We define topological 2-groupoids and study locally compact 2-groupoids with 2-Haar systems. We consider quasi-invariant measures on the sets of 1-arrows and unit space and build the corresponding vertical and horizontal modular functions. For a given 2-Haar system, we construct the vertical and horizontal full \Cst -algebras of a 2-groupoid and show that they are independent of the choice of the 2-Haar system, up to strong Morita equivalence. We make a correspondence between their bounded representations on Hilbert spaces and those of the 2-groupoid on Hilbert bundles. We show that representations of certain closed 2-subgroupoids are induced to representations of the 2-groupoid and use regular representation to build the vertical and horizontal reduced \Cst -algebras of the 2-groupoid. We establish strong Morita equivalence between \Cst -algebras of the 2-groupoid of composable pairs and those of the 1-arrows and unit space. We describe the reduced \Cst -algebras of r-discrete principal 2-groupoids and find their ideals and masa's.

Let $M$ be a module over a commutative ring and let $\Spec (M)$ be the collection of all prime submodules of $M$. We topologize $\Spec (M)$ with quasi-Zariski topology and, for a subset $T$ of $\Spec (M)$, we introduce a new graph $G(\tau ^{*}_{T})$, called the \textit {quasi-Zariski topology-graph}. It helps us to study algebraic (respectively, topological) properties of $M$ (respectively, $\Spec (M)$) by using graph theoretical tools. Also, we study the annihilating-submodule graph and investigate the relation between these two graphs.

We show that there are cubic irrationals whose real fifth roots can be constructed by marked ruler and compass.

In this paper, we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this result to establish a necessary, though not sufficient, condition for the solvability of the simultaneous quadratic diophantine equations \[ Q_j(x_1,\ x_2,\ x_3,\ x_4)=0,\quad j=1,\ 2, \] and give a method of obtaining their complete solution. In general, when these two equations have a rational solution, they represent an elliptic curve, but we show that there are several cases in which their complete solution may be expressed by a finite number of parametric solutions and/or a finite number of primitive integer solutions. Finally, we relate the solutions of the quartic equation \[ y^2=t^4+a_1t^3+a_2t^2+a_3t+a_4 \] to the solutions of a pair of quadratic diophantine equations, and thereby obtain new formulae for deriving rational solutions of the aforementioned quartic equation starting from one or two known solutions.

We will call a hyperbolic polygon \textit {cyclic}, \textit {horocyclic}, or \textit {equidistant} if its vertices lie on a metric circle, a horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Such convex $n$-gons are parametrized by the subspaces of $(\mathbb {R}^+)^n$ that contain their side length collections, and area and circumcircle or ``collar'' radius determine symmetric, smooth functions on these spaces. We give formulas for and bounds on the derivatives of these functions and make some observations on their behavior. Notably, the monotonicity properties of area and circumcircle radius exhibit qualitative differences on the collection of centered vs non-centered cyclic polygons, where a cyclic polygon is \textit {centered} if it contains the center of its circumcircle in its interior.

In this paper, we study the Ruan cohomologies of $X^s$ and $X^{sf}$, the natural compactifications of $V^s$ and $V^{sf}$, where $V^s$ and $V^{sf}$ are the two small resolutions of \[ V=\{(x,y,z,w)\mid xy-zw=0\}/\mu _r(1,-1,0,0),\quad r>1, \] the finite group quotient of the singular conifold. There is an additive isomorphism between the Chen-Ruan cohomologies $\phi :H^*_{CR}(X^s)\to H^*_{CR}(X^{sf})$. We study the three-point orbifold Gromov-Witten invariants of the exceptional curves $\Gamma ^s$ on $X^s$ and $\Gamma ^{sf}$ on $X^{sf}$ and show that the corresponding Ruan cohomology ring structures on the Chen-Ruan cohomologies of $X^s$ and $X^{sf}$, defined by these three-point functions, are isomorphic to each other under the map $\phi $ and the identification $[\Gamma ^s]\leftrightarrow -[\Gamma ^{sf}]$.

In this paper, we develop the geometry of bounded Fr\'{e}chet manifolds. We prove that a bounded Fr\'{e}chet tangent bundle admits a vector bundle structure. But, the second order tangent bundle $T^2M$ of a bounded Fr\'{e}chet manifold $M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. As an application, we prove the existence and uniqueness of an integral curve of a vector field on $M$.

Let $\mathcal {S}_k(N)$ be the space of holomorphic cusp forms of weight $k$, level $N$ and let $\mathcal {B}_k(N)$ be an orthogonal basis of $\mathcal {S}_k(N)$ consisting of newforms. Let $L(s, \textup {sym}^2 f)$ be the symmetric square $L$-function of $f\in \mathcal {B}_k(N)$. In this paper, the lower bound of the higher moment of $L(1/2,\textup {sym}^2 f)$ is established, i.e., for any even positive number $r$, \[ \sum _{f\in \mathcal {B}_k(N)}\omega _f^{-1}L\bigg (\frac 12, \textup {sym}^2 f\bigg )^r \gg (\log N)^{{r(r+1)}/{2}} \] holds for $N\rightarrow \infty $.

