Let $\{\xi_j\}_{j\in\mathbb{Z}}$ be a sequence of random variables which belong to the domain of attraction of a linear fractional stable motion $\{\Delta_{H,\alpha}(t)\}$ with infinite variance. We study the convergence of weighted sums $I_n(f):=A_n\sum_{j\in\mathbb{Z}}f({j}/{n})\xi_j$ with a suitable scaling $A_n$, to $I(f):=\int_{-\infty}^{\infty}f(u)d\Delta_{H,\alpha}(u)$ in distribution under suitable assumptions on a class of deterministic functions $f$. We also show that if $\{f_t, t\ge 0\}$ are the kernel functions from the ``moving average'' representation of a linear fractional stable motion with another index $H'$, then $\{I_n(f_t)\}$ converges to a linear fractional stable motion $\{ \Delta _{H+H'-1/\alpha, \alpha}(t)\}$.

We show that all types of self-adjoint perturbations of a semi-bounded operator $A$ (purely singular, mixed singular, and regular) can be described and studied from a unique point of view in the framework of the extension theory as well as in the framework of the additive perturbation theory. We also show that any singular finite rank perturbation $\widetilde{A}$ can be approximated in the norm resolvent sense by regular finite rank perturbations of $A$. An application is given to the study of Schr\"{o}dinger operators with point interactions.

Let $H$ be a reproducing kernel Hilbert space contained in a wider space $L^2(X,\mu)$. We study the Hilbert-Schmidt property of Hankel operators $H_g$ on $H$ with bounded symbol $g$ by analyzing the behavior of the iterated Berezin transform. We determine symbol classes $\mathcal{S}$ such that for $g\in \mathcal{S}$ the Hilbert-Schmidt property of $H_g$ implies that $H_{\bar{g}}$ is a Hilbert-Schmidt operator as well and there is a norm estimate of the form $\|H_{\bar{g}}\|_{\text{HS}}\leq C\cdot \| H_g\|_{\text{HS}}$. Finally, applications to the case of Bergman spaces over strictly pseudo convex domains in $\mathbb{C}^n$, the Fock space, the pluri-harmonic Fock space and spaces of holomorphic functions on a quadric are given.

For a number field $k$ and a prime number $p$, let $k_{\infty}$ be a $\textbf{Z}_{\textrm{p}}$-extension of $k$ and $X_{\infty}(k)$ the Galois group over $k_{\infty}$ of the maximal abelian unramified $p$-extension of $k_{\infty}$. We first give a sufficient condition, bearing on the norm index of units in the layers of $k_{\infty}$, for $X_{\infty}(k)$ to be finite. When the prime $p$ is 2 and $X_{\infty}(k)\simeq \textbf{Z}/2\textbf{Z}\times \textbf{Z}/2\textbf{Z}$, we study the structure of the Galois group of the maximal unramified $p$-extension of $k_{\infty}$, improving on some previous results in the case of quadratic fields.

We consider the existence of a non-trivial weak solution for the equation $$ \left\{ \begin{array}{@{}ll} -\Delta_p u= f(x,u) & \text{in } \ \Omega\,, \\ u=0 & \text{on } \ \partial\Omega\,, \end{array}\right. $$ where $f$ satisfies $f(x,u)=a u_+^{p-1} -bu_-^{p-1} + o(|u|^{p-1})$ ($p>1$) at 0 or $\infty$. By using Morse theory and calculating the critical groups, we show the existence of a non-trivial weak solution to the equation under mild auxiliary conditions.

In this note we uncover some analytical properties of the Weierstrass pair $\{g,\eta\}$ representing an oriented and connected maximal surface in Minkowski space $\mathbb{L}^3$, proving that $g$ is holomorphic with values in the unit disk $\mathbb{D}$ and $\eta$ is a holomorphic 1-form that never vanishes.

Let $L$ be an $n$-component Brunnian link and $F$ a genus $g$ closed surface containing $L$. Then, we show that $g>(n+3)/3$.

We are concerned with the existence and uniqueness of the \textit{classical} solution to IBVP for a nonlinear equation of a suspended string with uniform density to which a monotonous nonlinear time-independent outer force works. For this purpose we derive the higher order energy estimates, and employ the Galerkin method combining with the compactness argument along the refined method of [Sath]. We need the regularity theory of Nirenberg type for the suspended string operator $L$.

