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This paper extends the framework of soft set theory to directed graphs by introducing and analyzing the concepts of the normal product and restricted normal product of soft directed graphs. Building on Molodtsov’s foundational work in developing soft set theory to address uncertainty in data, this study presents new methods for modeling and understanding complex systems where uncertainty plays a significant role. Soft directed graphs, which enhance traditional graph models by incorporating parameters and uncertain relationships, serve as the foundation for this investigation. The normal product, defined as a combination of two soft directed graphs based on their respective parameter sets, and the restricted normal product, which combines soft directed graphs only where their parameter sets intersect, provide a comprehensive framework for these new operations. This paper also establishes the structural properties of these products, ensuring they are well-defined and retain the key features of soft directed graphs. Furthermore, we derive combinatorial identities related to vertex and arc counts, as well as degree sums, offering deeper insights into the composition and behavior of these graph products. [ABSTRACT FROM AUTHOR]

This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N, which assign a fixed number of 1's to some vertices and assign higher numbers to other vertices by adding up their neighbors' assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient conditions are obtained for superlinear and sublinear stepping stone solution growth. Furthermore this paper demonstrates the existence of a basis of graphs which characterizes superlinear growth. In the unbounded case, properly sublinear and superlinear stepping stone solution growth are obtained. [ABSTRACT FROM AUTHOR]

A reaction system is a computational model motivated by the functioning of living cells. In previous studies, Ehrenfeucht et al. have classified the functions specified by minimal reaction systems. Several studies have focused on the reaction system ranks of functions specified by minimal reaction systems, where the rank pertains to the minimum size of a specifying reaction system. This paper focuses on reaction system rank for a class of union-additive functions specified by minimal reaction systems introduced by Salomaa, which are closed under composition. More precisely, we establish a sufficient condition for the equality of reaction system ranks for functions belonging to the class. Then we study the reaction system ranks of such functions with signature sizes one and two, as well as those with one-to-one signatures, specifically over ternary and quaternary alphabets. Lastly, we establish a lower upper bound of reaction system ranks of such functions with one-to-one signatures for | S | ≥ 5. [ABSTRACT FROM AUTHOR]

Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach. [ABSTRACT FROM AUTHOR]

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The method of hypergraph containers, which was introduced several years ago by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph H, the threshold at which the distribution of edges in a typical H-free graph with a given number of edges undergoes a transition from 'random-like' to 'structured'. On the other hand, for nonmonotone hereditary graph properties, the standard version of this method does not allow one to establish even the existence of such a threshold. In this paper we introduce a refinement of the container method that takes into account the asymmetry between edges and non-edges in a sparse member of a hereditary graph property. As an application, we determine the approximate structure of a typical graph with n vertices, m edges, and no induced copy of the 4-cycle, for each function m D m.n/ satisfying n4=3.log n/4 6 m n2. We show that almost all such graphs G have the following property: the vertex set of G can be partitioned into an 'almost independent' set (a set with o.m/ edges) and an 'almost-clique' (a set inducing a subgraph with density 1-a The lower bound on m is optimal up to a polylogarithmic factor, as standard arguments show that if n then almost all such graphs are 'random-like'. As a further consequence, we deduce that the random graph G.n; p/ conditioned to contain no induced 4-cycles undergoes phase transitions at p=n -2/3+o(1) p=n -2/3+o(1). [ABSTRACT FROM AUTHOR]

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution. [ABSTRACT FROM AUTHOR]

In problems such as variable selection and graph estimation, models are characterized by Boolean logical structure such as the presence or absence of a variable or an edge. Consequently, false-positive error or false-negative error can be specified as the number of variables/edges that are incorrectly included or excluded in an estimated model. However, there are several other problems such as ranking, clustering, and causal inference in which the associated model classes do not admit transparent notions of false-positive and false-negative errors due to the lack of an underlying Boolean logical structure. In this paper, we present a generic approach to endow a collection of models with partial order structure, which leads to a hierarchical organization of model classes as well as natural analogs of false-positive and false-negative errors. We describe model selection procedures that provide false-positive error control in our general setting, and we illustrate their utility with numerical experiments. [ABSTRACT FROM AUTHOR]

