Score vectors of tournaments

A tournament T on any set X is a dyadic relation such that for any x, y ∈ X (a) (x, x) ? T and (b) if x ≠ y then (x, y) ∈ T iff (y, x) ? T. The score vector of T is the cardinal valued function defined by R(x) = |{y ∈ X : (x, y) ∈ T}|. We present theorems for infinite tournaments analogous to Landau's necessary and sufficient conditions that a vector be the score vector for some finite tournament. Included also is a new proof of Landau's theorem based on a simple application of the “marriage” theorem.     

On additive principles of zero utility

In the first part a fifth order approximation for the premium of a small risk is given; this formula is used to show that additivity for all x implies linear or exponential utility. The same conclusion is obtained from the weaker assumption of additivity for only one value of x (second part). For the proof one considers a functional equation for the utility function (that has been derived earlier); it has a two-parametric family of solutions, some of which are quite exotic.     

Properties of the density for a three-dimensional stochastic wave equation

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let pt,x(y) be the density of the law of the solution u(t,x) of such an equation at points (t,x)∈]0,T]×R³. We prove that the mapping (t,x)?pt,x(y) owns the same regularity as the sample paths of the process {u(t,x),(t,x)∈]0,T]×R³} established in [R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press]. The proof relies on Malliavin calculus and more explicitly, the integration by parts formula of [S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./Springer-Verlag, Bombay, 1984] and estimates derived from it.     

Periodic and almost periodic solutions of volterra integral differential equations with infinite memory

Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝₀^∞ [dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation.     

Optimal Transportation with an Oscillation-Type Cost: The One-Dimensional Case

The main result of this paper is the existence of an optimal transport map T between two given measures μ and ν, for a cost which considers the maximal oscillation of T at scale δ, given by ω _δ (T) :?=??sup_|x???y|?<?δ |T(x)???T(y)|. The minimization of this criterion finds applications in the field of privacy-respectful data transmission. The existence proof unfortunately only works in dimension one and is based on some monotonicity considerations.     

A note on Abel polynomials and rooted labeled forests

A special case of the Abel polynomials counts rooted labeled forests. This interpretation is used to obtain a combinatorial proof of the formula expressing xⁿ as a sum of these polynomials.     

Characterizations of Lie derivations of triangular algebras

Let T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if δ:T→T is an R-linear map satisfying

δ([x,y])=[δ(x),y]+[x,δ(y)]     

Diffusions conditionnelles. I. Hypoellipticité partielle

This paper is devoted to the proof of certain results on conditional diffusions. By using the Malliavin calculus of variations, it is shown that, under minimal conditions on the vectors fields which define a two component diffusion (x_t, z_t) and their Lie brackets, a.s., for any T ? 0, the conditional law of x_t (t > 0), given $\text{B}$(z_s,s?T) a C^∞ density.     

The improper infinite derivatives of Takagi's nowhere-differentiable function

Let T be Takagi's continuous but nowhere-differentiable function. Using a representation in terms of Rademacher series due to N. Kono [Acta Math. Hungar. 49 (1987) 315-324], we give a complete characterization of those points where T has a left-sided, right-sided, or two-sided infinite derivative. This characterization is illustrated by several examples. A consequence of the main result is that the sets of points where T^′(x)=±∞ have Hausdorff dimension one. As a byproduct of the method of proof, some exact results concerning the modulus of continuity of T are also obtained.     

Uniformity and inexact version of a proximal method for metrically regular mappings

We study stability properties of a proximal point algorithm for solving the inclusion 0∈T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of our algorithm is uniform, in the sense that it is stable under small perturbations whenever the set-valued mapping T is metrically regular at a given solution. We present also an inexact proximal point method for strongly metrically subregular mappings and show that it is super-linearly convergent to a solution to the inclusion 0∈T(x).     

A Proof that extends hurwitz formula into the critical strip

Hurwitz formula for the generalized zeta function ζ(s,a) has been established under condition Re(s = σ < 0. Using the same contour integral, the proof proposed in this paper allows the extension of this formula into the critical strip, 0 < σ < 1. A similar result is obtained for the related function (1Γ(s)) ∫₀∞x^−s–1(e^−x)) dx, 0 < a ≤ 1.     

