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1.
Define a sequence (sn) of two-variable words in variables x,y as follows: s0(x, y) = x, sn+1(x,y)=[sn(x, y]y, sn(x,y)for n 0. It is shown that a finite group G is soluble if andonly if sn is a law of G for all but finitely many values ofn. 2000 Mathematics Subject Classification 20D10, 20D06.  相似文献   

2.
We study zeros of elliptic integrals I(h)=HhR(x,y)dxdy, whereH(x,y) is a real cubic polynomial with a symmetry of order three,and R(x,y) is a real polynomial of degree at most n. It turnsout that the vector space An formed by such integrals is a Chebishevsystem: the number of zeros of each elliptic integral I(h)Anis less than the dimension of the vector space An. 1991 MathematicsSubject Classification 34C10.  相似文献   

3.
The paper derives a formula for the second variation of thedisplacement function for polynomial perturbations of Hamiltoniansystems with elliptic or hyperelliptic Hamiltonians H(x, y)=y2U(x)in terms of the coefficients of the perturbation. As an application,the conjecture stated by C. Chicone that a specific cubic systemappearing in a deformation of singularity with two zero eigenvalueshas at most two limit cycles is proved.  相似文献   

4.
Let S(x,y) be the set S(x,y)= 1 n x : P(n) y, where P(n) denotesthe largest prime factor of n. We study , where f is a multiplicative function. When f=1and when f=µ, we widen the domain of uniform approximationusing the method of Fouvry and Tenenbaum and making explicitthe contribution of the Siegel zero. Soit S(x,y) l'ensemble S(x,y)= 1 n x : P(n) y, désigne le plus grand facteur premier den. Nous étudions , lorsque f est une fonction multiplicative. Quand f=1 et quand f=µ,nous élargissons le domaine d'approximation uniformeenutilisant la méthode développée par Fouvryet Tenenbaum et en explicitant la contribution du zérode Siegel. 1991 Mathematics Subject Classification: 11N25, 11N99.  相似文献   

5.
We consider the Sturm–Liouville equation y'(x)+{–q(x)}y(x)=0 (0x<) (1.1) with a boundary condition at x = 0 which can be either the Dirichletcondition y(0)=0 (1.2) or the Neumann condition y'(0)=0 (1.3) As usual, is the complex spectral parameter with 0 arg <2, and the potential q is real-valued and locally integrablein [0, ).  相似文献   

6.
We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.  相似文献   

7.
Consider the general nonlinear boundary-value problem (p(t)y' (t))' = p(t)q(t) f (t, y(t), y' (t)), t 1, g(y(1), y' (1))= 0, where the function f may be singular at the point y(1)= 0 and p(1) 0. We obtain conditions which guarantee existenceof positive and bounded solutions of the above problem. As anapplication we prove existence and uniqueness of rotationallysymmetric solutions to a nonlinear boundary-value problem, representingthe elastic deformation of a spherical cap.  相似文献   

8.
We characterize the mapping properties such as the boundedness,compactness and measure of noncompactness for those real weightfunctions , , u0, v0, for which the Hardy-type integral operatorof the form acts from to , when the parameters are restricted to the range 1 < max (r,s) min (p, q) < and the kernel k(x, y) 0 satisfies theOinarov condition (see (2) below). For the case k(x, y) = 1,we obtain lower and upper estimates of the approximation numbers,extending the result of [5].  相似文献   

9.
In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRn, is acontinuous vector function with continuous first derivativesfr/xs, r, s=1, 2, ..., n. Let Dxf(t, x), (t, x)IxRn, denote the Jacobi matrix of f(t,x), with respect to the variables x1, ..., xn. Let x(t, t0,x0), tI(t0, x0) denote the maximal solution of the system (1)through the point (t0, x0)IxRn. For two vectors x, yRn, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(ars) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t0I, x1,x2Rn [formula] holds for all t>t0 as long as both solutions x(t, t0, xi),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRn the off-diagonal elements of the nxn matrix Dxf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.  相似文献   

10.
On the Optimum Criterion of Polynomial Stability   总被引:1,自引:0,他引:1  
The purpose of this note is to answer the question raised byNie & Xie (1987). Let f(x)=a0xn+a1xn–1+...+an be apositive-coefficient polynomial. The numbers 1=ai-1ai+2/aiai+1(i=1, ..., n–2) are called determining coefficients. Theoptimum criterion problem was posed as follows: for n3, findthe maximal number (n) such that the polynomial f(x) is stableif i < (n) (1in–2). For n6, we show that (n)=ß,where ß is the unique real root of the equation x(x+1)2=1.  相似文献   

11.
We discuss the practical determination of stability regionswhen various fixed-stepsize Runge-Kutta (RK) methods, combinedwith continuous extensions, are applied to the linear delaydifferential equation (DDE) y'(t)= y(t)+µ(t–) (t) with fixed delay . It is significant that the delay is not limitedto an integer multiple of the stepsize, and that we considervarious continuous extensions. The stability loci obtained in practice indicate that the standardboundary-locus technique (BLT) can fail to map the RK DDE stabilityregion correctly. The aim of this paper is to present an alternativestability boundary algorithm that overcomes the difficultiesencountered using the BLT. This new algorithm can be used forboth explicit and implicit RK methods.  相似文献   