The existence of a Markov partition of a hyperbolic group automorphism generated by an integral matrix with determinant $\pm 1$ is established by Sinai (see [22]). After that, there are many articles to construct Markov partitions of group automorphisms generated by non-negative matrices satisfying Pisot condition by the tiling method from substitutions (see [1], [7], [16], [19], [5]). One of the purpose of this paper is to establish the construction method of a Markov partition for a group automorphism generated by a non-positive matrix satisfying ``negative Pisot'' condition. An anti-homomorphic extension of a substitution, called $AH$-substitution, is introduced in the paper. Owing to this new substitution, the Markov partition of the group automorphism from the non-positive integral matrix is constructed.

Let $(A,\frak{m})$ denote a Noetherian local ring with maximal ideal $\frak{m}$, $J$ an $\frak m$-primary ideal, $I_1,\ldots, I_s$ ideals of $A$; $M$ a finitely generated $A$-module. This paper will answer when mixed multiplicities of the multi-graded fiber cone $$ F_M(J,I_1,\ldots, I_s)=\bigoplus_{n_1,\ldots,n_s\geqslant 0}\dfrac{I_1^{n_1}\cdots I_s^{n_s}M}{JI_1^{n_1}\cdots I_s^{n_s}M} $$ are positive and characterize them in terms of the length of modules.

In [3], Faudree et~al.\ considered the proposition ``Every $\{X,Y\}$-free graph of sufficiently large order has a $2$-factor,'' and they determined those pairs $\{X,Y\}$ which make this proposition true. Their result says that one of them is $\{X,Y\}=\{K_{1,4},P_4\}$. In this paper, we investigate the existence of $2$-factors in $r$-connected $\{K_{1,k},P_4\}$-free graphs. We prove that if $r\ge 1$ and $k\ge 2$, and if $G$ is an $r$-connected $\{K_{1,k},P_4\}$-free graph with minimum degree at least $k-1$, then $G$ has a $2$-factor with at most $\max\{k-r,1\}$ components unless $(k-1) K_2 + (k-2) K_1 \subseteq G \subseteq (k-1) K_2 + K_{k-2}$. The bound on the minimum degree is best possible.

In 1978 B. Y. Chen and T. Nagano obtained the local classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we investigate their global classification.

The finiteness of non-symmetric and symmetric cohomologies associated with Jackson integrals of type $BC_n$ is studied. The explicit bases of the cohomologies are also given. These bases determine parameter-dependent Jackson integral, and it is shown that they satisfy holonomic systems of linear $q$-difference equations with respect to the parameters.

We study an isometric immersion $f:M \to \bar{M}$ with geodesic normal sections, where $\bar{M}$ is a semi-Riemannian space form. In Riemannian geometry, it is known that $f$ is helical, in particular, all geodesics of $M$ have the same proper order in $\bar{M}$. However this does not hold in general, when $\bar{M}$ is indefinite semi-Riemannian. We give sufficient conditions for an isometric immersion with geodesic normal sections to be helical.

A Cauchy-Euler type factorization property which is closely related with the Hyers-Ulam stability problem is introduced in the algebra of all linear self maps of a commutative algebra without order. Several examples of linear self maps with such a property are given in this note.

Metric properties of Denjoy's canonical continued fraction expansion are studied, and the natural extension of the underlying ergodic system is given. This natural extension is used to give simple proofs of results on mediant convergents obtained by W. Bosma in 1990.

A diffusion process with a random potential consisting of two independent self-similar processes with different indices for the right and the left hand sides of the origin is considered. The limiting behavior of the process as time goes to infinity is investigated.

In the upper bound graph of a poset $P$ with the vertex set $V(P)$, $xy$ is an edge if there exists an $m\in V(P)$ such that $x,y\leq_P m$. We obtain some properties of edge operations of UB-graphs. According to these properties, we consider transformations on split UB-graphs.

Let $M$ be an $m$-dimensional compact complex manifold and $\Omega$ a Kähler class of $M$. Assume that $M$ admits an $\Omega$-preserving $0$-pseudofree $S^1$-action and that $\Omega$ contains a Kähler metric of constant scalar curvature. Then using the fixed point formula for the Bando-Calabi-Futaki character obtained in [5], we can obtain information on the fixed point data of the $S^1$-action. Our main result is Theorem 2.

In this paper, we obtain functional relations for Witten zeta functions by using relations of double Lerch values. By these functional relations, we obtain new proofs of known results on the Tornheim double zeta values, the Euler-Zagier double zeta values, and their alternating and character analogues.