In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

Tensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that M ⊗R N has nonzero torsion unless M is very special. An important example of such a module N is the Frobenius power peR over a complete intersection domain R of characteristic p > 0.

We consider, for a given large prime p, the problem of covering a square [0, p] × [0, p] with discs center at the lattice point (x, y), x and y subject to condition xy ≡ 1 (mod p) and with radius r. We are concerned with the size of r.

Consider n-th order linear differential equations with meromorphic periodic coefficients of the form w(n) + Rn-1(ez)w(n-1) + … + R1(ez)w′ + R0(ez)w = 0, n ≥ 2, where Rv(t) (0 ≤ ν ≤ n – 1) are rational functions of t. Under certain assumptions, we prove oscillation theorems concerning meromorphic solutions, which contain necessary conditions for the existence of a meromorphic solution with finite exponent of convergence of the zero-sequence. We also discuss meromorphic or entire solutions whose zero-sequences have an infinite exponent of convergence, and give a new zero-density estimate for such solutions.

Some functions related to the complex dilogarithmic function

(in the notation of Lewin [9]) are known to occur in connection with algebraic K-theory and characteristic classes (see e.g. Bloch [1], Gelfand-MacPherson [7], Dupont [5], and the references given there). Recently MacPherson and Hain (see [10]) has announced results of a similar kind for some higher polylogarithmic functions. Also Ramakrishnan [11] and [12] has recently studied the classical polylogarithms, which for |z| < 1 are given by

Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$ -primary ideals of A. Unlike $p_g$ -ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.

The concepts of Λ-shellability of locally finite posets as well as of extendable sequentially Koszul algebras will be introduced. It will be proved that the divisor poset of a homogeneous semigroup ring is Λ-shellable if and only if the semigroup ring is extendable sequentially Koszul. Examples of extendable sequentially Koszul semigroup rings contain all monomial ASL’s (algebras with straightening laws) and all second squarefree Veronese subrings.

We study the deformation theory of elliptic fiber bundles over curves in positive characteristics. As applications, we give examples of nonliftable elliptic surfaces in characteristics 2 and 3, which answer a question of Katsura and Ueno. Also, we construct a class of elliptic fibrations, whose liftability is equivalent to a conjecture of Oort concerning the liftability of automorphisms of curves. Finally, we classify deformations of bielliptic surfaces.

We characterize those positive measure µ’s on the higher dimensional unit ball such that “two-weighted inequalities” hold for holomorphic functions and their derivatives. Characterizations are given in terms of the Carleson measure conditions. The results of this paper also distinguish between the fractional and the tangential derivatives.

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

Let L be a nonnegative self-adjoint operator on L2 (X), where X is a space of homogeneous type. Assume that L generates an analytic semigroup e–tl whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier F(L) is bounded on for 0 < p < 1, the Hardy space associated to operator L, when F is a suitable function.

In this paper, firstly, we show the Bondi-mass type estimate of solutions of Calabi flow on closed 4-manifolds. Secondly, in our applications, we obtain the long time existence on closed 4-manifolds. In particular, we are able to show the asymptotic convergence of a subsequence of solutions of the Calabi flow on closed Einstein 4-manifolds.

Let D and C denote the open unit disk and the unit circle in the complex plane, respectively; and let f be a function from D into the Riemann sphere Ω. An arc γ⊂D is said to be an arc at p∈C if γ∪{p} is a Jordan arc; and, for each t (0<t<1), the component of γ∩{z: t≤|z|<1} which has p as a limit point is said to be a terminal subarc of γ. If γ is an arc at p, the arc-cluster set C(f, p,γ) is the set of all points a∈Ω for which there exists a sequence {zk}a⊂γ with zk→p and f(zk)→a.

Starting from the reciprocity law for Gaussian sums attached to an integral quadratic form we prove functional equations for a new kind of Dirichlet series in two variables. For special values of one variable they are of Hecke type with respect to the other variable. With Weil’s converse theorem we derive automorphic functions which generalize Siegel’s genus invariant and the automorphic functions of Cohen and Zagier.

In 1952, the late Professor T. Nakayama succeeded in constructing the Galois theory for finite dimensional simple ring extensions [7]. And, we believe, the theory was essentially due to the following proposition: If a simple ring A is Galois and finite over a simple subring B then A is B′-A-completely reducible for any simple intermediate ring B′ of A/B [7, Lemmas 1.1 and 1.2]. Moreover, as was established in [5], Nakayama’s idea was still efficient in considering the infinite dimensional Galois theory of simple rings.

We give a discussion on the properties of Δa(x) (− 1 < a < 0), which is a generalization of the error term Δ(x) in the Dirichlet divisor problem. In particular, we study its oscillatory nature and investigate the gaps between its sign-changes for −½ ≤ a < 0.

We study the complete Kähler–Einstein metric of certain Hartogs domains ${\rm\Omega}_{s}$ over bounded homogeneous domains in $\mathbb{C}^{n}$. The generating function of the Kähler–Einstein metric satisfies a complex Monge–Ampère equation with Dirichlet boundary condition. We reduce the Monge–Ampère equation to an ordinary differential equation and solve it explicitly when we take the parameter $s$ for some critical value. This generalizes previous results when the base is either the Euclidean unit ball or a bounded symmetric domain.

Let g(f), S(f), g*λ(f) be the Littlewood-Paley g function, Lusin area function, and Littlewood-Paley g*λ(f) function of f, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a BMO function f, one of the above functions is finite for a single point in ℝn, then it is finite a.e. on ℝn, and BMO boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, BMO, and Lipschitz spaces). One of us improved his g*λ(f) result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel μρ(f), μρs(f) and μλ*,ρ(f). In this paper, we show that the same results hold also in the case of rough kernel satisfying Lp-Dini type condition.

Let X = {X(A); A ∈ Q} be a Lévy’s Brownian motion with the basic time parameter space Q, where Q is taken to be the n-dimensional metric space Qn,k of constant curvature (2 ≤ n ≤ ∞, — ∞ < k: < ∞), i.e., Q is one of

(a) Euclidean space for k = 0, (b) sphere for k > 0 and

(c) real hyperbolic space for K < 0.

The increment X(A) — X(B) is, by definition, Gaussian in distribution and has mean 0 and variance d(A, B), the distance between A and B. The existence of such a Gaussian random field is well known ([3], [4], [16] and [23]).

The existence of condenser potentials was established in the framework of Dirichlet spaces by Beurling and Deny, cf. Deny [5] or Landkof [10], simply by choosing the potential of minimal energy within a certain convex set. This same idea works for non-symmetric Dirichlet spaces, cf. Bliedtner [3].