Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

Two strong limit theorems for processes with independent increments

Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), ?∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that $\text{(ξ(t) ? ξ(0))}\text{H(t)}\text{→ 0}$ for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is ξ. It addresses the question: Under what conditions does $\text{(2T)}{}^{\mathrm{?1}}\text{∫}{}_{\mathrm{?T}}{}^{\mathrm{T}}\text{X(t)}\text{X(t + τ)}\text{dt}$ converge as T → ∞ for all τ for almost all sample functions?     

Point processes of exits by bivariate Gaussian processes and extremal theory for the χ2-process and its concomitants

Let ζ(t), η(t) be continuously differentiable Gaussian processes with mean zero, unit variance, and common covariance function r(t), and such that ζ(t) and η(t) are independent for all t, and consider the movements of a particle with time-varying coordinates (ζ(t), η(t)). The time and location of the exists of the particle across a circle with radius u defines a point process in R³ with its points located on the cylinder {(t, u cos θ, u sin θ); t ≥ 0, 0 ≤ θ < 2π}. It is shown that if r(t) log t → 0 as t → ∞, the time and space-normalized point process of exits converges in distribution to a Poisson process on the unit cylinder. As a consequence one obtains the asymptotic distribution of the maximum of a χ²-process, χ²(t) = ζ²(t) + η²(t), P{sup_0≤t≤Tχ²(t) ≤ u²} → e^?τ if $\text{T(}\text{?r″(0)}\text{2π}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u ×}\text{exp}\text{(}\text{?u}{}^2\text{2}\text{) → τ}$ as T, u → ∞. Furthermore, it is shown that the points in R³ generated by the local ?-maxima of χ²(t) converges to a Poisson process in R³ with intensity measure (in cylindrical polar coordinates) (2πr²)^?1dtdθdr. As a consequence one obtains the asymptotic extremal distribution for any function g(ζ(t), η(t)) which is “almost quadratic” in the sense that $\text{g}{}^1\text{(r}\text{cos}\text{θ, r}\text{sin}\text{θ) =}\text{1}\text{2}\text{(r}{}^2\text{? g(r}\text{cos}\text{θ, r}\text{sin}\text{θ))}$ has a limit $\text{g}{}^1\text{(θ)}$ as r → ∞. Then if $\text{T(}\text{?r″(0)}\text{2π}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{u}\text{exp}\text{(}\text{?u}{}^2\text{2}\text{) → τ}$ as T, u → ∞.     

The zeros of the Artin l-series of a cubic field on the critical line

Denote by ψ(s) the Artin L-series of the non-abelian cubic field K. Let N₀(T) be the number of zeros of $\text{ψ(}\text{1}\text{2}\text{+ it)}$ for 0 < t ≤ T. Then for each ? > 0, there exists a positive constant T₀, such that for T > T₀:

$\text{N}{}_0\text{(T) > T}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(}\text{log}\text{2}\text{3}\text{T(}\text{loglog}\text{T)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext><mtext>+?</mtext>}}\text{)}{}^{\mathrm{?1}}$

     

On the Łojasiewicz numbers

Let f be a holomorphic function of two complex variables with an isolated critical point at $\text{0∈}\text{C}{}^2$. We give some necessary conditions for a rational number to be the smallest θ>0 in the ?ojasiewicz inequality |gradf(z)|?C|z|^θ for z near $\text{0∈}\text{C}{}^2$. To cite this article: E. Garc??a Barroso, A. P?oski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

A sharp inequality for Sobolev functionsUne inégalité dans un espace de Sobolev

Let N?5, a>0, $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$, $\text{2}{}^1\text{=}\text{2N}\text{N?2}$, $\text{2}{}^{\mathrm{\#}}\text{=}\text{2(N?1)}\text{N?2}$ and 6u6²=|?u|₂²+a|u|₂². We prove there exists an α₀>0 such that, for all $\text{u∈}\text{H}{}^1\text{(Ω)?{0}}$,

This inequality implies Cherrier's inequality. To cite this article: P.M. Girão, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 105–108     

How small are the increments of a Wiener process?

