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Let X ⊆ ℙn+1 be an integral hypersurface of degree d. We show that each locally Cohen–Macaulay instanton sheaf ℰ on X with respect to 풪X ⊗풪ℙn+1(1) in the sense of [V. Antonelli and G. Casnati, Instanton sheaves on projective schemes, J. Pure Appl. Algebra 227 (2023) 107246, Definition 1.3] yields the existence of Steiner bundles 풢 and ℱ on ℙn+1 of the same rank r and a morphism φ: 풢(−1) →ℱ∨ such that the form defining X to the power rk(ℰ) is exactly det(φ). We inspect several examples for low values of d, n and rk(ℰ). In particular, we show that the form defining a smooth integral surface in ℙ3 is the pfaffian of some skew-symmetric morphism φ: ℱ(−1) →ℱ∨, where ℱ is a suitable Steiner bundle on ℙ3 of sufficiently large even rank. [ABSTRACT FROM AUTHOR]

This paper lies in the context of the studies of eigenfunctions of graphs having minimum cardinality of support. One of the tools is the weight‐distribution bound, a lower bound on the cardinality of support of an eigenfunction of a distance‐regular graph corresponding to a nonprincipal eigenvalue. The tightness of the weight‐distribution bound was previously shown in general for the smallest eigenvalue of a Grassmann graph. However, a characterisation of optimal eigenfunctions was not obtained. Motivated by this open problem, we consider the class of strongly regular Grassmann graphs and give the required characterisation in this case. We then show the tightness of the weight‐distribution bound for block graphs of affine designs (defined on the lines of an affine space with two lines being adjacent when intersect) and obtain a similar characterisation of optimal eigenfunctions. [ABSTRACT FROM AUTHOR]

A procedure is presented for finding the shortest network connecting three given undirected points, subject to a curvature constraint on both the path joining two of the points and the path that connects to the third point. The problem is a generalisation of the Fermat–Torricelli problem and is related to a shortest curvature-constrained path problem that was solved by Dubins. The procedure has the potential to be applied to the optimal design of decline networks in underground mines. [ABSTRACT FROM AUTHOR]

We generalize Atsuyama's result on the geometry of symmetric spaces of type EVI to the case of fields of characteristic zero. We relate the possible mutual positions of two points with the classification of balanced symplectic ternary algebras (also known as Freudenthal triple systems). [ABSTRACT FROM AUTHOR]

The Steiner (k , k ′) -eccentricity of a given k ′ -subset of a graph G is the maximum Steiner distance over all k -subsets of V (G) which contain the given k ′ -subset, an extension of standard Steiner k -eccentricity. In this paper, we mainly study the Steiner (3 , 2) -eccentricity of trees. The σ -transformation which does not increase the average Steiner (3 , 2) -eccentricity is depicted. Using σ -transformation, several lower and upper bounds for the average Steiner (3 , 2) -eccentricity of trees are given. [ABSTRACT FROM AUTHOR]

In 1991, Shalaby conjectured that any Zn, where n≡1 or 3(mod8),n≥11, admits a strong Skolem starter. In 2018, the authors fully described and explicitly constructed the infinite "cardioidal" family of strong Skolem starters. No other infinite family of these combinatorial designs was known to date. Statements regarding the products of starters, proven in this paper give a new way of generating strong or skew Skolem starters of composite orders. This approach extends our previous result by generating new infinite families of these starters that are not cardioidal. [ABSTRACT FROM AUTHOR]

Let V $V$ be a vector space over the finite field Fq ${{\mathbb{F}}}_{q}$. A q $q$‐Steiner system, or an S(t,k,V)q $S{(t,k,V)}_{q}$, is a collection ℬ ${\rm{{\mathcal B}}}$ of k $k$‐dimensional subspaces of V $V$ such that every t $t$‐dimensional subspace of V $V$ is contained in a unique element of ℬ ${\rm{{\mathcal B}}}$. A large set of q $q$‐Steiner systems, or an LS(t,k,V)q $LS{(t,k,V)}_{q}$, is a partition of the k $k$‐dimensional subspaces of V $V$ into S(t,k,V)q $S{(t,k,V)}_{q}$ systems. In the case that V $V$ has infinite dimension, the existence of an LS(t,k,V)q $LS{(t,k,V)}_{q}$ for all finite t,k $t,k$ with 1

