Self-similar fractals: An algorithmic point of view

This paper studies the self-similar fractals with overlaps from an algorithmic point of view.A decidable problem is a question such that there is an algorithm to answer"yes"or"no"to the question for every possible input.For a classical class of self-similar sets{E b.d}b,d where E b.d=Sn i=1(E b,d/d+b i)with b=(b1,...,b n)∈Qn and d∈N∩[n,∞),we prove that the following problems on the class are decidable:To test if the Hausdorff dimension of a given self-similar set is equal to its similarity dimension,and to test if a given self-similar set satisfies the open set condition(or the strong separation condition).In fact,based on graph algorithm,there are polynomial time algorithms for the above decidable problem.     

On the Diophantine Equation sum from k=1 to m((k~n)=(m+1)~n)

P. Erds has conjectured [1] that the Diophantine equation 1~n+2~n+…+m~n=(m+1)~n (1) has no positive integer solutions except that n=1, m=2. It is true when m≤10~(10) [3]. A generalized form of (1) has been investigated in [1] [2], and various     

NOTE ON THE DIOPHANTINE EQUATION

<正> Dr.Erd(?)s conjectured that the Diophantine equation(1)x~x y~y=z~zhas no integer solution, if,x>1,y>1.z>1.In the present note,Ⅰshall prove that his conjecture is correct ouly (?)(x,y)=1 and(1)has intinitely many solutions when (x,y)>1.     

ON THE DIOPHANTINE EQUATION X4-Dy2=1(II)

For the Diophantine equation x^4 — Dy^2 = 1 (1) where D>0 and is not a perfect square, we prove the following theorems in this paper. Theorem 1. If D\[{\not \equiv }\]7 (mod 8)，D=p1p2...ps，s≥2，where pi（i = 1，…，s) are distincyt primes，p1≡1(mod 4) such that either 2p1=a^2+b^2，а≡\[ \pm \]3(mod 8)，b三\[ \pm \]3(mod 8) or there is a j(2≤j≤s), for which Legendre symbal \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\],and pi≡7(mod8) (i=2,..., s) or pi≡3(mod 8) (i=2,..., s), then (1) has no solutions in positive integer x,y. Theorem 2. If D=p1...ps,s≥2, where pi（i = 1，…，s) are distinct primes, and pi≡3(mod 4)（i = 1，…，s), then (1) has no solutions in positive integer x, y. Theorem 3. The equation (1) with D=2p1...ps has no solutions in positive integer x, y, if (1) p1≡(mod 4), pi≡7(mod 8) (i = 2, ???, s), snch that either 2p1 = a^2+b^2 a≡\[ \pm \]3(mod 8)，b≡\[ \pm \]3(mod 8)or there is a j (2≤j≤s)，for which \[\left( {\frac{{{p_j}}}{{{p_1}}}} \right) = - 1\]; or (2) p1≡5(mod8),pi≡3(mod8) (i = 2,..., s)； or ⑶p1≡5(mod8)，pi≡7(mod 8) (i=2，…，s). Corollary of theorem 3. If D = 2pq, p≡5(mod 8), q≡3(mod 4), where p, q are distinct primes, then (1) has no solutions in positive integer x, y. Theorem 4. If D=2p1...ps, pi≡3(mod 4)(0 = 1,...,s), then (1) has no solutions In positive integer x, y.     

STABLE SOLUTION OF TIME DOMAIN INTEGRAL EQUATION METHODS USING QUADRATIC B-SPLINE TEMPORAL BASIS FUNCTIONS

This paper is concerned with stable solutions of time domain integral equation （TDIE） methods for transient scattering problems with 3D conducting objects. We use the quadratic B-spline function as temporal basis functions, which permits both the induced currents and induced charges to be properly approximated in terms of completeness. Because the B-spline function has the least support width among all polynomial basis functions of the same order, the resulting system matrices seem to be the sparsest. The TDIE formula-tions using induced electric polarizations as unknown function are adopted and justified. Numerical results demonstrate that the proposed approach is accurate and efficient, and no late-time instability is observed.     

Liouville Type Results for a p-Laplace Equation with Negative Exponent

Positive entire solutions of the equation Δpu=u~(-q) in R~N(N ≥ 2)where 1p≤N,q0,are classified via their Morse indices.It is seen that there is a critical power q=q_c such that this equation has no positive radial entire solution that has finite Morse index when qq c but it admits a family of stable positive radial entire solutions when 0q≤q_c.Proof of the stability of positive radial entire solutions of the equation when 1p2 and 0q≤q_c relies on Caffarelli–Kohn–Nirenberg's inequality.Similar Liouville type result still holds for general positive entire solutions when 2p≤N and qq_c.The case of 1p2 is still open.Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p=2 obtained previously.Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.     

On Radon-Penrose transformation and k-Cauchy-Fueter operator

It is known that there is a 1-1 correspondence between the first cohomology of the sheaf O(-k-2) over the projective space and the solutions to the k-Cauchy-Fueter equations on the quaternionic space Hn.We find an explicit Radon-Penrose type integral formula to realize this correspondence:given a -closed(0,1)form f with coefficients in the(-k-2)th power of the hyperplane section bundle H-k-2,there is an integral representation Pf such that ι*(Pf) is a solution to the k-Cauchy-Fueter equations,where ι is an embedding of the quaternionic space Hn into C4n.     

Exponential polynomials as solutions of certain nonlinear difference equations

Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + L(z, f ) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 + q(z)f (z + 1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f(z)n + q(z)e Q(z) f(z + c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.     

