A Method for Classification of Doubly Resolvable Designs and Its Application

This article presents the principal results of the Ph.D. thesis Investigation and classification of doubly resolvable designs by Stela Zhelezova (Institute of Mathematics and Informatics, BAS), successfully defended at the Specialized Academic Council for Informatics and Mathematical Modeling on 22 February 2010.The resolvability of combinatorial designs is intensively investigated because of its applications. This research focuses on resolvable designs with an additional property - they have resolutions which are mutually orthogonal. Such designs are called doubly resolvable. Their specific properties can be used in statistical and cryptographic applications.Therefore the classification of doubly resolvable designs and their sets of mutually orthogonal resolutions might be very important. We develop a method for classification of doubly resolvable designs. Using this method and extending it with some theoretical restrictions we succeed in obtaining a classification of doubly resolvable designs with small parameters. Also we classify 1-parallelisms and 2-parallelisms of PG(5,2) with automorphisms of order 31 and find the first known transitive 2-parallelisms among them. The content of the paper comprises the essentials of the author’s Ph.D. thesis

On the resolutions of cyclic Steiner triple systems with small parameters

The paper presents useful invariants of resolutions of cyclic $STS(v)$ with $v\le 39$, namely of all resolutions of cyclic $STS(15)$, $STS(21)$ and $STS(27)$, of the resolutions with nontrivial automorphisms of cyclic $STS(33)$ and of resolutions with automorphisms of order $13$ of cyclic $STS(39)$

Free Resolutions Associated to Representable Matroids

As a matroid is naturally a simplicial complex, one can study its combinatorial properties via the associated Stanley-Reisner ideal and its corresponding free resolution. Using results by Johnsen and Verdure, we prove that a matroid is the dual to a perfect matroid design if and only if its corresponding Stanley-Reisner ideal has a pure free resolution, and, motivated by applications to their generalized Hamming weights, characterize free resolutions corresponding to the vector matroids of the parity check matrices of Reed-Solomon codes and certain BCH codes. Furthermore, using an inductive mapping cone argument, we construct a cellular resolution for the matroid duals to finite projective geometries and discuss consequences for finite affine geometries. Finally, we provide algorithms for computing such cellular resolutions explicitly

Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

On the seven non-isomorphic solutions of the fifteen schoolgirl problem

In this paper we give a simple and effective tool to analyze a given Kirkman triple system of order 15 and determine which of the seven well-known non-isomorphic KTS(15)s it is isomorphic to. Our technique refines and improves the lacing of distinct parallel classes introduced by F. N. Cole, by means of the notion of residual triple defined by G. Falcone and the present author in a previous paper. Unlike Cole's original lacing scheme, our algorithm allows one to distinguish two KTS(15)s also in the harder case where the two systems have the same underlying Steiner triple system. In the special case where the common STS is #19, an alternative method is given by means of the 1-factorizations of the complete graph K_8 associated to the two KTSs. Moreover, we present three new visual solutions to the schoolgirl problem, and we catalogue most of the classical (or interesting) solutions in the literature in terms of what KTS(15)s they are isomorphic to. This paper provides background on a classical topic, while shedding new light on the problem as well

MoS2-Carbon Nanodots as a New Electrochemiluminescence Platform for Breast Cancer Biomarker Detection

In this work, we present the combination of two different types of nanomaterials, 2D molybdenum disulfide nanosheets (MoS2-NS) and zero-dimensional carbon nanodots (CDs), for the development of a new electrochemiluminescence (ECL) platform for the early detection and quantification of the biomarker human epidermal growth factor receptor 2 (HER2), whose overexpression is associated with breast cancer. MoS2-NS are used as an immobilization platform for the thiolated aptamer, which can recognize the HER2 epitope peptide with high affinity, and CDs act as coreactants of the anodic oxidation of the luminophore [Ru(bpy)3]2+. The HER2 biomarker is detected by changes in the ECL signal of the [Ru(bpy)3]2+/CD system, with a low detection limit of 1.84 fg/mL and a wide linear range. The proposed method has been successfully applied to detect the HER2 biomarker in human serum samplesThis work has been financially supported by the Spanish Ministry of Economy and Competitiveness (PID2020-116728RB-I00, PID2020-116661RB-I00, CTQ2015-71955-REDT (ELECTROBIONET)), Community of Madrid (TRANSNANOAVANSENS, S2018/NMT-4349, and FotoArt (P2018/NMT4367), project S2018/NMT-4291 TEC2SPACE), MINECO (project CSIC13-4E-1794) and EU (FEDER, FSE). IMDEA Nanociencia receives support from the “Severo Ochoa” Programme for Centres of Excellence in R&D (MINECO, Grant CEX2020-001039-S)

Spatiotemporal Mapping of Electrochemical Diffusion Layers

Although pure electrochemical techniques can provide substantial knowledge about electrochemical reaction mechanisms, they lack the ability to provide direct molecular structure information about the species involved. This inability to extract molecular information can result in mechanistic ambiguity with respect to the identity of the reaction intermediates. This information can be obtained by coupling in situ spectroscopic methodologies with electrochemical techniques. This is known as spectroelectrochemistry. A particularly problematic area of study with spectroelectrochemical techniques is the analysis of the mass transport of species within the diffusion layer of the electrode. The visualization of the diffusion layer surrounding electrodes allows for the unambiguous determination of electrode processes. However, a high degree of spatial and temporal resolution is needed as the diffusion layer in a typical electrochemical reaction extends to a thickness of hundreds of microns in tens of seconds. While traditional infrared spectroelectrochemical techniques have been valuable for the study of electrochemical processes, they do not provide the spatial and/or the temporal resolution that is needed to examine the diffusion layers produced at electrodes. This thesis focuses on the development of an IR technique that couples synchrotron based IR radiation (SIR) with electrochemistry, allowing for the concentrations of species present within the diffusion layer of an electrode to be mapped during an electrochemical reaction. The reduction of ferricyanide and oxidation of hydroquinone (HQ) are used as test redox systems to study the ability of SIR to map electrochemical diffusion layers. The resulting diffusion coefficients of ferricyanide, ferrocyanide, hydroquinone and benzoquinone are extracted using the IR method developed here and compared to those determined independently by hydrodynamic linear sweep voltammetry (HLSV). The diffusion coefficients of all species as determined by SIR diffusion layer mapping will be shown to be consistent with the diffusion coefficients determined by HLSV. This validates the ability of SIR diffusion layer mapping to monitor electrochemically generated diffusion layers