There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree~2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations of Saito-Kurokawa lifts.

The Bellman function method is an efficient device which enables relating certain types of estimates arising in probability and harmonic analysis to the existence of the associated special function satisfying appropriate majorization and concavity. This technique has gained considerable interest in recent years and led to many interesting results concerning the boundedness of wide classes of singular integrals, Fourier multipliers, maximal functions and other related objects. The objective of this survey is to describe the Bellman function approach to certain classical results for martingale transforms. We present the detailed study of the weak-type and moment estimates, and develop some arguments which allow us to simplify and extend the statements, originally proven by Burkholder and others.

We use polynomial Dirac spinors associated to Euclidean Dirac-type operators and separation of variables to investigate the spectral theory of certain spherical Dirac-type operators. While the spectral theories of our main examples, the spherical Dirac and Laplace-Beltrami operators, are known, this is the first time they are treated together, in a unified manner. In particular, the multiplicities of these spectra, a topic difficult to negotiate in many previous treatments, are presented in simple closed form.

In this paper, we investigate rationality of the Barnes numbers, and we find explicit good bounds for their denominators. In addition, we give Fourier expansion of Barnes polynomials and, from this study, we connect generalized Barnes numbers to values of Dirichlet $L$-function at non-negative integers.

In this paper we get asymptotic formulas for eigenvalues and eigenfunctions of discontinuous two-point boundary value problems with the eigenparameter in the boundary conditions with transmission conditions at the two points of discontinuity.

When our problem is continuous the obtained results coincide with the corresponding results in [{\bf3}].

In this paper we obtain a parametric solution of the quintic diophantine equation $ab(a+b)(a^2+ab+b^2)=cd(c+d)(c^2+cd+d^2)$. We use this solution to obtain parametric solutions of two diophantine systems concerning fifth powers, namely, the system of simultaneous equations $x_1+x_2+x_3 = y_1+y_2+y_3 =0$, $x_1^5+x_2^5+x_3^5 = y_1^5+y_2^5+y_3^5$, and the system of equations given by $\sum_{i=1}^5x_i^k =\sum_{i=1}^5 y_i^k$, $k=1,\,2,\,4,\,5$.

The category $W$ is archimedean $l$-groups $G$ with distinguished weak order unit $e_G$, with unit-preserving $l$-group homomorphisms. For $G$ in $W$, $G^*$ denotes the convex sub-$l$-group generated by $e_G$. If $G$ satisfies [any isomorphism of $H^*$ with $G^*$ extends to an embedding of $H$ in $G$], then $G$ is called $\ast$-maximum. ($H^*$ and $G^*$ are isomorphic if and only if the unit intervals $[0, e_H]$ and $[0, e_G]$ are isomorphic MV-algebras.) This paper analyzes the property ``$\ast$-maximum'': Several characterizations are given. It is shown that any $\ast$-maximum $G$ has quasi-F Yosida space; this applies to prove a conjecture of the author and J\. Martinez about ``rings of $\omega_1$-quotients.'' It is shown that each $G$ in $W$ has a $\ast$-maximum hull, and this hull is described.

We consider a complete locally $m$-convex $^*$-algebra with continuous involution, which is also a ``perfect'' projective limit, and describe its multiplier algebra, {under a weaker topology}, making it a locally $C^*$-algebra. The same is applied in the case of certain locally convex $H^*$-algebras.

This paper studies the properties of the number of isolated vertices in a random graph where vertices arrive one-by-one at times $1,2,\ldots$\,. They are connected by edges to the previous vertices independently with the same probability. Assuming that the probability of an edge tends to zero, we establish the asymptotics of large, normal, and moderate deviations for the stochastic process of the number of the isolated vertices considered at times inversely proportional to that probability. In addition, we identify the most likely trajectory for that stochastic process to follow conditioned on the event that at a large time the graph is found with a large number of isolated vertices.

By using the critical point theory, the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for difference equations containing both advance and retardation is obtained. The proof is based on the mountain pass lemma in combination with periodic approximations. Our results extend the results of 2007.

The logarithm of the Estrada index has been recently proposed as a spectral measure to characterize the robustness of complex networks. We derive novel analytic lower bounds for the logarithm of the Estrada index based on the Laplacian spectrum and the mixing times of random walks on the network. The main techniques employed are some inequalities, such as the thermodynamic inequality in statistical mechanics, a trace inequality of von Neumann, and a refined harmonic-arithmetic mean inequality.

In this work we use elementary methods to discuss the question of the minimal number of points with bad reduction over $\pl$ for elliptic curves $E/k(T)$ which are non-constant, respectively have non-constant $j$-invariant.

The 2- and 4-dissections of some infinite products are established in this paper. As corollaries of our results, we derive the 4-dissections of some continued fractions appearing in Ramanujan's notebooks and their reciprocals.

This paper describes the $N$-dimensional vectorial Sturm-Liouville problem with coupled boundary conditions. We first derive the asymptotic expressions of large eigenvalues for the vectorial Sturm-Liouville operator with smooth coefficients. In addition, the regularized trace formula for the operator is calculated with residue techniques in complex analysis. These formulae are then used to obtain some results of inverse eigenvalue problems in the spirit of Ambarzumyan.