We give a new and very concise proof of the existence of a holomorphic continuation for a large class of twisted multivariable zeta functions. To do this, we use a simple method of “decalage” that avoids using an integral representation of the zeta function. This allows us to derive explicit recurrence relations between the values at $T$–tuples of negative integers. This also extends some earlier results of several authors where the underlying polynomials were products of linear forms.

It is shown that Seifert 4-manifolds with hyperbolic bases are geometric in the sense of Thurston if and only if the monodromies are periodic. This result will be used to prove virtually geometric Seifert 4-manifolds with hyperbolic bases are geometric and thus give a classification of such manifolds in terms of finite covers.

We find properties of the sets $M_y^{-1}$ of all points on a compact orientable Alexandrov surface $S$, the distance functions of which have a common maximum at $y\in S$. For example, the components of $M_y^{-1}$ are arcwise connected and their number is at most $max\{1,10g-5\}$, where $g$ is the genus of $S$. A special attention receives the case of local tree components of $M_y^{-1}$, providing a relationship to the unit tangent cone at $y$.

For a polynomial $p$ on $C^n$, the variety $V_p=\{z\in C^n;p(z)=0\}$ will be considered. Let $\mathrm{Exp}(V_p)$ be the space of entire functions of exponential type on $V_p$, and ${\mathrm{Exp}}^{\prime }(V_p)$ its dual space. We denote by $\partial p$ the differential operator obtained by replacing each variable $z_j$ with $\partial /\partial z_j$ in $p$, and by ${\mathit{O}}_{\partial p}(C^n)$ the space of holomorphic solutions with respect to $\partial p$. When $p$ is a reduced polynomial, we shall prove that the Fourier-Borel transformation yields a topological linear isomorphism: ${\mathrm{Exp}}^{\prime }(V_p)\to {\mathit{O}}_{\partial p}(C^n)$. The result has been shown by Morimoto, Wada and Fujita only for the case $p(z)=z_1^2+\cdots +z_n^2+\lambda (n\ge 2)$.

We study the exponential growth of the numbers of particles for a branching symmetric $\alpha$-stable process in terms of the principal eigenvalue of an associated Schrödinger operator. Here the branching rate and the branching mechanism can be state-dependent. In particular, the branching rate can be a measure belonging to a certain Kato class and is allowed to be singular with respect to the Lebesgue measure. We calculate the principal eigenvalues and give some examples.

We review the method due to Gel’fand and Fuks to show the finite dimensionality of the cohomology ring of the Lie algebra of formal contact vector fields. We apply this method to prove the relative cohomology of it with respect to formal Poisson vector fields is trivial.

This article treats a one parameter family of equations of motion of vibrating membrane whose energy functionals converge to the Dirichlet integral as the parameter $\epsilon$ tends to zero. It is proved that both weak solutions satisfying energy inequality and generalized minimizing movements converge to a unique solution to the d’Alembert equation.

Let $f:S⟶B$ be a non locally trivial relatively minimal fibred surface. We prove a lower bound for the slope of $f$ depending increasingly from the relative irregularity of $f$ and the Clifford index of the general fibres.

Let ${\mathrm{Bd}}_H(R^m)$ be the hyperspace of nonempty bounded closed subsets of Euclidean space $R^m$ endowed with the Hausdorff metric. It is well known that ${\mathrm{Bd}}_H(R^m)$ is homeomorphic to the Hilbert cube minus a point. We prove that natural dense subspaces of ${\mathrm{Bd}}_H(R^m)$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space ${\ell }_2$. For each , let

${\nu }_k^m=\{x=(x_i{)}_{i=1}^m\in R^m:x_i\in R\setminus Q$ except for at most $k$ many $i\}$,

where ${\nu }_k^{2k+1}$ is the $k$-dimensional Nöbeling space and ${\nu }_0^m=(R\setminus Q{)}^m$. It is also proved that the spaces ${\mathrm{Bd}}_H({\nu }_0^1)$ and ${\mathrm{Bd}}_H({\nu }_k^m)$, , are homeomorphic to ${\ell }_2$. Moreover, we investigate the hyperspace ${\mathrm{Cld}}_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. It is shown that a nonseparable component $\mathit{H}$ of ${\mathrm{Cld}}_H(R)$ is homeomorphic to the Hilbert space ${\ell }_2(2^{{\aleph }_0})$ of weight $2^{{\aleph }_0}$ in case where $\mathit{H}$ ∌ $R,[0,\infty ),(-\infty ,0]$.

In this paper, we shall prove in Theorem 1 that Gauss’ famous closed formula for the values of the digamma function at rational arguments is equivalent to the well-known finite expression for the $L(1,\chi )$, which in turn gives rise to the finite expression for the class number of quadratic fields. We shall also prove several equivalent expressions for the arithmetic function $N(q)$ introduced by Lehmer and reveal the relationships among them.

We present a Frobenius type theorem for a system of nonlinear partial differential equations. Typical application is the normal form theory of a singular vector field. The construction of a singular solution is closely related with a Riemann-Hilbert factorization.

In this article we discuss how the existence of Kaehler metrics with constant scalar curvature on the projectivization of a holomorphic vector bundle over a Kaehler manifold $M$ is related to a moment map condition for the action of the automorphism group of $M$ on the moduli of vector bundles.

Let $P(I)$ be the set of all operator monotone functions defined on an interval $I$, and put $P_{+}(I)=\{h\in P(I):h(t)\geqq 0,h\ne 0\}$ and $P_{+}^{-1}(I)=\{h:h$ is increasing on $I,h^{-1}\in P_{+}(0,\infty )\}$. We will introduce a new set $L{P}_{+}(I)=\{h:h(t)>0$ on $I,logh\in P(I)\}$ and show $L{P}_{+}(I)·P_{+}^{-1}(I)\subset P_{+}^{-1}(I)$ for every right open interval $I$. By making use of this result, we will establish an operator inequality that generalizes simultaneously two well known operator inequalities. We will also show that if $p(t)$ is a real polynomial with a positive leading coefficient such that $p(0)=0$ and the other zeros of $p$ are all in $\{z:\mathrm{Rz}\leqq 0\}$ and if $q(t)$ is an arbitrary factor of $p(t)$, then $p(A{)}^2\leqq p(B{)}^2$ for $A,B\geqq 0$ implies $A^2\leqq B^2$ and $q(A{)}^2\leqq q(B{)}^2$.