A generalization of Kolmogorov’s theorem to biorthogonal systems

The fundamental Kolmogorov’s theorem about divergent trigonometric Fourier series is generalized to bounded biorthonormal systems defined on a separable metric space with Borel regular outer measure. Sharp lower bounds at points and on sets of positive measure are obtained for the arithmetic means of the symmetrized Lebesgue functions of biorthonormal systems defined on an arbitrary measure space. Earlier, similar results were obtained by the author for orthogonal systems on an interval.     

Arithmetic properties arising from Ramanujan’s theta functions

We prove some interesting arithmetic properties of theta function identities that are analogous to q-series identities obtained by Michael D. Hirschhorn. In addition, we find infinite family of congruences modulo powers of 2 for representations of a non-negative integer n as \(\triangle _1+4\triangle _2\) and \(\triangle +k\square \).     

Phantom holomorphic projections arising from Sturm’s formula

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.     

Triple products of Coleman’s families

We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. We discuss a problem-solving approach in number theory, which is based on the use of generating functions and their connection with modular forms. In particular, the critical values of various L-functions of modular forms produce nontrivial but computable solutions of arithmetical problems. Namely, for a prime number we consider three classical cusp eigenforms

Partition identities arising from Somos’s theta function identities

In this paper, we establish certain partition identities for theta function identities of level 14 discovered by M. Somos.     

Partition identities arising from Ramanujan’s formulas for multipliers

We find new partition identities arising from Ramanujan’s formulas of multipliers. Several of the identities are for overpartitions, overpartition pairs, and \(\ell \)-regular partitions.     

A Note on the characterization of Shannon-Wiener’s measure of information

Summary In [1] the author treated a characterization problem of the ShannonWiener measure of information for continuous probability distributions defined over an abstract measure space (R, S, m), wherem is a σ-finite measure over a σ-field S of subsets ofR, whose rangeM(S) is such thatM(S)=[0, ∞]. This condition on the range of the basic measure, however, can slightly be altered such thatM(S)=[0,1], and this modification is useful for characterization of the Kullback-Leibler mean information. In the present paper, it is shown that the characterization procedure of [1] can be applicable to continuous probability distributions defined on a finite measure space.     

New modified optimal families of King’s and Traub-Ostrowski’s methods

Based on quadratically convergent Schröder’s method, we derive many new interesting families of fourth-order multipoint iterative methods without memory for obtaining simple roots of nonlinear equations by using the weight function approach. The classical King’s family of fourth-order methods and Traub-Ostrowski’s method are obtained as special cases. According to the Kung-Traub conjecture, these methods have the maximal efficiency index because only three functional values are needed per step. Therefore, the fourth-order family of King’s family and Traub-Ostrowski’smethod are the main findings of the present work. The performance of proposed multipoint methods is compared with their closest competitors, namely, King’s family, Traub-Ostrowski’s method, and Jarratt’s method in a series of numerical experiments. All the methods considered here are found to be effective and comparable to the similar robust methods available in the literature.     

A canonical module characterization of Serre’s (R1)

In this short note, we give a characterization of domains satisfying Serre’s condition (R₁) in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author [9 Yanagawa, K. (2015). Dualizing complexes of seminormal a?ne semigroup rings and toric face rings. J. Algebra 425:367–391.[Crossref], [Web of Science ®] , [Google Scholar]] where the normality is described in terms of the “shape” of the canonical module.     

A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited

Monatshefte für Mathematik - In this paper, we revisit the analyses of Stern (Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, W. Fr. Kaestner,...     

A convexity property and a new characterization of Euler’s gamma function

Our main results are:

1. Let α ≠ 0 be a real number. The function (Γ ? exp) ^α is convex on ${\mathbf{R}}$ if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x ₀ = 1.4616... denotes the only positive zero of ${\psi = \Gamma'/\Gamma}$ .

1. Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$

If there are a number b and a sequence of positive real numbers (a _n ) ${(n \in \mathbf{N})}$ with ${{\rm lim}_{n\to\infty} a_n =0}$ such that for every n the function ${(f \circ {\rm exp})^{a_n}}$ is Jensen convex on (b, ∞), then f is the gamma function.     

Analogues of Ramanujan’s partition identities and congruences arising from his theta functions and modular equations

In this paper, we study the partition function $p_{[c^{l}d^{m}]}(n)$ defined by $\sum_{n=0}^{\infty}p_{[c^{l}d^{m}]}(n)q^{n}=(q^{c};q^{c})_{\infty}^{-l}(q^{d};q^{d})_{\infty}^{-m}$ and prove some analogues of Ramanujan??s partition identities. We also deduce some interesting partition congruences.     

Generation of cosine families via Lord Kelvin’s method of images

We show that generation theorems for cosine families related to one-dimensional Laplacians in C[0, ∞] may be obtained by Lord Kelvin’s method of images, linking them with existence of invariant subspaces of the basic cosine family. This allows us to deal with boundary conditions more general than those considered before (Bátkal and Engel in J Differ Equ 207:1–20, 2004; Chill et al. in Functional analysis and evolution equations. The Günter Lumer volume, Birkhauser, Basel, pp 113–130, 2007; Xiao and Liang in J Funct Anal 254:1467–1486, 2008) and to give explicit formulae for transition kernels of related Brownian motions on [0, ∞). As another application we exhibit an example of a family of equibounded cosine operator functions in C[0, ∞] that converge merely on C ₀(0, ∞] while the corresponding semigroups converge on the whole of C[0, ∞].     

New analogues of Clausen?s identities arising from the theory of modular forms

Around 1828, T. Clausen discovered that the square of certain hypergeometric function can be expressed as a hypergeometric function. Special cases of Clausen?s identities were later used by S. Ramanujan in his derivation of 17 series for 1/π. Since then, there were several attempts to find new analogues of Clausen?s identities with the hope to derive new classes of series for 1/π. Unfortunately, none were successful. In this article, we will present three new analogues of Clausen?s identities. Their discovery is motivated by the study of relations between modular forms of weight 2 and modular functions associated with modular groups of genus 0.