We study the geometry underlying the Wilson loop diagram approach to calculating scattering amplitudes in Supersymmetric Yang Mills (SYM) N=4. In particular, we study the smallest non-trivial multi-propagator case, consisting of 2 propagators on 6 vertices. We do this by translating the integrals of the theory to the combinatorics of the positive geometry each diagram represents, specifically identifying the positroid cells defined by each diagram and the homology of the subcomplex they collectively generate in GR,≥0(2, 6). We verify the conjecture that the spurious singularities of the volume functional do all cancel on the codimension 1 boundaries of these cells, in this case. We also show that how the spurious singularities cancel is actually much more complicated than previously understood. The direct calculation laid out in this paper identifies many intricacies and artifacts of the geometry of Wilson loop diagram that need further study. [ABSTRACT FROM AUTHOR]

The zeros of the nth D'Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind η -function. In this paper, we prove that these coefficients are non-vanishing for | x | > κ (n - 1) and κ ≈ 9.7225 . Numerical computations imply that 9.72245 is a lower bound for κ . This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions. [ABSTRACT FROM AUTHOR]

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K * λ K v * , the λ -fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K * λ K v * has a D-decomposition is called the λ -fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ -fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ -fold spectrum problem for each of the remaining 21 orientations. [ABSTRACT FROM AUTHOR]

In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals. [ABSTRACT FROM AUTHOR]

The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi. [ABSTRACT FROM AUTHOR]

The path choice models play a key role in transportation engineering, especially when coupled with an assignment procedure allowing link flows to be obtained. Their implementation could be complex and resource-consuming. In particular, such a task consists of several stages, including (1) the collection of a large set of data from surveys to infer users' path choices and (2) the definition of a model able to reproduce users' choice behaviors. Nowadays, stage (1) can be improved using floating car data (FCD), which allow one to obtain a reliable dataset of paths. In relation to stage (2), different structures of models are available; however, a compromise has to be found between the model's ability to reproduce the observed paths (including the ability to forecast the future path choices) and its applicability in real contexts (in addition to guaranteeing the robustness of the assignment procedure). Therefore, the aim of this paper is to explore the opportunities offered by FCD to calibrate a path/route choice model to be included in a general procedure for scenario assessment. The proposed methodology is applied to passenger and freight transport case studies. Significant results are obtained showing the opportunities offered by FCD in supporting path choice simulation. Moreover, the characteristics of the model make it easily applicable and exportable to other contexts. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind and is hence not overly technical. We consider answers to questions like the following: How many sum-free subsets are there in a given abelian group G? What are its sum-free subsets of maximum cardinality? What is the maximum cardinality of these sum-free subsets? What does a typical sum-free subset of G look like? [ABSTRACT FROM AUTHOR]

Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a single differential equation within a dynamical system. The system in question comprises two first-order nonlinear ordinary differential equations of a particular kind. The method proposed hinges on applying elements of combinatorics to the traditional mathematical investigation of a function with a single independent variable. This approach enables the exact determination of the different qualitative scenarios in which the desired solution changes, under the assumption that the function values monotonically diminish from a specified value down to a finite zero. This paper outlines the creation and decomposition of the monotone stability region associated with the solution under consideration. [ABSTRACT FROM AUTHOR]

Mathematical modeling of the emergent dynamics of gene regulatory networks (GRN) faces a double challenge of (a) dependence of model dynamics on parameters, and (b) lack of reliable experimentally determined parameters. In this paper we compare two complementary approaches for describing GRN dynamics across unknown parameters: (1) parameter sampling and resulting ensemble statistics used by RACIPE (RAndom CIrcuit PErturbation), and (2) use of rigorous analysis of combinatorial approximation of the ODE models by DSGRN (Dynamic Signatures Generated by Regulatory Networks). We find a very good agreement between RACIPE simulation and DSGRN predictions for four different 2- and 3-node networks typically observed in cellular decision making. This observation is remarkable since the DSGRN approach assumes that the Hill coefficients of the models are very high while RACIPE assumes the values in the range 1-6. Thus DSGRN parameter domains, explicitly defined by inequalities between systems parameters, are highly predictive of ODE model dynamics within a biologically reasonable range of parameters. [ABSTRACT FROM AUTHOR]

