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1.
We prove the global Hölder continuity of convex solutions uC3() of the equation of prescribed positive Gauss curvature in a bounded convex domain with C1, for some (0,1]. We also obtain better regularity for the trace of u on . In the special case =1 we show that and u|C0,2/3(). We also investigate the global continuity of solutions in C1 domains and construct an example showing that global continuity need not hold in general convex domains.Supported by an Australian Research Council Senior Fellowship.Mathematics Subject Classification (2000): Primary 35J60; Secondary 53A05, 53C42  相似文献   

2.
The paper studies singular eigenvalue problems for the equation y (n) +p(x)y=0 with boundary conditions imposed on the derivatives y (i) at the points x=a and x=. We look for singular problems which are analogous to regular problems on a finite interval. It is characterized when each eigenfunction has a finite number of zeros and when the spectrum is discrete or continuous, respectively.  相似文献   

3.
We consider solutions of the class of ODEs y=6y 2x , which contains the first Painlevé equation (PI) for =1. It is well known that PI has a unique real solution (called a tritronquée solution) asymptotic to and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative 1.  相似文献   

4.
The properties of stationary solutions of the one-dimensional fractional Einstein--Smoluchowski equation with a potential of the form x 2m+2, m=1,2,..., and of the Riesz spatial fractional derivative of order , 12, are studied analytically and numerically. We show that for 1<2, the stationary distribution functions have power-law asymptotic approximations decreasing as x –(+2m+1) for large values of the argument. We also show that these distributions are bimodal.  相似文献   

5.
In terms of hyperelliptic functions, we integrate a two-particle Hamiltonian with quartic potential and additional linear and nonpolynomial terms in the Liouville integrable cases 1:6:1 and 1:6:8.  相似文献   

6.
Extended Rotation and Scaling Groups for Nonlinear Evolution Equations   总被引:1,自引:0,他引:1  
A (1+1)-dimensional nonlinear evolution equation is invariant under the rotation group if it is invariant under the infinitesimal generator V=x u u x . Then the solution satisfies the condition u x=–x/u. For equations that do not admit the rotation group, we provide an extension of the rotation group. The corresponding exact solution can be constructed via the invariant set R 0={u: u x=xF(u)} of a contact first-order differential structure, where F is a smooth function to be determined. The time evolution on R 0 is shown to be governed by a first-order dynamical system. We introduce an extension of the scaling groups characterized by an invariant set that depends on two constants and n1. When =0, it reduces to the invariant set S 0 introduced by Galaktionov. We also introduce a generalization of both the scaling and rotation groups, which is described by an invariant set E 0 with parameters a and b. When a=0 or b=0, it respectively reduces to R 0 or S 0. These approaches are used to obtain exact solutions and reductions of dynamical systems of nonlinear evolution equations.  相似文献   

7.
We solve an asymptotic problem in the geometry of numbers, where we count the number of singular n×n matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as T, the number is asymptotic to for n3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. M. Schmidt.  相似文献   

8.
We consider Dyson's hierarchical model on a d-dimensional hierarchical lattice and define a renormalization group (RG) transformation for complex values of d as a map in the space of sequences of coupling constants determining the model Hamiltonian. We show that d=4 is a bifurcation value of this transformation for the RG transformation parameter equal to 1+2/d, and we construct a non-Gaussian RG-invariant Hamiltonian in terms of the (4–d)-expansion. We establish that the (–3/2)- and (4–d)-expansion coefficients for a non-Gaussian fixed point in the dimension d=3 have the same asymptotic representation as the size of the elementary cell tends to infinity, thus confirming that both the expansions describe the same nontrivial fixed point in the dimension three.  相似文献   

9.
We show that real k-structures coincide for k=1,2 on all formal groups for which multiplication by 2 is an epimorphism. This enables us to give explicit polynomial generators for the Morava K(n)-homology of BSpin and BSO for n=1,2.  相似文献   

10.
We consider the Hamiltonian H (K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H0(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0.  相似文献   

11.
We obtain the analytic expression for the total cross section of the reaction e e +l l + (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e e ++ in the energy domain below the Z 0 peak.  相似文献   

