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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]

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]

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]

This paper concerns the analysis of large quantum states. It is a notoriously difficult problem to quantify separability of quantum states, and for large quantum states, it is unfeasible. Here we posit that when quantum states are large, we can deduce reasonable expectations for the complex structure of non-classical multipartite correlations with surprisingly little information about the state. We show, with pegagogical examples, how known results from combinatorics can be used to reveal the expected structure of various correlations hidden in the ensemble described by a state. [ABSTRACT FROM AUTHOR]

Originating from the unified treatment of probability distributions, the α-determinant is a one-parameter interpolation of the determinant and permanent of a matrix. While it generally does not have the invariance properties of the determinant, the irreducible representations of the associated general linear (Lie) groups define interesting invariants that unveil a new and rich mathematical theory with abundant unsolved problems, related to symmetric functions, representation theory, combinatorics, number theory, probability and other areas. [ABSTRACT FROM AUTHOR]

Escape the Math Room is a fun, interactive, sequence of puzzles designed to encourage mathematical collaboration, team work, and ingenuity. Using advanced mathematical topics in an accessible way, Escape the Math Room appropriately challenges students ranging from high school to advanced undergraduate. The activity can be adjusted by adding or removing hints and explanations to accommodate the audience. This activity can be used on its own for a mathematical event, in class on a "free day" or as a team building exercise. [ABSTRACT FROM AUTHOR]

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Schürmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behavior of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Schürmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of non-crossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Schürmann's aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively, Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature. [ABSTRACT FROM AUTHOR]

The German Mathematical Society (DMV) has awarded Felix Otto from the Max Planck Institute for Mathematics in Leipzig the Cantor Medal of the year 2024. Noga Alon from Princeton University and Adi Shamir have received the Wolf Prize of the year 2024 for their contributions to combinatorics and theoretical computer science. Peter Sarnak has been awarded the Shaw Prize for Mathematical Research by the American Mathematical Society (AMS) for his advancements in group theory. This year's Nemmers Prize from the AMS has been awarded to Luigi Ambrosio from the Scuola Normale Superiore in Pisa. Two mathematicians are among the new Alexander von Humboldt Professors: Arend Bayer, who will be transferring from the University of Edinburgh to the Free University of Berlin, and Daniel Král, who will be conducting research at the University of Leipzig and the Max Planck Institute for Mathematics in the Sciences. Mitsubishi Electric has set the Guinness World Record for the fastest robot to solve a Rubik's Cube using AI. [Extracted from the article]

Summary: Imagine being tasked with measuring the distance between two points in a city, but you can only use a taxicab's odometer which displays only whole-number values (the integer part) of its distance traveled so far—such is the display of modern digital measuring devices. This coarse measurement between two points might be called, tongue in cheek, a taxicab metric, but clearly of a different sort from the taxicab metric on a discrete grid that we encounter in combinatorics. We will explore how we might apply probability to our taxicab's inherent randomness, and along with limit theorems, indeed measure arbitrary distances with any desired accuracy. [ABSTRACT FROM AUTHOR]

This paper extends Bombieri and Pila's estimate of lattice points on curves to arbitrary finite sets by incorporating considerations of minimal separation and the doubling constant. We derive the estimate by establishing the ℓ 2 decoupling inequality for non-degenerate curves in ℝ n . Additionally, we review the curve-lifting method introduced in Bombieri and Pila's work and establish the estimate of lattice points in the neighborhood of a planar curve. [ABSTRACT FROM AUTHOR]

The problem of interpreting heterogeneous and multiprecision geophysical data is considered from the point of view of combinatorics within the framework of the approximation approach. The discrete nature of information about physical fields and their sources imposes a number of limitations on the possibilities of adequate data interpretation. Combinatorial methods of discrete mathematics make it possible to describe accurately the criteria for selecting data to approximate the elements of anomalous potential fields, as well as for controlling the quality of the approximate solution of inverse problems, both linear and nonlinear. [ABSTRACT FROM AUTHOR]

We find an explicit expression of the associated primes of monomial ideals as a colon by an element v , using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals. An algorithm to compute v is given using Macaulay2. For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph. For ideals of Borel type, the monomial v takes a simpler form, and we classify when v is unique. [ABSTRACT FROM AUTHOR]