This paper is a contribution to the classification of all pairs (T,G) $({\mathscr{T}},G)$, where T ${\mathscr{T}}$ is a triple system and G $G$ is a block‐transitive but not flag‐transitive automorphism group of T ${\mathscr{T}}$. We prove that if the socle of G $G$ is a sporadic or alternating group, then one of the following holds: (i)T ${\mathscr{T}}$ is a TS(10,2) $TS(10,2)$ and G≅A5 $G\cong {A}_{5}$;(ii)T ${\mathscr{T}}$ is a TS(10,4) $TS(10,4)$ and G≅S5 $G\cong {S}_{5}$;(iii)T ${\mathscr{T}}$ is a TS(55,28) $TS(55,28)$ and G≅A11 $G\cong {A}_{11}$ or S11 ${S}_{11}$;(iv)T ${\mathscr{T}}$ is a TS(55,λ) $TS(55,\lambda)$ with λ∈{4,8,16} $\lambda \in \{4,8,16\}$ and G≅M11 $G\cong {M}_{11}$. [ABSTRACT FROM AUTHOR]

A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$ -colouring of the complete $k$ -uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás' theorem by any 'expansive' $k$ -uniform hypergraph on $n$ vertices (that is, a $k$ -uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gyárfás' result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$ -partite $r$ -uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$ , there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$ , then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$). We prove our results in this dual setting. [ABSTRACT FROM AUTHOR]

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that 2-term homotopy Lie triple systems is equivalent to Lie triple 2-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple 2-systems. We show that skeletal Lie triple 2-systems can be classified the third cohomology group, and strict Lie triple 2-systems are equivalent to crossed modules of Lie triple systems. [ABSTRACT FROM AUTHOR]

A Steiner quadruple system of order v + 1 with resolvable derived designs (every derived Steiner triple system of order v at a point is resolvable), abbreviated as RDSQS (v + 1) , has been used to construct a large set of Kirkman triple systems of order 3v. In this paper, an RDSQS (v + 1) is reduced to an overlarge set of Kirkman triple systems of order v with an additional property (OLKTS + (v) ), which plays the same role in constructing an LKTS(3v) as an RDSQS (v + 1) . Some recursive constructions of OLKTS + s are also established. As a result, some infinite classes of OLKTS + s and LKTSs are obtained. [ABSTRACT FROM AUTHOR]

In [3], the authors adapted the edge-graceful graph labeling definition into block designs. In this article, we adapt the line-graceful graph labeling definition into block designs and define a block design (V; B) with |V| = ν as line-graceful if there exists a function f: B → {0; 1;: ::; v - 1} such that the induced mapping f+: V → Zv given by f+(x) = ∑A∊B: xA∊BA f(A) (mod v) is a bijection. In this article, the cases that are incomplete in terms of block-graceful labelings, are completed in terms of line-graceful labelings. Moreover, we prove that there exists a line-graceful Steiner quadruple system of order 2n for all n ≤ 3 by using a recursive construction. [ABSTRACT FROM AUTHOR]

A triple system is cancellative if it does not contain three distinct sets A , B , C such that the symmetric difference of A and B is contained in C. We show that every cancellative triple system H that satisfies a particular inequality between the sizes of H and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon. [ABSTRACT FROM AUTHOR]

A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$ -colouring of the complete $k$ -uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$. We prove a result which says that if one replaces $K_n^k$ in Gyárfás' theorem by any 'expansive' $k$ -uniform hypergraph on $n$ vertices (that is, a $k$ -uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gyárfás' result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$ -partite $r$ -uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$ , there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$ , then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$). We prove our results in this dual setting. [ABSTRACT FROM AUTHOR]

