关于指数丢番图方程的Hugh Edgar问题

p~m=q~n 2 m>1, n>1 (1)where p and q are primes, has the solution 3~3=5~2 2.Hall asked a question how many solutions else are there? Later, Hugh Edgar asked another question how many solutions(m,n) does the Diophantine equation p~m-q~n=2~h;m>1,n≥1,(2) have for primes p, q and integer h? In the references at el., Hall's and Hugh Edgar's problems were partly solved. In this paper, we have proved the following results: THEOREM 1. If p=qt~2±4 or q=pt~2±4, then the equation p~m-q~n=4 has no integer solution in which m>1, n>1. THEOREM 2. If(p,q)=(3,5), (5,3), (5,7), (7,5)(mod 8), then Eq.(2) has no integer solution in which h≥2, except that 3~2-5~1=2~2, 5~3-11~2=2~2 and     

关于丢番图方程ax~2+by~2=p~z

0 Introduction Let p be a prime, and let a, b∈Z_(>0), a>b>1 and (a, b)=1. If the equation X~2+abY~2=p~z, (X,Y)=1, Z>0 (1) has an integer solution (X,Y, Z), then there exists a unique integer solution (X_p, Y_p, Z_p,)which satisfies X_p>0, Y_p>0 and Z_p 0) satisfy the equation ax~2+by~2=2~z,z>2 (2) and x|~*a, y|~*b, where symbol x|~*a means that a is divided exactly by each prime factor of x.     

Erds猜想,Pell序列和丢番图方程

1. Introduction and Results 1.1 Erds conjecture In 1939, Erds conjectured that the equation (_m~n)=y~k, n>m>1, k≥3, (1)has no integer solution, where(_m~n)is a combinatorial number. In 1951, Erds himself proved that the conjecture is right when m>3, leaving the cases m=2 and m=3 unsolved (see Ref.[1]). In 1984, we proved that (_2~n) (n>2) is never a 2kth power if k>1. Since equation     

Positive integers possessing a weak order

For a positive integerm,we call a positive integertwhich possesses a weak order(modm),if there doesnot exist a positive integer nsuch thattn 1≡ t(modm). Letg(m)be the number of positive integerstforwhich1≤ t≤ mandtpossess a weak order(modm).Forx≥3,letG(x) =∑m≤xg(m).In 1981, V.S. Joshi[1]first proved the following as-ymptotic formula:G(x) =αx2 O(x(logx)3), (1)whereα=2ζ(12)∑n∞ =1C(nn2), C(n) =∏p npp 1.In 1986, Yang Zhaohua[2]improved (1) toG(x) =αx2 O(x(logx)2). (2) The key ne…     

费马大定理：Abel猜想的一个证明

1 IntroductionIn 1823 Abel made a conjecture in a particular case of Fermat's last theorem:If n>2and x,y and z are nonzero integers such thatx~n+y~n=z~n(1)then none of x,y or z can be a prime power(see P.Ribenboim[1],p.25).It is clear that we may assume,without loss of generality,that(x,y)=(x,z)=(y,z)=1and 0相似文献   

一般的Kalton型定理

In locally convex space theory,there is a nice result due to N.J.Kalton as follows(see [1] and [2]):Let (E,F) be a dual pair of vector spaces and suppose that τis an (E,F)-polartopology on E such that (E,τ) is separable.Then if ∑x_j is σ(E,F)-subseriesconvergent,then it is τ-subseries convergent.     

