Multiplicity of Periodic Bouncing Solutions for Sublinear Damped Variation Systems via Nonsmooth Variational Methods

Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems-x=g(t)x+f(t,x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique established by Wu Xian and Wang Shaomin.Both of them imply the condition "f≥0" required in some previous papers can be weakened,furthermore,one of them also implies the condition about ?F(t,x)/?t required in some previous papers,such as "|?F(t,x)/?t|=σ₀F(t,x)" and "|?F(t,x)/?t|...     

What does Part 2 of the Fundamental Theorem of Calculus mean?

We defined the indefinite integral as an anti-derivative,and defined the definite integral as the limit of Riemann sums.Both of them are very different and seem to be little in common.Part 1of the Fundamental Theorem of Calculus shows how indefinite integration and definite integration are related.In the other words,it shows how anti-derivative and the area are related.Today,we’ll learn Part 2 of the Fundamental Theorem of Calculus.     

RESEARCH ANNOUNCEMENTS Reconstruction of Band-limited Signals From Uniform Random Samples

The Shannon sampling theorem is one of the foundations for modern signal processing.Assume that a signal f（t）∈L²（R）.σ>0 is a constant.Signal f（t） is calledσ-band-limitedif |ω|>σ,F（ω） = f_-∞^∞f（t）e^-iω≧tdt = 0.The Shannon sampling theorem says that aσ-band-limitedf（t） can be reconstructed exactly by its all sampling points at the equal interval h≤π/σ.The reconstructed formula is     

A Note on First Order Differential Equation

In this note, a theorem and its three corollaries on solution of the first order ordinary differential equation are given. Theorem Suppose that b, F∈C,a∈C~1,b(y)≠0. If a(t) and b(t) satisfy the equality a′(t)b(t)=1, (1) then the first order differential equation y′=b(y)F(x,a(y)) (2) has a solution y=f(u) (3) where u=u(x) is a solution of the equatien     

非线性分数次边值问题改进的ADM解法(英文)

We use the modified Adomian decomposition method（ADM） for solving the nonlinear fractional boundary value problem {D（^α₀） + u（x） = f（x, u（x））, 0 < x < 1, 3 < α≤ 4 u（0） = α₀ , u’’ （0） = α₂ u（1） = β₀ , u’’（1） = β₂} （1） where D（₀^α）+u is Caputo fractional derivative and α₀,α₂,β₀,β₂ is not zero at all,and f:[0,1]×R→ R is continuous.The calculated numerical results show reliability and efficiency of the algorithm given.The numerical procedure is tested on linear and nonlinear problems.     

A Covering Lemma on the Unit Sphere and Application to the Fourier-Laplace Convergence

A covering lemma on the unit sphere is established and then is applied to establish an almost everywhere convergence test of Marcinkiewicz type for the Fourier-Laplace series on the unit sphere which can be stated as follows： Theorem Suppose f ∈ L（En-1）, n≥ 3. If f satisfies the condition 1/θ^n-1∫D（x,θ）｜f（y）-f（x）｜dy=O（1/｜logθ｜）,as θ→0＋, at every point x in a set E of positive measure in Σn-1, then the Cesàro means of critical order ,n-2/2 of the Fourier-Laplace series of f converge to f at almost every point x in E.     

保持两个等价关系的变换半群的Green关系

Let Tx be the full transformation semigroup on a set X. For a non-trivial equivalence F on X, let
TF（X） = {f ∈ Tx ： arbieary （x, y） ∈ F, （f（x）,f（y）） ∈ F}.
Then TF（X） is a subsemigroup of Tx. Let E be another equivalence on X and TFE（X） = TF（X） ∩ TE（X）. In this paper, under the assumption that the two equivalences F and E are comparable and E lohtain in F, we describe the regular elements and characterize Green＇s relations for the semigroup TFE（X）.     

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

On the rigidity theorems for lagrangian translating solitons in pseudo-Euclidean space I

Let f be a smooth strictly convex solution of
defined on a domain Ω C R^n, where ai, bi and c are constants. Then the graph Mvf of △f is a spacelike translating soliton for mean curvature flow in pseudo-Euclidean space with the translating vector（al, a2, . ., an; bz, b2, , bn）. In this paper, we will use alCfine technique to show a Berustein Theorem： if the graph Mvf is complete, then f（x） must be a quadratic polynomial and Mvf is an ailine n-plane.     

On quasi-reduced quadratic forms

With the help of continued fractions, we plan to list all the elements of the set Q△ = {aX2 ＋ bXY ＋ cY2 ： a,b, c ∈Z, b2 - 4ac = △ with 0 ≤ b 〈 √△}of quasi-reduced quadratic forms of fundamental discriminant △. As a matter of fact, we show that for each reduced quadratic form f = aX2 ＋ bXY ＋ cY2 = （a, b, c） of discriminant △〉0（and of sign σ（f） equal to the sign of a）, the quadratic forms associated with f and defined by {〈a＋bu＋cu2,b＋2cu.c〉},with 1≤σ（f）u≤b/2｜c｜ （whenever they exist）, 〈c,-b-2cu,a＋bu＋cu2〉 with b/2｜c｜≤σ（f）u≤[w（f）]=[b＋√△/2｜c｜], are all different from one another and build a set I（f） whose cardinality is #I（f）={1＋[ω（f）],when（2c）｜b,[ω（f）],when （2c）｜b. If f and g are two different reduced quadratic forms, we show that I（f） ∩ I（g） = Ф. Our main result is that the set Q△ is given by the disjoint union of all I（f） with f running through the set of reduced quadratic forms of discriminant △〉0. This allows us to deduce a formula for #（Q△） involving sums of partial quotients of certain continued fractions.     

Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form { b1（x,u1）/ t-div（a（x,t,u1,Du1））＋div（Ф1（u1））＋f1（x,u1,u2）=O in Q, b2（x,u2）/ t-div（a（x,t,u2,Du2））＋div（Ф2（u2））＋f2（x,u1,u2）=O in Q in the framework of weighted Sobolev spaces, where b（x,u） is unbounded function on u, the Carath6odory function ai satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function Фi is assumed to be continuous on ]R and not belong to （Lloc1（Q））N.     

THE EXISTENCE AND NON-EXISTENCE OF SIGN-CHANGING SOLUTIONS TO BI-HARMONIC EQUATIONS WITH A p-LAPLACIAN

We investigate the bi-harmonic problem ■ where Δ² u=Δ（Δu）,Δ_pu=div(|▽u|^p-2▽u) with p> 2.Ω is a bounded smooth domain in R^N,N≥1.By using a special function space with the constraint ∫_Ω udx=0,under suitable assumptions on f and g（x,u）,we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem.Recent results from the literature are extended.     

On Holder Dependence of the Parameterized Hartman–Grobman Theorem

The well-known Hartman-Grobman Theorem says that a C~1 hyperbolic diffeomorphism F can be locally linearized by a homeomorphismΦ.For parameterized systems F_θ,known results show that the corresponding homeomorphismsΦ_θexist uniquely in a functional space equipped with the supremum norm and depend continuously on the parameterθ.In this paper,we further extend the results to Holder dependence ofΦ_θonθby Pugh's strategy,but introducing a kind of special Holder norm instead of the usual supremum norm in the proof to control the linear parts of F_θ.This requires a new Holder linearization result for every F_θ.     

The definite integral

<正>1.What does the definite integral mean?The definite integral of f(x)fromato b is defined the limit of the sum as n→∞.That is limn→∞∑n i=1f(ξi)·Δxi.We divide the interval[a,b]into n subintervals of equal widthΔx=(b-a)n.Let x0=a,x1,x2,…,xn=b be the endpoints of these subintervals and we chooseξiis any point in the ith subinterval,that is,xi-1≤ξi≤xi,then,the sum∑n f(ξi)·Δxiis called a Rie-     

The estimate for mean values on prime numbers relative to \frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }}

If n is a positive integer,let f （n） denote the number of positive integer solutions （n 1,n 2,n 3） of the Diophantine equation 4/n=1/n₁ + 1/n₂ + 1/n₃.For the prime number p,f （p） can be split into f 1 （p） + f 2 （p）,where f i （p） （i=1,2） counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献   

A GENERALIZATION OF SIMPSON''''S RULE

Let TA(f)=integral form n= to 1/2(P_~n(x) + P_b~n(x))dx and let TM(f)=integral form n= to P_((+b)/2)~(n+1)(x)dx, where P_c~n denotes theTaylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of theTrapezoidal rule and the midpoint rule, respectively. and are each exact for all polynomial of degree ≤n+1.We let L(f) = αTM(f) + (1-α)TA(f), where α =(2~(n+1)(n+1))/(2~(n+1)(n+1)+1), to obtain a numerical integrationrule L which is exact for all polynomials of degree≤n+3 (see Theorem l). The case n = 0 is just the classicolSimpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacingP_(+b)/2)~(n+1)(x) by the Hermite cabic interpolant at a and b. we obtain some known formulae by a different ap-proach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by takingpiecewise polynomials of odd degree, each piece being the Taylor polynomial off at a and b. respectively. Ofcourse all of our formulae can be compounded over subintervals of [a, b].     

W0^1p（x） Versus C^1 Local Minimizers for a Functional with Critical Growth

Let Ω IR^N, （N ≥ 2） be a bounded smooth domain, p is Holder continuous on Ω^-,
1 〈 p^- ：= inf pΩ（x） ≤ p＋ = supp（x） Ω〈∞,
and f：Ω^-× IR be a C^1 function with f（x,s） ≥ 0, V （x,s） ∈Ω × R^＋ and sup ∈Ωf（x,s） ≤ C（1＋s）^q（x）, Vs∈IR^＋,Vx∈Ω for some 0〈q（x） ∈C（Ω^-） satisfying 1 〈p（x） 〈q（x） ≤p^＊ （x） -1, Vx ∈Ω ^- and 1 〈 p^- ≤ p^＋ ≤ q- ≤ q＋. As usual, p＊ （x） = Np（x）/N-p（x） if p（x） 〈 N and p^＊ （x） = ∞- if p（x） if p（x） 〉 N. Consider the functional I： W0^1,p（x） （Ω） →IR defined as
I（u） def= ∫Ω1/p（x）｜△｜^p（x）dx-∫ΩF（x,u^＋）dx,Vu∈W0^1,p（x）（Ω）,
where F （x, u） = ∫0^s f （x,s） ds. Theorem 1.1 proves that if u0 ∈ C^1 （Ω^-） is a local minimum of I in the C1 （Ω^-） ∩C0 （Ω^-）） topology, then it is also a local minimum in W0^1,p（x） （Ω）） topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation （P） defined below.     

全纯函数的分担值与正规族

Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.     

POSITIVE SOLUTIONS TO THE p-LAPLACIAN WITH SINGULAR WEIGHTS＊

By the Mountain Pass Theorem,we study existence and multiplicity of positive solutions of p-laplacian equation of the form -△pu =λf(x,u),the nonlinearity f(x,u) grows as uσ at infinity with a singular ...     

ON APPROXIMATION PROPERTIES OF NON-CONVOLUTION TYPE NONLINEAR INTEGRAL OPERATORS-Dedicated to Professor Paul L. Butzer on his 80th Birthday

In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]