Let $R$ be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of $R$-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local Tate cohomology through some natural isomorphisms. We investigate vanishing of formal local cohomology modules. Also, we give a characterization of Cohen-Macaulay complexes.

We study the reduced Beurling spectra $sp_{{\cal A},V}(F)$ of functions $F \in L^1_{\rm loc} (\jj,X)$ relative to certain function spaces ${\cal A}\subset L^{\infty}(\jj,X)$ and $V\subset L^1 (\r)$ and compare them with other spectra including the weak Laplace spectrum. Here $\jj$ is $\r_+$ or $\r$ and $X$ is a Banach space. If $F$ belongs to the space $ \f^{\p}_{ar}(\r,X)$ of absolutely regular distributions and has uniformly continuous indefinite integral with $0\notin sp_{\A,\f(\r)} (F)$ (for example, if F is slowly oscillating and $\A$ is $\{0\}$ or $C_0 (\jj,X)$), then $F$ is ergodic. If $F\in \f^{\p}_{ar}(\r,X)$ and $M_h F (\cdot)= (1/h) \int_0^h F(\cdot+s)\,ds$ is bounded for all $h > 0$ (for example, if $F$ is ergodic) and if $sp_{C_0(\r,X),\f} (F)=\emptyset$, then ${F}*\psi \in C_0(\r,X)$ for all $\psi\in \f(\r)$. We show that Tauberian theorems for Laplace transforms follow from results about reduced spectra. Our results are more general than previous ones, and we demonstrate this through examples.

In this paper, we will introduce real graph algebras and develop the theory to the point of being able to calculate the $K$-theory of such algebras. The $K$-theory situation is significantly more complicated than in the case for complex graph algebras. To develop the long exact sequence to compute the $K$-theory of a real graph algebra, we need to develop a generalized theory of crossed products for real C*-algebras for groups with involution. We also need to deal with the additional algebraic intricacies related to the period-8 real $K$-theory using united $K$-theory. Ultimately, we prove that the $K$-theory of a real graph algebra is recoverable from the $K$-theory of the corresponding complex graph algebra.

In this paper we introduce the notion of a generalized weak biFrobenius algebra (or briefly GWBF-algebra) which is designed to cover both weak Hopf algebras and bi-Frobenius algebras. Then we study properties of various modular elements of GWBF-algebras. As an application we mainly show Radford's formula for S$^{4}$ in the setting of GWBF-algebras.

While infinitely many examples of two triads of integers with equal sums of squares and cubes are known, until now there is no published example of three triads of integers with this property. In this paper we obtain, in parametric terms, three triads of integers with equal sums of squares and cubes and show that infinitely many similar parametric solutions for such triads can be obtained.

The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for $\gamma_k=\gamma_k(1)$. Some extensions are briefly described, as well as the relevance to expansions of Dirichlet $L$ functions.

The aim of this article is to establish some conditions under which the attractors of iterated function systems become dendrites. We associate to an attractor of an iterated function system a graph and we prove that, for a large class of iterated function systems, their attractors are dendrites if and only if the associated graph is a tree. We also give some examples of such sets.

Let $G_N$ be either a symplectic, unitary or Hermitian orthogonal group of rank $2N=2m+2n$ with $m\leq n$. We show that the restriction of a Siegel-Hilbert Eisenstein series on $G_N$ to the diagonally embedded group $G_m\times G_n$ has a nontrivial cuspidal component in the smaller variable. As a consequence, we explicitly construct classes of Siegel-Hilbert cuspforms with rational-valued Fourier coefficients.

This paper provides an asymptotic estimate for the expected number of real zeros of algebraic polynomials $P_n (x)=a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-1}x^{n-1} $, where $a_j$'s ($j=0,1,2,\cdots,n-1$) are a sequence of normal standard independent random variables with the symmetric property $a_j \equiv a_{n-1-j}$. It is shown that the expected number of real zeros in this case still remains asymptotic to $(2/\pi)\log n$. In the previous study, it was shown for the case of random trigonometric polynomials this expected number of real zeros is halved when we assume the above symmetric properties.

Let $\varphi$ be an elliptic automorphism of the open unit disc of order~$k$ and rotation parameter $w$, and let $C_{\varphi}$ be the composition operator on the Hardy space $H^2$ induced by $\varphi$. Then there are orthogonal projections $P_{s}$, $s=0, 1,\ldots, k-1$ with identical norm such that $C_{\varphi}=\sum_{s=0}^{k-1} w^s P_{s}$.

The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

Recall that a group $G$ is a Camina group if every nonlinear irreducible character of $G$ vanishes on $G \setminus G^{\p}$. Dark and Scoppola classified the Camina groups that can occur. We present a different proof of this classification using Theorem~2, which strengthens a result of Isaacs on Camina pairs. Theorem~2 is of independent interest.

In this paper, we establish a theorem concerning value distribution of certain difference polynomials of meromorphic functions, which extends [{\bf14}, Theorem 2] and [{\bf16}, Theorem 1.2]. Applying this result, we prove some uniqueness theorems of meromorphic functions whose certain difference polynomials share a non-zero polynomial, which extends [{\bf18}, Theorems 1.1 and 1.2] and [{\bf23}, Theorem 6], where the meromorphic functions are of finite order.

In this paper we prove that for any prime $p\ge 11$, $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} +4p^2\sum_{1\le i\lt j\le p-1}\frac{1}{ij}\pmod{p^7} $$ holds. This is a generalization of the famous Wolstenholme's theorem which asserts that ${2p-1\choose p-1} \equiv 1 \ \pmod{\,p^3}$ for all primes $p\ge 5$. Our proof is elementary, and it does not use a standard technique involving the classic formula for power sums in terms of the Bernoulli numbers. Notice that the above congruence reduced modulo $p^6$, $p^5$ and $p^4$ yields related congruences obtained by Tauraso, Zhao and Glaisher, respectively.

We construct an algebra of nonlinear generalized tensor fields on manifolds in the sense of Colom\-beau, i.e., containing distributional tensor fields as a linear subspace and smooth tensor fields as a faithful subalgebra. The use of a background connection on the manifold allows for a simplified construction based on the existing scalar theory of full diffeomorphism invariant Colombeau algebras on manifolds, still having a canonical embedding of tensor distributions. In the particular case of the Levi-Civita connection on Riemannian manifolds, one obtains that this embedding commutes with pullback along homotheties and Lie derivatives along Killing vector fields only.

In this paper we answer the question of whether we can extend the current proof of Abhyankar's two point lemma to any positive weight in the negative. This is done by constructing suitable $w$-homogeneous polynomial pairs of positive weight that behave the same way under the Jacobian as their negatively weighted counterparts, but they do not possess the same pairwise properties.

We generalize the notion of $\m$-full and basically full ideals in the setting of tight closure and demonstrate some $\m$-full and basically full ideals in non-regular rings.