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1.
Ala Alnaser 《Journal of Number Theory》2008,128(9):2582-2590
Let p be an odd prime and γ(k,pn) be the smallest positive integer s such that every integer is a sum of s kth powers . We establish γ(k,pn)?[k/2]+2 and provided that k is not divisible by (p−1)/2. Next, let t=(p−1)/(p−1,k), and q be any positive integer. We show that if ?(t)?q then γ(k,pn)?c(q)k1/q for some constant c(q). These results generalize results known for the case of prime moduli.
Video abstract
For a video summary of this paper, please visit http://www.youtube.com/watch?v=zpHYhwL1kD0. 相似文献2.
We find the greatest value p and least value q such that the double inequality L p (a, b)?<?T(a, b)?<?L q (a, b) holds for all a, b?>?0 with a?≠ b, and give a new upper bound for the complete elliptic integral of the second kind. Here ${T(a,b)=\frac{2}{\pi}\int\nolimits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}d\theta}$ and L p (a, b)?=?(a p+1?+?b p+1)/(a p ?+?b p ) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively. 相似文献
3.
We consider, for p∈(1,2) and q>1, self-similar singular solutions of the equation vt=div(|∇v|p−2∇v)−vq in Rn×(0,∞); here by self-similar we mean that v takes the form v(x,t)=t−αw(|x|t−αβ) for α=1/(q−1) and β=(q+1−p)/p, whereas singular means that v is non-negative, non-trivial, and for all x≠0. That is, we consider the ODE problem
(0.1) 相似文献
4.
Jin Ho Kwak 《Discrete Mathematics》2008,308(11):2156-2166
In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say (p1,p2,…,pk are distinct odd primes and ai>0 for each i?1), there are t distinct reflexible regular embeddings of the complete bipartite graph Kn,n up to isomorphism, where t=1 if a=0, t=2k if a=1, t=2k+1 if a=2, and t=3·2k+1 if a?3. And, there are s distinct self-Petrie dual regular embeddings of Kn,n up to isomorphism, where s=1 if a=0, s=2k if a=1, s=2k+1 if a=2, and s=2k+2 if a?3. 相似文献
5.
A.G. Ramm 《Journal of Mathematical Analysis and Applications》2006,316(2):764-767
Equation (−Δ+k2)u+f(u)=0 in D, u|∂D=0, where k=const>0 and D⊂R3 is a bounded domain, has a solution if is a continuous function in the region |u|?a, piecewise-continuous in the region |u|?a, with finitely many discontinuity points uj such that f(uj±0) exist, and uf(y)?0 for |u|?a, where a?0 is an arbitrary fixed number. 相似文献
6.
Jia-Feng Tang 《Journal of Mathematical Analysis and Applications》2007,334(1):517-527
In this paper, we study the differential equations of the following form w2+R(z)2(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w2+P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w2+(z−z0)P2(z)2(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w2+A(z−z0)2(w′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab2=1, a2=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B2=C and q′(z)=±P(z). 相似文献
7.
An even-order three-point boundary value problem on time scales 总被引:1,自引:0,他引:1
Douglas R Anderson Richard I Avery 《Journal of Mathematical Analysis and Applications》2004,291(2):514-525
We study the even-order dynamic equation (−1)nx(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x(Δ∇)i(a)=0 and x(Δ∇)i(c)=βx(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(b−a)<c−a for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale. 相似文献
8.
Vitali Liskevich I.I. Skrypnik 《Journal of Mathematical Analysis and Applications》2008,338(1):536-544
We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a0(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|p−2+u|u|q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron. 相似文献
9.
G. Gripenberg S.-O. Londen J. Prüss 《Mathematical Methods in the Applied Sciences》1997,20(16):1427-1448
It is proved that there is a (weak) solution of the equation ut=a*uxx+b*g(ux)x+f, on ℝ+ (where * denotes convolution over (−∞, t)) such that ux is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t−α, b(t)=t−β, and g(ξ)∼sign(ξ)∣ξ∣γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
10.
The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c1q|u|−k?f(u)u?c2q|u|+k and f′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate. 相似文献
11.
We study generalized solutions of the nonlinear wave equation
utt−uss=au+−bu−+p(s,t,u), 相似文献
12.
In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: ut=Δu+up+uq(x*,t) in B(R) where p,q>0,B(R)={x∈RN:|x|<R} and x*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.
- (i)
- If or p=q>2, then single point blow-up occurs whenever solutions blow up.
- (ii)
- If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.
- (iii)
- If p?1<q, total blow-up occurs whenever solutions blow up.
- (iv)
- If max(p,q)?1, every solution exists globally in time.
13.
Elói Medina Galego 《Journal of Mathematical Analysis and Applications》2008,338(1):653-661
We first introduce the notion of (p,q,r)-complemented subspaces in Banach spaces, where p,q,r∈N. Then, given a couple of triples {(p,q,r),(s,t,u)} in N and putting Λ=(q+r−p)(t+u−s)−ru, we prove partially the following conjecture: For every pair of Banach spaces X and Y such that X is (p,q,r)-complemented in Y and Y is (s,t,u)-complemented in X, we have that X is isomorphic Y if and only if one of the following conditions holds:
- (a)
- Λ≠0, Λ divides p−q and s−t, p=1 or q=1 or s=1 or t=1.
- (b)
- p=q=s=t=1 and gcd(r,u)=1.
14.
Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. For integers k≥0, t≥2, denote by K((t−1)×p,p+k) the complete t-partite graph that has t−1 partite sets of size p and one partite set of size p+k. Let K(s,t,p,k) be the set of graphs obtained from K((t−1)×p,p+k) by adding a set S of s edges to the partite set of size p+k such that 〈S〉 is bipartite. If s=1, denote the only graph in K(s,t,p,k) by K+((t−1)×p,p+k). In this paper, we shall prove that for k=0,1 and p+k≥s+2, each graph G∈K(s,t,p,k) is chromatically unique if and only if 〈S〉 is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph K+((t−1)×p,p+k) is chromatically unique for k=0,1 and p+k≥3. 相似文献
15.
M. Loayza 《Journal of Differential Equations》2006,229(2):509-528
We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u0 is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u0?γdΩ for some γ>0. We find a unique solution in the space . 相似文献
16.
We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C1(R×Rn,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b1|q|2?K(t,q)?b2|q|2, W is superlinear at the infinity and f is sufficiently small in L2(R,Rn). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations. 相似文献
17.
In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(2|u|)u=0 on [0,π]×R under the boundary conditions a1u(t,0)−b1ux(t,0)=0, a2u(t,π)+b2ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system. 相似文献
18.
In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α1y(b)+β1y′(b)=γ1, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C1(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L1(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined. 相似文献
19.
James S.W. Wong 《Journal of Mathematical Analysis and Applications》2004,291(1):180-188
We prove oscillation and nonoscillation theorems for the second order linear differential equation (E) y″+q(t)y=0, where q(t)?0 and locally integrable on These results are extensions of earlier results of Huang [J. Math. Anal. Appl. 210 (1997) 712-723]. Furthermore, we show that the oscillation criterion established for Eq. (E) can be extended to the delayed differential equation y″+q(t)y(σ(t))=0, where σ(t)?t and limt→∞σ(t)=∞. 相似文献
20.
We find the greatest value p and least value q in (0,1/2) such that the double inequality G(pa+(1−p)b,pb+(1−p)a)<I(a,b)<G(qa+(1−q)b,qb+(1−q)a) holds for all a,b>0 with a≠b. Here, G(a,b), and I(a,b) denote the geometric, and identric means of two positive numbers a and b, respectively. 相似文献