The analytic solution for the power series expansion of Heun function

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric ₂F₁${}_2{F}_1$, ₁F₁${}_1{F}_1$ and ₀F₁${}_0{F}_1$ functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.     

Optimal control for distributed linear systems subjected to null-controllability

In this article, we apply the notion of hierarchic control on a distributed system in which the state is governed by a parabolic equation. This notion assumes that we have two controls where one will be the Leader and the other, the Follower. The first control is of controllability type subjected to a constraint, while the second expresses that the state does not move too far from a given state. The results are achieved by means of an observability inequality of the Carleman type, which is ‘adapted’ to the constraint.     

On the number of hypercubic bipartitions of an integer

For n≤2^k$n\le 2^k$ we study the maximum number of edges of an induced subgraph on n$n$ vertices of the k$k$-dimensional hypercube _Qk$Q_k$. In the process we revisit a well-known divide-and-conquer maximin recurrence f(n)=max(min(_n1,_n2)+f(_n1)+f(_n2))$f\left(n\right)=max(min(n_1,n_2)+f\left(n_1\right)+f\left(n_2\right))$ where the maximum is taken over all proper bipartitions n=_n1+_n2$n=n_1+n_2$. We first use known results to present a characterization of those bipartitions n=_n1+_n2$n=n_1+n_2$ that yield the maximum f(n)=min(_n1,_n2)+f(_n1)+f(_n2)$f\left(n\right)=min(n_1,n_2)+f\left(n_1\right)+f\left(n_2\right)$. Then we use this characterization to present the main result of this article, namely, for a given n∈N$n\in \mathbb{N}$, the determination of the number h(n)$h\left(n\right)$ of these bipartitions that yield the said maximum f(n)$f\left(n\right)$. We present recursive formulae for h(n)$h\left(n\right)$, a generating function h(x)$h\left(x\right)$, and an explicit formula for h(n)$h\left(n\right)$ in terms of a special representation of n$n$.     

Closed forms for 4-parameter families of integrals

We compute closed forms for two multiparameter families of definite integrals, thus obtaining combinatorial formulas. As a consequence, a surprising formula is derived between a definite integral and an improper integral for the same parametric function.     

关于代换系统拓扑序列熵的注记

Goodman证明了对两符号等长代换系统, 如果代换规则中0和1对应的词只有一个位置不同,那么对应的代换系统为null的, 即此系统沿着任意正整数序列的序列熵均为0. 在本文中, 我们针对系统的结构特征, 通过考察因子系统, 给出了此经典结果的另外一种证明. 同时, 对此类代换系统沿着给定序列的复杂性, 我们得到了比Goodman更为精确的估计.     

On the solution of a third order linear homogeneous difference equation with variable coefficents

A closed form solution of a third order linear homogeneous difference equation with variable coefficients is presented. From it, the solution for the special case of an equation with constant coefficients is also obtained.     

Sums of primes and quadratic linear recurrence sequences

Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expressible by the sum of a prime number and an element of u has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form p ＋ [（2 ＋ √3）n], where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be ＂too small＂ for most prime numbers p.