The survival probability of critical and subcritical branching processes in finite state space Markovian environment

Let ${\left(Z_n\right)}_{n\ge 0}$ be a branching process in a random environment defined by a Markov chain ${\left(X_n\right)}_{n\ge 0}$ with values in a finite state space $\mathbb{X}$. Let ${\mathbb{P}}_i$ be the probability law generated by the trajectories of ${\left(X_n\right)}_{n\ge 0}$ starting at $X_0=i\in \mathbb{X}.$ We study the asymptotic behaviour of the joint survival probability ${\mathbb{P}}_i\left(Z_n>0,X_n=j\right)$, $j\in \mathbb{X}$ as $n\to +\infty$ in the critical and strongly, intermediate and weakly subcritical cases.     

Stable-like fluctuations of Biggins’ martingales

Let ${\left(W_n\left(\theta \right)\right)}_{n\in {\mathbb{N}}_0}$ be Biggins’ martingale associated with a supercritical branching random walk, and let $W\left(\theta \right)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1\left(\theta \right)$ belongs to the domain of normal attraction of an $\alpha$-stable distribution for some $\alpha \in (1,2)$, then, as $n\to \infty$, there is weak convergence of the tail process ${(W\left(\theta \right)?W_{n?k}\left(\theta \right))}_{k\in {\mathbb{N}}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $\alpha$-stable marginals.     

On the centre of mass of a random walk

For a random walk $S_n$ on ${\mathbb{R}}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n=n^{?1}{\sum }_{i=1}^n{S}_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, $G_n$ is recurrent if $d=1$ and transient if $d\ge 2$. In the transient case we show that $G_n$ has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which $G_n$ is transient in $d=1$.     

Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises

We establish the exponential convergence with respect to the $L^1$-Wasserstein distance and the total variation for the semigroup corresponding to the stochastic differential equation $d{X}_t=d{Z}_t+b\left(X_t\right)dt,$where ${\left(Z_t\right)}_{t\ge 0}$ is a pure jump Lévy process whose Lévy measure $\nu$ fulfills $\underset{x\in {\mathbb{R}}^d,\left|x\right|\le {\kappa }_0}{inf}[\nu \wedge ({\delta }_{x}1\nu )]\left({\mathbb{R}}^d\right)>0$for some constant ${\kappa }_0>0$, and the drift term $b$ satisfies that for any $x,y\in {\mathbb{R}}^d$, $〈b\left(x\right)?b\left(y\right),x?y〉\le \left\{\begin{array}{cc}{\Phi }_1\left(\right|x?y\left|\right)|x?y|,\hfill & |x?y|<l_0;\hfill \\ ?K_2|x?y{|}^2,\hfill & |x?y|\ge l_0\hfill \end{array}\right.$with some positive constants $K_2,l_0$ and positive measurable function ${\Phi }_1$. The method is based on the refined basic coupling for Lévy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of the process ${\left(X_t\right)}_{t\ge 0}$.     

Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.     

Construction of extremal Type II Z2k-codes

We give methods for constructing many self-dual ${\mathbb{Z}}_m$-codes and Type II ${\mathbb{Z}}_{2k}$-codes of length 2n starting from a given self-dual ${\mathbb{Z}}_m$-code and Type II ${\mathbb{Z}}_{2k}$-code of length 2n, respectively. As an application, we construct extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 24 for $k=4,5,\dots ,20$ and extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 32 for $k=4,5,\dots ,10$. We also construct new extremal Type II ${\mathbb{Z}}_4$-codes of lengths 56 and 64.     

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence and concentration behavior of minimizers for $i_V(c)=\underset{u\in S_c}{\inf }?I_V(u)$, here $S_c=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c>0\}$ and

$I_V(u)=\frac{1}{2}\underset{{\mathbb{R}}^N}{\int }(a|\mathrm{?}u{|}^2+V(x)|u{|}^2)+\frac{b}{4}{(\underset{{\mathbb{R}}^N}{\int }|\mathrm{?}u{|}^2)}^2?\frac{1}{p}\underset{{\mathbb{R}}^N}{\int }|u{|}^p,$

where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of $i_V(c)$ for $2<p<2^{?}$ when $V(x)\ge 0$, $V(x)\in {L^{\infty }}_{loc}({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }?V(x)=+\infty$. For the case $p\in (2,\frac{2N+8}{N})\backslash \{4\}$, we prove that the global constraint minimizers $u_c$ of $i_V(c)$ behave like

