Our aim in this paper is to discuss minimally fine limits at infinity for locally $p$-precise functions in the half space of $R^n$ with vanishing boundary limits. We also give a measure condition for a set to be minimally thin at infinity.

The Ruelle invariants for non-singular flows of a 3-dimensional manifold and diffeomorphisms of the disc are described by invariant fiber measures, which are families of probability measures on the fibers of the projectivized bundle invariant under the holonomies among almost all fibers. The dynamical properties of invariant fiber measures are also given, which show the benefit of this description.

We study universal holomorphic functions on a Stein manifold $M$ with projective compactification. Let $\left\{{\varphi }_n\right\}$ be a sequence of holomorphic automorphisms of $M$. We prove that if $\left\{{\varphi }_n^{-1}\right\}$ is $A$ run-away, then the set of all universal functions with respect to $\left\{{\varphi }_n\right\}$ in $\mathcal{A}\left(K\right)$ for all compact subsets $K$ with a certain property is the intersection of countable number of open dense subsets in the space of all holomorphic functions on $M$. We also note that there is a close connection between the direction of run-awayness and a family of compact sets for which there exists a universal function.

For any countable directed graph $E$ we describe the primitive ideal space of the corresponding generalized Cuntz-Krieger algebra $C^{*}\left(E\right)$.

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems.

We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

In a complex domain $V\subset C^n$, let $P$ be a linear holomorphic partial differential operator and $K$ be its characteristic hypersurface. When the localization of $P$ at $K$ is a Fuchsian operator having a non-negative integral characteristic index, it is proved, under some conditions, that every holomorphic solution to $Pu=0$ in $V\setminus K$ has a holomorphic extension in $V$. Besides, it is applied to the propagation of singularities for equations with non-involutive double characteristics.

Motivated by the classification problem of atomic degenerations, in our series of papers, we make a systematic study for splitting deformations of degenerations of complex curves. We provide various new methods to construct splitting deformations, and deduce many splitting criteria of degenerations, which will be applied to the classification of atomic degenerations. Roughly, our criteria are separated into two types; in the first type the criteria are expressed in terms of the configuration of a singular fiber, and in the second type, in terms of sub-divisors of a singular fiber. In both types, our constructions are 'visible', in that we can view how the singular fiber is deformed. In the present paper, we demonstrate splitting criteria of the first type.

In this paper, we prove a result on uniqueness of meromorphic functions sharing three values counting multiplicity. As applications of this, many known results can be improved. Examples are provided to show that the results in this paper are best possible.

In this paper first we prove that if $f$ and $g$ are two permutable transcendental entire functions satisfying $f=f_1\left(h\right)$ and $g=g_1\left(h\right)$, for some transcendental entire function $h$, rational function $f_1$ and a function $g_1$, which is analytic in the range of $h$, then $F\left(g\right)\subset F\left(f\right)$. Then as an application of this result, we show that if $f\left(z\right)=p\left(z\right)e^{q\left(z\right)}+c$, where $c$ is a constant, $p$ a nonzero polynomial and $q$ a nonconstant polynomial, or $f\left(z\right)={\int }^{z}p\left(z\right)e^{q\left(z\right)}dz$, where $p,$$q$ are nonconstant polynomials, such that $f\left(g\right)=g\left(f\right)$ for a nonconstant entire function $g$, then $J\left(f\right)=J\left(g\right)$.

We obtain the exact value of Voiculescu's invariant $k_{\infty }^{-}\left(\tau \right)$, which is an obstruction of the existence of quasicentral approximate units relative to the Macaev ideal in perturbation theory, for a tuple $\tau$ of operators in the following two classes: (1) creation operators associated with a subshift, which are used to define Matsumoto algebras, (2) unitaries in the left regular representation of a finitely generated group.

In this paper we study the homotopy types of mod $p$ truncated configuration spaces. In particular, we investigate a finite dimensional configuration space model for the based loop space of mod $p$ lens space as an application.

The purposes of this paper are: (a) to study the behavior of the principal distributions around an isolated umbilical point on a real-analytic surface of a certain type, which contains all the real-analytic, special Weingarten surfaces;(b) to present one condition such that if a real-analytic surface satisfies the condition, then the index of an isolated umbilical point on the surface is less than or equal to one.

We classify equivariantly Gorenstein $\log$ del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite groups.

The Massera Theorem for almost periodic solutions of linear periodic ordinary differential equations of the form $(*)x^{\prime }=A\left(t\right)x+f\left(t\right)$, where $f$ is almost periodic, is stated and proved. Furthermore, it is extended to abstract functional differential equations $(**)x^{\prime }=Ax+F\left(t\right)x_t+f\left(t\right)$, where $A$ is the generator of a compact semigroup, $F$ is periodic and $f$ is almost periodic. The main techniques used in the proofs involve a new variation of constants formula in the phase space and a decomposition theorem for almost periodic solutions.

We introduce the class of generalized double tilted artin algebras and prove that it coincides with the class of artin algebras whose $\mathrm{AR}$-quiver admits a faithful generalized standard almost directed component. A homological characterization of faithful generalized standard almost directed components is also established.

A squarefree module over a polynomial ring $S=k[x_1,\text{...},x_n]$ is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals more systematically.

The category $\mathbf{Sq}$ of squarefree modules is equivalent to the category of finitely generated left $\Lambda$-modules, where $\Lambda$ is the incidence algebra of the Boolean lattice $2^{\{1,\text{...},n\}}$. The derived category $D^b\left(\mathbf{Sq}\right)$ has two duality functors $\mathbf{D}$ and $\mathbf{A}$. The functor $\mathbf{D}$ is a common one with $H^i\left(\mathbf{D},\left(M^{\bullet }\right)\right)={\mathrm{Ext}}_S^{n+i}\left(M^{\bullet },{\omega }_S\right)$, while the Alexander duality functor $\mathbf{A}$ is rather combinatorial. We have a strange relation $\mathbf{D}\circ \mathbf{A}\circ \mathbf{D}\circ \mathbf{A}\circ \mathbf{D}\circ \mathbf{A}\cong {\mathbf{T}}^{2n}$, where $\mathbf{T}$ is the translation functor. The functors $\mathbf{A}\circ \mathbf{D}$ and $\mathbf{D}\circ \mathbf{A}$ give a non-trivial autoequivalence of $D^b\left(\mathbf{Sq}\right)$. This equivalence corresponds to the Koszul duality for $\Lambda$, which is a Koszul algebra with ${\Lambda }^{!}\cong \Lambda$. Our $\mathbf{D}$ and $\mathbf{A}$ are also related to the Bernstein-Gel'fand-Gel'fand correspondence.

The local cohomology $H_{I_{\Delta }}^i\left(S\right)$at a Stanley-Reisner ideal $I_{\Delta }$ can be constructed from the squarefree module $Ex{t}_S^i(S/I_{\Delta },{\omega }_S)$. We see that Hochster's formula on the ${\mathbf{Z}}^n$-graded Hilbert function of $H_{\mathfrak{m}}^i(S/I_{\Delta })$ is also a formula on the characteristic cycle of $H_{I_{\Delta }}^{n-i}\left(S\right)$ as a module over the Weyl algebra $A=k\langle x_1,\text{...},x_n,{\partial }_1,\text{...},{\partial }_n\rangle$(if $\mathrm{char}\left(k\right)=0$).