We use a projective groups representation $\rho$ of the unimodular group $\mathcal{G} \times \hat{\mathcal{G}}$ on $L^2(\mathcal{G}$) to define Gabor wavelet transform of a function $f$ with respect to a window function $g$, where $\mathcal{G}$ is a locally compact abelian group and $\hat{\mathcal{G}}$ its dual group. Using these transforms, we define a weighted Banach $\mathcal{H}^{1, \rho}_w(\mathcal{G})$ and its antidual space $\mathcal{H}^{{1}^{\sim}, \rho}_w(\mathcal{G})$, $w$ being a moderate weight function on $\mathcal{G} \times \hat{\mathcal{G}}$. These spaces reduce to the well known Feichtinger algebra $S_0(\mathcal{G})$ and Banach space of Feichtinger distribution $S'_0(\mathcal{G})$ respectively for $w\equiv 1$. We obtain an atomic decomposition of $\mathcal{H}^{1, \rho}_w(\mathcal{G})$ and study some properties of Gabor multipliers on the spaces $L^2(\mathcal{G}), \mathcal{H}^{1, \rho}_w(\mathcal{G})$ and $\mathcal{H}^{{1}^{\sim}, \rho}_w(\mathcal{G})$. Finally, we prove a theorem on the compactness of Gabor multiplier operators on $L^2(\mathcal{G})$ and $\mathcal{H}^{1, \rho}_w(\mathcal{G})$, which reduces to an earlier result of Feichtinger [Fei 02, Theorem 5.15 (iv)] for $w=1$ and $\mathcal{G}=R^d$.

W.Kirsch and I.Veselić proved a generalized Wegner estimate for Schrödinger operators with generalized alloy type potentials at negative energies for each fixed position of impurities. In this paper, a similar estimate is proven treating also the position of impurities as random variables.

It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z$, when the ceiling of the walk $ S_n $ is located at $ j$. The random reward (or scenery) $ \xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits.

We show a criterion to establish the existence of solutions with prescribed Gevrey wave front sets for pseudodifferential operators of finite order.

In this paper we will consider whether there exists a time periodic solution of the Navier-Stokes equations for infinite channels in $\mathbb{R}^n(n=2,3)$. H. Beirão da Veiga [4] treated such a problem. This paper is the special case of his paper and we argue the relation between the existence of stationary and time periodic solutions of the Navier-Stokes equations.

The main purpose of this article is to establish the existence result concerning to the problem $-\Delta u(x)+c(x)u(x)=a(x)f(u(x))$, $x\in \mathbb{R}^{N}$, $N>2$, $u(x)\rightarrow 0$ as $\left\vert x\right\vert \rightarrow \infty$. Similary problems have been also studied. The proofs of the existence are based on the maximum principle and sub and super solutions method.

The invariant Cantor set of Smale's horseshoe map is imbedded into a connented and not arcwise connected, invariant, compact set with zero Lebesgue measure.

Let $G$ be a finite subgroup of $GL(4,\mathbb{Q} )$. Let $G$ act on the rational function field $\mathbb{Q}(x_1,x_2,x_3,x_4)$ by $\mathbb{Q}$-automorphism defined by the linear action of variables.Linear Noether's problem asks whether the fixed field $\mathbb{Q} (x_1,x_2,x_3,x_4)^G$ is rational (i.e. purely transcendental) over $\mathbb{Q}$. So far some partial results have been known, but in this paper we will give the almost complete results of this problem. One of motivations of this problem is the relation to the inverse Galois problem.

Let $G$ be a simple linear algebraic group defined over $\mathbb{C}$ and $P$ a parabolic subgroup of it. Let $(M, E_P, \omega)$ be a holomorphic parabolic geometry of type $G/P$ over a smooth complex projective variety $M$. We prove that $(M, E_ , \omega)$ is holomorphically isomorphic to the standard parabolic geometry $(G/P, G, \omega_0)$ whenever $M$ is rationally connected. We then show that this is indeed the case if $M$ has Picard number one and contains a (possibly singular) rational curve. This last result is a generalization of the main result of [3], where we dealt with the case $G = PGL(d, \mathbb{C})$, $G/P = \mathbb{P}^{d-1}_{\mathbb{C}}$.

Using a variation from the construction of the Ornstein-Uhlenbeck process on canonical path-space $C([0,1]; \mathbb{R})$ in terms of the Brownian sheet, we obtain a large class of processes, adapted to the Brownian filtration, which admit the one dimensional marginals of a martingale.

Let $H$ be an ample line bundle on a non-singular projective surface $X$, and $M(H)$ the coarse moduli scheme of rank-two $H$-semistable sheaves with fixed Chern classes on $X$. We show that if $H$ changes and passes through walls to get closer to $K_X$, then $M(H)$ undergoes natural flips with respect to canonical divisors. When $X$ is minimal and $\kappa(X)\geq 1$, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. Remark that so-called Thaddeus-type flips somewhat differ from flips with respect to canonical divisors.

Dolgopyat [4] showed that a class of Axiom A flows has exponential decay of correlations for smooth observables, and Baladi-Vallée [2] gave a nice interpretation of it on suspension semiflows of one-dimensional expanding countable Markov maps. Avila-Gouëzel-Yoccoz [1] extends the result of Baladi-Vallée to higher dimensional systems.

In this paper we show that a class of non-Markov semiflows also has exponential decay of correlations.

We prove that such exponential decay can be shown on an open dense condition for the suspensions of piecewise expanding maps.