A New Approach to the Creation and Propagation of Exponential Moments in the Boltzmann Equation

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v _*|^β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v _*|^?1(v ? v _*)·σ and σ ∈ 𝕊 ^d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 ^d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|^β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.     

Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ_* > 0 such that for λ < λ_*, the problem has no positive solution; for λ = λ_*, it has at least one positive solution; and for λ > λ_*, it has at least two positive solutions.     

On a monotone empirical Bayes test in a positive exponential family

This paper studies a monotone empirical Bayes test δ _n ^* for testing H ₀:θ≤θ ₀ against H ₁:θ>θ ₀ for the positive exponential family f(x|θ)=c(θ)u(x)exp?(?x/θ), x>0, using a weighted quadratic error loss. We investigate the convergence rate of the empirical Bayes test δ _n ^* . It is shown that the regret of δ _n ^* converges to zero at a rate O(n ^?1), where n is the number of past data available when the present testing problem is considered. Errors regarding the rate of convergence claimed in Gupta and Li (J. Stat. Plan. Inference, 129: 3–18, 2005) are addressed.     

On a problem of P. Hall for Engel words

Let θ be a word in n variables and let G be a group; the marginal and verbal subgroups of G determined by θ are denoted by θ(G) and θ ^*(G), respectively. The following problems are generally attributed to P. Hall:

1. (I)
   If π is a set of primes and |G : θ ^*(G)| is a finite π-group, is θ(G) also a finite π-group?
    
2. (II)
   If θ(G) is finite and G satisfies maximal condition on its subgroups, is |G : θ ^*(G)| finite?
    
3. (III)
   If the set \({\{\theta(g_1,\ldots,g_n) \;|\; g_1,\ldots,g_n\in G\}}\) is finite, does it follow that θ(G) is finite?
    

We investigate the case in which θ is the n-Engel word e _n  = [x, _n y] for \({n\in\{2,3,4\}}\) .

     

Empirical bayes testing for mean life time of Rayleigh distribution

Consider a Rayleigh distribution withpdfp(x|θ) = 2xθ ^{- 1} exp(- x ²/θ) and mean lifetime μ = √πθ/2. We study the two-action problem of testing the hypothesesH ₀: μ≤ μ₀ againstH ₁: μ > μ₀ using a linear error loss of |μ- μ ₀ | via the empirical Bayes approach. We construct a monotone empirical Bayes test δ _n ^* and study its associated asymptotic optimality. It is shown that the regret of δ _n ^* converges to zero at a rate $\frac{{\ln ^2 n}}{n}$ , wheren is the number of past data available when the present testing problem is considered.     

Asymptotic Behaviour of Solutions of a Multidimensional Moving Boundary Problem Modeling Tumor Growth

We study a moving boundary problem modeling the growth of multicellular spheroids or in vitro tumors. This model consists of two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure in the tumor's body, respectively. The driving mechanism of the evolution of the tumor surface is governed by Darcy's law. Finally surface tension effects on the moving boundary are taken into account which are considered to counterbalance the internal pressure. To put our analysis on a solid basis, we first state a local well-posedness result for general initial data. However, the main purpose of our study is the investigation of the asymptotic behaviour of solutions as time goes to infinity. As a result of a centre manifold analysis, we prove that if the initial domain is sufficiently close to a Euclidean ball in the C ^m+μ-norm with m ≥ 3 and μ ∈ (0,1), then the solution exists globally and the corresponding domains converge exponentially fast to some (possibly shifted) ball, provided the surface tension coefficient γ is larger than a positive threshold value γ_*. In the case 0 < γ < γ_* the radially symmetric equilibrium is unstable.     

