共查询到20条相似文献,搜索用时 15 毫秒
1.
This paper is concerned with non-trivial solvability in p-adicintegers of systems of two and three additive forms. Assumingthat the congruence equation axk + byk + czk d (modp) has asolution with xyz 0(modp) we have proved that any system oftwo additive forms of odd degree k with at least 6k + 1 variables,and any system of three additive forms of odd degree k withat least 14k + 1 variables always has non-trivial p-adic solutions,provided p does not divide k. The assumption of the solubilityof the congruence equation above is guaranteed for example ifp > k4. In the particular case of degree k = 5 we have proved the followingresults. Any system of two additive forms with at least n variablesalways has non-trivial p-adic solutions provided n 31 and p> 101 or n 36 and p > 11. Furthermore any system of threeadditive forms with at least n variables always has non-trivialp-adic solutions provided n 61 and p > 101 or n 71 andp > 11. 2000 Mathematics Subject Classification 11D72, 11D79. 相似文献
2.
The free boundary between v > 0 and v = 0 for the porousmedium equation v1= |v|2 +nv2v can remain stationary for somepositive waiting-time and then start moving. It is of interestto know the way in which any part of the boundary first moves.It is already known that if the waiting time is given by purelylocal considerations then the boundary speed is continuous,i.e. the initial speed is zero. For cases where the boundary moves before the time found bysimply considering v(x, 0) close to the boundary, a perturbationanalysis indicates that it starts moving with positive speed.Two-dimensional problems show the possible formation of verticesin the boundary. At these points the normal velocity jumps fromzero to some positive value. 相似文献
3.
Let M denote a connected complete Riemannian manifold (possiblywith a convex boundary), the Riemannian distance function froma fixed point and V C2 (M) such that dµV eV d xis a probability measure. For any K 0, we prove that K/2 isthe infimum over all > 0 such that RicM HessV Kand imply the log-Sobolevinequality for the Dirichlet form µV(| f |2). 相似文献
4.
This paper studies existence problems in L1 for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L1type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5. 相似文献
5.
ELLIOTT C. M.; HERRERO M. A.; KING J. R.; OCKENDON J. R. 《IMA Journal of Applied Mathematics》1986,37(2):147-154
In this paper we consider the limit m+ of solutions of the porous-mediumequation ut = · (umu) (xRN), with N > 1. We conjecturethat, for initial data with a unique maximum, the evolutionis characterized by the onset of a mesa region,in which the solution is nearly spatially independent, surroundedby a region in which u is nearly equal to its initial value.The transition between these regions occurs near a surface whichis identified with the free boundary in a certain Stefan problemwhich can be studied using variational inequalities. Moreover,singular-perturbation theory can be used to describe the structureof the transition region. 相似文献
6.
Let K be a field and let V be a vector space of dimension 2mover K. Let V denote the exterior algebra of V and kV its kthexterior power for 0k2m. Let f be a non-degenerate alternatingbilinear form defined on VxV. The symplectic group Sp2m(K) isthe group of all isometries of f and it acts as a group of vectorspace automorphisms on kV. In the case that K is algebraicallyclosed and 1km, it is known that kV contains a composition factorcorresponding to the fundamental weight k of a root system oftype Cm. We shall refer to the irreducible module for Sp2m(K)given by this composition factor as a fundamental module. 相似文献
7.
Let H=–+V(x) be a Schrödinger operator on L2(R4),H0=–. Assume that |V(x)|+| V(x)|C x – for some>8. Let be the wave operators. It is known that W± extend to bounded operators in Lp(R4)for all 1p, if 0 is neither an eigenvalue nor a resonance ofH. We show that if 0 is an eigenvalue, but not a resonance ofH, then the W± are still bounded in Lp(R4) for all psuch that 4/3<p<4. 相似文献
8.
Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces 总被引:1,自引:0,他引:1
We consider a fully practical finite-element approximationof the following system of nonlinear degenerate parabolic equations: (u)/(t) + . (u2 [(v)]) - (1)/(3) .(u3 w)= 0, w = - c u - u-+ a u-3 , (v)/(t) + . (u v [(v)]) - v - .(u2 v w) = 0. The above models a surfactant-driven thin-film flow in the presenceof both attractive, a>0, and repulsive, >0 with >3,van der Waals forces; where u is the height of the film, v isthe concentration of the insoluble surfactant monolayer and(v):=1-v is the typical surface tension. Here 0 and c>0 arethe inverses of the surface Peclet number and the modified capillarynumber. In addition to showing stability bounds for our approximation,we prove convergence, and hence existence of a solution to thisnonlinear degenerate parabolic system, (i) in one space dimensionwhen >0; and, moreover, (ii) in two space dimensions if inaddition 7. Furthermore, iterative schemes for solving the resultingnonlinear discrete system are discussed. Finally, some numericalexperiments are presented. 相似文献
9.
