Combinatorial Techniques Research Articles

On the antichain tree property

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.

Design and In silico Screening of Combinatorial Library of New Herbicidal Analogs of Cycloalka[d]quinazoline-2,4dione−Benzoxazinones Inhibiting Protoporphyrinogen IX Oxidase

We virtually design here an important new weedkiller Cycloalka[d]quinazoline-2,4dione−Benzoxazinones (CQB), inhibitors of Protoporphyrinogen IX Oxidase (PPO). Based on computer-assisted combinatorial chemistry techniques, docking, 3D-QSAR and pharmacophore models we first enumerate, focus and in silico screen a virtual library of CQB analogs substituted at positions R1, R2 and R3. By docking inhibitors into the target active site from the crystal structure (PDB ID: 1SEZ) of PPO in complex with a CQB ligand, 3D models of 29 PPO:CQBx complexes with known observed activity (Kiexp) were prepared to establish a quantitative structure–activity (QSAR) model and linear correlation between relative Gibbs free energy (GFE) of receptor-ligand complex formation (ΔΔGcom) and Kiexp: pKiexp = -0.1664 ×ΔΔGcom + 8.306 (1); R2 = 0.94. A 3D QSAR pharmacophore model (PH4) derived from the QSAR directed our effort to design novel CQB analogs. During the design, an initial virtual library of 118 CQB was focused down and PH4 screened to identify 28 promising novel analogs. Their Ki (Kipre) values were predicted by means of equation (1). The most active analog namely CQB22 display Kipre 22 times superior to that of the reported most active training set ligand 17i. Our survey proposes this compounds to the synthesis and to the assessment on herbicidal.

A Combinatorial Proof of the Existence of Dense Subsets in ℝ without the “Steinhaus” like Property

This paper proves the existence of nonmeasurable dense sets with additional properties using combinatorial techniques.

On‐Chip Multiwavelength Single‐Mode Lasers with CdSe Nanoribbons‐Embedded Microcavities

Integrated multiwavelength nanolasers are important in the practical development of optical circuitry for optical information processing and optical data storage. Herein, a new method for on‐chip integrated lasers is proposed, which can integrate single‐mode lasers with multiple wavelengths at room temperature. A four‐wavelength single‐mode nanoribbon laser chip is fabricated for demonstration. The mode of different laser units is controlled by the thickness of dielectric microcavity layers to select the resonant modes of excitation. Therefore, multiwavelength lasing can be coupled out based on the integrated cavity which is realized using combinatorial etching technique and can integrate tens of different‐wavelength lasers on a chip. The lasing modes of this device are all single‐mode lasing because of the short cavity length. Its lowest lasing threshold is only 5.4 kW cm−2. The gain material and the dielectric microcavity can be controlled and fabricated separately, which have strong controllability and flexibility. This new laser chip paves the way for a new class of hybrid nanolasers for chip‐integrated applications.

RedDB, a computational database of electroactive molecules for aqueous redox flow batteries

An increasing number of electroactive compounds have recently been explored for their use in high-performance redox flow batteries for grid-scale energy storage. Given the vast and highly diverse chemical space of the candidate compounds, it is alluring to access their physicochemical properties in a speedy way. High-throughput virtual screening approaches, which use powerful combinatorial techniques for systematic enumerations of large virtual chemical libraries and respective property evaluations, are indispensable tools for an agile exploration of the designated chemical space. Herein, RedDB: a computational database that contains 31,618 molecules from two prominent classes of organic electroactive compounds, quinones and aza-aromatics, has been presented. RedDB incorporates miscellaneous physicochemical property information of the compounds that can potentially be employed as battery performance descriptors. RedDB’s development steps, including: (i) chemical library generation, (ii) molecular property prediction based on quantum chemical calculations, (iii) aqueous solubility prediction using machine learning, and (iv) data processing and database creation, have been described.