关于丢番图方程x~4—Dy~2=1

对于丢番图方程x~4-Dy~2=1,D>0且不是平方数,我们证明了在1)D=pq,p,q 是不时的奇素数,(?)p/q(?)=-1,且Pell 方程x~2-Dy~2=1的基本解ε=x_0+y_0D~(1/2)满足r|x_0+1,r≡3((mod4)是某个素数,2)D=2pq,p≡q≡5(mod8)时,均无正整数解。不难验证,适合1)的D 有无穷多个,如文末的两个推论。     

关于丢番图方程1 p~a=2~bq~c 2~dp~eq~f

It is well known that Diophantine equation is as follows:1 p~a=2~bq~c 2~dp~eq~f (1) where a, b, c, d, e, f are non-negative integcrs. In 1983, L. J. Alex and L. L. Foster (Rocky Mr. J. Math., 13(1983),     

Number of Negative Entries in A2≤0

Let A be a real matrix or a sign pattern of order n. N_ (A) denotes the number of negative entries in A. In 1972 R DeMarr and A Steger conjectured: If A is a real matrix of order n such that A^2≤0, then N- (A^2)≤( n - 1)^2 1. Now the conjecture is proved to be true when A is reducible or a matrix of order n≤3 and some sufficient conditions for N- (A^2)≤(n - 1)^2 1 are given. It is also proved that N_ (A^2)≤n^2 -4n 5 when A is a reducible combinatorially symmetric sign pattern such that A^2≤0, and the extreme sign patterns are characterized.     

双邻近与配邻近空间的某些性质

1 Contiguity and Contiguity Topology Defintion 1.1 Let X be a non-void set.A contiguity for X is a non-void family U of subsets of X×X such that (a) each member of U contains the diagonal △; (b) if u∈U and u(?)v(?)X×X, then v∈U;and (c) if u and v are members of U, then u ∩v∈U. We also call pair (X, U) is a contiguous space. (where U is a contiguity for X).     

Fuzzy数矩阵与Fuzzy数向量集(英文)

In this paper, we introduce the definition of a fuzzy number matrix which is a generalization of classical matrix, and then discuss some of its properties. Definition 1.A fuzzy set on a real number field R is said to be a Fuzzy number if it satisfies μ_(?)(0)=1. A Fuzzy number is said to be a Fuzzy singletion number if and only if we consider the following list which consists of Fuzzy numbers (?)_(ij) to be a Fuzzy number matrix. is a Fuzzy set on U_(m×n), the ccllection of all m×n matrixs, and its mem-     

Designing of optimal double loop networks

The double loop network G(N; r, s) has N vertices and 2N directed edges. A natural question is how to choose r and s such that G(N; r, s) has diameter as short as possible for a given N. In 1993, Li, Xu and Zhang proposed a method of constructing double loop networks with the minimum diameter for the case of r=1.The method is developed to construct such networks that none of their minimum diameters can be reached at r=1.As a by-product, a flaw in an assertation by Esqu et al. is pointed out.     

Number of Negative Entries in A2≤0

Let A be a real matrix or a sign pattern of order n. N-(A) denotes the number of negative entries in A. In 1972 R DeMarr and A Steger conjectured: If A is a real matrix of order n such that A2≤0, then N_(A2)≤(n-1)2 1. Now the conjecture is proved to be true when A is reducible or a matrix of order n≤3 and some sufficient conditions for N_(A2)≤(n-1)2 1 are given. It is also proved that N_(A2)≤n2-4n 5 when A is a reducible combinatorially symmetric sign pattern such that A2≤0, and the extreme sign patterns are characterized.     

Multi-agent consensus algorithm for second-order systems with a time-varying reference state

Two second-order consensus algorithms with a time-vary reference state without relative velocity measurements are proposed in a directed topology. Necessary and sufficient conditions are presented to ensure second-order consensus. It is shown that all the coupling strengths and the eigenvalues of the Laplacian matrix play important roles in reaching consensus. Specially when all nonzero eigenvalues of the Laplacian matrix are real,consensus can be achieved if and only if the coupling strengths are positive and the directed topology has a spanning tree for the first algorithm,and for the second one,consensus can be achieved if and only if the coupling strengths are positive. Finally,simulation examples are presented to verify the theoretical analysis.     

