Combinatorial Set Research Articles

Theory and Methods of Euclidian Combinatorial Optimization: Current Status and Prospects

Euclidean combinatorial optimization problems are considered as discrete optimization problems on a set of combinatorial configurations mapped into an arithmetic Euclidean space. Modern methods of Euclidean combinatorial optimization are reviewed. The properties of the corresponding images of combinatorial sets are described. A theory of continuous functional representations and convex extensions is proposed for solving this class of problems. Areas of practical application and promising research areas are indicated.

Revenue maximization in Stackelberg Pricing Games: beyond the combinatorial setting

In a Stackelberg Pricing Game a distinguished player, the leader, chooses prices for a set of items, and the other players, the followers, each seek to buy a minimum cost feasible subset of the items. The goal of the leader is to maximize her revenue, which is determined by the sold items and their prices. Most previously studied cases of such games can be captured by a combinatorial model where we have a base set of items, some with fixed prices, some priceable, and constraints on the subsets that are feasible for each follower. In this combinatorial setting, Briest et al. and Balcan et al. independently showed that the maximum revenue can be approximated to a factor of H_ksim log k, where k is the number of priceable items. Our results are twofold. First, we strongly generalize the model by letting the follower minimize any continuous function plus a linear term over any compact subset of {mathbb {R}}^n_{ge 0}; the coefficients (or prices) in the linear term are chosen by the leader and determine her revenue. In particular, this includes the fundamental case of linear programs. We give a tight lower bound on the revenue of the leader, generalizing the results of Briest et al. and Balcan et al. Besides, we prove that it is strongly NP-hard to decide whether the optimum revenue exceeds the lower bound by an arbitrarily small factor. Second, we study the parameterized complexity of computing the optimal revenue with respect to the number k of priceable items. In the combinatorial setting, given an efficient algorithm for optimal follower solutions, the maximum revenue can be found by enumerating the 2^k subsets of priceable items and computing optimal prices via a result of Briest et al., giving time O(2^k|I|^c) where |I| is the input size. Our main result here is a W[1]-hardness proof for the case where the followers minimize a linear program, ruling out running time f(k)|I|^c unless mathsf {FPT} =mathsf {W[1]} and ruling out time |I|^{o(k)} under the Exponential-Time Hypothesis.

Use of a segmented cell for the combinatorial development of platinum group metal-free electrodes for polymer electrolyte fuel cells

The development of novel platinum group metal (PGM)-free catalysts is a challenging task, particularly when coupled with the integration into a cathode catalyst layer (CCL). The optimization of such PGM-free electrode structures is often non-linear and iterative, making it a demanding task due to the high number of parameters that affect performance. To accelerate both materials discovery and electrode development, this work demonstrates the application of a high-resolution segmented cell (SC) with 121 segments of 0.413 cm2 size for combinatorial high-throughput PGM-free catalyst screening and CCL optimization. The approach utilizes three flow-field strips with active areas of 4.45 cm2 each, distributed over 11 segments. Electrodes with identical catalyst material and fabrication method result in reproducible data with typically less than 10% variation between each segment and match in performance with differential single cell data. High throughput testing of combinatorial sample sets, with varied PGM-free electrocatalyst materials or electrode composition, demonstrated the ability to rapidly discern high performing outliers, and establish or confirm trends in electrode optimization. The results indicated significant performance benefits of electrodes with ionomer content of 45 wt% over lower values, and of water rich inks with 82% water content over those with 50%.

Coefficients of Wronskian Hermite polynomials

AbstractWe study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on thep‐star.