Let G be a group and let k > 2 be an integer, such that (k2 3)(k 1) < |G|/15 if G is finite. Supposethat the condition |A2| k(k + 1)/2 + (k 3)/2 is satisfiedby every it-element subset A G. Then G is abelian. The proofuses the structure of quasi-invariant sets. 相似文献
10.
In this paper we investigate finite element approximations ofnonlinear elliptic equations in three dimensions. By applyingand extending the results of Lopez-Marcos and Sanz-Serna, weprove that the finite element approximation on a mesh of sizeh, has a solution Uk which converges to an exact solution ofthe differential equation as h0. This solution is unique withina suitably defined stability ball Bh. For the particular nonlinearequation u + (u + up) we show that the size of Bh depends uponh only if p > 5 when it tends to zero as h 0. In this casewe prove the existence of spurious solutions Vh of the Galerkinapproximation which become unbounded in the maximum norm ash0. The stability ball Bh then acts to separate the convergentand the spurious solutions. We present the results of some numericalexperiments to substantiate our claims. 相似文献
11.
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)||yn1||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 2s1. 相似文献
12.
This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation ut = (um) up in Rn x (0,) with (1 2/n)+ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)k with n + mn(2 n + nm)/(2[2 m + mn(1 m)]) k < 2/(1 m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of ut = (um) with an additional logarithmicanomalous decay exponent in time as t . 相似文献
13.
The Markov-type inequality
is proved for all real algebraic polynomials f of degree atmost n having at most k, with 0 k n, zeros (counting multiplicities)in the open unit disk of the complex plane, and for all p >0, where c(p) = cp + 1(l + p–2) with some absolute constantc > 0. This inequality has been conjectured since 1983 whenthe L case of the above result was proved. It improves and generalizesmany earlier results. Up to the multiplicative constant c(p)>0 the above inequality is sharp. A sharp Bernstein-type analoguefor real trigonometric polynomials is also established, whichis interesting on its own, and plays a key role in the proofof the Markov-type inequality. 相似文献
14.
Let k be a non-negative integer. Suppose that f and g are nonconstantentire functions and that a and b (b a(k) are small functionsrelated to f and g such that (a,f) + (a, g) > 1. Iff(k) b and gk b assume the same zeros with the same multiplicities,then f g unless (f a(k))(g(k) a(k)) = (b a(K))2. The problem is related to C. C. Yang's question. A correspondingresult was proved for the case where a 0, b 1, k 1 and theorder of f and g is finite. 相似文献
15.
It is well known that some numerical methods for initial valueproblems admit spurious limit sets. Here the existence and behaviourof spurious solutions of Runge-Kutta, linear multistep and predictor-correctormethods are studied in the limit as the step-size h0. In particular,it is shown that for ordinary differential equations definedby globally Lipschitz vector fields, spurious fixed points andperiod 2 solutions cannot exist for h arbitrarily small, whilstfor locally Lipschitz vector fields, spurious solutions mayexist for h arbitrarily small, but must become unbounded ash0. The existence of spurious solutions is also studied forvector fields merely assumed to be continuous, and an exampleis given, showing that in this case spurious solutions may remainbounded as h0. It is shown that if spurious fixed points orperiod 2 solutions of continuous problems exist for h arbitrarilysmall, then as h0 spurious solutions either converge to steadysolutions of the underlying differential equation or divergeto infinity. A necessary condition for the bifurcation spurioussolutions from h=0 is derived. To prove the above results forimplicit Runge-Kutta methods, an additional assumption on theiteration scheme used to solve the nonlinear equations definingthe method is needed; an example of a Runge-Kutta method whichgenerates a bounded spurious solution for a smooth problem withh arbitrarily small is given, showing that such an assumptionis necessary. It is also shown that an Adams-Bashforth/Adams-Moultonpredictor-corrector method in PCm implementation can generatespurious fixed point solutions for any m. 相似文献
16.