Let $\text{γ}{}_{\mathrm{т}}\text{=(8(}\text{logTa}{}^{-1}{}_{\mathrm{T}}\text{+log log T)}\text{π}{}^2\text{a}{}_{\mathrm{T}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, 0<a_T?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of

, under varying conditions on a_T and T/a_T. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given:

$\text{lim}{}_{\mathrm{h\to 0}}\text{inf}{}_{\mathrm{0?t?1}}\text{sup}{}_{\mathrm{0?s?h}}\text{(}\text{8 log h}{}^{-1}\text{π}{}^2\text{h}\text{)}\text{1}\text{2}\text{|W(t+s)?W(t)| =}{}^{\mathrm{a.s.}}\text{1.}$

and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process.     

Approximation to complex ξ from within a wedge

It is shown that for any complex ξ ? $\text{Q}$[i] and any angles θ₁ < θ₂ ≤ θ₁ + π there exists a constant C such that $\text{|ξ ?}\text{P}\text{Q}\text{| <}\text{C}\text{|Q|}{}^2$ and $\text{θ}{}_1\text{<}\text{arg}\text{(ξ ?}\text{P}\text{Q}\text{) < θ}{}_2$.     

Attractivity of the origin for the equation ẍ + f(t,x, ẋ) ¦ ẋ ¦αẋ + g(x) = 0

Sufficient conditions for asymptotic stability and global attractivity of the origin are obtained in the case of “unbounded damping”: $\text{f(t, x,}\text{x}\text{?}\text{) > a > 0}$. Restrictions on the unboundedness in t of $\text{f(t, x,}\text{x}\text{?}\text{)}$ are specified for either attractivity or nonattractivity of the origin. Roughly speaking, if there exists a nondecreasing function h(t) such that $\text{f(t, x,}\text{x}\text{?}\text{) < h(t)}$ and $\text{∝}{}^{\mathrm{\infty }}\text{dt}\text{h(t)}\text{=∞}$, the origin is an attractor; if $\text{f(t, x,}\text{x}\text{?}\text{) > h(t)}$ and $\text{∝}{}^{\mathrm{\infty }}\text{dt}\text{h(t)}\text{< ∞}$, it is not. These results are compared with several ones previously obtained by other authors. An attractivity criterion is given for the linear equation and a sufficient attractivity condition is obtained for dampings not bounded away from zero: $\text{0 < f(t, x,}\text{x}\text{?}\text{) < a}$.     

Numerical solution of a nonlinear integro-differential equation

Galerkin's method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t ? τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x < 1, 0 < t < T.An error estimate in a suitable norm will be derived for the difference u ? u^h between the exact solution u and the approximant u^h. It turns out that the rate of convergence of u^h to u as h → 0 is optimal. This result was confirmed by the numerical experiments.     

A third-order optimum property of the maximum likelihood estimator

Let θ⁽ⁿ⁾ denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T⁽ⁿ⁾ there exists q such that the risk of θ⁽ⁿ⁾ + n^?1q(θ⁽ⁿ⁾) exceeds the risk of T⁽ⁿ⁾ by an amount of order o(n^?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If $\text{q}{}^1$ is chosen such that $\text{θ}{}^{\mathrm{(n)}}\text{+ n}{}^{\mathrm{?1}}\text{q}{}^1\text{(θ}{}^{\mathrm{(n)}}\text{)}$ is unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$, then this estimator minimizes the risk up to an amount of order o(n^?1) in the class of all estimators which are unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$.The results are obtained under the assumption that T⁽ⁿ⁾ admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.     

The visually distinct configurations of k sets

Two sets of sets, C₀ and C₁, are said to be visually equivalent if there is a 1-1 mapping m from C₀ onto C₁ such that for every S, T?C₀, S ∩ T=0 if and only if m(S)∩ m(T)=0 and S?T if and only if m(S)?m(T). We find estimates for V(k), the number of equivalence classes of this relation on sets of k sets, for finite and infinite k. Our main results are that for finite k, $\text{1}\text{2}\text{k}{}^2\text{-k log k <log V (k)<ak}{}^2\text{+βk+log k}$, where α and β are approximately 0.7255 and 2.5323 respectively, and there is a set N of cardinality $\text{1}\text{2}\text{(k}{}^2\text{+k)}$ such that there are V(k) visually distinct sets of k subsets of N.     