This study investigates the existence of triple weak solutions for a system of nonlinear elliptic equations with a fourth-order operator. The problem arises in the mathematical modeling of complex physical phenomena. [ABSTRACT FROM AUTHOR]

We propose a heuristic method of using network centralities for constructing small-weight Steiner trees in this paper. The Steiner tree problem in graphs is one of the practical NP-hard combinatorial optimization problems. Given a graph and a set of vertices called terminals in the graph, the objective of the Steiner tree problem in graphs is to find a minimum weight Steiner tree that is a tree containing all the terminals. Conventional construction methods make a Steiner tree based on the shortest paths between terminals. If these shortest paths are overlapped as much as possible, we can obtain a small-weight Steiner tree. Therefore, we proposed to use network centralities to distinguish which edges should be included to make a small-weight Steiner tree. Experimental results revealed that using the vertex or the edge betweenness centralities contributes to making small-weight Steiner trees. [ABSTRACT FROM AUTHOR]

Two resolutions of the same 3-design are said to be orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. If a Steiner quadruple system has two mutually orthogonal resolutions, the design is called doubly resolvable and denoted by DRSQS. In this paper, we define almost doubly resolvable candelabra quadruple system and then to get a recursive construction of DRSQS, i.e., for n ≥ 16 , if there is a DRSQS(n), then there exists a DRSQS (14 n - 12) and a DRSQS (16 n - 12) . [ABSTRACT FROM AUTHOR]

We prove that the set of |$n$| -point configurations for which the solution to the planar Steiner problem is not unique has the Hausdorff dimension at most |$2n-1$| (as a subset of |$\mathbb{R}^{2n}$|⁠). Moreover, we show that the Hausdorff dimension of the set of |$n$| -point configurations for which at least two locally minimal trees have the same length is also at most |$2n-1$|⁠. The methods we use essentially rely upon the theory of subanalytic sets developed in [ 1 ]. Motivated by this approach, we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replaced by an arbitrary analytic Riemannian manifold |$M$|⁠. In this setup, we argue that the set of configurations possessing two locally-minimal trees of the same length either has dimension equal to |$n \dim M - 1$| or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to the above-mentioned results, we study the set of |$n$| -point configurations for which there is a unique solution to the Steiner problem in |$\mathbb{R}^{d}$|⁠. We show that this set is path-connected. [ABSTRACT FROM AUTHOR]

Many modern day cloud hosted applications such as virtual reality, real time games require low latency data access and computation to improve response time. So it is essential to bring the computation and data storage edge servers closer to the user's geographical location to improve response times and save bandwidth. In particulars, in online gaming and on demand video services, the required application data present at cloud servers need to be placed on the edge servers to provide low latency app-functionalities. The transfer of huge amount of data from cloud server to edge server incurs high cost and time penalties. Thus, we need an efficient way to solve edge data distribution (EDD) problem which distribute the application data to the edge servers that minimizes transfer cost. In this work, we provide a refined formulation of an optimal approach to solve the EDD problem using integer linear programming (ILP) technique. Due to the time complexity limitation of the ILP approach, we propose an O(k) approximation algorithm based on network Steiner tree estimation (EDD-NSTE) for estimating solutions to dense large-scale EDD problem. The proposed approach is analyzed to be 11/6 approximation which is better than the state-of-the-art 2 approximation EDD-A approach. The experimental evaluation through simulation using real world EUA data set demonstrate that the EDD-NSTE outperform state-of-the-art approach and other representative approaches. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular, uniform hypergraph H$\mathcal {H}$ with small maximum codegree has an almost‐perfect matching. We extend this result by obtaining a conflict‐free matching, where conflicts are encoded via a collection C$\mathcal {C}$ of subsets C⊆E(H)$C\subseteq E(\mathcal {H})$. We say that a matching M⊆E(H)$\mathcal {M}\subseteq E(\mathcal {H})$ is conflict‐free if M$\mathcal {M}$ does not contain an element of C$\mathcal {C}$ as a subset. Under natural assumptions on C$\mathcal {C}$, we prove that H$\mathcal {H}$ has a conflict‐free, almost‐perfect matching. This has many applications, one of which yields new asymptotic results for so‐called 'high‐girth' Steiner systems. Our main tool is a random greedy algorithm which we call the 'conflict‐free matching process'. [ABSTRACT FROM AUTHOR]