MALTA is part of the Depleted Monolithic Active Pixel sensors designed in Tower 180 nm CMOS imaging technology. A custom telescope with six MALTA planes has been developed for test beam campaigns at SPS, CERN, with the ability to host several devices under test. The telescope system has a dedicated custom readout, online monitoring integrated into DAQ with realtime hit map, time distribution and event hit multiplicity. It hosts a dedicated fully configurable trigger system enabling to trigger on coincidence between telescope planes and timing reference from a scintillator. The excellent time resolution performance allows for fast track reconstruction, due to the possibility to retain a low hit multiplicity per event which reduces the combinatorics. This paper reviews the architecture of the system and its performance during the 2021 and 2022 test beam campaign at the SPS North Area. [ABSTRACT FROM AUTHOR]

In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley-Lieb algebra, arising as the generic centralizer algebra EndUq(gl2)(V⊕k/, where V = V(0) ⊕ V(1) is the direct sum of the trivial and natural module for the quantized enveloping algebra Uq(gl2). It is a proper subalgebra of the Motzkin algebra (the Uq(sl2)-centralizer) of Benkart and Halverson. We prove a version of Schur-Weyl duality for the new algebras, and describe their generic representation theory. [ABSTRACT FROM AUTHOR]

The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart-Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang-Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang-Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant K-group of semi-infinite flag manifolds. The generalized quantum Yang-Baxter moves give rise to a "sijection" (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula. [ABSTRACT FROM AUTHOR]

For G a finite group acting linearly on A 2 , the equivariant Hilbert scheme Hilb r [ A 2 / G ] is a natural resolution of singularities of Sym r (A 2 / G) . In this paper, we study the topology of Hilb r [ A 2 / G ] for abelian G and how it depends on the group G. We prove that the topological invariants of Hilb r [ A 2 / G ] are periodic or quasipolynomial in the order of the group G as G varies over certain families of abelian subgroups of G L 2 . This is done by using the Białynicki-Birula decomposition to compute topological invariants in terms of the combinatorics of a certain set of partitions. [ABSTRACT FROM AUTHOR]

This paper introduces the R package clustAnalytics, which comprises a set of criteria for assessing the significance and stability of communities in networks found by any clustering algorithm. clustAnalytics works with graphs of class igraph from the R-package igraph, extended to handle weighted and/or directed graphs. clustAnalytics provides a set of community scoring functions, and methods to systematically compare their values to those of a suitable null model, which are of use when testing for cluster significance. It also provides a non parametric bootstrap method combined with similarity metrics derived from information theory and combinatorics, useful when testing for cluster stability, as well as a method to synthetically generate a weighted network with a ground truth community structure based on the preferential attachment model construction, producing networks with communities and scale-free degree distribution. [ABSTRACT FROM AUTHOR]

This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect 2, analogous to Richards's formula for defect 2 blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding "q-decomposition numbers", i.e. the canonical basis coefficients in the level-1 q-deformed Fock space of type A^{(2)}_{2n}; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic 2n+1. Along the way, we prove some general results on q-decomposition numbers. This paper represents the first substantial progress on canonical bases in type A^{(2)}_{2n}. [ABSTRACT FROM AUTHOR]

We investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on τ -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study. [ABSTRACT FROM AUTHOR]

A matching of a graph is a set of its edges that pairwise do not have common vertices. An important parameter of graphs, which is used in mathematical chemistry, is the Hosoya index, defined as the number of their matchings. Previously, the problems of maximizing this index were considered and completely solved for -vertex trees with two, three and four leaves for any sufficiently large . In the present paper, a similar problem is completely solved for 5-leaved trees with and for 6-leaved trees with . [ABSTRACT FROM AUTHOR]