McCoy环的扩张(英文)

A ring R is said to be right McCoy if the equation f(x)g(x)=0,where f(x)and g(x)are nonzero polynomials of R[x],implies that there exists nonzero s∈R such that f(x)s=0.It is proven that no proper(triangular)matrix ring is one-sided McCoy.It is shown that for many polynomial extensions,a ring R is right McCoy if and only if the polynomial extension over R is right McCoy.     

Resonance Problem for a Class of Duffing’s Equations

Consider the Duffing's equation+g(x)=f(t),(1)where g∈C(R,R)and f∈P≡{f∈C(R,R);f is ω-periodic for some ω>0}.The functiong is said to be resonant if there exists f∈P such that eq.(1) has no bounded solutions on[0,∞).Using a generalized version of the Poincare-Birkhoff fixed point theorem,theauthors establish conditions on g which guarantee the following result holds:for any f∈Pwith period ω,there exists K≥0 such that eq.(1) has infinitely many kω-periodic solutionsfor every integer k≥K.In such a case,g is clearly non-resonant.     

ON ESTIMATIONS OF TRIGONOMETRIC SUMS OVER PRIMES IN SHORT INTERVALS (Ⅲ)

In this paper the following result is proved: There is an absolute positive integer such that for every large odd integer N the Diophantine equation with prime variables N=p_1+p_2+p_3, N/3-U相似文献   

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

On entire solutions of some type of nonlinear difference equations

In this article, the existence of finite order entire solutions of nonlinear difference equations f~n+ P_d(z, f) = p_1 e~(α1 z)+ p_2 e~(α2 z) are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n-2), p_1, p_2 are small meromorphic functions of ez, and α_1, α_2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Quasi-periodic solutions with prescribed frequency in a nonlinear Schrdinger equation

In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 σ 1, and each given integer b 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

Heron三角形与Diuphantine方程

In this paper, we study the quantic Diophantine equation (1) with elementary geometry method, therefore all positive integer solutions of the equation (1) are obtained, and existence of Heron triangle whose median lengths are all positive integer are discussed here.     

Equality-constrained minimization of polynomial functions

This paper investigates the equality-constrained minimization of polynomial functions. Let R be the field of real numbers, and R[x1,..., xn] the ring of polynomials over R in variables x1,..., xn. For an f ∈ R[x1,..., xn] and a finite subset H of R[x1,..., xn], denote by V(f : H) the set {f( ˉα) | ˉα∈ Rn, and h( ˉα) =0, ? h ∈ H}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum inf V(f : H) of V(f : H) is a root of some polynomial in U whenever inf V(f : H) = ±∞.The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.     

An extended fast algorithm for constructing the Dixon resultant matrix

In recent years,the Dixon resultant matrix has been used widely in the re-sultant elimination to solve nonlinear polynomial equations and many researchers havestudied its efficient algorithms.The recursive algorithm is a very efficient algorithm,butwhich deals with the case of three polynomial equations with two variables at most.Inthis paper,we extend the algorithm to the general case of n 1 polynomial equations in nvariables.The algorithm has been implemented in Maple 9.By testing the random polyno-mial equations,the results demonstrate that the efficiency of our program is much betterthan the previous methods,and it is exciting that the necessary condition for the existenceof common intersection points on four general surfaces in which the degree with respectto every variable is not greater than 2 is given out in 48×48 Dixon matrix firstly by ourprogram.     

AN EFFECTIVE CONTINUOUS ALGORITHM FOR APPROXIMATE SOLUTIONS OF LARGE SCALE MAX-CUT PROBLEMS

An effective continuous algorithm is proposed to find approximate solutions of NP-hardmax-cut problems.The algorithm relaxes the max-cut problem into a continuous nonlinearprogramming problem by replacing n discrete constraints in the original problem with onesingle continuous constraint.A feasible direction method is designed to solve the resultingnonlinear programming problem.The method employs only the gradient evaluations ofthe objective function,and no any matrix calculations and no line searches are required.This greatly reduces the calculation cost of the method,and is suitable for the solutionof large size max-cut problems.The convergence properties of the proposed method toKKT points of the nonlinear programming are analyzed.If the solution obtained by theproposed method is a global solution of the nonlinear programming problem,the solutionwill provide an upper bound on the max-cut value.Then an approximate solution to themax-cut problem is generated from the solution of the nonlinear programming and providesa lower bound on the max-cut value.Numerical experiments and comparisons on somemax-cut test problems(small and large size)show that the proposed algorithm is efficientto get the exact solutions for all small test problems and well satisfied solutions for mostof the large size test problems with less calculation costs.     

Solution to an Extremal Problem on Bigraphic Pairs with a Z_3-connected Realization

Let S =(a_1...,a_m;b_1,...,b_n),where a_1,...,a_m and b_1,...,b_n are two nonincreasing sequences of nonnegative integers. The pair S =(a_1,..., a_m; b_1,..., b_n) is said to be a bigraphic pair if there is a simple bipartite graph G =(X U Y, E) such that a_1...,a_m and b_1,...b_n are the degrees of the vertices in X and Y, respectively. Let Z3 be the cyclic group of order 3. Define a(Z_3,m,n) to be the minimum integer k such that every bigraphic pair S =(a_1,..., a_m; b_1,..., b_n) with a_m,b_n≥2 and σ(S) = a_1+…+a_m≥k has a Z_3-connected realization. For n =m, Yin [Discrete Math.,339, 2018-2026(2016)] recently determined the values of σ(Z_3,m,m) for m≥ 4. In this paper, we completely determine the values of a(Z_3,m,n) for m ≥n≥4.