The Diophantine Equations x2± y4=±z6 and x2+y8= z3

In this article we determine all solutions to the equationx ^p+y ^q=z ^r, (p,q,r){(2,4,6), (2,6,4), (4,6,2), (2,8,3)} in coprime integers x,y,z. First we determine a set of curves of genus 2, such that every solution corresponds to a rational point on one of these curves. Then we determine the rational points on these curves using either covers of rank 0 elliptic curves or a method known as effective Chabauty which works if theMordell–Weil rank of the jacobian is smaller than the dimension.     

The number of all transformation groups on an infinite set

Let X be an infinite set, T(X)={f∈X~X:f is a bijection}, C={G:G is a transformation group on X}={G:G is a subgroup of T(X)}.Then |C|=2~(2~(|x|)). Inproof of this result, the AC is used.     

A new method for constructing infinite families of k-tight optimal double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. DLN has been widely used in the designing of local area networks and distributed systems. In this paper, a new method for constructing infinite families of k-tight optimal DLN is presented. For k = 0,1,…,40, the infinite families of k-tight optimal DLN can be constructed by the new method, where the number nk(t,a) of their nodes is a polynomial of degree 2 in t and contains a parameter a. And a conjecture is proposed.     

Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

The Laplacian on β-sets via the method of averages

In this paper, we show how the symmetric Laplacian on β-sets with $0<\beta <\frac{1}{2}$, together with its associated Dirichlet form and harmonic functions, can be defined entirely in terms of average values of a function over basic sets. We have generalized Strichartz’s and our previous results by using the method of averages.     

矩阵方程组A1XB1=C1,A2XB2=C2的相容性与通解

本文推广了R.Penrose关于矩阵方程组AX=C,XB=D的工作,给出了矩阵方程组 A_1XB_1=C_1,A_2XB_2=C_2的相容性条件,给出了通解表达式以及唯一解的充要条件。     

Local structure of graphs with λ=μ=2,a 2=4

Several properties of graphs with ==2,a ₂=4 are studied. It is proved that such graphs are locally unions of triangles, hexagons or heptagons. As a consequence, a distance regular graph with intersection array (13,10,7;1,2,7) does not exist.     

The Continued Fraction Solution of the Matrix Equation AX-XB=C

Let K[λ] denote the polynomial ring over the number field K.Suppose thatf(λ)and g(λ)are coprime in K[λ] and deg f(λ)≥1.By the Euclidean algorithmwe obtain the continued fraction     

The Realization of 4-dimensional 3-Lie Algebras

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.     

不定方程 x~3+y~3=z~2与 x~3+y~3=z~4

在 x,y 互素的条件下,本文给出不定方程 x~3+y~3=z~2所有的整数解,并证明不定方程 x~3+y~3=z~4无 xyz≠0之整数解.     

A new method of proof of Filippov’s theorem based on the viability theorem

Filippov??s theorem implies that, given an absolutely continuous function y: [t ₀; T] ?? ? ^d and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x ₀, there exists a solution x(·) to the differential inclusion x??(t) ?? F(t, x(t)) starting from x ₀ at the time t ₀ and satisfying the estimation $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ where the function ??(·) is the estimation of dist(y??(t), F(t, y(t))) ?? ??(t). Setting P(t) = {x ?? ? ⁿ : |x ?y(t)| ?? r(t)}, we may formulate the conclusion in Filippov??s theorem as x(t) ?? P(t). We calculate the contingent derivative DP(t, x)(1) and verify the tangential condition F(t, x) ?? DP(t, x)(1) ?? ?. It allows to obtain Filippov??s theorem from a viability result for tubes.     

A factorization of an integral 2×2 matrix via a rational method

While a 2×2 integral matrixA with characteristic polynomialx ²-m,m=2,3 (4), and square free, cannot always be expressed as the product of two integral symmetric matrices, it can be expressed as the product of two rational matrices, simultaneously similar to two integral symmetric matrices. This can be achieved in various ways. The paper deals with numerical examples which depend on the 2-part of the ideal class-group attached tox ²-m (see earlier work by the author, especially Acta Arith. 24, 151–156 (1973)).Dedicated to professor E. Hlawka on the occasion of his seventieth birthdayThe term rational is used here also to refer to not algebraic.     

A matrix-free implementation of Riemannian Newton’s method on the Stiefel manifold

Newton’s method for unconstrained optimization problems on the Euclidean space can be generalized to that on Riemannian manifolds. The truncated singular value problem is one particular problem defined on the product of two Stiefel manifolds, and an algorithm of the Riemannian Newton’s method for this problem has been designed. However, this algorithm is not easy to implement in its original form because the Newton equation is expressed by a system of matrix equations which is difficult to solve directly. In the present paper, we propose an effective implementation of the Newton algorithm. A matrix-free Krylov subspace method is used to solve a symmetric linear system into which the Newton equation is rewritten. The presented approach can be used on other problems as well. Numerical experiments demonstrate that the proposed method is effective for the above optimization problem.     

A New Proof of Diophantine Equation （^n2） = （^m4）

By using algebraic number theory and p-adic analysis method, we give a new and simple proof of Diophantine equation （^n2） = （^m4）