Let $p$ and $q$ be integers such that $p>q\ge 2$ and $q$ divides $p$. Let $\varphi \left(q\right)$ be the Euler number of $q$. We exhibit a Zariski $\varphi \left(q\right)$-ple, distinguished by the Alexander polynomial, whose curves are tame torus curves of type $(p,q)$, with $q$ smooth irreducible components of degree $p$,and one single singular point topologically equivalent to the Brieskorn-Pham singularity $v^q+u^{q{p}^2}=0$.

Popa proved that strongly amenable subfactors of type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ with the same type $\mathrm{I}\mathrm{I}$ and type $\mathrm{I}\mathrm{I}\mathrm{I}$ principal graphs are completely classified by their standard invariants. In this paper, we present a different proof of this classification theorem based on Connes and Haagerup's arguments on the uniqueness of the injective factor of type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$.

Let $(X,L)$ be a quasi-polarized variety defined over the complex number field. Then there are several invariants of $(X,L)$, for example, the sectional genus and the $\mathrm{\Delta }$-genus. In this paper we introduce the $i$-th $\mathrm{\Delta }$-genus ${\mathrm{\Delta }}_i(X,L)$ for every integer $i$ with $0\le i\le n=\dim X$. This is a generalization of the $\mathrm{\Delta }$-genus. Furthermore we study some properties of ${\mathrm{\Delta }}_i(X,L)$ and we will propose some problems.

Let us consider the following nonlinear singular partial differential equation in the complex domain with two independent variables $(t,x)\in {\mathbf{C}}^2$. When the equation is of totally characteristic type, this equation was solved in [2] and [9] under certain Poincaré condition. In this paper, the author will prove the uniqueness of the solution under the assumption that $u(t,x)$ is holomorphic in for some $r>0$, $\theta >0$, $R>0$ and that it satisfies $u(t,x)=O\left(\right|t{|}^a)$ (as $t0$) uniformly in $x$ for some $a>0$. The result is applied to the problem of removable singularities of the solution.

In this paper, we shall develop the linear causal analysis for the system consisting of two flows in a real inner product space and give an algorithm for calculating the non-linear filter for a discrete stochastic system which is given by two discrete time stochastic processes, to be called a signal process and an observation process, based upon the theory of ${\mathrm{K}\mathrm{M}}_2\mathrm{O}$-Langevin equations.

For a given finite dimensional $k$-algebra $A$ which admits a presentation in the form $R/G$, where $G$ is an infinite group of $k$-linear automorphisms of a locally bounded $k$-category $R$, a class of modules lying out of the image of the "push-down" functor associated with the Galois covering $R\to R/G$, is studied. Namely, the problem of existence and construction of the so called non-regularly orbicular indecomposable $R/G$-modules is discussed. For a $G$-atom $B$ (with a stabilizer $G_B$), whose endomorphism algebra has a suitable structure,a representation embedding ${\mathrm{\Phi }}^{B(f,s)}{}_{|}:I_n-{\mathrm{s}\mathrm{p}\mathrm{r}}_{l\left(s\right)}\left(k{G}_B\right)\to \mathrm{m}\mathrm{o}\mathrm{d}(R/G)$, which yields large families of non-regularly orbicular indecomposable $R/G$-modules,is constructed (Theorem 2.2). An important role in consideration is played by a result interpreting some class of $R/G$-modules in terms of Cohen-Macaulay modules over certain skew grup algebra (Theorem 3.3). Also, Theorems 4.5 and 5.4, adapting the generalized tensor product construction and Galois covering scheme, respectively, for Cohen-Macaulay modules context, are proved and intensively used.

Let $𝒳$ be a space of homogeneous type. Assume that $L$ has a bounded holomorphic functional calculus on $L^2\left(\mathrm{\Omega }\right)$ and $L$ generates a semigroup with suitable upper bounds on its heat kernels where $\mathrm{\Omega }$ is a measurable subset of $𝒳$. For appropriate bounded holomorphic functions $b$, we can define the operators $b\left(L\right)$ on $L^p\left(\mathrm{\Omega }\right)$, $1\le p\le \mathrm{\infty }$. We establish conditions on positive weight functions $u,v$ such that for each $p$, , there exists a constant $c_p$ such that $${\int }_{\mathrm{\Omega }}\left|b\right(L\left)f\right(x\left){|}^{p}u\right(x\left)d\mu \right(x)\le c_p|\left|b\right|{|}_{\mathrm{\infty }}^p{\int }_{\mathrm{\Omega }}\left|f\right(x\left){|}^{p}v\right(x\left)d\mu \right(x)$$for all $f\in L^p\left(vd\mu \right)$.

Applications include two-weight $L^p$ inequalities for Schrödinger operators with non-negative potentials on ${\mathbf{R}}^n$ and divergence form operators on irregular domains of ${\mathbf{R}}^n$.

In [BG], it is proved that the Whitehead length of a space $Z$ is less than or equal to the nilpotency of $\mathrm{\Omega }Z$. As for rational spaces, those two invariants are equal. We show this for a 1-connected rational space $Z$ by giving a way to calculate those invariants from a minimal model for $Z$. This also gives a way to calculate the nilpotency of an homotopy associative rational $H$-space.

In this note we are going to generalize Prudnikov's method of using a double integral to deduce relations between the Riemann zeta-values, so as to prove intriguing relations between double zeta-values of depth 2. Prior to this, we shall deduce the most well-known relation that expresses the sum ${\sum }_{j=1}^{m-2}\zeta (j+1)\zeta (m-j)$ in terms of ${\zeta }_2(1,m)$.

In this paper we consider the one-dimensional quantum random walk $X_n^{\varphi }$ at time $n$ starting from initial qubit state $\varphi$ determined by $2\times 2$ unitary matrix $U$. We give a combinatorial expression for the characteristic function of $X_n^{\varphi }$. The expression clarifies the dependence of it on components of unitary matrix $U$ and initial qubit state $\varphi$. As a consequence, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that $X_n^{\varphi }/n$ converges weakly to a limit $Z^{\varphi }$ as $n\to \mathrm{\infty }$, where $Z^{\varphi }$ has a density $1/\pi (1-x^2)\sqrt{1-2x^2}$ for $x\in (-1/\sqrt{2},1/\sqrt{2})$. Moreover we discuss some known simulation results based on our limit theorems.

We provide an extended interpretation of the zeta regularized product in [D]. This allows us to get regularized product expressions of Hölder's double sine function and its companion, i.e. the double and triple trigonometric functions. The expressions may reasonably explain the ladder structure among these multiple trigonometric functions. We also introduce the notion of finite ladders of functions which helps us understand the meaning behind these regularizations.

We show that (1) stick plus $\mathrm{c}\mathrm{o}\mathrm{v}\left(ℳ\right)>{\mathrm{\aleph }}_1$ implies the existence of a destructible gap and (2) $\clubsuit$ plus $\mathrm{c}\mathrm{o}\mathrm{f}\left(ℳ\right)={\mathrm{\aleph }}_1$ implies the existence of a destructible gap.

We show that any saturated generic graph satisfying some property (*) is strictly stable or $\omega$-stable. As a corollary, we obtain that any saturated generic pseudoplane is strictly stable or $\omega$-stable.