$u_c(x)\approx \frac{c}{|Q_p{|}_2}{\left(\frac{m_c}{c}\right)}^{\frac{N}{2}}{Q}_p(\frac{m_c}{c}x?z_c),$

for some $z_c\in {\mathbb{R}}^N$ when c is large, where $Q_p$ is, up to translations, the unique positive solution of $?\frac{N(p?2)}{4}\mathrm{\Delta }{Q}_p+\frac{2N?p(N?2)}{4}{Q}_p=|Q_p{|}^{p?2}{Q}_p$ in ${\mathbb{R}}^N$ and $m_c={(\frac{\sqrt{a^2{D}_1^2?4b{D}_2{i}_0(c)}+a{D}_1}{2b{D}_2})}^{\frac{1}{2}}$, $D_1=\frac{Np?2N?4}{2N(p?2)}$ and $D_2=\frac{2N+8?Np}{4N(p?2)}$.     

Packed words and quotient rings

The coinvariant algebra is a quotient of the polynomial ring $\mathbb{Q}[x_1,\dots ,x_n]$ whose algebraic properties are governed by the combinatorics of permutations of length n. A word $w=w_1\dots w_n$ over the positive integers is packed if whenever $i>2$ appears as a letter of w, so does $i?1$. We introduce a quotient $S_n$ of $\mathbb{Q}[x_1,\dots ,x_n]$ which is governed by the combinatorics of packed words. We relate our quotient $S_n$ to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring.     

Construction and enumeration of left dihedral codes satisfying certain duality properties

Let ${\mathbb{F}}_q$ be the finite field of q elements and let $D_{2n}=〈x,y|x^n=1,y^2=1,yxy=x^{n?1}〉$ be the dihedral group of 2n elements. Left ideals of the group algebra ${\mathbb{F}}_q[D_{2n}]$ are known as left dihedral codes over ${\mathbb{F}}_q$ of length 2n, and abbreviated as left $D_{2n}$-codes. Let $\gcd (n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over ${\mathbb{F}}_q$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over ${\mathbb{F}}_q$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over ${\mathbb{F}}_q$, and present several numerical examples to illustrative our applications.     

Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases

We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:

$\{\begin{array}{cc}\hfill & ?{\epsilon }^2\mathrm{\Delta }u+V(x)u+{\lambda }_1|u{|}^{2}u+{\lambda }_2(K?|u{|}^2)u=0\text{ in }{\mathbb{R}}^3,\hfill \\ \hfill & u>0,u\in H^1({\mathbb{R}}^3),\hfill \end{array}$

where ε is a small positive parameter, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, ? denotes the convolution, $K(x)=\frac{1?3{\cos }^2?\theta }{|x{|}^3}$ and $\theta =\theta (x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector x. Under certain assumptions on $({\lambda }_1,{\lambda }_2)\in {\mathbb{R}}^2$, we construct a family of positive solutions $u_{\epsilon }\in H^1({\mathbb{R}}^3)$ which concentrates around the local minima of V as $\epsilon \to 0$. Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation.     

Realization of PBW-deformations of type An quantum groups via multiple Ore extensions

The notion of multiple Ore extension is introduced as a natural generalization of Ore extensions and double Ore extensions. For a PBW-deformation ${\mathfrak{B}}_q(\mathfrak{sl}(n+1,\mathbb{C}))$ of type ${\mathbb{A}}_n$ quantum group, we explicitly obtain the commutation relations of its root vectors, then show that it can be realized via a series of multiple Ore extensions, which we call a ladder Ore extension of type $(1,2,?,n)$. Moreover, we analyze the quantum algebras ${\mathfrak{B}}_q(\mathfrak{g})$ with $\mathfrak{g}$ of type ${\mathbb{D}}_4$, ${\mathbb{B}}_2$ and ${\mathbb{G}}_2$ and give some examples and counterexamples that can be realized by a ladder Ore extension.     

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem

$?\mathrm{\Delta }u=\lambda a(x)e^u?4\pi N{\delta }_0\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a smooth and bounded domain in ${\mathbb{R}}^2$ such that $0\in \mathrm{\Omega }$, $a(x)$ is a positive smooth function, N is a positive integer and $\lambda >0$ is a small parameter. Here ${\delta }_0$ defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution $u_{\lambda }$ blowing up at 0 and satisfying $\lambda {\int }_{\mathrm{\Omega }}a(x)e^{u_{\lambda }}\to 8\pi (N+1)$ as $\lambda \to 0^{+}$.