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.     

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.     

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     

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.     

(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.     

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.     

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.     

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.     

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 the Cauchy Problem for $$\mathop {2b}\limits^ \to$$ -Parabolic Systems with Growing Coefficients

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.     

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.     

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.     

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.     

Sufficiency-Type Stability and Stabilization Criteria for Linear Time-Invariant Systems with Constant Point Delays

A set of criteria of asymptotic stability for linear and time-invariant systems with constant point delays are derived. The criteria are concerned with -stability local in the delays and -stability independent of the delays, namely, stability with all the characteristic roots in Res–<0 for all delays in some defined real intervals including zero and stability with characteristic roots in Res<–<0 as 0⁺ for all possible values of the delays, respectively. The results are classified in several groups according to the technique dealt with. The used techniques include both Lyapunov's matrix inequalities and equalities and Gerschgorin's circle theorem. The Lyapunov's inequalities are guaranteed if a set of matrices, built from the matrices of undelayed and delayed dynamics, are stability matrices. Some extensions to robust stability of the above results are also discussed.     

Nilpotent commuting varieties of reductive Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p0, and =LieG. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let =() denote the nilpotent variety of , and ^nil():={(x,y)×|[x,y]=0}, the nilpotent commuting variety of . Our main goal in this paper is to show that the variety ^nil() is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme _n Hilb ⁿ (²) is irreducible over any algebraically closed field. Mathematics Subject Classification (2000) 20G05     

Structure of Equations Solvable by the Inverse Scattering Transform for the Schrödinger Operator

We propose a new approach for deriving nonlinear evolution equations solvable by the inverse scattering transform. The starting point of this approach is consideration of the evolution equations for the scattering data generated by solutions of an arbitrary nonlinear evolution equation that rapidly decrease as x±. Using this approach, we find all nonlinear evolution equations whose integration reduces to investigation of the scattering-data evolution equations that are differential equations (in either ordinary or partial derivatives). In this case, the evolution equations for the scattering data themselves are linear and moreover solvable in the finite form.     

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.     

Commutation Relations in an Indefinite-Metric Space

We describe the irreducible regular representations of the algebra of operators a and b defined by [a,b]=1 and ba=a ⁺ b ⁺ in an arbitrary nondegenerate closed indefinite-metric space. We find the relation of this algebra to the generalized Heisenberg algebra.     

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.