A combinatorial formula for rank 2 cluster variables

Let r be any positive integer, and let x ₁,x ₂ be indeterminates. We consider the sequence {x _n } defined by the recursive relation $$x_{n+1} =\bigl(x_n^r +1\bigr)/{x_{n-1}}$$ for any integer n. Finding a combinatorial expression for x _n as a rational function of x ₁ and x ₂ has been an open problem since 2001. We give a direct elementary formula for x _n in terms of subpaths of a specific lattice path in the plane. The formula is manifestly positive, providing a new proof of a result by Nakajima and Qin.     

Remarks on pseudo-contractive mappings

Let X be a Banach space, B a closed ball centered at the origin in X, and T: B → X a pseudo-contractive mapping (i.e., (λ ? 1) ∥x ? y∥ ? ∥(λI ? T)(x) ? (λI ? T) (y)∥ for all x, y?B and λ > 1). It is shown here that the antipodal boundary condition: T(x) = ?T(?x) for all x?δB assures existence of a fixed point of T in B provided that the ball B has the fixed point property with respect to non-expansive self-mappings. Also included are some fixed point theorems which involve the Leray-Schauder condition.     

Some eigenvalue results for maximal monotone operators

We study the eigenvalue problem of the form

0∈Tx−λCx,     

Regularity and integrability of rotating shallow-water equations

We consider classical shallow-water equations for a rapidly rotating fluid layer. The Poincaré/Kelvin linear propagator describes fast oscillating waves for the linearized system. We show that solutions of the full nonlinear shallow-water equations can be decomposed as U(t,x₁,x₂) + Ũ(t,x₁,x₂) + W’(t,x₁,x₂) + r, where Ũ is a solution of the quasigeostrophic (QG) equation. Here r is a remainder, which is uniformly estimated from above by a majorant of order 1/f₀. The vector field W’(t,x₁,x₂) describes the rapidly oscillating ageostrophic (AG) component. This component is exactly solved in terms of Poincaré/Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations. The mathematically rigorous control of the error r, based on estimates of small divisors, is used to prove the existence, on a long time interval T^*, of regular solutions to classical shallow-water equations with general initial data (T^* → +∞, as 1/f₀ → 0).     

High level subcritical branching processes in a random environment

A subcritical branching process in a random environment is considered under the assumption that the moment-generating function of a step of the associated random walk Θ(t), t ≥ 0, is equal to 1 for some value of the argument ? > 0. Let T _x be the time when the process first attains the half-axis (x,+∞) and T be the lifetime of this process. It is shown that the random variable T _x /lnx, considered under the condition T _x < +∞, converges in distribution to a degenerate random variable equal to 1/Θ′(?), and the random variable T/ ln x, considered under the same condition, converges in distribution to a degenerate random variable equal to 1/Θ′(?) ? 1/Θ′(0).     

Existence and uniqueness of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(x^′(t))+g(t,x(t))=e(t).     

The solution to the BCS gap equation and the second-order phase transition in superconductivity

The existence and the uniqueness of the solution to the BCS gap equation of superconductivity are established in previous papers, but the temperature dependence of the solution is not discussed. In this paper, in order to show how the solution varies with the temperature, we first give another proof of the existence and the uniqueness of the solution and point out that the unique solution belongs to a certain set. Here this set depends on the temperature T. We define another certain subset of a Banach space consisting of continuous functions of both T and x. Here, x stands for the kinetic energy of an electron minus the chemical potential. Let the solution be approximated by an element of the subset of the Banach space above. Second, we show, under this approximation, that the transition to a superconducting state is a second-order phase transition.     

Positive solutions of singular problems with sign changing Carathéodory nonlinearities depending on x′

We consider the singular boundary value problem for the differential equation x″+f(t,x,x′)=0 with the boundary conditions x(0)=0, w(x(T),x′(T))+?(x)=0. Here f is a Carathéodory function on which may by singular at the value x=0 of the phase variable x and f may change sign, w is a continuous function, and ? is a continuous nondecreasing functional on C⁰([0,T]). The existence of positive solutions on (0,T] in the classes AC¹([0,T]) and C⁰([0,T])∩AC¹_loc((0,T]) is considered. Existence results are proved by combining the method of lower and upper functions with Leray-Schauder degree theory.     

Maps on matrices that preserve the spectrum

Let T be a continuous map of the space of complex n×n matrices into itself satisfying T(0)=0 such that the spectrum of T(x)-T(y) is always a subset of the spectrum of x-y. There exists then an invertible n×n matrix u such that either T(a)=uau^-1 for all a or T(a)=ua^tu^-1 for all a. We arrive at the same conclusion by supposing that the spectrum of x-y is always a subset of the spectrum of T(x)-Tt(y), without the continuity assumption on T.