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

On effective determination of symmetric-square lifts

Let F be the symmetric-square lift with Laplace eigenvalue λ _F (Δ) = 1+4µ². Suppose that |µ| ≤ Λ. We show that F is uniquely determined by the central values of Rankin-Selberg L-functions L(s, F ? h), where h runs over the set of holomorphic Hecke eigen cusp forms of weight κ ≡ 0 (mod 4) with κ?^?+?, t9 = max {4(1+4θ)/(1?18θ), 8(2?9θ)/3(1?18θ)} for any 0 ≤ θ < 1/18 and any ∈ > 0. Here θ is the exponent towards the Ramanujan conjecture for GL₂ Maass forms.     

Small Stochastic Perturbations of Random Maps with Position Dependent Probabilities

Abstract

A position dependent random map is a dynamical system consisting of a collection of maps such that, at each iteration, a selection of a map is made randomly by means of probabilities which are functions of position. Let f* be an invariant density of the position dependent random map T. We consider a model of small random perturbations 𝔗_? of the random map T. For each ? > 0, 𝔗_? has an invariant density function f ^?. We prove that f ^? → f* as ? → 0.     

Metric discrepancy results for alternating geometric progressions

The law of the iterated logarithm for discrepancies of {θ ^k x} is proved for θ < ?1. When θ is not a power root of rational number, the limsup equals to 1/2. When θ is an odd degree power root of rational number, the limsup constants for ordinary discrepancy and star discrepancy are not identical.     

On the influence of the prior distribution in image reconstruction

Two measures of the influence of the prior distribution p(θ) in Bayes estimation are proposed. Both involve comparing with alternative prior distributions proportional to p(θ) ^s , for s  ≥  0. The first one, the influence curve for the prior distribution, is simply the curve of parameter values which are obtained as estimates when the estimation is made using p(θ) ^s instead of p(θ). It measures the overall influence of the prior. The second one, it the influence rate for the prior, is the derivative of this curve at s = 1, and quantifies the sensitivity to small changes or inaccuracies in the prior distribution. We give a simple formula for the influence rate in marginal posterior mean estimation, and discuss how the influence measures may be computed and used in image processing with Markov random field priors. The results are applied to an image reconstruction problem from visual field testing and to a stylized image analysis problem.     

Characterizations of *-Cancellation Ideals of an Integral Domain

Let D be an integral domain and * a star-operation on D. For a nonzero ideal I of D, let I ^{* _f} = ?{J* | (0) ≠ J ? I is finitely generated} and I ^{* _w} = ? _{P∈* f -Max(D)} ID _P . A nonzero ideal I of D is called a *-cancellation ideal if (IA)* = (IB)* for nonzero ideals A and B of D implies A* =B*. Let X be an indeterminate over D and N _* = {f ∈ D[X] | (c(f))* =D}. We show that I is a * _w -cancellation ideal if and only if I is * _f -locally principal, if and only if ID[X] _{N *} is a cancellation ideal. As a corollary, we have that each nonzero ideal of D is a * _w -cancellation ideal if and only if D _P is a principal ideal domain for all P ∈ * _f -Max(D), if and only if D[X] _{N *} is an almost Dedekind domain. We also show that if I is a * _w -cancellation ideal of D, then I ^{* _w}  = I ^{* _f}  = I _t , and I is * _w -invertible if and only if I ^{* _w}  = J _v for a nonzero finitely generated ideal J of D.     

Locally adaptive fitting of semiparametric models to nonstationary time series

We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function μ(·) and a p-dimensional function θ(·)=(θ⁽¹⁾(·),…,θ^(p)(·))′ that parametrizes the time-varying spectral density f_θ(·)(λ). Whereas the mean function is estimated by a usual kernel estimator, each component of θ(·) is estimated by a nonlinear wavelet method. According to a truncated wavelet series expansion of θ⁽ⁱ⁾(·), we define empirical versions of the corresponding wavelet coefficients by minimizing an empirical version of the Kullback–Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of θ⁽ⁱ⁾(·). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor.     