Some Natural Phenomena and Sign Combinatorial Spaces

The literature describes many natural phenomena associated with combinatorial numbers including golden number that is represented by Fibonacci numbers. This suggests that the laws of combinatorics are inherent in nature. To explain such natural phenomena as the presence of combinatorial numbers in nature, the formation of fractal structures and symmetry in biology one applies the properties of sign combinatorial spaces. In the combinatorial sets ordered by certain rules the numerical sequences that specify in them the number of combinatorial configurations also contain combinatorial numbers including Fibonacci numbers. In addition, these sets are characterized by the symmetry. In these sets symmetry is modeled by a finite sequence of numbers that define the number of combinatorial configurations in subsets. Their values increase to the largest and then decrease (or decrease to the smallest and then increase). The symmetry plane passing through the largest (or smallest) number of the sequence divides it into two parts whose values from the center decrease (or increase) evenly, but these parts are not necessarily mirror symmetrical. They are characterized by both approximate and exact symmetry. Sign combinatorial spaces, the point of which is combinatorial configurations of different types exist in two states: convolved (rest) and developed (dynamics). Axioms are introduced for them. As in combinatorial sets, fractals and symmetries of different kinds are formed in the process of these spaces unfolding. The axioms of sign combinatorial spaces are valid for some natural ones including the biological. Therefore studying symmetry and fractals in combinatorics we can explain how they are formed in biology. The question of how the symmetry arises in developed biological spaces has not been answered yet. Knowing the symmetry formation in combinatorial sets, one can explain the symmetry formation in biology.

The Tethering Assay: A Simple Method for the Characterization of mRNA-Fate Regulators.

In trypanosomatids, posttranscriptional controls are very important in regulation of individual gene expression. These are achieved through combinatorial sets of RNA-binding proteins (RBPs) which recognize RNA regulatory motifs or regions of secondary structure within RNAs. To analyze the potential functional impact of an RBP on their mRNA targets, we have applied a robust technique called tethering assay. In this method, the protein under study is attached to an mRNA reporter through an artificial RNA-protein interaction. Therefore, the functional activity of a protein can be analyzed independently of its intrinsic ability to bind to RNA. By making use of a cell line expressing a chloramphenicol acetyltransferase (CAT) reporter mRNA, we have characterized dozens of novel mRNA-fate regulators in cultured Trypanosoma brucei. After induction of the candidate fusion protein, the effect on the reporter expression is determined by a rapid CAT assay. The protocol is simple and typically takes one working day for analysis of a single protein and controls. In this chapter, we provide a description of materials and methods for the tethering method and should allow the assay to be successfully deployed in any laboratory with minimal user training.

The Fractal Nature of Combinatorial Sets and Finding Formulas for Combinatorial Numbers

The fractal structure of combinatorial sets, which is formed in ordering of combinatorial configurations, is considered. Using the fractal properties of the sets, an approach to solving enumeration problems in combinatorics is developed. Arithmetic sequences are used to find combinatorial numbers.

CYCLIC PERMUTATIONS IN THE METHODS OF COMBINATORIAL OPTIMIZATION ON THE BASIS OF CYCLIC TRANSFERS

Permutations sets are very often considered in theoretical and applied research in the field of combinatorics andcombinatorial optimization. By now many properties of permutations have been investigated, in particular those associatedwith the cyclic structure of permutations. To solve optimization problems on combinatorial sets, a number of methodshave been developed and successfully used. Among them, there are methods for finding exact, approximate, and heuristicsolutions. Among the search methods for an approximate solution, search methods in the neighborhood are widely used.Among the combinatorial search methods in the neighborhood, it is necessary to distinguish a group of methods that usescyclic transfers. A general search scheme using cyclic transfers is known. It is based on the search for a cyclic transferin the auxiliary graph. But the features of the tasks being solved sometimes do not allow constructing an auxiliary graphand using the existing search methods. In such cases, an alternative approach is proposed. This is a general strategy forsolving problems using cyclic transfers, which is a modification of the strategy of Thompson and Orlin. We propose theconstruction of cyclic transfers based not on the use of an auxiliary graph, but based on the set of cyclic permutationsand its properties to obtain cyclic transfers. To demonstrate the strategy for solving combinatorial optimization problemsdescribed in the article, one of the vehicle routing problems was solved. Computational experiments were conducted todemonstrate the effectiveness of the proposed strategy. Their results are given in the article.

Frequent Causal Pattern Mining: A Computationally Efficient Framework For Estimating Bias-Corrected Effects.