The average height of planted plane trees with M leaves

The number of planted plane trees with n nodes, m leaves, and height h is computed. Assuming that all n-node planted plane trees with m leaves are equally likely, it is shown that the average height $\text{h}\text{(n, m)}$ is asymptotically given for all ε > 0 and fixed $\text{? =}\text{m}\text{n}$, 0 < ? < 1, by

$\text{h}\text{(n,m)=(φn(ρ}{}^{-1}\text{? 1))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ 1.5 ? ρ}{}^{-1}\text{+ O(}\text{ln}\text{(n)}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2-\epsilon </mtext>}}\text{)}$

     

Lifting of BV functions with values in S1

We show that for every $\text{u∈}\text{BV}\text{(Ω;S}{}^1\text{)}$, there exists a bounded variation function $\text{?∈}\text{BV}\text{(Ω;}\text{R}\text{)}$ such that u=e^i? a.e. on $\text{Ω}$ and |?|_BV?2|u|_BV. The constant 2 is optimal in dimension n>1. To cite this article: J. Dávila, R. Ignat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On convergence rates and the expectation of sums of random variables

Let X_i be iidrv's and S_n=X₁+X₂+…+X_n. When EX²₁<+∞, by the law of the iterated logarithm $\text{(S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{)}\text{(n log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{→0 a.s.}$ for some constants α_n. Thus the r.v. $\text{Y=sup}{}_{\mathrm{n?1}}\text{[|S}{}_{\mathrm{n}}\text{?α}{}_{\mathrm{n}}\text{|?(δn log n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}{}^{\mathrm{+}}$ is a.s.finite when δ>0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply it to show, without the prior assumption EX²₁<+∞ that EY^h<+∞ if and only if $\text{E}\text{|X}{}_1\text{|}{}^{\mathrm{2+h}}\text{[log|X}{}_1\text{|]}{}^{-1}\text{<+∞}$ for 0<h<1 and δ> $\text{h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$, whereas $\text{E}\text{Y}{}^{\mathrm{h}}\text{=+∞}$ whenever h>0 and $\text{0<δ<h}\text{E}\text{(X}{}_1\text{?}\text{E}\text{X}{}_1\text{)}{}^2$.     

A combinatorial interpretation for two transformations of series that commute with compositional inversion

For a formal power series $\text{g(t) =}\text{1}\text{[1 + ∑}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{h}{}_{\mathrm{n}}\text{t}{}^{\mathrm{n}}\text{]}$ with nonnegative integer coefficients, the compositional inverse $\text{f}\text{(t) = t · f(t)}$ of $\text{g}\text{(t) = t · g(t)}$ is shown to be the generating function for the colored planted plane trees in which each vertex of degree i + 1 is colored one of h_i colors. Since the compositional inverse of the Euler transformation of f(t) is the star transformation [[g(t)]^?1 ? 1]^?1 of g(t), [2], it follows that the Euler transformation of f(t) is the generating function for the colored planted plane trees in which each internal vertex of degree i + 1 is colored one of h_i colors for i > 1, and h₁ ? 1 colors for i = 1.     

Branching brownian motion with absorption

We consider a branching diffusion {Z_t}_t?0 in which particles move during their life time according to a Brownian motion with drift -μ and variance coefficient σ², and in which each particle which enters the negative half line is instantaneously removed from the population. If particles die with probability c dt+o(dt) in [t,t+dt] and if the mean number of offspring per particle is m>1, then Z_t dies out w.p.l. if $\text{μ?μ}{}_0\text{≡{2σ}{}^2\text{c(m?1)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. If μ<μ₀, then itZ_t grows exponentially with positive probability. Our main concern here is with the critical case where μ=μ₀. Even though $\text{E}\text{{}\text{Z}{}_{\mathrm{T}}\text{}∽}\text{const.}\text{T}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}$ in this case, we find that $\text{P}\text{{}\text{Z}{}_{\mathrm{T}}\text{>0}}$ is only exp$\text{{–}\text{const.}\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+0(}\text{log}\text{T)}{}^2\text{}}$, and conditionally on {Z_T>0} there are with high probability much fewer particles alive at time T than $\text{E}\text{{Z}{}_{\mathrm{T}}\text{|Z}{}_{\mathrm{T}}\text{0}}$.     

Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–StokesNon-blowup at large times and stability for global solutions to the Navier–Stokes equations

Suppose there exists a global solution u to the incompressible Navier–Stokes equations, such that $\text{u∈}\text{C}{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{1/2}}\text{)}$. We prove that its $\text{H}\text{?}{}^{\mathrm{1/2}}$ norm goes to 0 at infinity. We next use this fact to control the $\text{L}{}^2{}_{\mathrm{t}}\text{(}\text{H}\text{?}{}^{\mathrm{3/2}}\text{)}$ norm of u, and finally we prove that such a solution is stable. To cite this article: I. Gallagher et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 289–292.