Suppose $\varphi$ is an analytic self-map of open unit disk $\mathbf{D}$ and $\psi$ is an analytic function on $\mathbf{D}$. Then a weighted composition operator induced by $\varphi$ with weight $\psi$ is given by $(W_{\psi, \varphi}f)(z)= \psi(z)f(\varphi(z))$, for $z \in \mathbf{D}$ and $f$ analytic on $\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\psi, \varphi}$. In terms of fixed points in the closed unit disk $\overline{\mathbf{D}}$, conditions under which $W_{\psi, \varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\varphi}$ are given in terms of the angular derivative $\varphi^{\prime}(\zeta)$ where $\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\beta_1), H^q(\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

We study Borel systems and continuous systems of measures, with a focus on mapping properties: compositions, liftings, fibred products and disintegration. Parts of the theory we develop can be derived from known work in the literature, and in that sense this paper is of an expository nature. However, we put the above notions in the spotlight and provide a self-contained, purely measure-theoretic, detailed and thorough investigation of their properties, and in that aspect our paper enhances and complements the existing literature. Our work constitutes part of the necessary theoretical framework for categorical constructions involving measured and topological groupoids with Haar systems, a line of research we pursue in separate papers.

As a particular example of a general theorem presented in \cite{me}, there is a conditional expectation from the tensor product of Cuntz algebras, $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$, onto the Cuntz algebra $\mathcal{O}_{d_1d_2}$. Motivated by this example, we examine the embedding of $\mathcal{O}_{d_1d_2}$ in $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$, first by examining the index of the conditional expectation mentioned, and then by expressing $\mathcal{O}_{d_1}\otimes\mathcal{O}_{d_2}$ as a concrete Paschke crossed product by an endomorphism and then abstractly as the image of a faithful representation of the Stacey crossed product of $\mathcal{O}_{d_1d_2}$ by the same endomorphism.

By means of the inverse series relations due to Carlitz \cite{carlitz}, we establish several transformation formulae for nonterminating $q$-series, which will systematically be employed to review identities of the Rogers-Ramanujan type moduli 5, 7, 8, 10, 14 and 27.

Triangles having rational sides $a, b, c$ and rational area $Q$ are called \textit{Heron triangles}. Associated to each Heron triangle is the quartic \[ v^2=u(u-a)(u-b)(u-c). \] The Heron formula states that $Q=\sqrt{P(P-a)(P-b)(P-c)}$ where $P$ is the semi-perimeter of the triangle, so the point $(u, v)=(P, Q)$ is a rational point on the quartic. Also, the point of infinity is on the quartic. By a standard construction, it can be proved that the quartic is equivalent to the elliptic curve \[ y^2=(x+ a\, b)(x+ b \,c)(x+c\, a). \] The point $(P, Q)$ on the quartic transforms to \[ (x, y)= \bigg(\frac{-2 a b c }{a + b +c} ,\frac{ 4 Q a b c}{(a+ b+ c)^2}\bigg) \] on the cubic, and the point of infinity goes to $(0, a b c)$. Both points are independent, so the family of curves induced by Heron triangles has rank $\geq 2$. In this note we construct subfamilies of rank at least $3$, $4$ and $5$. For the subfamily with rank $\geq 5$, we show that its generic rank is exactly equal to $5$, and we find free generators of the corresponding group. By specialization, we obtain examples of elliptic curves over $\Q$ with rank equal to $9$ and $10$. This is an improvement of results by Izadi et al., who found a subfamily with rank $\ge 3$ and several examples of curves of rank $7$ over $\Q$.

We show that the following $K_0$-monoid properties of $C^*$-algebras in the class $\Omega$ are inherited by simple unital $C^*$-algebras in the class $TA\Omega$: \begin{enumerate} \item[(1)] almost divisible, \item[(2)] directly finite, \item[(3)] strong Riesz interpolation property. \end{enumerate}

We extend a recent inequality due to Fournier, Letac and Ruscheweyh to a class of inequalities involving a bound-preserving operator as a parameter.

This paper is devoted to the study of periodic solutions of the first-order vector differential equations $x'(t)+f(t,x(t))=0$. We first introduce the concepts of positive, negative and mixed-type solutions. Then, by using a fixed point theorem in cones, we obtain some existence and multiplicity results of such solutions. Furthermore, we also present some examples to illustrate our main results.

We investigate the asymptotic solutions of the planar dynamic systems and the second order equations on a time scale by using a new version of Levinson's asymptotic theorem. In this version the error estimate is given in terms of the characteristic (Riccati) functions which are constructed from the phase functions of an asymptotic solution. It means that the improvement of the approximation depends essentially on the asymptotic behavior of the Riccati functions. We describe many different approximations using the flexibility of this approach. As an application we derive the analogue of D'Alembert's formula for the one dimensional wave equation in a discrete time.