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A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G. In this paper we introduce the general position polynomial as ∑ i ≥ 0 a i x i , where a i is the number of distinct general position sets of G with cardinality i. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs K(n, 2), with unimodal general position polynomials are presented. [ABSTRACT FROM AUTHOR]

The well-known Erdős-Hajnal conjecture states that for any graph $F$ , there exists $\epsilon \gt 0$ such that every $n$ -vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon }$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$ , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$ , for every $S \subseteq \{0,1,2,3,4\}$ , we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$ , whether the growth rate is polynomial or polylogarithmic. Some open problems remain. [ABSTRACT FROM AUTHOR]

A finite classical polar space of rank \(n\) consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that \(n\) is the maximal dimension of such a subspace. A \(t\)-Steiner system in a finite classical polar space of rank \(n\) is a collection \(Y\) of totally isotropic \(n\)-spaces such that each totally isotropic \(t\)-space is contained in exactly one member of \(Y\). Nontrivial examples are known only for \(t=1\) and \(t=n-1\). We give an almost complete classification of such \(t\)-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.Mathematics Subject Classifications: 51E23, 05E30, 33C80Keywords: Association schemes, codes, polar spaces, Steiner systems

The article "Normal Form of Multi-Tours of Binary Trees" discusses the minimal filling problem for a finite metric space, focusing on binary trees and multi-tours. It introduces the concept of multi-tours, irreducible multi-tours, and the normal form of multi-tours. The article also presents theorems and lemmas related to multi-tours, providing insights into the weight of minimal fillings and the bound on the multiplicity of irreducible multi-tours. The study concludes with acknowledgments, funding information, and a declaration of no conflicts of interest. [Extracted from the article]

Conformational polymorphs of organic molecular crystals represent a challenging test for quantum chemistry because they require careful balancing of the intra- and intermolecular interactions. This study examines 54 molecular conformations from 20 sets of conformational polymorphs, along with the relative lattice energies and 173 dimer interactions taken from six of the polymorph sets. These systems are studied with a variety of van der Waals-inclusive density functionals theory models; dispersion-corrected spin-component-scaled second-order Møller–Plesset perturbation theory (SCS-MP2D); and domain local pair natural orbital coupled cluster singles, doubles, and perturbative triples [DLPNO-CCSD(T)]. We investigate how delocalization error in conventional density functionals impacts monomer conformational energies, systematic errors in the intermolecular interactions, and the nature of error cancellation that occurs in the overall crystal. The density functionals B86bPBE-XDM, PBE-D4, PBE-MBD, PBE0-D4, and PBE0-MBD are found to exhibit sizable one-body and two-body errors vs DLPNO-CCSD(T) benchmarks, and the level of success in predicting the relative polymorph energies relies heavily on error cancellation between different types of intermolecular interactions or between intra- and intermolecular interactions. The SCS-MP2D and, to a lesser extent, ωB97M-V models exhibit smaller errors and rely less on error cancellation. Implications for crystal structure prediction of flexible compounds are discussed. Finally, the one-body and two-body DLPNO-CCSD(T) energies taken from these conformational polymorphs establish the CP1b and CP2b benchmark datasets that could be useful for testing quantum chemistry models in challenging real-world systems with complex interplay between intra- and intermolecular interactions, a number of which are significantly impacted by delocalization error. [ABSTRACT FROM AUTHOR]