This paper describes a new method of classification based on spectral analysis. The motivations behind developing the new model were the failures of the classical spectral cluster analysis based on combinatorial and normalized Laplacian for a set of real-world datasets of textual documents. Reasons of the failures are analysed. While the known methods are all based on usage of eigenvectors of graph Laplacians, a new classification method based on eigenvalues of graph Laplacians is proposed and studied. [ABSTRACT FROM AUTHOR]

The phased-mission reliability of unmanned aerial vehicle (UAV) swarm refers to its capability to successfully complete the missions of each phase under specified conditions for a specified period. In order to study the reliability of phased-mission in UAV swarm, this paper firstly studies the reliability of a single UAV under fault coverage. Then, considering the mission characteristics of UAV swarm, the consecutive k-out-of-n system is studied to model and predict the reliability of UAV swarm phase mission. Some importance measures are introduced to analyze the influence of UAV in different positions on the reliability of the whole system. Finally, numerical examples of linear and circular UAV swarms are given to demonstrate and verify the correctness of the model. The reliability modeling established in this paper can predict the phased-mission reliability of UAV swarm scientifically. [ABSTRACT FROM AUTHOR]

The leftmost column entries of RNA arrays I and II count the RNA numbers that are related to RNA secondary structures from molecular biology. RNA secondary structures sometimes have mutations and wobble pairs. Mutations are random changes that occur in a structure, and wobble pairs are known as non-Watson–Crick base pairs. We used topics from RNA combinatorics and Riordan array theory to establish connections among combinatorial objects related to linear trees, lattice walks, and RNA arrays. In this paper, we establish interesting new explicit bijections (one-to-one correspondences) involving certain subclasses of linear trees, lattice walks, and RNA secondary structures. We provide an interesting generalized lattice walk interpretation of RNA array I. In addition, we provide a combinatorial interpretation of RNA array II as RNA secondary structures with n bases and k base-point mutations where ω of the structures contain wobble base pairs. We also establish an explicit bijection between RNA structures with mutations and wobble bases and a certain subclass of lattice walks. [ABSTRACT FROM AUTHOR]

The present Special Issue of Designs, Codes and Cryptography is dedicated to Aart Blokhuis, one of the leading mathematicians in finite geometry and combinatorics. References 1 Ball S, Blokhuis A, Mazzocca F. Maximal arcs in Desarguesian planes of odd order do not exist. The third editor of this special issue, Michel Lavrauw, was a PhD student of Aart between 1997 and 2001. Another challenging problem at the time was to prove the non-existence of maximal arcs in Galois planes of odd order; this was achieved by Aart and his coauthors Simeon Ball and Francesco Mazzocca in [[1]]. [Extracted from the article]

The notion of correlation immune functions has been introduced by Siegenthaler (1984) in symmetric cryptography in the framework of stream ciphers. At the conference CRYPTO’91 by Camion et al., it has been pointed out that this notion existed in statistics and combinatorics. It has recently been highlighted that such functions also play an important role in a new framework related to side-channel attack counter-measures. Since then, the interest in correlation immune Boolean functions has been renewed, and new challenges regarding these functions have appeared. Specifically, low Hamming weight correlation immune functions have been selected as useful for counter-measures to side-channel attacks. Despite their importance, the literature is not abundant in this research direction. Two very interesting articles in which such correlation immune functions were nicely explored, given this novel use of them. Carlet initiated the first one in 2013, and the second one is due to Carlet and Chen (2018). This paper deals with correlation immune Boolean functions aiming to produce more candidates of those processing low Hamming weights. We shall focus on correlation immune Boolean functions with Hamming weights power of 2 (which offer a flexibility to control the correlation immunity aspects) and present several methods of designing them. Some design methods are efficient and could be employed to derive such functions. Consequently, given two positive integers $n$ and $m$ , we derive new effective constructions of correlation immune Boolean functions with Hamming weight power of 2. Furthermore, an upper bound on the correlation immunity of the newly constructed $n$ -variable Boolean functions with Hamming weight $2^{m}$ was determined for $n-m\ge 0$. Besides, exact values and lower bounds on the maximum correlation immunity of those functions are explored and discussed, mainly when the values of $n$ and $m$ are very close. This paper also exhibits explicit examples of those correlation immune functions that illustrate our methods. [ABSTRACT FROM AUTHOR]