SCIPOG: Seeding and constraint support in IPOG strategy for combinatorial t-way testing to generate optimum test cases

Combinatorial t-way technique is efficient in generating test data and addressing the problem of combinatorial explosion. When constructing a test case, numerous literatures classified t-way strategies into two basic approaches as either One-test-at-a-time approach (OTAT) and One-parameter-at-a-time approach (OPAT). At least three major challenge groups can be encountered when creating test cases. The first one is provision of parameters seeding support that will improve the software quality. The second involves automatically obtaining data regarding parameter constraints and identifying interactions between systemcomponents. The last one is the execution speed and the test suite size. However, in all the existing OPAT t-way strategies, given that the system is loaded with this information, testing present-day software systems is made difficult or impossible. This study presents an effective combinatorial t-way test case generation strategy named Seeding and Constraint support in In-Parameter-Order Generalized (SCIPOG), to develop an improved paired testing approach. However, the study examines the present state-of-the-art and compares several OPAT strategies found in the literature. Moreover, experiments are discussed as part of this process to demonstrate the correctness of the implementation. When statistically analyzing the findings, two non-parametric tests—the Wilcoxon Rank and Friedman tests–were run. SCIPOG, however, produced competitive results. Finally, SCIPOG showed the efficiency of the two proposed methods, which are seeding and constraint support in IPOG strategy.

An Exposition of Integral Powers of Polynomial with Binomial Coefficients

This article exposed the disposition of powers of a polynomial function with binomial coefficients. The relationship was established by the deft deployment of summation notation, combinatorial techniques, and differential operators. The result of the investigation shows clearly that the disposition of integrals powers of polynomials with binomial coefficients generates even positive integer factorial values (2k)!

Nearly sharp Lang–Weil bounds for a hypersurface

AbstractWe improve to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb {F}_q$ -rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we slice the given hypersurface with a random plane.

Location of Points on Plane and the Order of Disposition of Sum of Powers of Cardinal Coordinates

This article exposed the structure of all orders of the sum of powers of cardinal coordinates with the explicitly determined computational disposition of similarity, between the coefficient of binomial expansion and the sum of powers of pairs of cardinal points on the plane. The proofs were achieved by a logical deployment of combinatorial techniques, laws of indices on powers of cardinal bases, and a comparison of corresponding results. The results proved conclusively that the sum of the powers of cardinal points is equal to the coefficients of the Binomial expansion with respect to the Pascal triangle pattern and entries.

SoftMatch: Comparing Scanpaths Using Combinatorial Spatio-Temporal Sequences with Fractal Curves.

Recent studies matching eye gaze patterns with those of others contain research that is heavily reliant on string editing methods borrowed from early work in bioinformatics. Previous studies have shown string editing methods to be susceptible to false negative results when matching mutated genes or unordered regions of interest in scanpaths. Even as new methods have emerged for matching amino acids using novel combinatorial techniques, scanpath matching is still limited by a traditional collinear approach. This approach reduces the ability to discriminate between free viewing scanpaths of two people looking at the same stimulus due to the heavy weight placed on linearity. To overcome this limitation, we here introduce a new method called SoftMatch to compare pairs of scanpaths. SoftMatch diverges from traditional scanpath matching in two different ways: firstly, by preserving locality using fractal curves to reduce dimensionality from 2D Cartesian (x,y) coordinates into 1D (h) Hilbert distances, and secondly by taking a combinatorial approach to fixation matching using discrete Fréchet distance measurements between segments of scanpath fixation sequences. These matching “sequences of fixations over time” are a loose acronym for SoftMatch. Results indicate high degrees of statistical and substantive significance when scoring matches between scanpaths made during free-form viewing of unfamiliar stimuli. Applications of this method can be used to better understand bottom up perceptual processes extending to scanpath outlier detection, expertise analysis, pathological screening, and salience prediction.