12.
We evaluate finite-temperature equilibrium correlators for thermal time ordered Bose fields to good approximations by new methods of functional integration in d=1,2,3 dimensions and with the trap potentials V(r)0. As in the translationally invariant cases, asymptotic behaviors fall as to longer-range condensate values for and only for d=3 in agreement with experimental observations; but there are generally significant corrections also depending on due to the presence of the traps. For d=1, we regain the exact translationally invariant results as the trap frequencies 0. In analyzing the attractive cases, we investigate the time-dependent c-number Gross–Pitaevskii (GP) equation with the trap potential for a generalized nonlinearity –2c||2n and c<0. For n=1, the stationary form of the GP equation appears in the steepest-descent approximation of the functional integrals. We show that collapse in the sense of Zakharov can occur for c=0 and nd2 and a functional E NLS[]0 even when V(r)0. The singularities typically arise as -functions centered on the trap origin r=0.  相似文献   

13.
Let X/Fp be an Artin–Schreier curve defined by the affine equation y p y=f(x) where f(x)Fp[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP1(X/Fp). We use our method to prove that if p>d2 then NP1(X/Fp)(p–1)/d/(p–1). If p>2d4, we give a sufficient condition for the equality to hold.  相似文献   

14.
We propose a manifestly invariant renormalization scheme for N=1 non-Abelian supersymmetric gauge theories.  相似文献   

15.
We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. We first write formulas for moves 33 and 24 based on the results in our two previous works and then study moves 15 in detail. Based on this, we obtain the formula for a four-dimensional manifold invariant. As an example, we present a detailed calculation of our invariant for the sphere S 4; in particular, the complex does turn out to be acyclic.  相似文献   

16.
For -parabolic dissipative systems and systems with growing coefficients as |x| in the presence of degeneracies in the initial hyperplane, we investigate the fundamental matrix of solutions and the solvability of the Cauchy problem.  相似文献   

17.
We construct strong solutionsu, p/of the general nonhomogeneous Stokes equations -u + p=f inG, ·u=g inG, u= on in an exterior domainG n (n3) with boundary of class C2. Our approach uses a localization technique: With the help of suitable cut-off functions and the solution of the divergence equation ·=g inG, = 0 on , the exterior domain problem is reduced to the entire space problem and an interior problem.  相似文献   

18.
We consider the (q, ) numeration system, with basis q2 and the set of digits {, +1,,q+–1} where –(q–1)0. We study properties of numbers where some digits do not occur. This is analogous to the Cantor set {0.a1a2ai{0,2}}. We compute an asymptotic equivalent of the nth moment of the Cantor (q, D)-distribution which can be described as the numbers 0. w1w2 with wiD{,,q+–1}, and each such letter can occur with the same probability 1/CardD. Furthermore, we consider n random strings according to the distribution and the expected minimum of them. We find a recursion which we solve asymptotically.This author was supported by the CNRS/NRF-project no 10959. Part of this work was done during the first authors visit to the John Knopfmacher Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand, Johannesburg, South Africa.This author was supported by the CNRS/NRF-project no 10959.  相似文献   

19.
We give a refined analysis of the Hölder regularity for the limit functions arising from a nonlinear pyramid algorithm for robust removal of non-Gaussian noise proposed by Donoho and Yu [6,7,17]. The synthesis part of this algorithm can be interpreted as a nonlinear triadic subdivision scheme where new points are inserted based on local quadratic polynomial median interpolation and imputation. We introduce the analogon of the Donoho–Yu scheme for dyadic refinement, and show that its limit functions are in C for >log4(128/31)=1.0229.... In the triadic case, we improve the lower bound of >log2(135/121)=0.0997... previously obtained in [6] to >log3(135/53)=0.8510.... These lower bounds are relatively close to the anticipated upper bounds of log2(16/7)=1.1982... in the dyadic, respectivly 1 in the triadic cases, and have been obtained by deriving recursive inequalities for the norm of second rather than first order differences of the sequences arising in the subdivision process.  相似文献   

20.
For X,Y,>0, let and define I 8(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y 2/X 3=O(), =O(Y 3/X 3) and X=O (Y 2), one has I 8(X,Y,)=O(X 2 Y 2+X min (X {3/2} Y 3, X {11/2} Y {–1})+X min ({1/3} X 2 Y 3, X {14/3} Y {1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.  相似文献   

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