On deficiency problems for graphs

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$ , the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer t such that the join $G*K_t$ has property $\mathcal P$ . In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$ -factor (for any fixed $r\geq 3$ ). In this paper, we resolve their problem fully. We also give an analogous result that forces $G*K_t$ to contain any fixed bipartite $(n+t)$ -vertex graph of bounded degree and small bandwidth.

KOLMOGOROV BOUNDS FOR THE NORMAL APPROXIMATION OF THE NUMBER OF TRIANGLES IN THE ERDŐS–RÉNYI RANDOM GRAPH

We bound the error for the normal approximation of the number of triangles in the Erdős–Rényi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein bounds obtained by Barbour et al. [(1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B 47: 125–145], resolving a long-standing open problem. The proofs are based on a new variant of the Stein–Tikhomirov method—a combination of Stein's method and characteristic functions introduced by Tikhomirov [(1976). The rate of convergence in the central limit theorem for weakly dependent variables. Vestnik Leningradskogo Universiteta 158–159, 166].

Method of developing an increase plan energy efficiency of the portal regional electrotechnical complex

Припортовый региональный электротехнический комплекс является инфраструктурой техноценологического типа. Ввиду энерговооружённости и значительной установленной электрической мощности приёмников электрической энергии его экономическая эффективность в большей степени обусловлена энергетической эффективностью. Одним из направлений повышения энергетической эффективности является энергосбережение, предполагающее выполнение энергетических обследований объектов и оценку потенциала энергосбережения. Практическая реализация потенциала энергосбережения, основанного на мероприятиях организационного характера, заключается в составлении и выполнении плана повышения энергоэффективности. Это предусматривает последовательное и постепенное, без технического перевооружения сокращение объектами своего электропотребления таким образом, чтобы к заданному времени достичь требуемых целевых показателей. Предлагается разработать методику составления плана повышения энергоэффективности, учитывающего техноценологические свойства припортового регионального электротехнического комплекса и обеспечивающего максимальную вероятность достижения цели. Теоретической основой данной методики является ранговый анализ и комбинаторная теория ранговой динамики, позволяющие на системном уровне описывать вероятностные закономерности электропотребления систем техноценологического типа. The portal regional electrotechnical complex is a technocenological type infrastructure. In view of the power supply and the significant installed electrical capacity of the receivers of electrical energy, its economic efficiency is largely due to energy efficiency. One of the directions for increasing energy efficiency is energy saving, which involves performing energy inspections of facilities and assessing the potential for energy saving. Practical implementation of the energy saving potential based on organizational measures is to draw up and implement a plan to improve energy efficiency. This provides for a consistent and gradual, without technical re-equipment, reduction of their electricity consumption by objects in such a way as to achieve the required target indicators by a given time. It is proposed to develop a methodology for drawing up a plan for increasing energy efficiency, taking into account the technocenological properties of the portal regional electrotechnical complex and ensuring the maximum probability of achieving the goal. The theoretical basis of this methodology is rank analysis and the combinatorial theory of rank dynamics, which allow describing the probabilistic patterns of power consumption of systems of technocenological type at the system level.

The positive and negative effects of using transdermal nonsteroidal anti-inflammatory drugs: chemoreactome analysis

Transdermal nonsteroidal anti-inflammatory drugs (NSAIDs) are actively used for mild and moderate pain syndrome, muscle contusions and sprains, sports injuries, and the widest range of musculoskeletal diseases. The transdermal administration of NSAIDs aims to create sufficiently high drug concentrations in the lesion focus, provided that the side effects associated with its systemic action are maximally reduced.Objective: to comparatively simulate the effects of transdermal NSAIDs.Material and methods. Chemoreactome profiling of six NSAIDs (meloxicam, diclofenac, ibuprofen, ketoprofen, nimesulide, and dexketoprofen) was performed. The pharmacological capabilities of molecules were analyzed within the framework of a chemoreactome methodology, by comparing the chemical structure of the studied molecule with those of millions of other molecules, the pharmacological properties of which had already been studied in experimental and clinical studies. Training the artificial intelligence algorithms based on the big data available in in the databases PubChem/PHARMGKB, HMDB, STRING, and others was done with multi-level training quality control in the cross validation framework according to the combinatorial theory of solvability and the theory of feature value classification.Results and discussion. Meloxicam versus other NSAIDs accumulates primarily in the muscles and skin and, to a much lesser extent, in heart tissues, lymphocytes, gonads, and cartilage. This drug showed the greatest dose-dependent decongestant effect in the model of edema induced by croton oil. Analysis of the systemic effects of NSAIDs indicated that meloxicam might affect the metabolism of vitamins A, D, PP, and B6 to a lesser extent than other NSAIDs.Conclusion. The chemoreactomе analysis has demonstrated that meloxicam as a gel causing minimal side effects can be used effectively and long.