Let A > 0 be an integer. The equation x5y5 = Az5 wasfirst studied by Dirichlet and Lebesgue. Lebesgue conjecturedin 1843 that if A has no prime divisors of the form 10k+1, theequation has no solutions except the visible ones. Partial resultswere obtained by Lebesgue and by Terjanian in 1987. The purposeof the paper is to prove Lebesgue's conjecture. The main toolused is the method known as the elliptic Chabauty method. 相似文献
17.
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 ThueSiegel 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
18.
Let f, g: (Rn, 0) (Rp, 0) be two C map-germs. Then f and gare C0-equivalent if there exist homeomorphism-germs h and lof (Rn, 0) and (Rp, 0) respectively such that g = l f h1.Let k be a positive integer. A germ f is k-C0-determined ifevery germ g with jk g(0) = jk f(0) is C0-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C0-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle Jk(N,P) with fiber Jk(n, p). Let z in Jk(n, p) and let f be suchthat z = jkf(0). Define
Whether (f) < k depends only on z, not on f. We can thereforedefine the set
Let Wk(N, P) be the subbundle of Jk(N, P) with fiber Wk(n, p).Mather has constructed a finite Whitney (b)-regular stratificationSk(n, p) of Jk(n, p) Wk(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if Sk(N,P) denotes the corresponding stratification of Jk(N, P) Wk(N, P) and f C(N, P) is a C map such that jkf is multitransverseto Sk(N, P), jkf(N) Wk(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rn, 0) (Rp, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U Rp such that jkf(U 0) Wk (Rn, Rp = and suchthat jkf is multitransverse to Sk (Rn, Rp) at any finite setof points S U 0. Moreover, the inequality controlsthe rate jkf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27. 相似文献
19.
Let f: (Rn,0) (Rp,0) be a C map-germ. We define f to be finitely,or -, A-determined, if there exists an integer m such that allgerms g with jmg(0) = jmf(0), or if all germs g with the sameinfinite Taylor series as f, respectively, are A-equivalentto f. For any integer k, 0 k < , we can consider A' sCkcounterpart (consisting of Ck diffeomorphisms) A(k), and wecan define the notion of finite, or -,A(k)-determinacy in asimilar manner. Consider the following conditions for a C germf: (ak) f is -A(k)-determined, (bk) f is finitely A(k)-determined,(t) , (g) there exists a representative f : U Rp defined on some neighbourhood U of 0 in Rn such thatthe multigerm of f is stable at every finite set , and (g') every f' with j f'(0)=j f(0) satisfiescondition (g). We also define a technical condition which willimply condition (g) above. This condition is a collection ofp+1 Lojasiewicz inequalities which express that the multigermof f is stable at any finite set of points outside 0 and onlybecomes unstable at a finite rate when we approach 0. We willdenote this condition by (e). With this notation we prove thefollowing. For any C map germ f:(Rn,0) (Rp,0) the conditions(e), (t), (g') and (a) are equivalent conditions. Moreover,each of these conditions is equivalent to any of (ak) (p+1 k < , (bk) (p+1 k < ). 1991 Mathematics Subject Classification:58C27. 相似文献
20.
Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case 总被引:2,自引:0,他引:2
** Email: Paul.Houston{at}mcs.le.ac.uk*** Email: Janice.Robson{at}comlab.ox.ac.uk**** Email: Endre.Suli{at}comlab.ox.ac.uk We develop a one-parameter family of hp-version discontinuousGalerkin finite element methods, parameterised by [1,1], for the numerical solution of quasilinear elliptic equationsin divergence form on a bounded open set d, d 2. In particular,we consider the analysis of the family for the equation ·{µ(x, |u|)u} = f(x) subject to mixed DirichletNeumannboundary conditions on . It is assumed that µ is a real-valuedfunction, µ C( x [0, )), and thereexist positive constants mµ and Mµ such that mµ(t s) µ(x, t)t µ(x, s)s Mµ(t s) for t s 0 and all x . Using a result from the theory of monotone operators for any valueof [1, 1], the corresponding method is shown to havea unique solution uDG in the finite element space. If u C1() Hk(), k 2, then with discontinuous piecewise polynomials ofdegree p 1, the error between u and uDG, measured in the brokenH1()-norm, is (hs1/pk3/2), where 1 s min {p+ 1, k}. 相似文献