László Fuchs’s contributions to commutative ring theory

In commemoration of the 90th birthday of Professor Laszlo Fuchs, this article gives a short account of some of his contributions to commutative ring theory.     

The effective application of a new approach to the generalized orienteering problem

The Orienteering Problem (OP) is an important problem in network optimization in which each city in a network is assigned a score and a maximum-score path from a designated start city to a designated end city is sought that is shorter than a pre-specified length limit. The Generalized Orienteering Problem (GOP) is a generalized version of the OP in which each city is assigned a number of scores for different attributes and the overall function to optimize is a function of these attribute scores. In this paper, the function used was a non-linear combination of attribute scores, making the problem difficult to solve. The GOP has a number of applications, largely in the field of routing. We designed a two-parameter iterative algorithm for the GOP, and computational experiments suggest that this algorithm performs as well as or better than other heuristics for the GOP in terms of solution quality while running faster. Further computational experiments suggest that our algorithm also outperforms the leading algorithm for solving the OP in terms of solution quality while maintaining a comparable solution speed.     

Eisenstein series and Ramanujan-type series for 1/π

Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π.     

Cranks—really, the final problem

A survey of Ramanujan’s work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks—not mock theta functions.     

Synergistic Self-Assembly of Oxoanions and d-Block Metal Ions with Heteroditopic Receptors into Triple-Stranded Helicates

Cooperativity effects among the interconnected anion and cation binding sites can profoundly alter the performance of heteroditopic receptors in selective ion pair recognition, processes that are oftentimes pertinent to biological systems and chemical separations. This work reports the effect of the linker that connects both binding sites on self-assembly of heteroditopic receptors in the presence of divalent first-row transition metal salts in solution and solid phase. Introduction of backbone flexibility in the receptor results in the formation of triple-stranded ion-pair helicates with an extraordinary selectivity towards CuSO₄ through an anion-induced fit.     

Generalization of the theorems of Barndorff-Nielsen and Balakrishnan–Stepanov to Riesz spaces

Positivity - In a Dedekind complete Riesz space, E, we show that if $$(P_n)$$ is a sequence of band projections in E then $$begin{aligned} limsup limits _{nrightarrow infty } P_n - liminf...     

Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi

We evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers?CRamanujan continued fraction play central roles in our evaluations.     

Weighted divisor sums and Bessel function series, IV

One fragment (p.?335) published with Ramanujan??s Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.