Tritronquée Solutions of Perturbed First Painlevé Equations

We consider solutions of the class of ODEs y=6y ²–x , 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.     

Extended Rotation and Scaling Groups for Nonlinear Evolution Equations

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.     

ε-Expansions in Dyson's Hierarchical Model

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.     

Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential

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.     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

Quantum Integrable and Nonintegrable Nonlinear Schrödinger Models for Realizable Bose–Einstein Condensation in d+1 Dimensions (d = 1, 2, 3)

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.     

Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

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.     

Universal Invariant Renormalization for the Supersymmetric Yang–Mills Theory

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

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003     

Singular Eigenvalue Problems for the Equation y (n) + λp(x)y = 0

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.     

Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: III. Moves 1 ↔ 5 and Related Structures

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.     

Slope Estimates of Artin–Schreier Curves

Let X/F_p be an Artin–Schreier curve defined by the affine equation y ^p –y=f(x) where f(x)F_p[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/F_p). We use our method to prove that if p>d2 then NP₁(X/F_p)(p–1)/d/(p–1). If p>2d4, we give a sufficient condition for the equality to hold.     

Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods

Let I: be a given bounded image function, where is an open and bounded domain which belongs to ⁿ. Let us consider n=2 for the purpose of illustration. Also, let S={x_i}_i be a finite set of given points. We would like to find a contour , such that is an object boundary interpolating the points from S. We combine the ideas of the geodesic active contour (cf. Caselles et al. [7,8]) and of interpolation of points (cf. Zhao et al. [40]) in a level set approach developed by Osher and Sethian [33]. We present modelling of the proposed method, both theoretical results (viscosity solution) and numerical results are given. AMS subject classification 49L25, 74G65, 68U10     

Quasianapole Interaction of Charged Fermions

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.     

Finite element approximation of a nonlinear cross-diffusion population model

Summary. We consider a fully discrete finite element approximation of the nonlinear cross-diffusion population model: Find u_i, the population of the i^th species, i=1 and 2, such that where ji and g_i(u₁,u₂):=(_i–_iiu_i–_iju_j)u_i. In the above, the given data is as follows: v is an environmental potential, c_i, a_i are diffusion coefficients, b_i are transport coefficients, _i are the intrinsic growth rates, and _ii are intra-specific, whereas _ij, ij, are interspecific competition coefficients. In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d3. Finally some numerical experiments in one space dimension are presented.Mathematics Subject Classification (2000): 65M60, 65M12, 35K55, 92D25Acknowledgements. Part of this work was carried out while the authors participated in the 2003 programme {\it Computational Challenges in Partial Differential Equations} at the Isaac Newton Institute, Cambridge, UK.     

Counting Singular Matrices with Primitive Row Vectors

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.     

Smoothness of Nonlinear Median-Interpolation Subdivision

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.     

Global continuity estimates for two dimensional graphs of prescribed Gauss curvature

We prove the global Hölder continuity of convex solutions uC³() of the equation of prescribed positive Gauss curvature in a bounded convex domain with C^1, 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|C^0,2/3(). 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     

On Differences of Semicubical Powers

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.     

On the finite sum representations of the Lauricella functions <Emphasis Type="Italic">F</Emphasis><Subscript>D</Subscript>

By using divided differences, we derive two different ways of representing the Lauricella function of n variables F_D⁽ⁿ⁾(a,b₁,b₂,...,b_n;c;x₁,x₂,...,x_n) as a finite sum, for b₁,b₂,...,b_n positive integers, and a,c both positive integers or both positive rational numbers with c–a a positive integer. AMS subject classification 33D45, 40B05, 40C99Jieqing Tan: Research supported by the National Natural Science Foundation of China under Grant No. 10171026 and Anhui Provincial Natural Science Foundation under Grant No. 03046102.Ping Zhou: Corresponding author. Research supported by NSERC of Canada.