Combinatorial and Multinomial Coefficients and its Computing Techniques for Machine Learning and Cybersecurity

Mathematical and combinatorial techniques with nonnegative integers are used as computing algorithms for the programs development to apply in artificial intelligence and cybersecurity. Methodological advances in combinatorics and mathematics play a vital role in artificial intelligence and machine learning for data analysis and artificial intelligence-based cybersecurity for protection of the computing systems, devices, networks, programs and data from cyber-attacks. In connection with these ideas, this article is prepared for applications in computing science and cybersecurity. This paper presents computing and combinatorial formulae such as theorems on factorials, binomial, and multinomial coefficients and probability and binomial distributions.

Identities involving some special numbers and polynomials on p-adic integral

In this paper, we get new identities involving Bernoulli, Daehee and Stirling numbers, and their representations by using p-adic integrals and combinatorial techniques.

Hrip1 mediates rice cell wall fortification and phytoalexins elicitation to confer immunity against Magnaporthe oryzae.

Magnaporthe oryzae is a potent fungus that adversely affects rice yield. Combinatorial techniques of prevention, toxic chemicals, and fungicide are used to remedy rice blast infection. We reported the role of Hrip1 in cell death elicitation and expression of systematic acquired resistance that could potentially stifle M. oryzae infection. In this study, transcriptome and metabolomic techniques were used to investigate the mechanism by which Hrip1 reprogramed the transcriptome of rice seedlings to confer immunity against M. oryzae. Our results showed that Hrip1 induces cell wall thickening and phytoalexin elicitation to confer immunity against M. oryzae infection. Hrip1 activates key lignin biosynthetic genes and myeloblastosis transcription factors that act as molecular switches for lignin production. Lignin content was increased by 68.46% and more after 48 h onwards in Hrip1-treated seedlings compared to the control treatment. Further analysis of cell wall morphology using the transmission electron microscopy technique revealed over 100% cell wall robustness. Hrip1 also induced the expression of 24 diterpene synthases. These include class I and II terpene synthases, cytochrome P450 subfamilies (OsCYP76M and OsCYP71Z), and momilactones synthases. The relationship between the expression of these genes and metabolic elicitation was analyzed using ultra-performance liquid chromatography–tandem mass spectrometry. Enhanced amounts of momilactones A and B, oryzalactone, and phytocassane A and G were detected in the Hrip1-treated leaves. We also identified seven benzoxazinoid genes (BX1-BX7) that could improve rice immunity. Our findings show that Hrip1 confers dual immunity by leveraging lignin and phytoalexins for physical and chemical resistance. This study provides novel insights into the mechanisms underlying Hrip1-treated plant immunity.

Random-bond antiferromagnetic Ising model in a field.

Using combinatorial optimization techniques we study the critical properties of the two- and three-dimensional Ising models with uniformly distributed random antiferromagnetic couplings (1≤J_{i}≤2) in the presence of a homogeneous longitudinal field, h, at zero temperature. In finite systems of linear size, L, we measure the average correlation function, C_{L}(ℓ,h), when the sites are either on the same sublattice, or they belong to different sublattices. The phase transition, which is of first order in the pure system, turns to mixed order in two dimensions with critical exponents 1/ν≈0.5 and η≈0.7. In three dimensions we obtain 1/ν≈0.7, which is compatible with the value of the random-field Ising model, but we cannot discriminate between second-order and mixed-order transitions.

Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus

Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. Harmonic forms are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular cohomology class is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed piecewise polynomial differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed polynomial forms, that is, arbitrary degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.

Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study

We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. We also present a comprehensive analysis of the geometric models' behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones.

Conversion of the free Cellvibrio japonicus xyloglucan degradation system to the cellulosomal mode.

