A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

Discrete optimization in public rail transport

Many problems arising in traffic planning can be modelled and solved using discrete optimization. We will focus on recent developments which were applied to large scale real world instances. Most railroad companies apply a hierarchically structured planning process. Starting with the definition of the underlying network used for transport one has to decide which infrastructural improvements are necessary. Usually, the rail system is periodically scheduled. A fundamental base of the schedule are the lines connecting several stations with a fixed frequency. Possible objectives for the construction of the line plan may be the minimization of the total cost or the maximization of the passengers’s comfort satisfying certain regulations. After the lines of the system are fixed, the train schedule can be determined. A criterion for the quality of a schedule is the total transit time of the passengers including the waiting time which should be minimized satisfying some operational constraints. For each trip of the schedule a train consisting of a locomotive and some carriages is needed for service. The assignment of rolling stock to schedule trips has to satisfy operational requirements. A comprehensible objective is to minimize the total cost. After all strategic and tactical planning the schedule has to be realized. Several external influences, for example delayed trains, force the dispatcher to recompute parts of the schedule on-line. A Web page with examples quoted in this survey can be found at http://www.math.tu-bs.de/mo/ismp.html.     

The application of automated reasoning to formal models of combinatorial optimization

Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy. In 1989 Helman [9] presented a common formalism that captures dynamic programming and branch and bound type algorithms. The formalism was later extended to include greedy algorithms. In this paper, we describe the application of automated reasoning techniques to the domain of our model, in particular considering some representational issues and demonstrating that proofs about the model can be obtained by an automated reasoning program. The long-term objective of this research is to develop a methodology for using automated reasoning to establish new results within the theory, including the derivation of new lower bounds and the discovery (and verification) of new combinatorial search strategies.     

Several improved asymptotic normality criteria and their applications to graph polynomials

As widely regarded, one of the most classical and remarkable tools to measure the asymptotic normality of combinatorial statistics is due to Harper's real-rooted method proposed in 1967. However, this classical theorem exists some obvious shortcomings, for example, it requests all the roots of the corresponding generating function, which is impossible in general.Aiming to overcome this shortcoming in some extent, in this paper we present an improved asymptotic normality criterion, along with several variant versions, which usually just ask for one coefficient of the generating function, without knowing any roots. In virtue of these new criteria, the asymptotic normality of some usual combinatorial statistics can be revealed and extended. Among which, we introduce the applications to matching numbers and Laplacian coefficients in detail. Some relevant conjectures, proposed by Godsil (Combinatorica, 1981) and Wang et al. (J. Math. Anal. Appl., 2017), are generalized and verified as corollaries.     

Polyethylene glycol phospholipids encapsulated silicon 2,3-naphthalocyanine dihydroxide nanoparticles (SiNcOH-DSPEPEG(NH2) NPs) for single NIR laser induced cancer combination therapy

Currently, the combination of photothermal therapy (PTT) and photodynamic therapy (PDT) has emerged as a powerful technique for cancer treatment. However, most examples of combined PTT and PDT reported use multi-component nanocomposites under excitation of separate wavelength, resulting in complex treatment process. In this work, a novel theranostic nanoplatform (SiNcOH-DSPE-PEG(NH₂) NPs) has been successfully developed by coating silicon 2,3-naphthalocyanine dihydroxide (SiNcOH) with DSPE-PEG and DSPE-PEG-NH₂ for photoacoustic (PA) imaging-guided PTT and PDT tumor ablation for the first time. The as-prepared single-agent SiNcOH-DSPE-PEG(NH₂) NPs not only have good water solubility and biocompatibility, but also exhibit high photothermal conversion efficiency and singlet oxygen generation capability upon 808 nm NIR laser irradiation. In addition, owing to their high absorption at NIR region, the SiNcOH-DSPE-PEG(NH₂) NPs can also be employed as an effective diagnostic nanoagent for photoacoustic (PA) imaging. In vitro and in vivo experimental results clearly indicated that the simultaneously combined PTT and PDT under the guidance of PA imaging with single NIR laser excitation can effectively kill cancer cells or eradicate tumor tissues. Taking facile synthesis and high efficiency in cancer treatment by SiNcOH-DSPE-PEG(NH₂) NPs into consideration, our study provides a promising strategy to realize molecular imaging-guided combination therapy.     

Neighbor Distinguishing Edge Colorings Via the Combinatorial Nullstellensatz Revisited

Consider a simple graph and its proper edge coloring c with the elements of the set . We say that c is neighbor set distinguishing (or adjacent strong) if for every edge , the set of colors incident with u is distinct from the set of colors incident with v. Let us then consider a stronger requirement and suppose we wish to distinguishing adjacent vertices by sums of their incident colors. In both problems the challenging conjectures presume that such colorings exist for any graph G containing no isolated edges if only . We prove that in both problems is sufficient. The proof is based on the Combinatorial Nullstellensatz, applied in the “sum environment.” In fact the identical bound also holds if we use any set of k real numbers instead of as edge colors, and the same is true in list versions of the both concepts. In particular, we therefore obtain that lists of length ( in fact) are sufficient for planar graphs.     

A practical and inexpensive ‘convertible’ isonitrile for use in multicomponent reactions

N-tert-Butylamides are readily converted into the corresponding carboxylic acids by simple nitrosation. The process, which occurs under mild nonaqueous conditions, leaves carboxylic esters untouched and transforms multicomponent reaction products into useful building blocks for further synthetic elaboration.