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1.
We find a formula for the sum of the first n squares:Sn = 12 22 32 42 … n2.We haveS1 = 1, S2 = 12 22 = 5, S3 = l2 22 32 = 14,S4 = 30, S5 = 55, S6 = 91.and so on. To make things a little simpler, we will also use the sum of the squares of the first zero terms , S0 = 02 = 0. Arranging these values in a table,we find thatSince the third differenes △3Sn are constant,these data values must follow a cubic pattern;that is, the formula for the sum of the squares of the first n integers is a cubic function,  相似文献   

2.
This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda's results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.  相似文献   

3.
§1. IntroductionTofindthegeneraltermformulaofcalculatingwasanoldestprobleminNumberTheory.Earlyin300B.C.,Archimedo,themathematicianofancientGreece,workedouttheformula∑nk=1k=n(n+1)2and∑nk=1k2=n(n+1)(2n+1)6.Sofar,peoplehavegotmanyformsofepressionofthege…  相似文献   

4.
In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.  相似文献   

5.
Existence of nonzero positive solutions  of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the well-known theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in  ordered Banach spaces.  相似文献   

6.
We establish upper bounds for a group of *-deviations of Faber sums on the classes of -integrals in a complex plane introduced by Stepanets.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 451–461, April, 2004.  相似文献   

7.
In general category, the group of self-equivalences of sum object can decompose as the product of its two subgroups under certain conditions and also several split short exact sequences are obtained. Subsequently, we apply the above results to the category of space under a space and get some results which are dual to those in fibrewise category.  相似文献   

8.
We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.  相似文献   

9.
1. Introduction A theorem of Carleson,as generalized by Duren characterizes those positive measure μ on the unit disc U={z∈C:|z|<1} for which the H~p norm domiates the L~q(μ) norm of elements of H~p. Later on, Hasting proved an analogous results with H~p replaced by A~p, the Bergman space of fuctions f  相似文献   

10.
DynamicsofPolynomialAutomorphismsofC~N¥ZhangWenjun(HenanUniversity,Kaifeng,P.R.Chian,475001)Abstract:Thispaperisassignedtodis?..  相似文献   

11.
We study the asymptotic behavior for the condensate solutions of the self-dual Chern–Simons–Higgs equation as the Chern–Simons parameter tends to zero. By using these estimates, we establish existence results for solutions of non-topological type.  相似文献   

12.
In this paper, we consider a Leslie–Gower predator–prey model with Holling type II functional response and density-dependent diffusion under zero Dirichlet boundary condition. Using degree theory, bifurcation theory, energy estimates and asymptotical behavior analysis, the existence and multiplicity of positive steady state solutions were shown under certain conditions on the parameters.  相似文献   

13.
We solve an abstract parabolic problem in a separable Hilbert space, using the projection-difference method. The spatial discretization is carried out by the Galerkin method and the time discretization, by the Crank–Nicolson scheme. On assuming weak solvability of the exact problem, we establish effective energy estimates for the error of approximate solutions. These estimates enable us to obtain the rate of convergence of approximate solutions to the exact solution in time up to the second order. Moreover, these estimates involve the approximation properties of the projection subspaces, which is illustrated by subspaces of the finite element type.  相似文献   

14.
Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler–Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.  相似文献   

15.
Motivated by the study on the uniqueness problem of the coupled model, in this paper, we revisit 2d incompressible Navier–Stokes equations in bounded domains. In fact, we establish some new smoothing estimates to the Leray solution based on the spectral analysis of Stokes operator. To understand well these estimates, on one hand, we establish some new Brezis–Waigner type inequalities in general domain and in any dimension and disclose the connection between both of them. On the other hand, we show that these new estimates can be applied to prove the existence and uniqueness of the weak solutions for two coupled models: Boussinesq system with partial viscosity (no dissipation for the temperature) and Fluid/Particle system, in two dimension and in bounded domains.  相似文献   

16.
In this article, we study the asymptotics of the positive solutions of the Euler–Lagrange system of the weighted Hardy–Littlewood–Sobolev in R n $$\begin{array}{ll} u(x) = \frac{1}{|x|^{\alpha}}\int\limits_{R^{n}} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy,\\ v(x) = \frac{1}{|x|^{\beta}}\int\limits_{R^{n}} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}} dy.\end{array}$$ A new iterative method is introduced to obtain the optimal weighted local integrability of u(x). By this new method, we establish the asymptotic estimates of the solutions around the origin and near infinity. With these new estimates, we complete the study of the asymptotic behavior of the solutions. We believe this new iterative method and the new type of the weighted local estimates can be used in many other cases.  相似文献   

17.
《Applied Mathematics Letters》2007,20(10):1031-1038
In this paper, a discrete periodic predator–prey system with type IV functional responses and time delay is investigated. Using Gaines and Mawhin’s continuation theorem from coincidence degree theory as well as some prior estimates, we get sufficient conditions for the existence of positive periodic solutions of the system. This is also the first time that the coincidence degree theory has been used to obtain multiple positive periodic solutions in discrete ecological systems.  相似文献   

18.
We prove Strichartz estimates for radial solutions of the Schrödinger and wave equations on Damek–Ricci spaces, and in particular on symmetric spaces of noncompact type and rank one, using the perturbative theory with potentials. The curvature of the noncompact manifold has an influence on the dispersive properties, and indeed we obtain Strichartz estimates with weights at spatial infinity, which are stronger than the standard ones in the flat case.  相似文献   

19.
A new method for obtaining computable estimates for the difference between exact solutions of elliptic variational inequalities and arbitrary functions in the respective energy space is suggested. The estimates are obtained by transforming the corresponding variational inequality without the use of variational duality arguments. These estimates are valid for any function in the energy class and contain no constants depending on the mesh used to find an approximate solution. This method for linear elliptic and parabolic problems was earlier suggested by the author. The guaranteed error bounds we derive can be of two types. Estimates of the first type contain only one global constant, which is a constant in the Friedrichs type inequality. Estimates of the second type are based on the decomposition of Ω into convex subdomains and the Payne–Weinberger inequalities for these subdomains. Bibliography: 20 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 81–90.  相似文献   

20.
In this paper we study the asymptotic behaviour of solutions of the pantograph-type differnce equation, and obtain aymptotic estimates, which can imply asymptotic stability or stability of solutions  相似文献   

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