We provide two conditions, both in the spirit of classical regularity, that are equivalent to the existence of the steady state for a stochastic matrix. Our development of these characterizations sidesteps Perron-Frobenius theory for non-negative matrices, hinging instead on an elementary fixed point result that complements Banach's contraction mapping theorem.

The purpose of this paper is to give some properties of several Bernoulli polynomials to express the integral of those polynomials from 0 to 1 in terms of beta and gamma functions. From those properties, we derive some relations on the integral of the product of Bernoulli polynomials and new identities on the Bernoulli numbers.

The aim of this expository/pedagogical paper is to describe a Gram-Schmidt biorthogonalization method in such a way that it can be used as an introduction to the subject for undergraduate presentation. The task of biorthogonalization naturally arises when the scalar product of vectors formed are linear combinations of two sets of linearly independent vectors, as the case may be. If one wants the scalar product to have the usual form, the two sets of basis vectors should be biorthogonal. If they are not, the question of biorthogonalization arises. New is the detailed description of the biorthogonalzation method for teaching purposes as well as the comparison of this method with Schmidt's orthogonalization method in the case when the sets of linearly independent vectors are identical.

We introduce and study $t$-coverings in $E^n$, i.e., arrangements of proper translates of a convex body $K \subset E^n$ sufficient to cover $K$. First, we investigate relations between $t$-coverings of the whole of $K$ and $t$-coverings of its boundary only. Refining the notion of $t$-covering in several ways, we then derive, particularly for centrally symmetric convex bodies and $n = 2$, theorems which are interesting for the geometry of normed planes. These statements are related to respective generalizations of Ti\c{t}eica's and Miquel's theorem as well as to notions like Voronoi regions. We also compare $t$-coverings with coverings in the spirit of Hadwiger, using smaller homothetical copies of $K$ instead of proper translates. This is done via a slight modification of Boltyanski's and Hadwiger's notion of illumination. Finally, we give upper bounds on the cardinalities of $t$-coverings.

Necessary and sufficient conditions for \[ (ab)^\#=b^\#(a^\dag abb^\#)^\#a^\dag \] and \[ (ab)^\#= b^\dag(a^\#abb^\dag)^\#a^\# \] to hold in rings with involution are presented. Also, some equivalent conditions concerning the reverse order laws $(ab)^\#=b^\dag a^\#$ and $(ab)^\#=b^\#a^\dag$ are studied.

We prove qualitatively sharp heat kernel bounds in the setting of Fourier-Bessel expansions when the associated type parameter $\nu$ is half-integer. Moreover, still for half-integer~$\nu$, we also obtain sharp estimates of all kernels subordinated to the heat kernel. Analogous estimates for general $\nu > -1$ are conjectured. Some consequences concerning the related heat semigroup maximal operator are discussed.

Let $N$ be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let $Z$ be the center of $N$. Assume that $N=P\rtimes M$ such that $P$ and $M$ are simply connected, connected abelian Lie groups, $P$ is a maximal normal abelian subgroup of $N$, $M$ acts non-trivially on $P$ by automorphisms and $\dim P/Z=\dim M$. We study bandlimited subspaces of $L^2(N)$ which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of bandlimited subspaces of $L^2(N)$ which do not admit a single Parseval frame. We also provide some conditions under which continuous wavelets transforms related to the left regular representation admit discretization, by some discrete set $\Gamma\subset N$. Finally, we show some explicit examples in the last section.

Given a genus $3$ canonical curve $X=\{ F=0\}$ we derive a set of equations for an open affine set of the Jacobian $J(X)$. The law group on the Jacobian is also explicitly constructed and, as an application, a set of equations for Kummer's variety $K(X)$ is obtained.

We show that the number of integer points on an elliptic curve $y^2=f(x)$ with $X_0\lt x\le X_0+X$ is $\ll X^{1/2}$ where the implicit constant depends at most on the degree of $f(x)$. This improves on various bounds of Cohen \cite{SDC}, Bombieri and Pila \cite{BP} and of Pila \cite{PJ}, and others. In particular it follows that the number of positive integral solutions to $x^3+y^2=n$ is $\ll n^{1/6}$.

In this paper we deal with the existence and multiplicity of solutions for the $p$-Laplacian problems involving critical and supercritical Sobolev exponent via variational arguments. By means of the truncation combining with the Moser iteration, we extend the result obtained by Ricceri \cite{14} to the critical and supercritical case.