Our aging population increasingly suffers from multiple chronic diseases simultaneously, necessitating the comprehensive treatment of these conditions. Finding the optimal set of drugs for a combinatorial set of diseases is a combinatorial pattern exploration problem. Association rule mining is a popular tool for such problems, but the requirement of health care for finding causal, rather than associative, patterns renders association rule mining unsuitable. To address this issue, we propose a novel framework based on the Rubin-Neyman causal model for extracting causal rules from observational data, correcting for a number of common biases. Specifically, given a set of interventions and a set of items that define subpopulations (e.g., diseases), we wish to find all subpopulations in which effective intervention combinations exist and in each such subpopulation, we wish to find all intervention combinations such that dropping any intervention from this combination will reduce the efficacy of the treatment. A key aspect of our framework is the concept of closed intervention sets which extend the concept of quantifying the effect of a single intervention to a set of concurrent interventions. Closed intervention sets also allow for a pruning strategy that is strictly more efficient than the traditional pruning strategy used by the Apriori algorithm. To implement our ideas, we introduce and compare five methods of estimating causal effect from observational data and rigorously evaluate them on synthetic data to mathematically prove (when possible) why they work. We also evaluated our causal rule mining framework on the Electronic Health Records (EHR) data of a large cohort of 152000 patients from Mayo Clinic and showed that the patterns we extracted are sufficiently rich to explain the controversial findings in the medical literature regarding the effect of a class of cholesterol drugs on Type-II Diabetes Mellitus (T2DM).

Parasitism – Predation algorithm (PPA): A novel approach for feature selection

Maximizing the classification accuracy and minimizing the number of selected features are the two main incompatible objectives for using feature selection to overcome the curse of dimensionality. “Classification accuracy highly dependents on the nature of the features in a dataset which may contain irrelevant or redundant data. The main aim of feature selection is to eliminate these types of features to enhance the classification accuracy.” This work presents a new meta-heuristic optimization approach, called Parasitism-Predation Algorithm (PPA), which mimics the interaction between the predator (cats), the parasite (cuckoos) and the host (crows) in the crow–cuckoo–cat system model to overcome the problems of low convergence and the curse of dimensionality of large data. The proposed hybrid framework combines the relative advantages of cat swarm optimization (CSO), cuckoo search (CS) and crow search algorithm (CSA) to attain a combinatorial set of features to boost up the classification accuracy. Nesting, parasitism, and predation phases are supposed to help exploration ability and balance in the context of solving classification problems. In addition, Levy flight distribution is applied to help better diversity of conventional CSA and improve ability of exploration. Meanwhile, an effective fitness function is utilized to enable the proposed PPA-based feature selector using K-Nearest Neighbors algorithm (KNN) to attain a combinatorial set of features. The proposed PPA and four standard heuristic search algorithms are looked at to gauge how efficient the proposed option is. Additionally, eighteen classification datasets are deployed to gauges its efficacy. The results highlight that the algorithm proposed is both effective and competitive in terms of performance of classification and dimensionality reduction as opposed to other heuristic options.

Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.

A new perspective in understanding rainfall from satellites over a complex topographic region of India

Present study focuses on rainfall over Western Ghats (WG), a complex topographic region (elevation > 500 m) of India to evaluate and to better understand the satellite behavior in contrast with a flat region (FR) (elevation < 500 m) of central India from 1998 to 2016 using the combinatory data sets of TMPA and IMERG (satellite rainfall estimation). The categorical Intra Seasonal Oscillations (ISO) of Indian summer monsoon (ISM) namely, Madden Julian Oscillation (MJO) and Quasi Bi-Weekly Oscillation (QBWO) are tested in satellite and India Meteorological Department (IMD) gridded rainfall data sets to find out the satellite performance. As the accurate estimation of rainfall from satellites over higher elevation zones is challenging, here we propose a new perspective to select the rainfall products of satellite for better comparison with ground measurements. Considering the satellite’s best capability in detecting the cold clouds resulting from deep convection and its coupling with higher-level circulation, we show that the rainfall from satellites yield fruitful comparison with ground measurements when moist static stability, tropical easterly jet is above the climatological values.