Sur la meilleure constante dans l'inégalité de Sobolev

We prove that the best constant in the Sobolev inequality (H₁^p ⊂L_p^* with 1/p^* = 1/p − 1/n and 1 < p < n) is achieved on compact Riemannian manifolds, or only complete under some hypotheses. We also establish stronger inequalities where the norms are to some exponent which seems optimal. For the proof we show a general result of dominated convergence at a simple point of concentration.     

Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA * = B

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equations AX = B and AXA ^* = B. We first establish a variety of rank and inertia formulas for calculating the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian definite solutions of the matrix equations AX = B and AXA ^* = B, and then use them to characterize many qualities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations and their variations.     

Θ-type Calderón-Zygmund operators with non-doubling measures

Let µ be a Radon measure on ? ^d which may be non-doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr ⁿ for all x∈? ^d , r > 0 and for some fixed 0 < n ≤ d. In this paper, under this assumption, we prove that θ-type Calderón-Zygmund operator which is bounded on L ²(µ) is also bounded from L ^∞(µ) into RBMO(µ) and from H _atb ^1,∞ (µ) into L ¹(µ). According to the interpolation theorem introduced by Tolsa, the L ^p (µ)-boundedness (1 < p < ∞) is established for θ-type Calderón-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type Calderón-Zygmund operator with RBMO(µ) function are bounded on L ^p (µ) (1 < p < ∞).     

Complex interpolation between two weighted Bergman spaces on tubes over symmetric cones

We prove that the complex interpolation space [A_ν^p₀,A_ν^p₁]_θ, 0<θ<1, between two weighted Bergman spaces A_ν^p₀ and A_ν^p₁ on the tube in $\text{C}{}^{\mathrm{n}}$, n?3, over an irreducible symmetric cone of $\text{R}{}^{\mathrm{n}}$ is the weighted Bergman space A_ν^p with 1/p=(1?θ)/p₀+θ/p₁. Here, ν>n/r?1 and 1?p₀<p₁<2+ν/(n/r?1) where r denotes the rank of the cone. We then construct an analytic family of operators and an atomic decomposition of functions, which are related to this interpolation result. To cite this article: D. Békollé et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let (M, θ) be a pseudo-Hermitian space of real dimension 2n + 1, that is M is a CR-manifold of dimension 2n + 1 and θ is a contact form on M giving the Levi distribution \({HT(M) \subset TM}\). Let \({M^\theta \subset T^* M}\) be the canonical symplectization of (M, θ) and let M be identified with the zero section of M ^θ . Then M ^θ is a manifold of real dimension 2(n + 1) which admits a canonical foliation by surfaces parametrized by \({\mathbb{C} \ni t+i\sigma\mapsto \phi^{\theta}_{p}(t+i\sigma)=\sigma\theta_{g_t(p)}}\), where \({p \in M}\) is arbitrary and g _t is the flow generated by the Reeb vector field associated to the contact form θ. Let J be an (integrable) complex structure defined in a neighbourhood U of M in M ^θ . We say that the pair (U, J) is an adapted complex tube on M ^θ if all the parametrizations \({\phi^{\theta}_{p}(t+i\sigma)}\) defined above are holomorphic on \({(\phi^{\theta}_{p})^{-1}(U)}\). In this paper we prove that if (U, J) is an adapted complex tube on M ^θ , then the real function E on \({M^\theta\subset T^*M}\) defined by the condition \({\alpha=E (\alpha)\theta_{\pi(\alpha)}}\), for each \({\alpha \in M^\theta}\), is a canonical defining function for M which satisfies the homogeneous Monge–Ampère equation (dd ^c E)ⁿ⁺¹ = 0. We also prove that if M and θ are real analytic then the symplectization M ^θ admits an unique maximal adapted complex tube.     

Well-behaved Beurling primes and integers

In this paper, we study generalised prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately li(x) and ρx, respectively (where ρ is a positive constant), with error terms of order O(x^θ₁) and O(x^θ₂) for some θ₁,θ₂<1. We show that it is impossible to have both θ₁ and θ₂ less than .