For an integer $n$, let $d\left( n \right)$ denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum

$$S\left( x \right)\,:=\sum\limits_{m\le x,n\le x}{d\left( {{m}^{2}}+{{n}^{2}} \right)}$$

It is proved in the paper that, as $x\,\to \,\infty $,

$$S(x):={{A}_{1}}{{x}^{2}}\log x+{{A}_{2}}{{x}^{2}}+{{O}_{\in }}({{x}^{\frac{3}{2}+\in }}),$$

where ${{A}_{1}}$ and ${{A}_{2}}$ are certain constants and $\in $ is any fixed positive real number.

The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error $O({{x}^{\frac{5}{3}}}\,{{(\log \,x)}^{9}})$ claimed by Gafurov.

In this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense.

A number of interesting Diophantine equations are of the a form f(n) = f(a) f(b). Thus the case f(a) = aa has been studied by C. Ko [2] and W. H. Mills [3] and a class of non-trivial solutions has been found, though whether these give all the solutions is still unsettled. The case f(n) = n! has been mentioned by W. Sierpinski. Here the situation is that besides the trivial solutions m! = m! 1! and (m!-1)! m! = (m!)! and the special solution 10! =7! 6! no other solutions are known, nor are they known not to exist. In the present note we show that the equation in the title has no solutions. A sketch of a somewhat different proof that this equation has at most a finite number of solutions was recently communicated to the author by P. Erdös.

We study Calabi–Yau manifolds constructed as double coverings of ${{\mathbb{P}}^{3}}$ branched along an octic surface. We give a list of 87 examples corresponding to arrangements of eight planes defined over $\mathbb{Q}$. The Hodge numbers are computed for all examples. There are 10 rigid Calabi–Yau manifolds and 14 families with ${{h}^{1,2}}\,=\,1.$ The modularity conjecture is verified for all the rigid examples.

In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs, first introduced in 1978 by A. R. Camina. Camina’s work was inspired by the study of Frobenius groups. We show that if $(G,\,N,\,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient.

In this paper, we characterize Jordan $*$-derivations of a 2-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the following. Let $\delta :\,R\,\to \,R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R\,=\,U\,\oplus \,V$ such that both $U$ and $V$ are invariant under $\delta $. Moreover, $*$ is the identity map of $U$ and $\delta {{|}_{U}}$ is a derivation, and the Jordan $*$-derivation $\delta {{|}_{V}}$ is inner. We also prove the following. Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, char $R\,\ne \,2$, and let $\delta $ be a nonzero Jordan $*$-derivation of $R$. If $\delta $ is an elementary operator of $R$, then ${{\dim}_{C}}\,R\,<\,\infty $ and $\delta $ is inner.

We axiomatize the main properties of the classical Erdös-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

In this paper we study connections between topological games such as Rothberger, Menger, and compact-open games, and we relate these games to properties involving covers by ${{G}_{\delta }}$ subsets. The results include the following: (1) If TWO has a winning strategy in theMenger game on a regular space $X$, then $X$ is an Alster space. (2) If TWO has a winning strategy in the Rothberger game on a topological space $X$, then the ${{G}_{\delta }}$-topology on $X$ is Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.

In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of $\text{S}{{\text{L}}_{2\left( \mathbb{Z} \right)}}$ by means of the cross-ratio, weight 2 modular forms, quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant.

As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$.

If $f({{x}_{1}},...,{{x}_{k}})$ is a polynomial with complex coefficients, the Mahler measure of $f$ , $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus ${{\mathbb{T}}^{k}}$. We construct a sequence of approximations ${{M}_{n}}\,(f)$ which satisfy $-d{{2}^{-n}}\,\log \,2\,+\,\log \,{{M}_{n}}(f)\,\le \,\log \,M(f)\,\le \,\log \,{{M}_{n}}(f)$. We use these to prove that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials of fixed total degree $d$. Since ${{M}_{n}}\,(f)$ can be computed in a finite number of arithmetic operations from the coefficients of $f$ this also demonstrates an effective (but impractical) method for computing $M(f)$ to arbitrary accuracy.

It is shown that for totally real number fields the Main Conjecture in Iwasawa-Theory for p = 2 proposed by Fédérer implies the 2-primary part of the Birch-Tate-Conjecture in analogy with the case p odd.

In a recent paper, Murray Eden [5] generalized the simple identity for the Eulerian product,

1.1

and obtained the following infinite family of identities:

For A= 1,2, 3,…, let

1.2

where we assume throughout that |x| < 1, empty products equal unity and empty sums equal zero; then

1.3

As Eden noted, Fh(b;x) is the generating function of ph(m, n) which denotes the number of partitions of n into m parts, in which the largest part appears exactly h times and all other parts are distinct:

This paper is devoted to the study of Wirtinger-type inequalities for the Lebesgue $\Delta$-integral on an arbitrary time scale $\mathbb{T}$. We prove a general inequality for a class of absolutely continuous functions on closed subintervals of an adequate subset of $\mathbb{T}$. By using this expression and by assuming that $\mathbb{T}$ is bounded, we deduce that a general inequality is valid for every absolutely continuous function on $\mathbb{T}$ such that its $\Delta$-derivative belongs to $L_{\Delta }^{2}\,([a,\,b)\,\cap \,\mathbb{T})$ and at most it vanishes on the boundary of $\mathbb{T}$.

We classify the left-invariant metrics with nonnegative sectional curvature on $\text{SO}\left( 3 \right)$ and $U\left( 2 \right)$.

Let X be a closed convex subset of a Banach space and let T: X → X be a nonexpansive mapping, i.e.

In this section we continue to study the indecomposable matrix pairs adopting the same notation as in part I of this paper.

If the matrix A is regular and if it is symmetric or anti-symmetric such that

3.1

and if the matrix pair (X, A) is indecomposable then the corresponding representation space either is indecomposable or it is the direct sum of two indecomposable invariant subspaces. These are operator isomorphic if and only if the minimal polynomial mX of X is equal to (x-δ)μ where

3.2

at any rate there is even a decomposition of the representation space into the direct sum of two isotropic indecomposable invariant subspaces provided the characteristic of F is not 2.

McCarthy [4] generalized a necessary and sufficient condition for an arithmetic function to be totally multiplicative to the incidence algebra on a partially ordered set. Several equivalent conditions for an arithmetic function to be totally multiplicative are known [1], [2]. In this paper we generalize several of these (and some apparently new ones) to the regular convolution rings of Narkiewicz [5]. We also investigate the prime factorization of arithmetic functions in a certain subring of some of these regular convolution rings.

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group $F(2,m)\,(m>2)$ is a generalized torsion element.