12.
A famous Diophantine equation is given by yk=(x+1)(x+2)...(x+m). (1) For integers k2 and m2, this equation only has the solutionsx = –j (j = 1, ..., m), y = 0 by a remarkable result ofErds and Selfridge [9] in 1975. This put an end to the old questionof whether the product of consecutive positive integers couldever be a perfect power (except for the obviously trivial cases).In a letter to D. Bernoulli in 1724, Goldbach (see [7, p. 679])showed that (1) has no solution with x0 in the case k = 2 andm = 3. In 1857, Liouville [18] derived from Bertrand's postulatethat for general k2 and m2, there is no solution with x0 ifone of the factors on the right-hand side of (1) is prime. Byuse of the Thue–Siegel theorem, Erds and Siegel [10] provedin 1940 that (1) has only trivial solutions for all sufficientlylarge kk0 and all m. This was closely related to Siegel's earlierresult [30] from 1929 that the superelliptic equation yk=f(x) has at most finitely many integer solutions x, y under appropriateconditions on the polynomial f(x). The ineffectiveness of k0was overcome by Baker's method [1] in 1969 (see also [2]). In 1955, Erds [8] managed to re-prove the result jointly obtainedwith Siegel by elementary methods. A refinement of Erds' ideasfinally led to the above-mentioned theorem as follows.  相似文献   

13.
In this paper, we study the nonlinear one-dimensional periodic wave equation with x-dependent coefficients u(x)ytt−(ux(x)yx)+g(x,t,y)=f(x,t) on (0,πR under the boundary conditions a1y(0,t)+b1yx(0,t)=0, a2y(π,t)+b2yx(π,t)=0 ( for i=1,2) and the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t). Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. A main concept is the notion “weak solution” to be given in Section 2. For T is the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.  相似文献   

14.
Let f Z[x,y] be a reducible homogeneous polynomial of degree3. We show that f(x,y) has an even number of prime factors asoften as an odd number of prime factors.  相似文献   

15.
16.
Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ2Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ2S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V3 into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ3T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation
F(t)+t3G(1/t)=0,  相似文献   

17.
For ordinary differential equations satisfying a one-sided Lipschitzcondition with Lipschitz constant v, the solutions satisfy with l=hv, so that, in the case of Runge-Kutta methods, estimatesof the form ||yn||2k(l)||yn–1||2 are desirable. Burrage(1986) has investigated the behaviour of the error-boundingfunction k for positive l for the family of s-stage Gauss methodsof order 2s, and has shown that k(l)=exp 2l+O(l3) (l0) for s3.In this paper, we extend the analysis of k to any irreduciblealgebraically stable Runge-Kutta method, and obtain resultsabout the maximum order of k as an approximation to exp 2l.As a particular example, we investigate the function k for allalgebraically stable methods of order 2s–1.  相似文献   

18.
Irregularities of Point Distribution Relative to Convex Polygons III   总被引:1,自引:0,他引:1  
Suppose that P is a distribution of N points in the unit squareU=[0, 1]2. For every x=(x1, x2)U, let B(x)=[0, x1]x[0, x2] denotethe aligned rectangle containing all points y=(y1, y2)U satisfying0y1x1 and 0y2x2. Denote by Z[P; B(x)] the number of points ofP that lie in B(x), and consider the discrepancy function D[P; B(x)]=Z[P; B(x)]–Nµ(B(x)), where µ denotes the usual area measure.  相似文献   

19.
Suppose that K is a closed, total cone in a real Banach spaceX, that A:XX is a bounded linear operator which maps K intoitself, and that A' denotes the Banach space adjoint of A. Assumethat r, the spectral radius of A, is positive, and that thereexist x00 and m1 with Am(x0)=rmx0 (or, more generally, thatthere exist x0(–K) and m1 with Am(x0)rmx0). If, in addition,A satisfies some hypotheses of a type used in mean ergodic theorems,it is proved that there exist uK–{0} and K'–{0}with A(u)=ru, A'()=r and (u)>0. The support boundary of Kis used to discuss the algebraic simplicity of the eigenvaluer. The relation of the support boundary to H. Schaefer's ideasof quasi-interior elements of K and irreducible operators Ais treated, and it is noted that, if dim(X)>1, then thereexists an xK–{0} which is not a quasi-interior point.The motivation for the results is recent work of Toland, whoconsidered the case in which X is a Hilbert space and A is self-adjoint;the theorems in the paper generalize several of Toland's propositions.  相似文献   

20.
A Class of Infinite Dimensional Simple Lie Algebras   总被引:1,自引:0,他引:1  
Let A be an abelian group, F be a field of characteristic 0,and , ß be linearly independent additive maps fromA to F, and let ker()\{0}. Then there is a Lie algebra L = L(A,, ß, ) = xA Fex under the product [ex, ey]]=(xy)ex+y+(ß) (x, y) ex+y. If, further, ß() = 1, and ß(A) = Z, thereis a subalgebra L+:=L(A+, , ß, ) = xA+ Fex, whereA+ = {xA|ß(x)0}. The necessary and sufficient conditionsare given for L' = [L, L] and L+ to be simple, and all semi-simpleelements in L' and L+ are determined. It is shown that L' andL+ cannot be isomorphic to any other known Lie algebras andL' is not isomorphic to any L+, and all isomorphisms betweentwo L' and all isomorphisms between two L+ are explicitly described.  相似文献   

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