Metric Learning for High-Throughput Combinatorial Data Sets.

Materials design and discovery through the high-throughput exploration of materials space has been recognized as a new paradigm in materials science. However, typical high-throughput exploration methods deliver high-dimensional and very diverse data sets that pose the challenge of extracting the key features and patterns that could guide the discovery process. Unraveling patterns is a nontrivial task as quite often the underlying physical phenomena are uncertain and latent variables governing the performance are mainly unknown. In this paper, we discuss challenges related to designing a data analytics tool for clustering high-throughput measurements performed on the compositional library of materials. The critical aspects of our methodology are (i) learning the similarity measures, as opposed to using fixed similarity measures (e.g., Euclidean distance, dynamic time warping), while (ii) imposing the similarity in the composition space. Our methodology is based on the multitask learning approach that is formulated to account for the composition neighborhoods that are specific to the compositional libraries. We demonstrate the advantages of our methodology for the library of cyclic voltammetry curves generated for model multimetal catalysts, as well as X-ray diffraction patterns from experimental studies. We also compare our approach with the current state-of-the-art methods used in similar problems. This work has important implications for designing high-throughput exploration including catalysts for electrochemical systems, such as fuel cells and metal-air batteries.

On the Combinatorics of Gentle Algebras

AbstractFor $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$. We use this to describe support $\unicode[STIX]{x1D70F}$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$-tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

Context-Specific Transcription Factor Functions Regulate Epigenomic and Transcriptional Dynamics during Cardiac Reprogramming.

Ectopic expression of combinations of transcription factors (TFs) can drive direct lineage conversion, thereby reprogramming a somatic cell's identity. To determine the molecular mechanisms by which Gata4, Mef2c, and Tbx5 (GMT) induce conversion from a cardiac fibroblast toward an induced cardiomyocyte, we performed comprehensive transcriptomic, DNA-occupancy, and epigenomic interrogation throughout the reprogramming process. Integration of these datasets identified new TFs involved in cardiac reprogramming and revealed context-specific roles for GMT, including the ability of Mef2c and Tbx5 to independently promote chromatin remodeling at previously inaccessible sites. We also find evidence for cooperative facilitation and refinement of each TF's binding profile in a combinatorial setting. A reporter assay employing newly defined regulatory elements confirmed that binding of a single TF can be sufficient for gene activation, suggesting that co-binding events do not necessarily reflect synergy. These results shed light on fundamental mechanisms by which combinations of TFs direct lineage conversion.

Властивості Евклідових задач лексикографічної комбінаторної оптимізації на розміщеннях

In the paper Euclidean problems of lexicographic combinatorial; optimization are discussed. These problems are to find lexicographically minimal (for minimization problems) or lexicographically maximal (for maximization problems) points among those that give the extremum of the objective function on a given Euclidean combinatorial set. The properties of linear and linear-fractional problems of lexicographic combinatorial optimization on a general set of arrangements are substantiated. The results obtained in the work are based on the previously known criteria of extremals of linear and linear-fractional functions on arrangements: any extremal is an element of certain set of polyarrangements (for linear problems the form of the extremal set is established explicitly, for linear-fractional problems the polyarrangement set is formed on the basis of some known extremal). In the paper we substantiate the form of points that are an lexicographic minimal and lexicographic maximal of linear function on the general set of arrangements. In particular if elements of multiset are in nondecreasing order, coefficients of objective function are in nonincreasing order and s is the least index such that corresponding coefficient of objective function is negative, tehn lexicographic minimal if formed as s – 1 first and k – s + 1 ( k is the dimension of space) last elements of multiset which are in nondecreasing order. For problems with linear-fractional function we obtain the method of forming solution of lexicographic combinatorial problem on arrangements, if any minimal (for minimization problems) or any maximal (for maximization problems) of objective function on given set of arrangements is known. In this case ordering of components of the extremal is carried out taking into account ordering for nonincreasing of coefficients of special linear function

Combinatorial Ricci Flow for Degenerate Circle Packing Metrics

Chow and Luo [3] showed in 2003 that the combinatorial analogue of the Hamilton Ricci flow on surfaces converges under certain conditions to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [3] was that the weights are nonnegative. Recently we have shown that the same statement on convergence can be proved under a weaker condition: some weights can be negative and should satisfy certain inequalities [4]. On the other hand, for weights not satisfying conditions of Chow — Luo’s theorem we observed in numerical simulation a degeneration of the metric with certain regular behaviour patterns [5]. In this note we introduce degenerate circle packing metrics, and under weakened conditions on weights we prove that under certain assumptions for any initial metric an analogue of the combinatorial Ricci flow has a unique limit metric with a constant curvature outside of singularities.

