It is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

The main aim of this paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and prove that an hyperplane of $c$ is isometric to the whole space if and only if it is 1-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to ${{\ell }_{1}}$ and give a complete description of the preduals of ${{\ell }_{1}}$ under the assumption that the standard basis of ${{\ell }_{1}}$ is weak$^{*}$-convergent.

In a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in ${{H}^{s}}$ for $s>1-\alpha /2$ and globally well-posed for $s>10\alpha -1/12$. In this paper we define an invariant probability measure $\mu$ on ${{H}^{s}}$ for $s<\alpha -1/2$, so that for any $\text{ }\!\!\varepsilon\!\!\text{ }>0$ there is a set $\Omega \subset {{H}^{s}}$ such that $\mu \left( {{\Omega }^{c}} \right)<\text{ }\!\!\varepsilon\!\!\text{ }$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense for $\frac{1-\alpha }{2}<\alpha -\frac{1}{2},i.e.,\alpha >\frac{2}{3}$ in an almost sure sense.

We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions that are applied in order to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations.

A magma$\left( M,\star \right)$ is a nonempty set with a binary operation. A double magma$\left( M,\star ,\bullet \right)$ is a nonempty set with two binary operations satisfying the interchange law$\left( w\star x \right)\bullet \left( y\star z \right)=\left( w\bullet y \right)\star \left( x\bullet z \right)$. We call a double magma proper if the two operations are distinct, and commutative if the operations are commutative. A double semigroup, first introduced by Kock, is a double magma for which both operations are associative. Given a non-trivial group $G$ we define a system of two magma $\left( G,\star ,\bullet \right)$ using the commutator operations $x\star y=\left[ x,y \right]\left( ={{x}^{-1}}{{y}^{-1}}xy \right)$ and $x\bullet y=\left[ y,x \right]$. We show that $\left( G,\star ,\bullet \right)$ is a double magma if and only if $G$ satisfies the commutator laws $\left[ x,y;x,z \right]=1$ and ${{\left[ w,x;y,z \right]}^{2}}=1$. We note that the first law defines the class of 3-metabelian groups. If both these laws hold in $G$, the double magma is proper if and only if there exist ${{x}_{0}},{{y}_{0}}\in G$ for which ${{\left[ {{x}_{0}},{{y}_{0}} \right]}^{2}}\ne 1$. This double magma is a double semigroup if and only if $G$ is nilpotent of class two. We construct a specific example of a proper double semigroup based on the dihedral group of order 16. In addition, we comment on a similar construction for rings using Lie commutators.

In this paper we define a space$VM{{O}_{P}}$ associated with a family $P$ of parabolic sections and show that the dual of $VM{{O}_{P}}$ is the Hardy space $H_{P}^{1}$. As an application, we prove that almost everywhere convergence of a bounded sequence in $H_{P}^{1}$ implies weak$^{\star }$ convergence

We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.

There are several notions of Ricci curvature tensor in Finsler geometry and spray geometry. One of them is defined by the Hessian of the well-known Ricci curvature. In this paper we will introduce a new notion of Ricci curvature tensor and discuss its relationship with the Ricci curvature and some non-Riemannian quantities. Using this Ricci curvature tensor, we shall have a better understanding of these non-Riemannian quantities.

A group $G$ is self dual if every subgroup of $G$ is isomorphic to a quotient of $G$ and every quotient of $G$ is isomorphic to a subgroup of $G$. It is minimal non self dual if every proper subgroup of $G$ is self dual but $G$ is not self dual. In this paper, the structure of minimal non self dual groups is determined.

Let ${{S}_{k}}\left( \Gamma \right)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL\left( 2,\mathbb{Z} \right)$. Let ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{g}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{h}}\left( n \right)$ be the $n$-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f\left( z \right)\in {{S}_{{{k}_{1}}}}\left( \Gamma \right),g\left( z \right)\in {{S}_{{{k}_{2}}}}\left( \Gamma \right)$, and $h\left( z \right)\in {{S}_{{{k}_{3}}}}\left( \Gamma \right)$, respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\text{ }{{\lambda }_{f}}{{\left( n \right)}^{4}}{{\lambda }_{g}}{{\left( n \right)}^{2}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{6}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{2}}{{\lambda }_{h}}{{\left( n \right)}^{4}}$, and ${{\text{ }\!\!\lambda\!\!\text{ }}_{g}}{{\left( {{n}^{3}} \right)}^{2}}$ twisted by the arithmetic function ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right)$.

Parts of the Brunn–Minkowski theory can be extended to hedgehogs, which are envelopes of families of affine hyperplanes parametrized by their Gauss map. F. Fillastre introduced Fuchsian convex bodies, which are the closed convex sets of Lorentz–Minkowski space that are globally invariant under the action of a Fuchsian group. In this paper, we undertake a study of plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the Fuchsian analogues of classical geometrical inequalities (analogues that are reversed as compared to classical ones).