A subset A of a group G is called product-free if there is no solution to a = b c with a , b , c all in A . It is easy to see that the largest product-free subset of the symmetric group S n is obtained by taking the set of all odd permutations, i.e. S n ∖ A n , where A n is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group A n also contains a product-free set of constant density. This conjecture was refuted by Gowers (whose result was subsequently improved by Eberhard), still leaving the long-standing problem of determining the largest product-free subset of A n wide open. We solve this problem for large n , showing that the maximum size is achieved by the previously conjectured extremal examples, namely families of the form { π : π (x) ∈ I , π (I) ∩ I = ∅ } and their inverses. Moreover, we show that the maximum size is only achieved by these extremal examples, and we have stability: any product-free subset of A n of nearly maximum size is structurally close to an extremal example. Our proof uses a combination of tools from Combinatorics and Non-abelian Fourier Analysis, including a crucial new ingredient exploiting some recent theory developed by Filmus, Kindler, Lifshitz and Minzer for global hypercontractivity on the symmetric group. [ABSTRACT FROM AUTHOR]

The Witt ring of a complex flag variety describes the interesting – i.e. torsion – part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types G_2 and F_4. The results also extend to flag varieties over other algebraically closed fields. [ABSTRACT FROM AUTHOR]

Let S T λ (τ) denote the set of symmetric transversals of a self-conjugate Young diagram λ which avoid the permutation pattern τ . Given two permutations τ = τ 1 τ 2 ... τ n of { 1 , 2 , ... , n } and σ = σ 1 σ 2 ... σ m of { 1 , 2 , ... , m } , the direct sum of τ and σ , denoted by τ ⊕ σ , is the permutation τ 1 τ 2 ... τ n (σ 1 + n) (σ 2 + n) ... (σ m + n) . We establish an exterior peak set preserving bijection between S T λ (321 ⊕ τ) and S T λ (213 ⊕ τ) for any pattern τ and any self-conjugate Young diagram λ . Our result is a refinement of part of a result of Bousquet-Mélou–Steingrímsson for pattern-avoiding symmetric transversals. As applications, we derive several enumerative results concerning pattern-avoiding reverse alternating involutions, including two conjectured equalities posed by Barnabei–Bonetti–Castronuovo–Silimbani. [ABSTRACT FROM AUTHOR]

The F1-ATPase enzyme is the smallest-known molecular motor that rotates in 120° steps, driven by the hydrolysis of ATP. It is a multi-subunit enzyme that contains three catalytic sites. A central question is how the elementary chemical reactions that occur in the three sites are coupled to mechanical rotation. Various models and coupling schemes have been formulated in an attempt to answer this question. They can be classified as 2-site (bi-site) models, exemplified by Boyer's binding change mechanism first proposed 50 years ago, and 3-site (tri-site) models such as Nath's torsional mechanism, first postulated 25 years ago and embellished 1 year back. Experimental data collated using diverse approaches have conclusively shown that steady-state ATP hydrolysis by F1-ATPase occurs in tri-site mode. Hence older models have been continually modified to make them conform to the new facts. Here, we have developed a pure mathematical approach based on combinatorics and conservation laws to test if proposed models are 2-site or 3-site. Based on this novel combinatorial approach, we have proved that older and modified models are effectively bi‒site models in that catalysis and rotation in F1-ATPase occurs in these models with only two catalytic sites occupied by bound nucleotide. Hence these models contradict consensus experimental data. The recent 2023 model of ATP hydrolysis by F1-ATPase has been proved to be a true tri-site model based on our novel mathematical approach. Such pure mathematical proofs constitute an important step forward for ATP mechanism. However, in what must be considered an aspect with great scientific potential, the power of such mathematical proofs has not been fully exploited to solve molecular biological problems, in our opinion. We believe that the creative application of pure mathematical proofs (for another example see Nath in Theory Biosci 141:249–260, 2022) can help resolve with finality various longstanding molecular-level issues that arise as a matter of course in the analysis of fundamental biological problems. Such issues have proved extraordinarily difficult to resolve by standard experimental, theoretical, or computational approaches. [ABSTRACT FROM AUTHOR]

We discuss patterns formed by terracotta tiles found on a housing terrasse in London, UK and note their relationship to Truchet tiles. We consider some of the mathematical questions that arise when combining the tiles into 2 × 2 arrays and their use in the creation of woven strips. In particular, we determine equivalence classes of patterns, and using graph theory and combinatorics we provide results regarding the number of woven strips that meet particular criteria. [ABSTRACT FROM AUTHOR]

The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on $ \mathbb{S}^n $ to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes. [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.

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]