In 1887, beneath a canopy of stars, Srinivasa Ramanujan commenced his brief existence on this planet. In a universe at the mercy of its entropy, against all odds, a genius was born. His destiny was mathematics, a subject born thousands of years earlier. The power of this discipline is not to be denied. After all, with our minds in the stars, we have placed footprints on the moon. The conquest of the moon was a triumph of applied mathematics; however, it was the landscape of pure mathematics that awaited Ramanujan. In time, he would explore it with passion, leaving footprints lasting for eternity.

Professor Bruce C.~Berndt has done a remarkable job of editing the notebooks Ramanujan left behind. In particular, Berndt's Chapter 9 of \textit {Ramanujan's notebooks Part} I provides a magnificent in-depth look at raw mathematical talent in action. The primary purpose of this article is to present three Chapter~9 related results, a series evaluation and two new functional equations, that Ramanujan either missed or his work on them was lost. The secondary purpose is to present what Berndt calls a ``corrected version'' \cite [page 233]{Berndt} of an incorrect Chapter~9 formula of Ramanujan. This series evaluation represents one of the few serious mistakes to be found in Ramanujan's work.

Dirichlet's theorem on the distribution of primes in arithmetic progressions states that every positive integer sequence $\{an+b\mid n\geq 0\}$ with $a$ and $b$ coprime contains infinitely many primes. In 1959, Golomb pointed out that, by taking such arithmetic progressions as a base for a topology $\mathcal D$ on the positive integers, the resulting topological space $(\mathbb Z^+$, $\mathcal D)$ is both Hausdorff and connected. More recently, Knopfmacher and Porubsky showed that $(\mathbb Z^+$, $\mathcal D)$ is a topological semigroup under multiplication. After revisiting this result, we show that Dirichlet's theorem is implied by the statement that the $\mathcal D$-closure of the primes contains the semiprimes.

We construct Abelian model structures on the category of chain complexes over a ring~$R$, from the notion of homological dimensions of modules. Given an integer, $n > 0$, we prove that the left modules over a ringoid~$\mathfrak {R}$ with projective dimension at most $n$ form the left half of a complete cotorsion pair. Using this result, we prove that there is a unique Abelian model structure on the category of chain complexes over $R$, where the exact complexes are the trivial objects and the complexes with projective dimension at most $n$ form the class of trivially cofibrant objects. In \cite {Rada}, the authors construct an Abelian model structure on chain complexes, where the trivial objects are the exact complexes, and the class of cofibrant objects is given by the complexes whose terms are all projective. We extend this result by finding a new Abelian model structure with the same trivial objects where the cofibrant objects are given by the class of complexes whose terms are modules with projective dimension at most $n$. We also prove similar results concerning flat dimension.

In this paper, we consider slowly non-dissipative reaction-diffusion equations and establish several estimates. In particular, we manage to control $L^p$ norms of the solution in terms of $W^{1,2}$ norms of the initial conditions, for every $p>2$. This is done by carefully combining preliminary estimates with Gronwall's inequality and the Gagliardo-Nirenberg interpolation theorem. By considering only positive solutions, we obtain upper bounds for the $L^p$ norms, for every $p>1$, in terms of the initial data. In addition, explicit estimates concerning perturbations of the initial conditions are established. The stationary problem is also investigated. We prove that $L^2$ regularity implies $L^p$ regularity in this setting, while further hypotheses yield additional estimates for the bounded equilibria. We close the paper with a discussion of the connection between our results and some related problems in the theory of slowly non-dissipative equations and attracting inertial manifolds.

In this paper, we describe a copy-and-paste method for constructing a class of infinite self-similar trees. A copy-paste tree is constructed by repeatedly attaching copies of a finite tree (called a \textit {generator}) to certain designated attachment vertices. We show that each generator has an associated nonnegative matrix which can be used to determine a formula for the growth function of the copy-paste tree. In our main theorem, we use results from Perron-Frobenius theory to show that every copy-paste tree has exponential growth, with growth rate equal to the spectral radius of its associated matrix.

In \cite {Sz13-1} we defined and described the {\it regular infinite or bounded} $p$-gonal prism tilings in $\SLR $ space. We proved that there exist infinitely many regular infinite $p$-gonal face-to-face prism tilings $\cT ^i_p(q)$ and infinitely many regular bounded $p$-gonal non-face-to-face prism tilings $\cT _p(q)$ for integer parameters $p,q$, $3 \le p$, $ {2p}/({p-2}) \lt q$. Moreover, in \cite {MSz14, MSzV13} we have determined the symmetry group of $\cT _p(q)$ via its index~2 rotational subgroup, denoted by $\mathbf {pq2_1}$ and investigated the corresponding geodesic and translation ball packings.

In this paper, we study the structure of the regular infinite or bounded $p$-gonal prism tilings and we prove that the side curves of their base figures are arcs of Euclidean circles for each parameter. Furthermore, we examine the non-periodic geodesic ball packings of congruent regular non-periodic prism tilings derived from the regular infinite $p$-gonal face-to-face prism tilings $\cT ^i_p(q)$ in $\SLR $ geometry. We develop a procedure to determine the densities of the above non-periodic optimal geodesic ball packings and apply this algorithm to them. We search for values of parameters $p$ and $q$ that provide the largest packing density. In this paper, we obtain greater density $0.626606\ldots $ for $(p, q) = (29,3)$ than the maximum density of the corresponding periodic geodesic ball packings under the groups $\mathbf {pq2_1}$.

In our work we use the projective model of $\SLR $ introduced by {Moln\'ar} in \cite {M97}.