin maximum principle and the transversality conditions to a class of optimal control problems for an evolution system of parabolic type through the analysis of proximal normals to the epigraph of suitable value functions. The paper extends previous results of the same authors to nonconvex target situations.This work was supported by MURST of Italy, Fondi 40%, Equazioni di Evoluzione ed Applicazioni Fisico-Matematiche, and Fondi 60%, University of Bari and University of Calabria.     

On the solutions of a parametric family of cubic Thue equations

In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact, we use Baker’s method find all solutions to the Thue equation

On the representation of unity by binary cubic forms

If is a binary cubic form with integer coefficients such that has at least two distinct complex roots, then the equation possesses at most ten solutions in integers and , nine if has a nontrivial automorphism group. If, further, is reducible over , then this equation has at most solutions, unless is equivalent under -action to either or . The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations for cubic and irreducible of positive discriminant . As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form and to Mordell's equation , where and are nonzero integers.

     

Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points

In this paper we study those cubic systems which are invariant under a rotation of radians. They are written as where is complex, the time is real, and , are complex parameters. When they have some critical points at infinity, i.e. , it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On the other hand when they have no critical points at infinity, i.e. there are examples exhibiting at least two limit cycles surrounding nine critical points. In this paper we give two criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds the origin. Our results apply to systems having a limit cycle that surrounds either 1, 5 or 9 critical points, the origin being one of these points. The key point of our approach is the use of Abel equations.

     

一种细分格式生成极限曲面的法向量的求法

在Levin给出的三角域上生成极限曲面的法向量求法基础上,给了同拟蝴蝶形细分在矩形域上生成极限曲面的情况,并得到了两个自由度,可以对法向量进行优化选取,这对讨论曲面的等距面有广泛的实际意义。     

Generalized Electrodynamics With Ternary Internal Structure

In Refs. [2]–[7] we suggested generalized dynamic equations of motion of relativistic charged particles inside electromagnetic fields. The dynamic equations had been formulated in terms of external as well as internal momenta. Evolution equations for external momenta, the Lorentz-force equations, had been derived from evolution equations for internal momenta. In this paper, along with relativistic dynamics we generalize electromagnetic fields within the scope of ternary algebras. The full theory is constructed in 4D euclidean space. This space possesses an advantage to build ternary mappings from three vectors onto one. The dynamics is given by non-linear evolution equations with cubic characteristic polynomial. In polar representation the internal momenta obey the Jacobi equations whereas external momenta obey the Weierstrass equations for elliptic functions. The generalized electromagnetic fields are defined by the triple fields where the first one has properties of the electric field and the other two have properties of the magnetic field. The field equations for the triple fields analogous to the Maxwell equations are suggested.     

求解非光滑方程组的三次正则化方法

考虑求解非光滑方程组的三次正则化方法及其收敛性分析.利用信赖域方法的技巧,保证该方法是全局收敛的.在子问题非精确求解和BD正则性条件成立的前提下,分析了非光滑三次正则化方法的局部收敛速度.最后,数值实验结果验证了该算法的有效性.     

基于样条的高阶矩阵微分方程近似解

矩阵微分方程经常出现在许多物理模型和工程技术模型中.利用矩阵样条构造形如{y(p)(x)=Ap-1(x)y(p-1)(x)+Ap-2(x)y(p-2)(x)+…+A1(x)y(1)(x)+A0(x)y(x)+B0(x),y(a)=ya,…,y(p-1)(a)=y(p-1)a,x∈[a,b];Ai(x),B0(x)∈C4[a,b],0≤i≤p-烅烄烆1的高阶矩阵线性微分方程初值问题的数值解.给出实现算法和数值解的近似误差估计以及数值实例.先将高阶矩阵微分方程转化为一阶矩阵微分方程,然后利用三次矩阵样条求出一阶矩阵线性微分方程的数值解,从而解决高阶微分方程问题.     

Computing centre conditions for certain cubic systems

We present necessary and sufficient conditions for a critical point of certain two-dimensional cubic differential systems to be a centre. Extensive use of the computer algebra system REDUCE is involved. The search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations. It has proved to be a difficult problem, and full conditions are known for very few classes of systems. Such conditions are also required in the investigation of Hilbert's sixteenth problem concerning the number of limit cycles of polynomial systems.     

Existence of a solution to a nonlinear equation

Equation (−Δ+k²)u+f(u)=0 in D, u_|∂D=0, where k=const>0 and D⊂R³ is a bounded domain, has a solution if is a continuous function in the region |u|?a, piecewise-continuous in the region |u|?a, with finitely many discontinuity points uj such that f(uj±0) exist, and uf(y)?0 for |u|?a, where a?0 is an arbitrary fixed number.     

An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.     

Limit cycles and invariant parabola in a kukles system of degree three

In this article,the authors consider a class of Kukles planar polynomial differential system of degree three having an invariant parabola.For this class of second-order differential systems,it is shown that for certain values of the parameters the invariant parabola coexists with a center.For other values it can coexist with one,two or three small amplitude limit cycles which are constructed by Hopf bifurcation.This result gives an answer for the question given in[4],about the existence of limit cycles for such class of system.     

Inductive Rings and Systems of Diophantine Equations

In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don＇t.     

具有一类三次曲线解的Kolmogorov三次系统的极限环的存在性

证明了具有三次曲线解y=-x(x-1)2 4/24的Kolmogorov三次系统是有存在极限环可能的.