Implementing wavelet transform with SAW elements

Wavelet transform not only has inherited and developed the partial idea of short-time Fourier transform, but also has overcome the shortcoming that the window of short-time Fourier does not vary with frequency. In order to clearly observe the variation of a signal with time, if this signal is of high frequency, and its period is short, short-time window should be adopted; if this signal is of low frequency, and its period is long, long-time window should be adopted. Therefore, wavelet transfor…     

论简化平面镜

0 IntroductionBy means of quaternion,we can easily describe the imaging regularities forplane mirrors systems in parallel ray paths.In convergent ray paths,however,thebiquaternion has to be used because the posistion of the image has to be consid-ered.The one plane mirror's operator N in biquaternion can be written asWhere (?) is the unit normal vector of the plane mirror and l is the perpendicular dis-tance from the origin to the plane mirror,ωis Clifford W.Symbol and ω~2=O.Obviously this shows that the image can be obtained by rotating the inver-sion-object through 180°around (?) and then removing it a distance of 21 along (?).If some plane mirrors systems' operators can be written in the form of Eq.(1),then these systems may be simplified into one plane mirror,which is called simpli-fied plane mirror.     

Bi-prism-based single-lens three dimensional digital image correlation system with a long working distance: Methodology and application in extreme high temperature deformation test

As a novel three dimensional digital image correlation(3 D DIC) method, the bi-prism-based single lens(BSL) 3 D DIC method has been proposed and developed in recent years. Making use of a bi-prism, this method is able to perform a 3 D DIC measurement using only a single camera. Thus, the integration level of a BSL 3 D DIC system could be much higher than that of the double-camera3 D DIC system. In this paper, using a small-angle bi-prism and a camera with a longer focal length, a special BSL 3 D DIC system with a long working distance is designed for measurements in extreme environments. The principle of the system is first studied,and practical methods are then proposed for the system set-up and the determination of system parameters. Then, feasibility of the measurement system is verified by out-of-plane rigid-body translation tests. Finally, the BSL 3 D DIC system is proven to be capable of combining with a high-temperature testing instrument to perform deformation tests in a high-temperature environment of up to 1500°C.     

Sloping-and-shaking——Multiway merging and sorting

Most traditional merging and merging-based sorting algorithms are based on 2 sorters or 2 comparators A new merging technique is developed, namely sloping-and-shaking multiway merging, and a corresponding mul-tiway sorting method based only on k-sorters is proposed The sloping-and-shaking merging algorithm merges k sorted lists into one, where k can be any prime number The merging process is not a series of recursive applications of 2-way morging It sorts the keys on the m × k plane in vertical and horizontal directions, then along sloping lines with various slope rates step by step Only k-sorters are needed in the merging or sorting process. The time needed to merge ksorted lists, with m of each, is ( k + [log2( m / k) ]) tk, and the time for sorting N keys is (1 + (p - 1) k + 1/2( p -1) (p - 2)[ log2k])tk, where p - logkN, and tk is the time to sort k keys. The proposed algorithms can be implemented either by hardwared sorting networks, or on general purpose parallel and vector machines The tradition     

关于分数算子的一个注记

Let f*g (z) be the convolution or Hadamard product of two functiom f(z) and g(z), that is, if f (z) =z+sum from n=2 to ∞a_nz~n and g(z) =z+sum from n=2 to ∞b_n z_n, then f*g(z)=z+sum from n=2 to ∞a_n b_n z~n (1) Let T denote the class of functions of the form     

On Locating Numbers of Graphs

Let G =( V,E) be a connected graph and W = { w_1,w_2,…,w_k} be an ordered subset of V( G).For any vertex v ∈V,the locating code of v with respect to W is the k-vector CW( v) = { d( v,w_1),d( v,w_2),…,d( v,w_k) },W is said to be a locating set of G if distinct vertices have the distinct locating code,and the locating number of G is defined as: Loc( G) = min{ | W| : W is a locating set of G}.We study the locating set and locating number of a graph G,obtain some bounds for the locating numbers of graphs,and determine the exact value of Loc( G) for some special classes of graphs,such as cycles,wheels,complete t-partite graph and some Cartesian products of paths and cycles. In addition,we also prove that Loc( T) ≥Δ-1 holds for all trees T with maximum degree Δ,and shows a tree T with Loc( T) = Δ-1.