This paper presents a robust quadratic convex model for the optimal scheduling of photovoltaic generators in unbalanced bipolar DC grids. The proposed model is based on Taylor's series expansion which relaxes the hyperbolic relation between constant power terminals and voltage profiles. Furthermore, the proposed model is solved in the recursive form to reduce the error generated by relaxations assumed. Additionally, uncertainties in PV generators are considered to assess the effectiveness of the proposed recursive convex. Several proposed scenarios for the numerical validations in a modified 21-bus test system were tested to validate the robust convex model's performance. All the simulations were carried out in the MATLAB programming environment using Yalmip and Gurobi solver. Initially, a comparative analysis with three combinatorial optimization methods under three PV generation scenarios was performed. These scenarios consider levels of 0, 50, and 100% capacity of the PV systems. The results demonstrate the effectiveness of the proposed recursively solved convex model, which always achieves the global optimum for three levels of capacity of the PV generators, with solutions of 95.423 kW, 31.525 kW, and 22.985 kW for 0%, 50%, and 100% of the capacity PV rating, respectively. In contrast, the combinatorial optimization methods do not always reach these solutions. Furthermore, the power loss for the robust model is comparable to the deterministic model, increasing by 1.65% compared to the deterministic model. [ABSTRACT FROM AUTHOR]

Combinatorics math problems are often used as a benchmark to test human cognitive and logical problem-solving skills. These problems are concerned with counting the number of solutions that exist in a specific scenario that is sketched in natural language. Humans are adept at solving such problems as they can identify commonly occurring structures in the questions for which a closed-form formula exists for computing the answer. These formulas exploit the exchangeability of objects and symmetries to avoid a brute-force enumeration of all possible solutions. Unfortunately, current AI approaches are still unable to solve combinatorial problems in this way. This paper aims to fill this gap by developing novel AI techniques for representing and solving such problems. It makes the following five contributions. First, we identify a class of combinatorics math problems which traditional lifted counting techniques fail to model or solve efficiently. Second, we propose a novel declarative language for this class of problems. Third, we propose novel lifted solving algorithms bridging probabilistic inference techniques and constraint programming. Fourth, we implement them in a lifted solver that solves efficiently the class of problems under investigation. Finally, we evaluate our contributions on a real-world combinatorics math problems dataset and synthetic benchmarks. [ABSTRACT FROM AUTHOR]

The simple exclusion process studied in this paper consists of a system of K particles moving on the vertex set of a finite undirected connected graph. If there is one, the particle at x chooses one of the deg(x) neighbors of its current location uniformly at random at rate ρx > 0, and jumps to that vertex if and only if it is empty. Though this model is a natural mathematical object, it is also motivated by applications in the control of robotic swarms. After expressing the stationary distribution and the limiting occupation times of the process, we study how the total number of particles affects the occupation times ratios at different vertices. Using a novel qualitative argument based on combinatorial techniques, we prove that, while the occupation time at x increases with both D(x) = deg(x)=ρx and the total number of particles, the limiting occupation time increases faster with the total number of particles at vertices with a low D(x). [ABSTRACT FROM AUTHOR]

A set C of vertices in a graph G = (V, E) is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala's contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes. [ABSTRACT FROM AUTHOR]

Operations in the theory of graphs has a substantial inuence in the analytical and factual dimensions of the domain. In the realm of chemical graph theory, topological descriptor serves as a comprehensive graph invariant linked with a specific molecular structure. The study on the Sombor index is initiated recently by Ivan Gutman. The triangle parallel graph comprises of the edges of subdivision graph along with the edges of the original graph. In this paper, we make use of combinatorial inequalities related with the vertices, edges and the neighborhood concepts as well as the other topological descriptors in the computations for the determination of bounds of Sombor index for certain corona products involving the triangle parallel graph. [ABSTRACT FROM AUTHOR]

Catalan sequence initially introduced by Euler but named after Eugene Catalan has found abundant applications in the fields of probability theory, number theory, analysis, combinatorics etc. Recently ˙Ilkhan formulated the matrix associated with Catalan numbers and studied its domain in c0(C~) and c(C~). In this paper, by employing Catalan matrix defined by Catalan sequence and convergence via ideal, we establish Catalan sequence spaces c I 0 (C~), c I (C~) and ℓ I ∞(C~), study the properties of these corresponding spaces and establish inclusion relations. [ABSTRACT FROM AUTHOR]

The Erdős-Hajnal conjecture is one of the most classical and well-known problems in extremal and structural combinatorics dating back to 1977. It asserts that in stark contrast to the case of a general n-vertex graph, if one imposes even a little bit of structure on the graph, namely by forbidding a fixed graph H as an induced subgraph, instead of only being able to find a polylogarithmic size clique or an independent set, one can find one of polynomial size. Despite being the focus of considerable attention over the years, the conjecture remains open. In this paper, we improve the best known lower bound of 2 Ω (log n) on this question, due to Erdős and Hajnal from 1989, in the smallest open case, namely when one forbids a P 5 , the path on 5 vertices. Namely, we show that any P 5 -free n-vertex graph contains a clique or an independent set of size at least 2 Ω (log n) 2 / 3 . We obtain the same improvement for an infinite family of graphs. [ABSTRACT FROM AUTHOR]

Mathematics is the most logical and unbiased scientific field there is. What is true in mathematics today will forever stay true, and mathematical arguments will always clearly indicate which side "wins". These are some of the characteristics of mathematics that I have liked very much since I was young. In this article, I will try to demonstrate the beauty of mathematics and show you that it appears in many situations in our daily lives--including places we might not expect. Then, I will give you a taste of two studies I conducted in a mathematical field called combinatorics, and I will explain how they are related to everyday problems. One of these problems is how to perform accurate mathematical analyses of large streams of data that we cannot save in a computer's memory. Finally, I will try to give you an idea of my life as a mathematician--the challenges and difficulties, alongside the immense enjoyment and satisfaction I derive from them. Join me to an extended dive into my work on the wonders of combinatorics. [ABSTRACT FROM AUTHOR]

Divide-and-conquer dividing by a half recurrences, of the form x n = a · x ⌈ n / 2 ⌉ + a · x ⌊ n / 2 ⌋ + p (n) , n ⩾ 2 , appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. These equations are usually "solved" by means of a Master Theorem that provides a bound for the growing order of xn, but not the solution's explicit expression. In this paper we give a finite explicit expression for this solution, in terms of the binary decomposition of n, when the independent term p(n) is a polynomial in ⌈n/2⌉ and ⌊n/2⌋. As an application, we obtain explicit formulas for several sequences of interest in phylogenetics, combinatorics, and computer science, for which no such formulas were known so far: for instance, for the Total Cophenetic index and the rooted Quartet index of the maximally balanced bifurcating phylogenetic trees with n leaves, and the sum of the bitwise AND operator applied to pairs of complementary numbers up to n. [ABSTRACT FROM AUTHOR]

Logical analysis of data (LAD), an approach to data analysis based on Boolean functions, combinatorics, and optimization, can be considered one of the methods of interpretable machine learning. A feature of LAD is that, among many patterns, different types of patterns can be identified, for example, prime, strong, spanned, and maximum. This paper proposes a decision-support approach to recognition by sharing different types of patterns to improve the quality of recognition in terms of accuracy, interpretability, and validity. An algorithm was developed to search for pairs of strong patterns (prime and spanned) with the same coverage as the training sample, having the smallest (for the prime pattern) and the largest (for the spanned pattern) number of conditions. The proposed approach leads to a decrease in the number of unrecognized observations (compared with the use of spanned patterns only) by 1.5–2 times (experimental results), to some reduction in recognition errors (compared with the use of prime patterns only) of approximately 1% (depending on the dataset) and makes it possible to assess in more detail the level of confidence of the recognition result due to a refined decision-making scheme that uses the information about the number and type of patterns covering the observation. [ABSTRACT FROM AUTHOR]

In recent years, integral inequalities are investigated due to their extensive applications in several domains. The aim of the paper is to investigate certain new fractional integral inequalities which include Hermite-Hadamard inequality and different forms of trapezoid type inequalities related to Hermite-Hadamard inequality for h-Godunova-Levin preinvex function. Moreover, we compare our obtained results with the existing work in the literature and are represented by corollaries. [ABSTRACT FROM AUTHOR]

Aiming at improving the accuracy of current tracking algorithms and tracking the target more quickly, a new target tracking method that is from multiscale analysis is presented, which is combined with combinatorial polygon matching. First, a target's range is estimated by analyzing multiple scales in the reference frame. Then, the pixels are judged as the target pixels within the target range determined in the previous step by calculating the relevant radial distance and judging the similarity of radial distance between target pixels to be detected and its background pixels. At the same time, the background modeling is updated according to the similarity between the incremental code of the background pixels and the corresponding pixels in the observed images. Last, the target trajectory is obtained by dynamic polygon matching. When encountering external interference, the performance of the improved algorithm is relatively stable. Compared with the existing methods, the proposed algorithm in this paper is verified by simulation, which is simple in operation and can track the target quickly and accurately. [ABSTRACT FROM AUTHOR]

Symmetric functions play an important role in several subjects of mathematics, such as algebraic combinatorics, representation theory of finite groups and algebraic geometry. In this paper, we study the solutions of the following system of equations related to bases of symmetric functions: where , are natural numbers, and . Our main theorems generalize several known results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of normalized distance Laplacian matrices for signed graphs corresponding to the two signed distances defined for signed graphs. We characterize balance in signed graphs using these matrices and compare the normalized distance Laplacian spectral radius of signed graphs with that of all-negative signed graphs. Also we characterize the signed graphs having maximum normalized distance Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

There is no known formula that counts the number of transitive relations on a set with n elements. In this paper, it is shown that no polynomial in n with integer coefficients can represent a formula for the number of transitive relations on a set with n elements. Several inequalities giving various useful recursions and lower bounds on the number of transitive relations on a set are also proved. [ABSTRACT FROM AUTHOR]

Let H = (V (H) , E (H)) be a hypergraph, where V (H) is a set of vertices and E (H) is a set of non-empty subsets of V (H) called edges. If all edges of H have the same cardinality r , then H is an r -uniform hypergraph. A hypergraph H ′ = (V (H ′) , E (H ′)) is called a hypersubgraph of a hypergraph H = (V (H) , E (H)) if V (H ′) ⊆ V (H) and E (H ′) ⊆ E (H). The v e r t e x - c o n n e c t i v i t y of hypergraph H , denoted by κ (H) , is the cardinality of a minimum vertex set S such that H − S is disconnected or is a trival hypergraph. We call H k - c o n n e c t e d if κ (H) ≥ k. Tian, Lai and Meng [Y. Z. Tian, H.-J. Lai, J. X. Meng, On the sizes of vertex- k -maximal r -uniform hypergraphs, Graphs and Combinatorics 35(5) (2019) 1001–1010] conjectered that, for sufficiently large n , every n -vertex r -uniform hypergraph with no (k + 1) -connected hypersubgraphs has at most n r − n − k r + n k − 2 k r edges. This upper bound is equal to 1 2 k (n − k) 2 + 2 3 k 2 (n − k) − 3 2 k − 2 9 (n − k) when r = 3. In this paper, we prove that for k ≥ 3 and n ≥ 2. 6 7 k , every n -vertex 3 -uniform hypergraph with no (k + 1) -connected hypersubgraphs has at most 3 5 k (n − k) 2 + 1 2 k 2 (n − k) − k (n − k) + 1 3 (n − k) edges. [ABSTRACT FROM AUTHOR]

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained by using the methods of multivariate analytic combinatorics, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion— $q$ -ary sequences with fixed Manhattan weight—is briefly discussed, and an application in coding for molecular timing channels is illustrated. [ABSTRACT FROM AUTHOR]