General Solution for Hamiltonians with Extended Cubic and Quartic Potentials

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.     

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.     

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     

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.     

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.     

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.     

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.     

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.     

Torsion in K‐Theory for Boundary Actions on Affine Buildings of Type ?n

Let be a torsionfree lattice in G=PGL(n+1, , where n 1 and is a nonArchimedean local field. Then acts on the Furstenberg boundary G/P, where P is a minimal parabolic subgroup of G. The identity element I in the crossedproduct C ^*algebra C(G/P) generates a class [I] in the K ₀ group of C(G/P) . It is shown that [I] is a torsion element of K ₀ and there is an explicit bound for the order of [I]. The result is proved more generally for groups acting on affine buildings of type à _n. For n=1, 2 the Euler–Poincaré characteristic () annihilates the class [I].     

Essential and Discrete Spectra of the Three-Particle Schrödinger Operator on a Lattice

We consider the system of three quantum particles (two are bosons and the third is arbitrary) interacting by attractive pair contact potentials on a three-dimensional lattice. The essential spectrum is described. The existence of the Efimov effect is proved in the case where either two or three two-particle subsystems of the three-particle system have virtual levels at the left edge of the three-particle essential spectrum for zero total quasimomentum (K=0). We also show that for small values of the total quasimomentum (K0), the number of bound states is finite.     

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.     

Integrals of Borcherds Forms

In his Inventiones papers in 1995 and 1998, Borcherds constructed holomorphic automorphic forms (f) with product expansions on bounded domains D associated to rational quadratic spaces V of signature (n2), starting from vector valued modular forms f of weight 1–n2 for SL₂(Z) which are allowed to have poles at the cusp and whose nonpositive Fourier coefficients are integers c (–m), m0. In this paper, we use the Siegel–Weil formula to give an explicit formula for the integral ((f)) of –log||(f)||² over X=\D, where || ||² is the Petersson norm. This integral is given by a sum for m0 of quantities c (–m)(m), where (m) is the limit as Im() of the mth Fourier coefficient of the second term in the Laurent expansion at s=n2 of a certain Eisenstein series E(s) of weight (n2)+1 attached to V. The possible role played by the quantity ((f)) in the Arakelov theory of the divisors Z (m) on X is explained in the last section.     

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.     

(q, δ)-Numeration Systems with Missing Digits

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.a₁a₂a_i{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 w_iD{,,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.     

Zeros of the partial sums of cos (z) and sin (z). I

We extend results of Szeg (1924) and Kappert (1996) on the location of the zeros of the normalized partial sums of cos(z) and sin(z), and their rates of convergence to the associated Szeg curves.     

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     

Nonideal Bose Gases: Correlation Inequalities and Bose Condensation

We consider two simple model systems describing effective repulsion in a nonideal Bose gas. The interaction Hamiltonians in these systems can be analytically represented as functions of the occupation number operators for modes with nonzero momenta (p0). One of these models contains an interaction term corresponding to repulsion of bosons with the mode p=0 and ensuring the thermodynamic superstability of the system; the other model does not contain such a term. We use the Bogoliubov–Dirac–Ginibre approximation and the method of correlation inequalities to prove that a Bose condensate can exist in these model systems. Because of the character of interaction, the condensate can be formed in the superstable case for any values of the spatial dimensions, temperature, and positive chemical potentials.     

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.     

Low-Frequency Spin Oscillation Branch in a System with Exchange Interaction

We find the spin oscillation branch with the frequency =(H/)(k/k _c)² in the paramagnetic phase for systems with exchange interaction in the presence of an external magnetic field .