A surface $\sum$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration, $G\left( \pi \right)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\text{ }\!\!\pi\!\!\text{ }$ is not an area form on the 2-sphere, then $G\left( \pi \right)$ is diffeomorphic to the cotangent bundle $T*\sum$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

We study the restriction of Bump–Friedberg integrals to affine lines $\left\{ \left( s+\alpha ,2s \right),s\in \mathbb{C} \right\}$. It has simple theory, very close to that of the Asai L-function. It is an integral representation of the product $L\left( s+\alpha ,\pi \right)L\left( 2s,{{\Lambda }^{2}},\pi \right)$, which we denote by ${{L}^{\operatorname{lin}}}\left( s,\pi ,\alpha \right)$ for this abstract, when $\pi$ is a cuspidal automorphic representation of $GL\left( k,\mathbb{A} \right)$ for $\mathbb{A}$ the adeles of a number field. When $k$ is even, we show that the partial $L$-function ${{L}^{\text{lin,S}}}\left( s,\text{ }\!\!\pi\!\!\text{ ,}\alpha \right)$ has a pole at $1/2$ if and only if $\pi$ admits a (twisted) global period. This gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that π has a twisted global period if and only if $L\left( \alpha +1/2,\text{ }\!\!\pi\!\!\text{ } \right)\ne 0$ and $L\left( 1,{{\Lambda }^{2}},\pi \right)=\infty $. When $k$ is odd, the partial $L$-function is holmorphic in a neighbourhood of $\operatorname{Re}\left( s \right)\ge 1/2$ when $\operatorname{Re}\left( \alpha \right)\,\,\text{is}\,\,\ge \text{0}\,$.

One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to $\mathbb{C}{{\mathbb{P}}^{2}}$ and a projection of the image curve froman appropriate point $p\in \mathbb{C}{{\mathbb{P}}^{2}}$ to the pencil of lines through $p$ . We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran, and I. Tyomkin.

Necessary and sufficient conditions are given for the existence of a graph decomposition of the Kneser Graph $K{{G}_{n,2}}$ and of the Generalized Kneser Graph $GK{{G}_{n,3,1}}$ into paths of length three.

Let $n$ be a positive even integer, and let $F$ be a totally real number field and $L$ be an abelian Galois extension which is totally real or $\text{CM}$ . Fix a finite set $S$ of primes of $F$ containing the infinite primes and all those which ramify in $L$ , and let ${{S}_{L}}$ denote the primes of $L$ lying above those in $S$ . Then $\mathcal{O}_{L}^{S}$ denotes the ring of ${{S}_{L}}$ -integers of $L$ . Suppose that $\psi$ is a quadratic character of the Galois group of $L$ over $F$ . Under the assumption of the motivic Lichtenbaum conjecture, we obtain a non-trivial annihilator of the motivic cohomology group $H_{\mathcal{M}}^{2}\left( \mathcal{O}_{L}^{S},\mathbb{Z}\left( n \right) \right)$ from the lead term of the Taylor series for the S-modified Artin $L$ -function $L_{L/F}^{S}\left( s,\psi\right)$ at $s=1-n$ .

Given a measure ${{\bar{\mu }}_{\infty }}$ on a locally symmetric space $Y=\Gamma \backslash G/K$ obtained as a weak-$*$ limit of probability measures associated with eigenfunctions of the ring of invariant differential operators, we construct a measure ${{\bar{\mu }}_{\infty }}$ on the homogeneous space $X=\Gamma \backslash G$ that lifts ${{\bar{\mu }}_{\infty }}$ and is invariant by a connected subgroup ${{A}_{1}}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then ${{\bar{\mu }}_{\infty }}$ is also the limit of measures associated with Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

This paper is concerned with the following elliptic system of Hamiltonian type

$$\left\{ \begin{align} & -\Delta u+V\left( x \right)u={{W}_{v}}\left( x,u,v \right),\,x\in {{\mathbb{R}}^{N}}, \\ & -\Delta v+V\left( x \right)v={{W}_{u}}\left( x,u,v \right),\,x\in {{\mathbb{R}}^{N}}, \\ & u,v\in {{H}^{1}}\left( {{\mathbb{R}}^{N}} \right), \\ \end{align} \right.$$

where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta +V,W\left( x,u,v \right)$ is periodic in $x$ and superlinear in $u$ and $v$ at infinity. We develop a direct approach to finding ground state solutions of Nehari–Pankov type for the above system. Our method is especially applicable to the case when

$$W\left( x,u,v \right)=\sum\limits_{i=1}^{k}{\,\int _{0}^{\left| \alpha iu+\beta iv \right|}\,{{g}_{i}}\left( x,\,t \right)t\text{d}t+\sum\limits_{j=1}^{l}{\int_{0}^{\sqrt{{{u}^{2}}+2{{b}_{juv+aj{{v}^{2}}}}}}{{{h}_{j}}\left( x,t \right)tdt,}}}$$

where ${{\alpha }_{i}},{{\beta }_{i}},{{a}_{j}},{{b}_{j}}\in \mathbb{R}$ with $\alpha _{i}^{2}+\beta _{i}^{2}\ne 0$, and ${{a}_{j}}>b_{j}^{2},{{g}_{i}}\left( x,t \right)$ and ${{h}_{j}}\left( x,t \right)$ are nondecreasing in $t\in {{\mathbb{R}}^{+}}$ for every $x\in {{\mathbb{R}}^{N}}$ and ${{g}_{i}}\left( x,0 \right)={{h}_{j}}\left( x,0 \right)=0$.

Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$, and $X$ is an $R$-module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules $\text{H}_{\mathfrak{a}}^{i}\left( X \right)$. Then we give some inequalities between the Betti numbers of $X$ and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of $X$ in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of $\text{H}_{\mathfrak{a}}^{i}\left( X \right)$ in terms of the flat dimensions of the modules $\text{H}_{\mathfrak{a}}^{j}\left( X \right),j\ne i$, and that of $X$.