A bidirectional inductive power transfer (BIPT) system of full-compensated series-series (SS) topology with full bridge converters on both primary and secondary sides is analyzed in this paper. The steady-state electrical characteristics of the BIPT system under triple-phase-shift control are obtained, based on which, the conditions for achieving the maximum transfer efficiency of the intermediate circuit and zero voltage switching of all switches are derived. Triple Phase-Shift Control (TPSC) strategy was proposed for the control of the two inner phase shifted of the primary and secondary side full bridge converters and the fundamental excitation voltage phase shift, which achieved the maximum transfer efficiency of the intermediate circuit and zero voltage switching of all switches. The proposed control method was verified through simulation. The results showed that the control strategy can realize the bidirectional energy transfer of the IPT system, the efficiency optimization of the intermediate link, and the zero-voltage turn-on of all switching devices under various load conditions. [ABSTRACT FROM AUTHOR]

Liu et al. (IEEE Trans Inf Theory 68:3096–3107, 2022) investigated a class of BCH codes C (q , q + 1 , δ , 1) with q = δ m a prime power and proved that the set B δ + 1 of supports of the minimum weight codewords supports a Steiner system S (3 , δ + 1 , q + 1) . In this paper, we give an equivalent formulation of B δ + 1 in terms of elementary symmetric polynomials and then construct a number of mutually disjoint Steiner systems S (3 , δ + 1 , δ m + 1) when m is even and a number of mutually disjoint G-designs G ( δ m + 1 δ + 1 , δ + 1 , δ + 1 , 3) when m is odd. In particular, the existence of three mutually disjoint Steiner systems S (3 , 5 , 4 m + 1) or three mutually disjoint G-designs G ( 4 m + 1 5 , 5 , 5 , 3) is established. [ABSTRACT FROM AUTHOR]

We give a new decomposition theorem in unit disk graphs (UDGs) and demonstrate its applicability in the fields of Structural Graph Theory and Parameterized Complexity. First, our new decomposition theorem shows that the class of UDGs admits an "almost" Contraction Decomposition Theorem. Prior studies on this topic exhibited that the classes of planar graphs [Klein, SICOMP, 2008], graphs of bounded genus [Demaine, Hajiaghayi and Mohar, Combinatorica 2010], and H-minor free graphs [Demaine, Hajiaghayi and Kawarabayashi, STOC 2011] admit a Contraction Decomposition Theorem. Even bounded-degree UDGs can contain arbitrarily large cliques as minors, and therefore our result is a significant advance in the study of contraction decompositions. Additionally, this result answers an open question posed by Hajiaghayi (www.youtube.com/watch?v=2Bq2gy1N01w) regarding the existence of contraction decompositions for classes of graphs beyond H-minor free graphs though under a relaxation of the original formulation. Second, we present a "parameteric version" of our new decomposition theorem. We prove that there is an algorithm that, given a UDG G and a positive integer k, runs in polynomial time and outputs a collection of \(\mathcal {O}(k)\) tree decompositions of G with the following properties. Each bag in any of these tree decompositions can be partitioned into \(\mathcal {O}(k)\) connected pieces (we call this measure the chunkiness of the tree decomposition). Moreover, for any subset S of at most k edges in G, there is a tree decomposition in the collection such that S is well preserved in the decomposition in the following sense. For any bag in the tree decomposition and any edge in S with both endpoints in the bag, either its endpoints lie in different pieces or they lie in a piece that is a clique. Having this decomposition at hand, we show that the design of parameterized algorithms for some cut problems becomes elementary. In particular, our algorithmic applications include single-exponential (or slightly super-exponential) algorithms for well-studied problems such as Min Bisection, Steiner Cut, s-Way Cut, and Edge Multiway Cut-Uncut on UDGs; these algorithms are substantially faster than the best-known algorithms for these problems on general graphs. [ABSTRACT FROM AUTHOR]