A node-based informed modularity strategy to identify organizational modules in anatomical networks

ABSTRACTThe study of morphological modularity using anatomical networks is growing in recent years. A common strategy to find the best network partition uses community detection algorithms that optimize the modularity Q function. Because anatomical networks and their modules tend to be small, this strategy often produces two problems. One is that some algorithms find inexplicable different modules when one inputs slightly different networks. The other is that algorithms find asymmetric modules in otherwise symmetric networks. These problems have discouraged researchers to use anatomical network analysis and boost criticisms to this methodology. Here, I propose a node-based informed modularity strategy (NIMS) to identify modules in anatomical networks that bypass resolution and sensitivity limitations by using a bottom-up approach. Starting with the local modularity around every individual node, NIMS returns the modular organization of the network by merging non-redundant modules and assessing their intersection statistically using combinatorial theory. Instead of acting as a black box, NIMS allows researchers to make informed decisions about whether to merge non-redundant modules. NIMS returns network modules that are robust to minor variation and does not require optimization of a global modularity function. NIMS may prove useful to identify modules also in small ecological and social networks.

A Node-based Informed Modularity Strategy to Identify Organizational Modules in Anatomical Networks

The use of anatomical networks to study the modular organization of morphological systems and their evolution is growing in recent years. A common strategy to find the best partition of anatomical networks into modules is to use a community detection algorithm that tries to optimize the modularity Q function. However, this strategy overlooks the fact that Q has a resolution limit for small modules, which is often the case in anatomical networks. This produces two problems. One is that some algorithms find inexplicable different modules when we input slightly different networks. The other is that algorithms find asymmetric modules in otherwise symmetric networks. These problems have discouraged researchers to use anatomical network analysis and boost criticisms to this methodology. Here, I propose a Node-based Informed Modularity Strategy (NIMS) to identify modules in anatomical networks that bypass resolution and sensitivity limitations by using a bottom-up approach. Starting with the local modularity around every individual node, NIMS returns the modular organization of the network by merging non-redundant modules and assessing their intersection statistically using combinatorial theory. Instead of acting as a black box, NIMS allows researcher to make informed decisions about whether to merge non-redundant modules. NIMS returns network modules that are robust to minor variation and does not require to optimize a global modularity function. NIMS may prove useful to identify modules also in small ecological and social networks.

Polynomials from Combinatorial -theory

We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $K$ -theoretic deformation of the quasi-key basis and also a lift of the $K$ -analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $K$ -analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $K$ -theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $K$ -analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations.

GenomeScope 2.0 and Smudgeplots: Reference-free profiling of polyploid genomes

An important assessment prior to genome assembly and related analyses is genome profiling, where the k-mer frequencies within raw sequencing reads are analyzed to estimate major genome characteristics such as genome size, heterozygosity, and repetitiveness. Here we introduce GenomeScope 2.0 (https://github.com/tbenavi1/genomescope2.0), which applies combinatorial theory to establish a detailed mathematical model of how k-mer frequencies are distributed in heterozygous and polyploid genomes. We describe and evaluate a practical implementation of the polyploid-aware mixture model that, within seconds, accurately infers genome properties across thousands of simulated and eleven real datasets spanning a broad range of complexity. We also present a new method called Smudgeplots (https://github.com/KamilSJaron/smudgeplot) to visualize and infer the ploidy and genome structure of a genome by analyzing heterozygous k-mer pairs. We successfully apply the approach to systems of known variable ploidy levels in the Meloidogyne genus and also the extreme case of octoploid Fragaria x ananassa.

Unique Rectification in $d$-Complete Posets: Towards the $K$-Theory of Kac-Moody Flag Varieties

The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's '$d$-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting.