Search

To face market volatility, reconfigurable manufacturing systems (RMS) aim to efficiently and cost-effectively react to changes. We focus on one characteristic of RMS: the scalability (ability to adapt the volume of throughput). In the literature, the only few indicators for scalability are not always formally defined and usually only consider a partial view of scalability. Moreover, most of them are actually more suited for the configuration planning rather than for the design. However, the design of the RMS has a high impact on its scalability. We propose the first combinatorial definition of this problem and a new measure to fully assess the scalability of a system at the design phase. This measure, based on a multi-objective approach, can assess the scalability of single-product manufacturing systems, analysing all configurations that it can implement. We present numerical experiments to compare this indicator with a state-of-the-art scalability indicator and with some classical production line design indicators, and we show that future research should focus on scalability as a specific criterion to optimise during the design of an RMS. In addition, the results obtained allow us to infer some managerial insights on the best levers to use when performing a reconfiguration for scalability purpose. [ABSTRACT FROM AUTHOR]

A connected matching in a graph G consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of G. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to determine an optimal connected matching in an edge-weighted graph, even in the planar bipartite case. We present two mixed integer programming formulations and a sophisticated branch-and-cut scheme to find weighted connected matchings in general graphs. The formulations explore different polyhedra associated to this problem, including strong valid inequalities both from the matching polytope and from the connected subgraph polytope. We conjecture that one attains a tight approximation of the convex hull of connected matchings using our strongest formulation, and report encouraging computational results over DIMACS Implementation Challenge benchmark instances. The source code of the complete implementation is also made available. [ABSTRACT FROM AUTHOR]

Monoids generated by elements of order two appear in numerous places in the literature. For example, Coxeter reflection groups in geometry, Kuratowski monoids in topology, various monoids generated by regular operations in language theory and so on. In order to initiate a classification of these monoids, we are interested in the subproblem of monoids, called strict Projection Involution Monoids (2-PIMs), generated by an involution and an idempotent. In this case we show, when the monoid is finite, that it is generated by a single equation (in addition to the two defining the involution and the idempotent). We then describe the exact possible forms of this equation and classify them. We recover Kuratowski's theorem as a special case of our study. [ABSTRACT FROM AUTHOR]

Let G be a graph with m edges, minimum degree $\delta $ and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan et al. ['Bisections of graphs without short cycles', Combinatorics, Probability and Computing 27 (1) (2018), 44–59] showed that if (i) G is $2$ -connected, or (ii) $\delta \ge 3$ , or (iii) $\delta \ge 2$ and the girth of G is at least 5, then G admits a bisection such that $\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$ , where $e(V_i)$ denotes the number of edges of G with both ends in $V_i$. Let $s\ge 2$ be an integer. In this note, we prove that if $\delta \ge 2s-1$ and G contains no $K_{2,s}$ as a subgraph, then G admits a bisection such that $\max \{e(V_1),e(V_2)\}\le (1/4+o(1))m$. [ABSTRACT FROM AUTHOR]

Given a finite group G acting freely on a compact metric space M, and ε > 0 , we define the G-Borsuk graph on M by drawing edges x ∼ y whenever there is a non-identity g ∈ G such that d x , g y ≤ ε . We show that when ε is small, its chromatic number is determined by the topology of M via its G-covering number, which is the minimum k such that there is a closed cover M = F 1 ∪ ⋯ ∪ F k with F i ∩ g (F i) = ∅ for all g ∈ G \ { 1 } . We are interested in bounding this number. We give lower bounds using G-actions on Hom-complexes, and upper bounds using a recursive formula on the dimension of M. We conjecture that the true chromatic number coincides with the lower bound, and give computational evidence. We also study random G-Borsuk graphs, which are random induced subgraphs. For these, we compute thresholds for ε that guarantee that the chromatic number is still that of the whole G-Borsuk graph. Our results are tight (up to a constant) when the G-index and dimension of M coincide. [ABSTRACT FROM AUTHOR]

We suggest a new random model for links based on meander diagrams and graphs. We then prove that trivial links appear with vanishing probability in this model, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained for a fixed number of crossings. A random meander diagram is obtained through matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of random links in this model, and, moreover, of the respective 3-manifolds that are link complements in 3-sphere. We use this for exploring geometric properties of a link complement. Specifically, we give expected twist number of a link diagram and use it to bound expected hyperbolic and simplicial volume of random links. The tools from combinatorics that we use include Catalan and Narayana numbers, and Zeilberger's algorithm. [ABSTRACT FROM AUTHOR]

We construct combinatorial analogs of 2d higher topological quantum field theories. We consider triangulations as vertices of a certain CW complex Ξ . In the "flip theory," cells of Ξ flip correspond to polygonal decompositions obtained by erasing the edges in a triangulation. These theories assign to a cobordism Σ a cochain Z on Ξ flip constructed as a contraction of structure tensors of a cyclic A ∞ algebra V assigned to polygons. The cyclic A ∞ equations imply the closedness equation (δ + Q) Z = 0 . In this context, we define combinatorial BV operators and give examples with coefficients in Z 2 . In the "secondary polytope theory," Ξ sp is the secondary polytope (due to Gelfand–Kapranov–Zelevinsky) and the cyclic A ∞ algebra has to be replaced by an appropriate refinement that we call an A ^ ∞ algebra. We conjecture the existence of a good Pachner CW complex Ξ for any cobordism, whose local combinatorics is described by secondary polytopes and the homotopy type is that of Zwiebach's moduli space of complex structures. Depending on this conjecture, one has an "ideal model" of combinatorial 2d HTQFT determined by a local A ^ ∞ algebra. [ABSTRACT FROM AUTHOR]

Let (X,\tau) be a Polish space with Borel probability measure \mu, and G a locally finite one-ended Borel graph on X. We show that G admits a Borel one-ended spanning tree generically. If G is induced by a free Borel action of an amenable (resp., polynomial growth) group then we show the same result \mu-a.e. (resp., everywhere). Our results generalize recent work of Timár, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved this in the probability measure preserving setting. We apply our theorem to find Borel orientations in even-degree graphs and measurable and Baire measurable perfect matchings in regular bipartite graphs, refining theorems that were previously only known to hold for measure preserving graphs. In particular, we prove that bipartite one-ended d-regular Borel graphs admit Baire measurable perfect matchings. [ABSTRACT FROM AUTHOR]

In this article, I first reconsider the personal links between Comenius and Leibniz, searching for how Comenian texts and ideas may have reached Leibniz. For that reason, I outline a constellation of thinkers within Central European Protestantism, since a vast majority of the important intellectuals with some noteworthy link both to Comenius and Leibniz were based there. Second, I investigate more closely one particular intellectual link between Comenius and Leibniz, namely their common combinatorial standpoint, because the striking parallels, connections and similarities between Comenius's and Leibniz's ideas on combinatorics, universal language and relational metaphysics have been largely overlooked thus far. [ABSTRACT FROM AUTHOR]

We consider the problem of counting the number of varieties in a family over Q with a rational point. We obtain lower bounds for this counting problem for some families over P1, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher-dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics. [ABSTRACT FROM AUTHOR]

In this interdisciplinary, conceptual article with implications in marketing financial products and services, we study real estate and capital markets characterized by a predatory paradigm and economic agents' dark financial profiles (DFPs). These are estimated by three orthogonal components—disconnection, irrationality, and deceit. We identify the best interactional patterns of borrower-lender profiles, ones that expectedly minimize the risk of default. We resort to discretized, predator–prey Lotka–Volterra equations where lenders act as predators and borrowers as prey, incorporating market trends and learning effects. To mathematically operationalize our framework, we use combinatorics with high, medium, and low levels of the three components of DFPs. We find 27 salient lender-borrower interactional scenarios and observe three different patterns: explosive, conducive, and implosive. Our theoretical findings indicate that equal (ir)rationality (in financial terms) between lenders and borrowers is a necessary but insufficient condition to maintain harmonious, long-term relationships. We use eutectic theory to map the agents' profiles by introducing another variable: Expected return [E(Rp)] versus risk [σ], using the Capital Asset Pricing Model (CAPM) as a base. We find six market segments: the inactive predators and prey, the loose, the greedy, the vulnerable, and the stable. We identify the optimal combination of borrowers–lenders interaction under risk, given market trends and learning effects. We propose a path for future research that would see the application of analytical tools such as factor analysis, k-means clustering algorithm, χ2 and non-parametric Kruskal–Wallis and Dunn's multiple comparison tests to verify differences among the hypothesized segments. [ABSTRACT FROM AUTHOR]

A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the edges. In 2006 Babai proved that for a connected directed Cayley graph on n vertices the directed diameter is bounded above by a polynomial in undirected diameter and log n . Moreover, Babai conjectured that a similar bound holds for vertex-transitive graphs. We prove this conjecture of Babai, in fact, it follows from a more general bound for connected relations of homogeneous coherent configurations. The main novelty of the proof is a generalization of Ruzsa's triangle inequality from additive combinatorics to the setting of graphs [ABSTRACT FROM AUTHOR]

For any integer h ⩾ 2 , a set of integers B = { b i } i ∈ I is a B h -set if all h-sums b i 1 + ... + b i h with i 1 < ... < i h are distinct. Answering a question of Alon and Erdős [2], for every h ⩾ 2 we construct a set of integers X which is not a union of finitely many B h -sets, yet any finite subset Y ⊆ X contains an B h -set Z with | Z | ⩾ ε | Y | , where ε : = ε (h) . We also discuss questions related to a problem of Pisier about the existence of a set A with similar properties when replacing B h -sets by the requirement that all finite sums ∑ j ∈ J b j are distinct. [ABSTRACT FROM AUTHOR]

The order scheduling and rack sequencing problem deals with the order picking process in robotic mobile fulfillment systems: automated guided vehicles lift and transport movable storage racks to picking stations to supply items requested by customer orders which are put together in cardboard boxes on a workbench of limited capacity. To efficiently operate the station, the sequence in which racks visit the station one after another and the intervals at which customer orders are scheduled must be coordinated. We present a dynamic programming algorithm for the order scheduling and rack sequencing problem at a single picking station minimizing the number of rack visits. Despite monolithic mixed integer linear programming formulations, our approach appears to be the first combinatorial solution method for the problem in literature. A computational study demonstrates the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

The regularity method was pioneered by Szemerédi for graphs and is an important tool in extremal combinatorics. Over the last two decades, several extensions to hypergraphs were developed which were based on seemingly different notions of quasirandom hypergraphs. We consider the regularity lemmata for three‐uniform hypergraphs of Frankl and Rödl and of Gowers, and present a new proof that the concepts behind these approaches are equivalent. [ABSTRACT FROM AUTHOR]

The complementary roles of visible light communication (VLC) and radio frequency (RF) in both coverage and capacity have made their combined use essential in wireless communication. While this combination offers some advantages over a sole media, it presents challenges with respect to the handover from VLC to RF, provision of seamless connections between VLC and RF, and the blockage of line-of-sight (LOS) in VLC. Previous works have not paid much attention to the quality of experience (QoE) of a user in VLC heterogeneous network (HetNet) and the effect of user's mobility on QoE. Thus, in this article, we consider a heterogeneous environment where a user moves between two-tier VLC-RF network. We propose a fuzzy logic approach termed the estimated quality of experience fuzzy logic (EQFL-VHO) scheme. This scheme fuses four input metrics variables, namely: estimated QoE in RF, handover failure probability, delay cost, and velocity of the user equipment (UE), to decide whether handover needs to be triggered. After fuzzification, the handover factor (HF) is used to determine whether handover to RF would enhance user's QoE or not. Our scheme couples the advantages of the Mamdani fuzzy logic system and QoE estimation to enhance system performance. Comparatively, simulation outcomes demonstrate that the EQFL-VHO scheme outperforms traditional methods including immediate VHO (I-VHO) and dwell VHO (D-VHO), as well as alternative fuzzy logic techniques like combinatorial fusion analysis (CFA) and Kalman Filter and Fuzzy Logic in Heterogeneous Wireless Networks (KAFUZ), in terms of QoE enhancement, reduced handover failures, and minimized delay costs. Additionally, our approach effectively diminishes handover initiation frequency. [ABSTRACT FROM AUTHOR]

Cellular deconvolution via bulk RNA sequencing (RNA-seq) presents a cost-effective and efficient alternative to experimental methods such as flow cytometry and single-cell RNA-seq (scRNA-seq) for analyzing the complex cellular composition of tumor microenvironments. Despite challenges due to heterogeneity within and among tumors, our innovative deep learning-based approach, DeSide, shows exceptional accuracy in estimating the proportions of 16 distinct cell types and subtypes within solid tumors. DeSide integrates biological pathways and assesses noncancerous cell types first, effectively sidestepping the issue of highly variable gene expression profiles (GEPs) associated with cancer cells. By leveraging scRNA-seq data from six cancer types and 185 cancer cell lines across 22 cancer types as references, our method introduces distinctive sampling and filtering techniques to generate a high-quality training set that closely replicates real tumor GEPs, based on The Cancer Genome Atlas (TCGA) bulk RNA-seq data. With this model and high-quality training set, DeSide outperforms existing methods in estimating tumor purity and the proportions of noncancerous cells within solid tumors. Our model precisely predicts cellular compositions across 19 cancer types from TCGA and proves its effectiveness with multiple additional external datasets. Crucially, DeSide enables the identification and analysis of combinatorial cell type pairs, facilitating the stratification of cancer patients into prognostically significant groups. This approach not only provides deeper insights into the dynamics of tumor biology but also highlights potential therapeutic targets by underscoring the importance of specific cell type or subtype interactions. [ABSTRACT FROM AUTHOR]

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell–Neuwirth for configuration spaces, and their existence is detected by a combinatorial property of an associated finite partially ordered set. This is consistent with Terao's fibration theorem connecting bundles of hyperplane arrangements to Stanley's lattice supersolvability. We obtain a combinatorially determined class of K (π , 1) toric and elliptic arrangements. Under a stronger combinatorial condition, we prove a factorization of the Poincaré polynomial when the Lie group is noncompact. In the case of toric arrangements, this provides an analogue of Falk–Randell's formula relating the Poincaré polynomial to the lower central series of the fundamental group. [ABSTRACT FROM AUTHOR]

Catalan–Stieltjes matrices and their corresponding Catalan-like numbers are very important in combinatorics. In this paper, we give a sufficient condition for the Catalan–Stieltjes matrix to be triangular strictly totally positive (Δ STP), that is, all its minors are positive except those which are zero due to the triangular structure. On this basis, we show that it is possible to compute the bidiagonal factorization of a class of Catalan–Stieltjes matrices with high relative accuracy (HRA). We also demonstrate that this class of matrices includes many well-known combinatorial triangles, such as the Catalan triangle of Aigner and the Bell triangle. Moreover, we construct an accurate algorithm to compute the bidiagonal factorization of some Catalan–Stieltjes matrices. The algorithm guarantees HRA for several linear algebraic computations associated with these matrices, like computing their singular values, their inverses, as well as the solutions of some linear systems. We show the accuracy and effectiveness of the proposed algorithm through numerical experiments. [ABSTRACT FROM AUTHOR]

The projective norm graphs are central objects to extremal combinatorics. They appear in a variety of contexts, most importantly they provide tight constructions for the Turán number of complete bipartite graphs K t , s with s > (t - 1) ! . In this note we deepen their understanding further by determining their automorphism group. [ABSTRACT FROM AUTHOR]

In spite of its widespread applicability in learning theory, probability, combinatorics, and in various other fields, the Mallows model has only recently been examined from revenue management perspectives. To our knowledge, the only provably-good approaches for assortment optimization under the Mallows model have recently been proposed by Désir et al. (Oper Res 69(4):1206–1227, 2021), who devised three approximation schemes that operate in very specific circumstances. Unfortunately, these algorithmic results suffer from two major limitations, either crucially relying on strong structural assumptions, or incurring running times that exponentially scale either with the ratio between the extremal prices or with the Mallows concentration parameter. The main contribution of this paper consists in devising a quasi-polynomial-time approximation scheme for the assortment optimization problem under the Mallows model in its utmost generality. In other words, for any accuracy level ϵ > 0 , our algorithm identifies an assortment whose expected revenue is within factor 1 - ϵ of optimal, without resorting to any structural or parametric assumption whatsoever. Our work sheds light on newly-gained structural insights surrounding near-optimal Mallows-based assortments and fleshes out some of their unexpected algorithmic consequences. [ABSTRACT FROM AUTHOR]

We use algorithmic methods from online learning to explore some important objects at the intersection of model theory and combinatorics, and find natural ways that algorithmic methods can detect and explain (and improve our understanding of) stable structure in the sense of model theory. The main theorem deals with existence of \epsilon-excellent sets (which are key to the Stable Regularity Lemma, a theorem characterizing the appearance of irregular pairs in Szemerédi's celebrated Regularity Lemma). We prove that \epsilon-excellent sets exist for any \epsilon < \frac {1}{2} in k-edge stable graphs in the sense of model theory (equivalently, Littlestone classes); earlier proofs had given this only for \epsilon < 1/{2^{2^k}} or so. We give two proofs: the first uses regret bounds from online learning, the second uses Boolean closure properties of Littlestone classes and sampling. We also give a version of the dynamic Sauer-Shelah-Perles lemma appropriate to this setting, related to definability of types. We conclude by characterizing stable/Littlestone classes as those supporting a certain abstract notion of majority: the proof shows that the two distinct, natural notions of majority, arising from measure and from dimension, densely often coincide. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with R red, G green, B blue edges, the number of rainbow triangles is at most 2 R G B , which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality. [ABSTRACT FROM AUTHOR]

Positroids are a family of matroids introduced by Postnikov in the study of non-negative Grassmannians. Postnikov identified several combinatorial objects in bijections with positroids, among which are bounded affine permutations. On the other hand, the notion of essential sets, introduced for permutations by Fulton, was used by Knutson in the study of the special family of interval rank positroids. We generalize Fulton's essential sets to bounded affine permutations. The bijection of the latter with positroids, allows the study of the relationship between them. From the point of view of positroids, essential sets are maximally dependent cyclic intervals. We define connected essential sets and prove that they give a facet description of the positroid polytope, as well as equations defining the positroid variety. We define a subset of essential sets, called core, which contains minimal rank conditions to uniquely recover a positroid. We provide an algorithm to retrieve the positroid satisfying the rank conditions in the core or any compatible rank condition on cyclic intervals. [ABSTRACT FROM AUTHOR]

The study of P -polynomial association schemes (distance-regular graphs) and Q -polynomial association schemes, and in particular P - and Q -polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of P - and Q -polynomial association schemes are described by Askey-Wilson orthogonal polynomials or their relatives. These polynomials are one-variable orthogonal polynomials. It seems that the new attempt to define and study higher rank P - and Q -polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Crampé-d'Andecy-Vinet-Zaimi, and then followed by Bannai-Kurihara-Zhao-Zhu. The general theory and some explicit examples of families of higher rank (multivariate) P - and/or Q -polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate Q -polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the association schemes obtained from attenuated spaces are bivariate Q -polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study. [ABSTRACT FROM AUTHOR]

Near maximum distance separable (NMDS) codes are important in finite geometry and coding theory. Self-dual codes are closely related to combinatorics, lattice theory, and have important application in cryptography. In this paper, we construct a class of q-ary linear codes and prove that they are either MDS or NMDS which depends on certain zero-sum condition. In the NMDS case, we provide an effective approach to construct NMDS self-dual codes which largely extend known parameters of such codes. In particular, we proved that for square q, almost q/8 NMDS self-dual q-ary codes can be constructed. [ABSTRACT FROM AUTHOR]

A partition is said to be ℓ-regular if none of its parts is a multiple of ℓ. Let b5′(n) denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of n. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of b5′(n). We provide full characterization of the parity of b5′(2n + 1), present several congruences modulo 4, and prove that the generating function of the sequence (b5′(5n + 1)) is lacunary modulo any arbitrary positive powers of 5. [ABSTRACT FROM AUTHOR]

It is evident that skew polynomials offer promising directions for developing cryptographic schemes. This paper focuses on exploring skew polynomials and studying their properties with the aim of exploring their potential applications in fields such as cryptography and combinatorics. We begin by deriving the concept of resultants for bivariate skew polynomials. Then, we employ the derived resultant to incrementally eliminate indeterminates in skew polynomial systems, utilising both direct and modular approaches. Finally, we discuss some applications of the derived resultant, including cryptographic schemes (such as Diffie–Hellman) and combinatorial identities (such as Pascal's identity). We start by considering a bivariate skew polynomial system with two indeterminates; our intention is to isolate and eliminate one of the indeterminates to reduce the system to a simpler form (that is, relying only on one indeterminate in this case). The methodology is composed of two main techniques; in the first technique, we apply our definition of a (bivariate) resultant via a Sylvester-style matrix directly from the polynomials' coefficients, while the second is based on modular methods where we compute the resultant by using evaluation and interpolation approaches. The idea of this second technique is that instead of computing the resultant directly from the coefficients, we propose to evaluate the polynomials at a set of valid points to compute its corresponding set of partial resultants first; then, we can deduce the original resultant by combining all these partial resultants using an interpolation technique by utilising a theorem we have established. [ABSTRACT FROM AUTHOR]

We provide computationally efficient methods for unranking derangements and ménage permutations. That is, we provide a polynomial-time algorithm to extract the k th such permutation under the lexicographic ordering. More generally, we show that there exists a polynomial-time algorithm for unranking words in lexicographic order whenever there exists a polynomial-time algorithm for counting the number of such words with a given prefix. We use rook theory to give a polynomial-time algorithm for counting the number of derangements and ménage permutations with a given prefix and in turn use this to give an unranking algorithm. This has applications to combinatorics, probability, modeling, and data compression. [ABSTRACT FROM AUTHOR]

In matching theory, one of the most fundamental and classical branches of combinatorics, canonical decompositions of graphs are powerful and versatile tools that form the basis of this theory. However, the abilities of the known canonical decompositions, that is, the Dulmage–Mendelsohn, Kotzig–Lovász, and Gallai–Edmonds decompositions, are limited because they are only applicable to particular classes of graphs, such as bipartite graphs, or they are too sparse to provide sufficient information. To overcome these limitations, we introduce a new canonical decomposition that is applicable to all graphs and provides much finer information. This decomposition also provides the answer to the longstanding absence of a canonical decomposition that is nontrivially applicable to general graphs with perfect matchings. We focus on the notion of factor‐components as the fundamental building blocks of a graph; through the factor‐components, our new canonical decomposition states how a graph is organized and how it contains all the maximum matchings. The main results that constitute our new theory are the following: (i) a canonical partial order over the set of factor‐components, which describes how a graph is constructed from its factor‐components; (ii) a generalization of the Kotzig–Lovász decomposition, which shows the inner structure of each factor‐component in the context of the entire graph; and (iii) a canonically described interrelationship between (i) and (ii), which integrates these two results into a unified theory of a canonical decomposition. These results are obtained in a self‐contained way, and our proof of the generalized Kotzig–Lovász decomposition contains a shortened and self‐contained proof of the classical counterpart. [ABSTRACT FROM AUTHOR]

AbstractSuppose that K is a field, S=K[x1,…,xn] or S=K[[x1,…,xn]]. We present a characterization of those monomial ideals I of S, for which S/I is a half-factorial ring. This characterization is related to the structure of the base field K and also depends on the combinatorics of a graph constructed from the monomial ideal I. We also show that if R[x] or R[[x]] is half-factorial for an arbitrary commutative ring R, then R is an integral domain. [ABSTRACT FROM AUTHOR]

The demand for reliability in multicore processors, which are ubiquitous in today's computers, smartphones, and various other smart devices, is growing. Several factors can influence the seamless operation of multicore processors. For instance, at high frequencies, challenges such as signal distribution and maintaining low power consumption are prominent. An overheated processor can result in various issues in a computer, including freezes, unexpected shutdowns, and errors while running programs. These issues can have serious consequences, particularly during critical tasks. Processor cores belong to the class of multifunctional elements. Their multifunctionality contributes to the structural flexibility of the processor, allowing functions to be redistributed between cores and ensuring the processor can continue to operate successfully even if individual core blocks fail. This flexibility significantly enhances processor reliability, as demonstrated by quantitative assessments of the fault tolerance of multicore processors. We use mathematical combinatoric methods for studying and quantitating different variants of cores and blocks in multicore processor. Number of conditions of work in view of the faults of individual blocks. With the results we also have a quantitative assessment and analysis of the flexibility and the likelihood of functioning (reliability) of multicore processors, which is the Central focus of this article. [ABSTRACT FROM AUTHOR]

This article analyzes the potential khipu-document "match" involving six Inka-style khipus from Peru's Santa Valley that record data similar to a 1670 colonial census of San Pedro de Corongo. Despite its potential as a breakthrough in khipu decipherment, crucial details surrounding these potential "Rosetta khipus" and their associated colonial document have been left out, overlooked, or confused in the literature. This article's objective is to synthesize and evaluate what is known about this khipu-document "match" while also using combinatorics to identify an optimal moiety alignment for the six lineage groups listed in the 1670 census. This novel moiety alignment differs from the one proposed in 2018 by Manuel Medrano and Gary Urton and is supported by additional structures found in the six khipus. This discovery could lead to a broader understanding of the social hierarchy of San Pedro de Corongo in 1670, as well as an expanded reading of the six Santa Valley khipus. Furthermore, this study presents the first-ever identification of recto and verso cord attachment knot orientations being used as a marked and unmarked sign, respectively. This article's findings provide a significant contribution toward the ongoing decipherment of nonnumerical khipu signs. [ABSTRACT FROM AUTHOR]

Summary: We provide two proofs of a theorem due to Baxter on the four-coloring of a cubic planar graph. We then consider an n-dimensional version of Baxter's theorem and relate it to Sperner's lemma. [ABSTRACT FROM AUTHOR]

Combinatorial analysis of Ag–Pd alloy thin films shows the formation of a solid solution with the face‐centered cubic structure for all investigated alloy compositions. Using mixed matter theory and the high field model, oxide growth upon applying an anodic potential is indicated. Electrochemical impedance spectroscopy supports the idea of oxide formation based on a change in behavior correlated to a sharp increase in oxide formation as indicated by cyclic voltammetry measurements. After polarization at 4 V, a stable total impedance and increasingly less‐pronounced phase shift are detected in the frequency range of 100 mHz to 10 kHz, which encompasses the common range of application of active implantable medical devices (AIMD). Upon fitting the measured impedance data with the Randles circuit, capacitances are determined in the nF cm−2 range for the investigated alloys and attributed to the formed mixed oxide layer. Charge transfer resistances are found to be in the kΩ cm−2 range. No correlation between alloy composition and capacitance or resistance can be detected so far. These findings indicate a fundamental applicability of Ag–Pd alloys for use in AIMD. [ABSTRACT FROM AUTHOR]

For a discrete poset X , McCord proved that the natural map | X | → X , from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, Živaljević defined the notion of order complex for a topological poset. For a large class of topological posets we prove the analog of McCord's theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. A familiar topological example is the Grassmann poset G n (R) of proper non-zero linear subspaces of R n + 1 partially ordered by inclusion. But our motivation in topological combinatorics is to apply the theorem to posets associated with tropical phased matroids over the tropical phase hyperfield, and in particular to elucidate the tropical version of the MacPhersonian Conjecture. This is explained in Sect. 2. [ABSTRACT FROM AUTHOR]

In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group W, every pair t,t' of distinct reflections lie in a unique maximal dihedral reflection subgroup of W. Our proof only relies on the combinatorics of words, in particular we do not use root systems at all. As an application, we deduce a new proof of a recent result of Delucchi-Paolini-Salvetti, stating that the poset [1,c]_T of generalized noncrossing partitions in any Coxeter group of rank 3 is a lattice. We achieve this by showing the more general statement that any interval of length 3 in the absolute order on an arbitrary Coxeter group is a lattice. This implies that the interval group attached to any interval [1,w]_T where w is an element of an arbitrary Coxeter group with \ell _T(w)=3 is a quasi-Garside group. [ABSTRACT FROM AUTHOR]

The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of F 2 r , focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for "generic" Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries. [ABSTRACT FROM AUTHOR]

Let G : = (G 1 , G 2 , G 3) be a triple of graphs on the same vertex set V of size n. A rainbow triangle in G is a triple of edges (e 1 , e 2 , e 3) with e i ∈ G i for each i and { e 1 , e 2 , e 3 } forming a triangle in V. The triples G not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities (α 1 , α 2 , α 3) such that if | E (G i) | > α i n 2 for each i and n is sufficiently large, then G must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Győri, He, Lv, Salia, Tompkins, Varga and Zhu. [ABSTRACT FROM AUTHOR]

In 2007, Andrews introduced Durfee symbols and k -marked Durfee symbols so as to give a combinatorial interpretation for the symmetrized moment function $\eta _{2k}(n)$ of ranks of partitions. He also considered the relations between odd Durfee symbols and the mock theta function $\omega (q)$ , and proved that the $2k$ th moment function $\eta _{2k}^0(n)$ of odd ranks of odd Durfee symbols counts $(k+1)$ -marked odd Durfee symbols of n. In this paper, we first introduce the definition of symmetrized positive odd rank moments $\eta _k^{0+}(n)$ and prove that for all $1\leq i\leq k+1$ , $\eta _{2k-1}^{0+}(n)$ is equal to the number of $(k+1)$ -marked odd Durfee symbols of n with the i th odd rank equal to zero and $\eta _{2k}^{0+}(n)$ is equal to the number of $(k+1)$ -marked Durfee symbols of n with the i th odd rank being positive. Then we calculate the generating functions of $\eta _{k}^{0+}(n)$ and study its asymptotic behavior. Finally, we use Wright's variant of the Hardy–Ramanujan circle method to obtain an asymptotic formula for $\eta _{k}^{0+}(n)$. [ABSTRACT FROM AUTHOR]

The sandpile group of a connected graph G, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of F 2 r , focusing on their poorly understood Sylow-2 component. We find the number of Sylow-2 cyclic factors for "generic" Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-2 cyclic factors. In the case of hypercubes, we give exact formulae for the largest n–1 Sylow-2 cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the 2-adic valuations of binomial sums via the combinatorics of carries. [ABSTRACT FROM AUTHOR]

We give an explicit description of two operad structures on the species composition p ∘ q , where q is any given positive operad, and where p is the NAP operad, or a shuffle version of the magmatic operad Mag . No distributive law between p and q is assumed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce saturation and semisaturation functions of sequences, and we prove a number of fundamental results about these functions. Given a forbidden sequence u with r distinct letters, we say that a sequence s on a given alphabet is u -saturated if s is r -sparse, u -free, and adding any letter from the alphabet to an arbitrary position in s violates r -sparsity or induces a copy of u. We say that s is u -semisaturated if s is r -sparse and adding any letter from the alphabet to s violates r -sparsity or induces a new copy of u. Let the saturation function Sat (u , n) denote the minimum possible length of a u -saturated sequence on an alphabet of size n , and let the semisaturation function Ssat (u , n) denote the minimum possible length of a u -semisaturated sequence on an alphabet of size n. For alternating sequences, we determine both the saturation function and the semisaturation function up to a constant multiplicative factor. We show for every sequence that the semisaturation function is always either O (1) or Θ (n). For the saturation function, we show that every sequence u has either Sat (u , n) ≥ n or Sat (u , n) = O (1). For every sequence with 2 distinct letters, we show that the saturation function is always either O (1) or Θ (n). [ABSTRACT FROM AUTHOR]

Abstract This work initiates a concept of reduced reverse degree based $$\mathcal {RR_D\,M}$$ RR D M -Polynomial for a graph, and differential and integral operators by using this $$\mathcal {RR_D\,M}$$ RR D M -Polynomial. In this study twelve reduced reverse degree-based topological descriptors are formulated using the $$\mathcal {RR_D\,M}$$ RR D M -Polynomial. The topological descriptors, denoted as $$\mathbb{T}\mathbb{D}$$ T D ’s, are numerical invariants that offer significant insights into the molecular topology of a molecular graph. These descriptors are essential for conducting QSPR investigations and accurately estimating physicochemical attributes. The structural and algebraic characteristics of the graphene and graphdiyne are studied to apply this methodology. The study involves the analysis and estimation of Reduced reverse degree-based topological descriptors and physicochemical features of graphene derivatives using best-fit quadratic regression models. This work opens up new directions for scientists and researchers to pursue, taking them into new fields of study.

AbstractCurrently, there is a rapid development of artificial intelligence systems that can solve and explain the solution of mathematical problems in the same way as students do. The problem of organizing interaction of artificial and human intelligence which does not lead to the degradation of the student’s thinking skills arises. The article proposes one of the ways to solve this issue through the use of systems such as “smart problems” using the example of a support system for a combinatorics course. In this system, the student sets problems and solves them himself without the help of a computer, but the system can check the solutions to problems without simulating human reasoning. Such a system cannot replace a person, but it becomes an intelligent interlocutor when setting and solving problems. Thus, the task of formulating a problem shows the way to bypass the gap formed with the advent of artificial intelligence. [ABSTRACT FROM AUTHOR]

This work initiates a concept of reduced reverse degree based RR D M -Polynomial for a graph, and differential and integral operators by using this RR D M -Polynomial. In this study twelve reduced reverse degree-based topological descriptors are formulated using the RR D M -Polynomial. The topological descriptors, denoted as T D 's, are numerical invariants that offer significant insights into the molecular topology of a molecular graph. These descriptors are essential for conducting QSPR investigations and accurately estimating physicochemical attributes. The structural and algebraic characteristics of the graphene and graphdiyne are studied to apply this methodology. The study involves the analysis and estimation of Reduced reverse degree-based topological descriptors and physicochemical features of graphene derivatives using best-fit quadratic regression models. This work opens up new directions for scientists and researchers to pursue, taking them into new fields of study. [ABSTRACT FROM AUTHOR]

Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expense to compute. In this paper, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a power series expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms. [ABSTRACT FROM AUTHOR]

Bioactive glass ceramics (BGC) play a pivotal role in bone tissue engineering, specifically addressing challenges associated with bone repair and regeneration. In this study, we developed silica-based 55S0P and 45S10P spherical BGC nanoparticles (NPs), without and with P 2 O 5 content, using a modified Stöber sol-gel process with base hydrolysis. The fabricated spherical NPs with pronounced textural features effectively contribute to enhanced surface roughness, a desirable parameter for improved biological activity in physiologically relevant solutions and tissue matrices. A comprehensive evaluation of the structural properties of these novel spherical BGC-NPs using diverse characterization techniques revealed distinct nano-textural features in the form of cracks and pores. The HR-TEM confirmed the nanoscale dimensions (170–190 nm) of particles with spherical surface morphology and also systematically analysed via DLS, FE-SEM studies. An augmented surface area with mesopores when phosphate added to the silica glass network was revealed by BET analysis. On the biological front, the study records the formation of an HCA layer on the surfaces of the BGC-NPs, the thickness of which increased with an increase in both phosphate content and immersion time (0, 3, 14 & 28 days) in the SBF solution. The zeta potential values attain a maximum from −22 mV to −23.6 mV for 55S0P and −21.5 mV to −26.4 mV for 45S10P BGC-NPs, and improved with increased immersion time, thereby favouring cellular adhesion for desired biological responses. In vitro, cell culture studies on MG-63 cells, after 24 and 72 h of incubation, demonstrated higher cell adhesion and proliferation, with pronounced biocompatibility. In addition, these BGC-NPs exhibited good hemocompatibility even at the highest test concentration (15 mg/ml) and demonstrated effective antibacterial activity against K. pneumoniae and E. coli. A combinatorial analysis of structural and biological properties revealed that the 45S10P BGC-NPs are more potent than 55S0P BGC-NPs for desired biological performances. In conclusion, the results suggest that the sol-gel fabricated BGC-NPs with enhanced nano-textural features are evidently suitable for applications in bone tissue engineering and regenerative medicine. [Display omitted] • Design and development of BGC-NPs (55S0P and 45S10P) via modified Stöber's method. • Confirmation of enhanced textural features by BET, FE-SEM and HR-TEM analyses. • Obtained high negative Zeta values supporting improved surface reactivity for bio-compatible applications. • Identification of HCA layer by Optical Absorption (OA), supported by XRD, FTIR & FE-SEM. • Enhanced cell viability, hemocompatibility and antibacterial activity with P 2 O 5 content for bone regenerative applications. [ABSTRACT FROM AUTHOR]

This article provides an overview of the truncated octahedron, a geometric shape that is part of the Archimedean solids. It discusses the historical and mathematical significance of the truncated octahedron, as well as its applications in architecture and crystallography. The article also introduces the concept of convex polytopes and explains how the truncated octahedron fits into this category. It explores the permutation group and its relationship to the permutohedron, and discusses Cayley graphs and their use in studying magic squares and solids. The text also explores the history and classification of magic squares, and presents examples of magic cubes and prisms. It introduces the concept of the magic permutohedron and describes a program used to find solutions for placing numbers on its vertices. The authors express curiosity about whether a similar property holds for the n-permutohedron for n≥4. The article concludes with acknowledgments and a note from the publisher. [Extracted from the article]

Background: Out-of-hospital cardiac arrest (OHCA) is associated with high mortality and cerebral disability in survivors. Current models of risk prediction and survival are mainly based on resuscitation duration. We examined the prognostic value of circulating biomarkers in predicting mortality and severe cerebral disability for OHCA survivors, alongside traditional clinical risk indicators. Methods: Biomarkers including BNP, troponin I, and galectin-3 were measured at hospital admission in resuscitated OHCA patients. Prognostic significance for mortality and cerebral disability involving circulating biomarkers, resuscitation duration, demographics, and laboratory and clinical characteristics was examined via univariate and multivariate Cox proportional hazards regression models. The incremental prognostic value of the index covariates was examined through model diagnostics, focusing on the Akaike information criterion (AIC) and Harrell's concordance statistic (c-statistic). Results: In a combinatorial analysis of 144 OHCA survivors (median follow-up 5.7 years (IQR 2.9–6.6)), BNP, galectin-3, arterial pH, and resuscitation time were significant predictors of all-cause death and severe cerebral disability, whereas troponin I levels were not. Multivariate regression, adjusting for BNP, arterial pH, and resuscitation time, identified galectin-3 as an independent predictor of long-term mortality. Multiple linear regression models also confirmed galectin-3 as the strongest predictor of cerebral disability. The incorporation of galectin-3 into models for predicting mortality and cerebral disability enhanced fit and discrimination, demonstrating the incremental value of galectin-3 beyond traditional risk predictors. Conclusions: Galectin-3 is a significant, independent long-term risk predictor of cerebral disability and mortality in OHCA survivors. Incorporating galectin-3 into current risk stratification models may enhance early prognostication and guide targeted clinical interventions. [ABSTRACT FROM AUTHOR]