Engineering repressors with coevolutionary cues facilitates toggle switches with a master reset.

Engineering allosteric transcriptional repressors containing an environmental sensing module (ESM) and a DNA recognition module (DRM) has the potential to unlock a combinatorial set of rationally designed biological responses. We demonstrated that constructing hybrid repressors by fusing distinct ESMs and DRMs provides a means to flexibly rewire genetic networks for complex signal processing. We have used coevolutionary traits among LacI homologs to develop a model for predicting compatibility between ESMs and DRMs. Our predictions accurately agree with the performance of 40 engineered repressors. We have harnessed this framework to develop a system of multiple toggle switches with a master OFF signal that produces a unique behavior: each engineered biological activity is switched to a stable ON state by different chemicals and returned to OFF in response to a common signal. One promising application of this design is to develop living diagnostics for monitoring multiple parameters in complex physiological environments and it represents one of many circuit topologies that can be explored with modular repressors designed with coevolutionary information.

The accessible chromatin landscape of the murine hippocampus at single-cell resolution.

Here we present a comprehensive map of the accessible chromatin landscape of the mouse hippocampus at single-cell resolution. Substantial advances of this work include the optimization of a single-cell combinatorial indexing assay for transposase accessible chromatin (sci-ATAC-seq); a software suite, scitools, for the rapid processing and visualization of single-cell combinatorial indexing data sets; and a valuable resource of hippocampal regulatory networks at single-cell resolution. We used sci-ATAC-seq to produce 2346 high-quality single-cell chromatin accessibility maps with a mean unique read count per cell of 29,201 from both fresh and frozen hippocampi, observing little difference in accessibility patterns between the preparations. By using this data set, we identified eight distinct major clusters of cells representing both neuronal and nonneuronal cell types and characterized the driving regulatory factors and differentially accessible loci that define each cluster. Within pyramidal neurons, we identified four major clusters, including CA1 and CA3 neurons, and three additional subclusters. We then applied a recently described coaccessibility framework, Cicero, which identified 146,818 links between promoters and putative distal regulatory DNA. Identified coaccessibility networks showed cell-type specificity, shedding light on key dynamic loci that reconfigure to specify hippocampal cell lineages. Lastly, we performed an additional sci-ATAC-seq preparation from cultured hippocampal neurons (899 high-quality cells, 43,532 mean unique reads) that revealed substantial alterations in their epigenetic landscape compared with nuclei from hippocampal tissue. This data set and accompanying analysis tools provide a new resource that can guide subsequent studies of the hippocampus.

Multivariate modeling of engineered nanomaterial features associated with developmental toxicity

Despite the increasing prevalence of engineered nanomaterials (ENMs) in consumer products, their toxicity profiles remain to be elucidated. ENM physicochemical characteristics (PCC) are known to influence ENM behavior, however the mechanisms of these effects have not been quantified. Further confounding the question of how the PCC influence behavior is the inclusion of structural and molecular descriptors in modeling schema that minimize the effects of PCC on the toxicological endpoints. In this work, we analyze ENM physico-chemical measurements that have not previously been studied within a developmental toxicity framework using an embryonic zebrafish model. In testing a panel of diverse ENMs to build a consensus model, we found nonlinear relationships between any singular PCC and bioactivity. By using a machine learning (ML) method to characterize the information content of combinatorial PCC sets, we found that concentration, surface area, shape, and polydispersity can accurately capture the developmental toxicity profile of ENMs with consideration to whole-organism effects.