In the paper a problem of assignment of tasks to machines is formulated and solved, where a criterion of data replication is used and a large size of data imposes additional constraints. This problem is met in practice when dealing with large genomic files or other types of vast data. The necessity of comparing all pairs of files within a big set of DNA sequencing results, which we collected, maintained, and analyzed within a national genomic project, brought us to the proposed results. This problem resembles that of generating a particular Steiner system, and a mechanism observed there is employed in one of our algorithms. Based on the problem complexity, we propose two heuristic algorithms, which work very well even for instances with tight constraints and a heterogeneous environment defined. In addition, we propose a simplified method, nevertheless capable of finding very good solutions and surpassing the algorithms in some special cases. The methods are validated in tests on a wide set of instances, where values of parameters reflect our real-world application and where their usefulness is proven. [ABSTRACT FROM AUTHOR]

We abstract and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph G=(V,E) and a set of demand pairs P⊆VxV, a reachability preserver is a sparse subgraph H that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an n-node graph and demand pairs of the form P⊆SxV for a small node subset S, there is always a reachability preserver on O(n+√n/P//S/) edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which O(n) size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the O(n0.6+ε) of Chlamatac, et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534-443] (n4/7+ε). [ABSTRACT FROM AUTHOR]

In this paper, we proved that each nonlinear non-global semi-Jordan triple higher derivable mapping on a 2-torsion free triangular algebra is an additive higher derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there is no block-transitive 4-(v, k, 1) designs and 6-(v, k, 1) designs with k divides v. [ABSTRACT FROM AUTHOR]

In this paper, first we give the notion of a crossed homomorphism on a Lie triple system with respect to an action on another Lie triple system. We also construct an L ∞ -algebra using the higher derived brackets, whose Maurer-Cartan elements are crossed homomorphisms on Lie triple systems. Consequently, we obtain the twisted L ∞ -algebra that controls deformations of a given crossed homomorphism on Lie triple systems. Next we construct a cohomology theory for a crossed homomorphism on Lie triple systems and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Moreover we consider the relationship between cohomology groups of crossed homomorphisms on Lie algebras and associated Lie triple systems. [ABSTRACT FROM AUTHOR]

For an integer k ≥ 2 , let (P n (k)) n be the k-generalized Pell sequence which starts with 0 , ... , 0 , 1 (k terms), and each term afterwards is given by P n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . In this paper, we determine all solutions of the Markov equation x 2 + y 2 + z 2 = 3 x y z , with x, y, and z being k-generalized Pell numbers. This paper continues and extends a previous work of Kafle, Srinivasan and Togbé, who found all Markov triples with Pell components. [ABSTRACT FROM AUTHOR]

The optical response in a hybrid optomechanical system containing a triple quantum well (TQW) is investigated theoretically. The optomechanical cavity is driven by a strong pump field and a weak probe field. We show that multiple electromagnetically induced transparency windows and optomechanically induced transparency (OMIT) window exist simultaneously for the weak probe field due to the Jaynes–Cummings coupling and optomechanical coupling respectively. Furthermore, the system absorption spectra can be tuned by changing the system coupling strength. Especially, when two external control lasers are applied to the TQW with frequency detuning, multiple transparency windows can be illustrated and adjusted by the external fields. Therefore, via changing the external coupling fields, we can realize manipulating the OMIT transparency window, tunable group delay and switch from fast light to slow light. Such a system may be much practical for the flexibility of TQW in the quantum information processing. [ABSTRACT FROM AUTHOR]

In this study, we provide the triple sequences space of fuzzy numbers identified by simple elliptic Orlicz function and discuss some properties like the space (l∞)³F (M,△ij ) is complete, (l∞)³F (M1,△ij ) ⊆ (l∞)³F(M0 M1,△ij ), (l∞)³F (M1,△ij ) ∩ (l∞)³F(M2,△ij ) ⊆ (l∞)³F (M1 + M²,△ij), (l∞)³F(M,△ij) ⊂ (l∞)³F(M,△Ij ), for 0 ≤ I < i . [ABSTRACT FROM AUTHOR]

