Let $G$ be a simple algebraic group, defined over a finite field $\gF_q$, with Frobenius map $F$. Let $X^{\bul}_f$ be the Hansen-Demazure-Deligne-Lusztig compactification of the Deligne-Lusztig variety $X_f$ of $G$ associated with a Coxeter element in the Weyl group $W_G$ of $G$, and let $X^{\bul}_{f,0}$ be the $\gF_q \delta$-structure on $X^{\bul}_f$ over the finite extension $\gF_q \delta$ of $\gF_q$ determined by $F^{\delta} : X^{\bul}_f \to X^{\bul}_f$, where $\delta$ is the smallest positive integer such that $F^{\delta}$ is the identity map on $W_G$. We shall give an affirmative answer to the Tate conjecture over finite fields for algebraic cycles on $X^{\bul}_{f,0}$ and related projective schemes.

It is known that the Stokes operator is not well-defined in $L^q$-spaces for certain unbounded smooth domains unless $q=2$. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see \cite{FKS1}, to the $n$-dimensional one, $n\geq 2$, replacing the space $L^q, 1\ltq\lt\infty$, by $\s{L}^q$ where $\s{L}^q = L^q\cap L^2$ for $q\geq 2$ and $\s{L}^q = L^q+L^2$ for $1\ltq\lt2$. In particular, we show that the Stokes operator is well-defined in $\s{L}^q$ for every unbounded domain of uniform $C^{1,1}$-type in $\R^n$, $n\geq 2$, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.

In this paper, we establish the CMO estimate for the higher-order commutator of Carlderón-Zygmund singular integral with rough kernel. This estimate enables us to get the CMO estimates for the higher-order commutators of a class of Marcinkiewicz integral operators $\mu_\Omega$, $\mu_{\Omega,\lambda}^{\ast}$ and $\mu_{\Omega,S}$ with rough kernels, which are corresponding to the Littlewood-Paley $g$ function, Littlewood-Paley $g_{\lambda}^{\ast}$ function and the Lusin area integral, respectively.

Given an initial condition $x(T)=A$, $x'(T)=B$ ($'=d/dt$, $0\lt T \lt \infty$, $0 \lt A \lt \infty$, $-\infty \lt B \lt \infty$) for the differential equation denoted in the title, we shall conclude that if $T$, $A$ are fixed arbitrarily, then there exists a number $B_{\ast}$ such that in every case of $B=B_{\ast}$, $B \lt B_{\ast}$, $B \gt B_{\ast}$ we determine analytical expressions of the solution of the initial value problem which shows asymptotic behavior of the solution. That is, these analytical expressions are valid in neighborhoods of ends of the domain of the solution. If $\lambda =-1$, then we shall treat the case $T=0$, since there exists the solution continuable to $t=0$.

We investigate several stabilities and a genericity of function germs with respect to the reticular t-P-K-equivalence.

In this paper we study a $C^{1+\alpha}$-partially hyperbolic diffeomorphism f of which restriction on one dimensional center unstable direction behaves as Manneville-Pomeau map. We show that f admits a unique ergodic SRB measure with polynomial upper bounds on correlations for Hölder continuous functions.

The moduli space of the solutions to the monopole equations over an oriented closed 3-manifold M carrying the geometric structure $\mathbf{R} \times H^2$ is studied. Solving the parallel spinor equation, we obtain an explicit solution to the monopole equations. The moduli space consists of a single point with the Seiberg-Witten invariant $\pm 1$. Further, the (anti-)canonical line bundle $K_M^\pm 1$ gives a monopole class of M.

We will derive linear differential relations satisfied by Wirtinger integrals by exploiting classical formulas of Jacobi’s theta functions. Although Wirtinger integrals are related to Gauss' hypergeometric functions, we will do that without referring to Gauss' hypergeometric differential equation. We believe that our method to derive them will be applicable to another definite integral defined on the torus which has as integrand a power product of not necessarily four theta functions, and which is not a lift of any definite integral defined on the complex projective line.

The class of relative Tchebychev hypersurfaces extends the class of affine hyperspheres in Blaschke geometry. This paper aims at finding examples for relative Tchebychev hypersurfaces among the translation hypersurfaces. A complete local classification is given. Specializing from relative to centroaffine geometry, the list reduces to paraboloids.