Statistical and renewal results for the random sequential adsorption model applied to a unidirectional multicracking problem

We work out a stationary process on the real line to represent the positions of the multiple cracks which are observed in some composites materials submitted to a fixed unidirectional stress $\varepsilon$. Our model is the one-dimensional random sequential adsorption. We calculate the intensity of the process and the distribution of the inter-crack distance in the Palm sense. Moreover, the successive crack positions of the one-sided process (denoted by $X_i^{\varepsilon }$, $i⩾1$) are described. We prove that the sequence $\{(X_i^{\varepsilon },Y_i^{\varepsilon }),1⩽i⩽n\}$ is a “conditional renewal process”, where $Y_i^{\varepsilon }$ is the value of the stress at which $X_i^{\varepsilon }$ forms. The approaches “in the Palm sense” and “one-sided process” merge when $n\to +\infty$. The saturation case $(\varepsilon =+\infty )$ is also investigated.     

The oscillation of the Poisson semigroup associated to parabolic Hermite operator

Let O(P_τ~L) be the oscillation of the Possion semigroup associated with the parabolic Hermite operator L = ?_t-?+|x|~2. We show that O(P_τ~L) is bounded from L~p(R~(n+1))into itself for 1 p ∞, bounded from L~1(R~(n+1)) into weak-L~1(R~(n+1)) and bounded from L_c~∞(R~(n+1)) into BMO(R~(n+1)). In the case p = ∞ we show that the range of the image of the operator O(P_τ~L) is strictly smaller than the range of a general singular operator.     

Duality of weighted multiparameter Triebel-Lizorkin spaces

In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces F_p~(α,q) (ω; R~(n_1)× R~(n_2)) based on the Frazier and Jawerth' method in [11]. This space was′firstly introduced in [18]. Then we establish its dual space and get that(F_p~(α,q))*= CMO_p~(-α,q') for 0 p ≤ 1.     

A note on exact convergence rate in the local limit theorem for a lattice branching random walk

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Z_n(z) the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Z_n(z) assuming a condition like "EN(log N)~(1+λ) ∞" for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.     

Global fluctuations for 1D log-gas dynamics

We study in this article the hydrodynamic limit in themacroscopic regime of the coupled system of stochastic differential equations,

(0.1)$d{\lambda }_t^i=\frac{1}{\sqrt{N}}d{W}_t^i?V^{\prime }\left({\lambda }_t^i\right)dt+\frac{\beta }{2N}\sum _{j\ne i}\frac{dt}{{\lambda }_t^i?{\lambda }_t^j},i=1,\dots ,N,$

with $\beta >1$, sometimes called generalized Dyson’s Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty$ is known to satisfy a mean-field Mc-Kean–Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE.The proof is very much indebted to the harmonic potential case treated in Israelsson (2001). Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process.     

Toeplitz quantization on Fock space

For Toeplitz operators $T_f^{(t)}$ acting on the weighted Fock space $H_t^2$, we consider the semi-commutator $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}$, where $t>0$ is a certain weight parameter that may be interpreted as Planck's constant ? in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit

($?$)$\underset{t\to 0}{\lim }?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t.$

It is well-known that ${6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t$ tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to $f,g\in \mathrm{BUC}({\mathbb{C}}^n)$ by Bauer and Coburn. We now further generalize (?) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\mathrm{VMO}\cap L^{\infty }$ of bounded functions having vanishing mean oscillation on ${\mathbb{C}}^n$. Our approach is based on the algebraic identity $T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}=?{(H_{\overline{f}}^{(t)})}^{?}{H}_g^{(t)}$, where $H_g^{(t)}$ denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (?) to vanish. For g we only have to impose ${\lim \sup }_{t\to 0}{6H_g^{(t)}6}_t<\infty$, e.g. $g\in L^{\infty }({\mathbb{C}}^n)$. We prove that the set of all symbols $f\in L^{\infty }({\mathbb{C}}^n)$ with the property that ${\lim }_{t\to 0}?{6T_f^{(t)}{T}_g^{(t)}?T_{fg}^{(t)}6}_t={\lim }_{t\to 0}?{6T_g^{(t)}{T}_f^{(t)}?T_{gf}^{(t)}6}_t=0$ for all $g\in L^{\infty }({\mathbb{C}}^n)$ coincides with $\mathrm{VMO}\cap L^{\infty }$. Additionally, we show that $\underset{t\to 0}{\lim }?{6T_f^{(t)}6}_t={6f6}_{\infty }$ holds for all $f\in L^{\infty }({\mathbb{C}}^n)$. Finally, we present new examples, including bounded smooth functions, where (?) does not vanish.     

A cohomological study of local rings of embedding codepth 3

The generating series of the Bass numbers ${\mu }_R^i={\mathrm{rank}}_k{\mathrm{Ext}}_R^i(k,R)$ of local rings $R$ with residue field $k$ are computed in closed rational form, in case the embedding dimension $e$ of $R$ and its depth $d$ satisfy $e?d\le 3$. For each such $R$ it is proved that there is a real number $\gamma >1$, such that ${\mu }_R^{d+i}\ge \gamma {\mu }_R^{d+i?1}$ holds for all $i\ge 0$, except for $i=2$ in two explicitly described cases, where ${\mu }_R^{d+2}={\mu }_R^{d+1}=2$. New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings.     

On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder $?M:={\beta }_{\mathrm{s}}^{?}M?M$ of the semialgebraic Stone–Cěch compactification ${\beta }_{\mathrm{s}}^{?}M$ of a semialgebraic set $M?{\mathbb{R}}^m$ in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of ${\beta }_{\mathrm{s}}^{?}M$ that admit a metrizable neighborhood in ${\beta }_{\mathrm{s}}^{?}M$ equals $M_{\mathrm{lc}}\cup ({\mathrm{Cl}}_{{\beta }_{\mathrm{s}}^{?}M}({\stackrel{￣}{M}}_{\le 1})?{\stackrel{￣}{M}}_{\le 1})$ where $M_{\mathrm{lc}}$ is the largest locally compact dense subset of M and ${\stackrel{￣}{M}}_{\le 1}$ is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets $\stackrel{?}{?}M$ and $\stackrel{?}{?}M$ of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ?M and that the differences $?M?\stackrel{?}{?}M$ and $\stackrel{?}{?}M?\stackrel{?}{?}M$ are also dense subsets of ?M. It holds moreover that all the points of $\stackrel{?}{?}M$ have countable systems of neighborhoods in ${\beta }_{\mathrm{s}}^{?}M$.     

Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0,π]

Our aim in this paper is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the space ${\mathbb{L}}_2^{(\alpha ,\beta )}$.