We introduce an extension of first-order logic that comes equipped with additional predicates for reasoning about an abstract state. Sequents in the logic comprise a main formula together with pre- and postconditions in the style of Hoare logic, and the axioms and rules of the logic ensure that the assertions about the state compose in the correct way. The main result of the paper is a realizability interpretation of our logic that extracts programs into a mixed functional/imperative language. All programs expressible in this language act on the state in a sequential manner, and we make this intuition precise by interpreting them in a semantic metatheory using the state monad. Our basic framework is very general, and our intention is that it can be instantiated and extended in a variety of different ways. We outline in detail one such extension: A monadic version of Heyting arithmetic with a wellfounded while rule, and conclude by outlining several other directions for future work. [ABSTRACT FROM AUTHOR]

The Steiner subratio is one of the Steiner-type ratios that show how much the weights of optimal connecting graphs in a metric space can differ. In this paper, an upper estimate for the Steiner subratio of Riemannian manifolds is obtained. [ABSTRACT FROM AUTHOR]

The Turán number T(n,5,3) $T(n,5,3)$ is the minimum size of a system of triples out of a base set X $X$ of n $n$ elements such that every quintuple in X $X$ contains a triple from the system. The exact values of T(n,5,3) $T(n,5,3)$ are known for n≤17 $n\le 17$. Turán conjectured that T(2m,5,3)=2m3 $T(2m,5,3)=2\left(\genfrac{}{}{0.0pt}{}{m}{3}\right)$, and no counterexamples have been found so far. If this conjecture is true, then T(2m+1,5,3)≥⌈m(m−2)(2m+1)∕6⌉ $T(2m+1,5,3)\ge \lceil m(m-2)(2m+1)\unicode{x02215}6\rceil $. We prove the matching upper bound for all n=2m+1>17 $n=2m+1\gt 17$ except n=27 $n=27$. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, Q be a prime ideal of R and F be a free R -module of finite rank. In this paper, we use hypergraphs to give a full characterization of prime submodules of F. Also we define a hypergraph P H Q (F) called the prime submodules hypergraph of F with respect to Q and use properties of Steiner systems for counting the number of Q -prime submodules of F when the number of cosets of Q in R is finite. [ABSTRACT FROM AUTHOR]

Let H$H$ be a k$k$‐uniform D$D$‐regular simple hypergraph on N$N$ vertices. Based on an analysis of the Rödl nibble, in 1997, Alon, Kim and Spencer proved that if k⩾3$k \geqslant 3$, then H$H$ contains a matching covering all but at most ND−1/(k−1)+o(1)$ND^{-1/(k-1)+o(1)}$ vertices, and asked whether this bound is tight. In this paper we improve their bound by showing that for all k>3$k > 3$, H$H$ contains a matching covering all but at most ND−1/(k−1)−η$ND^{-1/(k-1)-\eta }$ vertices for some η=Θ(k−3)>0$\eta = \Theta (k^{-3}) > 0$, when N$N$ and D$D$ are sufficiently large. Our approach consists of showing that the Rödl nibble process not only constructs a large matching but it also produces many well‐distributed 'augmenting stars' which can then be used to significantly improve the matching constructed by the Rödl nibble process. Based on this, we also improve the results of Kostochka and Rödl from 1998 and Vu from 2000 on the size of matchings in almost regular hypergraphs with small codegree. As a consequence, we improve the best known bounds on the size of large matchings in combinatorial designs with general parameters. Finally, we improve the bounds of Molloy and Reed from 2000 on the chromatic index of hypergraphs with small codegree (which can be applied to improve the best known bounds on the chromatic index of Steiner triple systems and more general designs). [ABSTRACT FROM AUTHOR]

We show that the base polytope PM of any paving matroid M can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of PM, starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni's work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent. [ABSTRACT FROM AUTHOR]