Cellulosomes are multi-enzyme complexes produced by specialised micro-organisms. The spatial proximity of synergistically acting enzymes incorporated in these naturally occurring complexes supports the efficient hydrolysis of lignocellulosic biomass. Several functional designer cellulosomes, incorporating naturally non-cellulosomal cellulases, have been constructed and can be used for cellulose saccharification. However, in lignocellulosic biomass, cellulose is tightly intertwined with several hemicelluloses and lignin. One of the most abundant hemicelluloses interacting with cellulose microfibrils is xyloglucan, and degradation of these polymers is crucial for complete saccharification. Yet, designer cellulosome studies focusing on the incorporation of hemicellulases have been limited. Here, we report the conversion of the free Cellvibrio japonicus xyloglucan degradation system to the cellulosomal mode. Therefore, we constructed multiple docking enzyme variants of C. japonicus endoxyloglucanase, β-1,2-galactosidase, α-1,6 xylosidase and β-1,4-glucosidase, using the combinatorial VersaTile technique dedicated to the design and optimisation of modular proteins. We individually optimised the docking enzymes to degrade the xyloglucan backbone and side chains. Finally, we show that a purified designer xyloglucanosome comprising these docking enzymes was able to release xyloglucan oligosaccharides, galactose, xylose and glucose from tamarind xyloglucan. KEY POINTS: • Construction of xyloglucan-degrading designer cellulosome. • Conversion of free Cellvibrio japonicus enzymes to cellulosomal mode. • Type of linker inserted between dockerin and enzyme module affects docking enzyme activity.

An algebraic approach to q-partial fractions and Sylvester denumerants

In 1857, Sylvester established an elegant theory that certain counting functions (which he termed denumerants) are quasi-polynomials by decomposing them into periodic and non-periodic parts. Each component of the decomposition, called a Sylvester wave, corresponds to a root of unity. Recently several researchers, using either combinatorial arguments or complex analytic techniques, obtained explicit formulas for the waves. In this work, we develop an algebraic approach to the Sylvester’s theory. Our methodology essentially relies on deriving q-partial fractions of the generating functions of the denumerants, and thereby obtains new explicit formulas for the waves. The formulas we obtain are expressed in terms of the degenerate Bernoulli numbers and a generalization of the Fourier–Dedekind sum. Further, we also prove certain reciprocity theorems of the generalized Fourier–Dedekind sums and a structure result on the top-order terms of the waves. The proofs rely on our evaluation operator and our far-reaching generalization of the Heaviside’s cover-up method for partial fractions. SageMath code for this work is available in the public domain.

The partial transpose and asymptotic free independence for Wishart random matrices, II

Using new combinatorial techniques, we significantly improve the previous results on asymptotic distributions and asymptotic free independence relations of partial transposes of Wishart random matrices. In particular, we give a necessary and sufficient condition for the asymptotic free independence of partial transposes of Wishart matrices with difference block sizes.

Breeding columnar apple-tree cultivars for supercompact growth habit

This work was based on the combinatorial breeding technique using donors of the columnar growth type (Co gene), sources of winter hardiness and high consumer qualities, and scab immunity donors (Rvi6 gene). The selection in the columnar apple-tree hybrid progeny was accompanied by the assessment of growth vigor, compactness degree, and seedling cultivability in accordance with the guidelines.Due to the increasing popularity of apple trees with the columnar crown type, the volume of breeding work for columnar growth habit at the Russian Research Institute of Fruit Crop Breeding (VNIISPK) over the past 9 years amounted to 133,582 flowers, or one third of the entire hybridization envisaged by the apple-tree breeding program. Meticulous selection of parent forms is carried out in order to incorporate a set of useful traits in the columnar growth habit genotype. Both cultivars and hybrid forms serve as sources and donors of the traits interesting for breeders. All the studied cultivars released by VNIISPK are dwarf: their height is within the range from 287 cm (cv. ‘Priokskoe’) to 198 cm (cv. ‘Orlovskaya Yesenia’), and they can be recommended for both commercial plantations and amateur gardens. The columnar form 34-34-98 is characterized by a unique degree of compactness (the ratio of the internode length to the shoot diameter) equal to 0.4 versus the average value of 1.6 for this indicator in the studied columnar cultivars. This form was involved in the release of 78 promising hybrid seedlings, the best of which (38.5%) were transferred to the garden for primary variety trials. A large yield of seedlings characterized by high compactness in the progeny of the columnar apple tree 34-34-98 makes it possible to recommend it for the development of new apple-tree genotypes for highly cost-effective orchards with high tree planting density.