In this paper using the Banach limit we have determined a Gibbs-like measure ${\mu }_h$ supported by a cookie-cutter set $E$ which is generated by a single cookie-cutter mapping $f$. For such a measure ${\mu }_h$ and $r\in (0,+\infty )$ we have shown that there exists a unique ${\kappa }_r\in (0,+\infty )$ such that ${\kappa }_r$ is the quantization dimension function of the probability measure ${\mu }_h$, and we established its functional relationship with the temperature function of the thermodynamic formalism. The temperature function is commonly used to perform the multifractal analysis, in our context of the measure ${\mu }_h$. In addition, we have proved that the ${\kappa }_r$-dimensional lower quantization coefficient of order $r$ of the probability measure is positive.

We consider a semilinear Neumann problem with a reaction which is resonant both at $\pm \infty$ and at zero with respect to any eigenvalue, possibly the same one. Using the reduction method and Morse theory, we show that the problem has at least two nontrivial smooth solutions.

We show that the Ozsváth–Szabó contact invariant $c^{+}\left(\xi \right)\in H{F}^{+}(-Y)$ of a contact $3$-manifold $(Y,\xi )$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$ and if $\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsváth and Szabó to determine the Heegaard–Floer homology of such $3$-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsváth–Stipsicz–Szabó obstruction to admitting a planar open book for a certain class of contact structures. We also define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. Using this technique, we do a sample calculation showing that the invariant can get infinitely many distinct values.

We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that if a commutative Noetherian complete local ring $R$ admits a test module of finite Gorenstein dimension, then $R$ is Gorenstein.

In this paper we study the discrete spectrum of tensor products of irreducible unitary representations of the universal covering group of $SU(1,1)$. As a consequence of these results, we show that the set of smooth vectors of the tensor product intersects trivially some of the representations in the discrete spectrum. These results illustrate aspects of the much larger program of branching laws for reductive groups.

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $E$ be a vector bundle on $X$. Let ${\mathcal{O}}_{{Gr}_r\left(E\right)}\left(1\right)$ be the tautological line bundle over the Grassmann bundle ${Gr}_r\left(E\right)$ parameterizing all the $r$-dimensional quotients of the fibers of $E$. We give necessary and sufficient conditions for ${\mathcal{O}}_{{Gr}_r\left(E\right)}\left(1\right)$ to be ample and nef, respectively. As an application, we compute the nef cone of ${Gr}_r\left(E\right)$. This yields a description of the nef cone of any flag bundle over $X$ associated to $E$.

Markovian semigroups on $L^2$-space with suitable conditions can be regarded as Markovian semigroups on $L^p$-spaces for $p\in [1,\infty )$. When we additionally assume the ergodicity of the Markovian semigroups, the rate of convergence on $L^p$-space for each $p$ is considerable. However, the rate of convergence depends on the norm of the space. The purpose of this paper is to investigate the relation between the rates on $L^p$-spaces for different $p$’s, to obtain some sufficient condition for the rates to be independent of $p$, and to give an example for which the rates depend on $p$. We also consider spectra of Markovian semigroups on $L^p$-spaces, because the rate of convergence is closely related to the spectra.

The Bohr–Jessen limit theorem states that for each $\sigma >\frac{1}{2}$, there exists an asymptotic probability distribution of $log\zeta (\sigma +\sqrt{-1}\cdot )$. Here $\zeta (\cdot )$ is the Riemann zeta function, and $log\zeta (\cdot )$ is a primitive function of $\zeta \text{'}/\zeta$ on some simply connected domain of $\mathbb{C}$. In this paper, we generalize this limit theorem to a functional limit theorem and show a similar limit theorem for a continuous process $\{log\zeta (\sigma +\sqrt{-1}\cdot ){\}}_{\sigma >1/2}$, which we call the Bohr–Jessen functional limit theorem.

In this paper we study Lagrangian-invariant objects (LI objects for short) in the derived category $D^b\left(A\right)$ of coherent sheaves on an abelian variety. For every element of the complexified ample cone $D_A$ we construct a natural phase function on the set of LI objects, which in the case $dimA=2$ gives the phases with respect to the corresponding Bridgeland stability. The construction is based on the relation between endofunctors of $D^b\left(A\right)$ and a certain natural central extension of groups, associated with $D_A$ viewed as a Hermitian symmetric space. In the case when $A$ is a power of an elliptic curve, we show that our phase function has a natural interpretation in terms of the Fukaya category of the mirror dual abelian variety. As a by-product of our study of LI objects we show that the Bridgeland component of the stability space of an abelian surface contains all full stabilities.