Steiner triple systems (STSs) have been classified up to order 19. Earlier estimations of the number of isomorphism classes of STSs of order 21, the smallest open case, are discouraging as for classification, so it is natural to focus on the easier problem of merely counting the isomorphism classes. Computational approaches for counting STSs are here considered and lead to an algorithm that is used to obtain the number of isomorphism classes for order 21: 14,796,207,517,873,771. [ABSTRACT FROM AUTHOR]

For a connected graph G of order p and a set S ⊆ V (G) , the Steiner distance of S is the minimum number of edges in a connected subgraph of G containing S. If n is an integer, 2 ≤ n ≤ p and a vertex v ∈ V (G) , the Steiner n -eccentricity of a vertex v of G , ex n (v) , is the maximum Steiner distance of all n -subsets of V (G) containing v. The Steiner n -radius of G , rad n (G) , is the minimum Steiner n -eccentricities of all vertices in G. We give bounds on rad n (G) in terms of the order of G and the minimum degree of G for all graphs and for graphs that contain no triangles. We shall also investigate the relation between the n -radius of a graph G and its complement Ḡ. [ABSTRACT FROM AUTHOR]

For a connected graph G of order p and a non-empty subset S of the vertex set of G, the n-Steiner distance of S is defined to be the smallest size among all connected sub-graphs whose vertex sets contain. In this paper, we obtain the Hosoya polynomials of Steiner n-distance of some windmill graphs. Moreover, the Steiner Wiener indices of certain windmill graphs are also obtained. [ABSTRACT FROM AUTHOR]

A line coloring of PG(n,q) $\text{PG}(n,q)$, the n $n$‐dimensional projective space over GF(q) $(q)$, is an assignment of colors to all lines of PG(n,q) $\text{PG}(n,q)$ so that any two lines with the same color do not intersect. The chromatic index of PG(n,q) $\text{PG}(n,q)$, denoted by χ′(PG(n,q)) $\chi ^{\prime} (\text{PG}(n,q))$, is the least number of colors for which a coloring of PG(n,q) $\text{PG}(n,q)$ exists. This paper translates the problem of determining the chromatic index of PG(n,q) $\text{PG}(n,q)$ to the problem of examining the existences of PG(3,q) $\text{PG}(3,q)$ and PG(4,q) $\text{PG}(4,q)$ with certain properties. In particular, it is shown that for any odd integer n $n$ and q∈{3,4,8,16},χ′(PG(n,q))=(qn−1)∕(q−1) $q\in \{3,4,8,16\},\chi ^{\prime} (\text{PG}(n,q))=({q}^{n}-1)\unicode{x02215}(q-1)$, which implies the existence of a parallelism of PG(n,q) $\text{PG}(n,q)$ for any odd integer n $n$ and q∈{3,4,8,16} $q\in \{3,4,8,16\}$. [ABSTRACT FROM AUTHOR]

We prove that for n \in \mathbb N and an absolute constant C, if p \geq C\log ^2 n / n and L_{i,j} \subseteq [n] is a random subset of [n] where each k\in [n] is included in L_{i,j} independently with probability p for each i, j\in [n], then asymptotically almost surely there is an order-n Latin square in which the entry in the ith row and jth column lies in L_{i,j}. The problem of determining the threshold probability for the existence of an order-n Latin square was raised independently by Johansson [ Triangle factors in random graphs , 2006], by Luria and Simkin [Random Structures Algorithms 55 (2019), pp. 926–949], and by Casselgren and Häggkvist [Graphs Combin. 32 (2016), pp. 533–542]; our result provides an upper bound which is tight up to a factor of \log n and strengthens the bound recently obtained by Sah, Sawhney, and Simkin [ Threshold for Steiner triple systems , arXiv: 2204.03964 , 2022]. We also prove analogous results for Steiner triple systems and 1-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a 1-factorization of a nearly complete regular bipartite graph. [ABSTRACT FROM AUTHOR]