Search

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]

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]

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]

We prove that with high probability G (3) (n , n - 1 + o (1)) contains a spanning Steiner triple system for n ≡ 1 , 3 (mod 6) , establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park. [ABSTRACT FROM AUTHOR]

We show that every intermediate β -transformation is topologically conjugate to a greedy β -transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate β -transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [22], and (2) obtain new metric and topological results on survivor sets of intermediate β -transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [18]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate β -transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [15], Tseng [35] and Färm, Persson and Schmeling [11]. [ABSTRACT FROM AUTHOR]

We investigate the mesoscopic electron transport through multiple quantum dot systems in a T-shaped configuration and analyse the effect of the number of quantum dots on transport observables. Analytically, the expressions for the transmission probability and current have been derived, in the presence of intradot Coulomb interaction, using non-equilibrium Green function techniques. The higher-order Green functions containing the Coulomb interaction term have been decoupled using the mean-field approximation. The analytical, as well as the numerical, calculations have been presented for single, double and triple quantum dot systems in T-shaped configuration and then the results have been generalised for N number of quantum dots in the same configuration. The transport observables, such as transmission coefficient, I–V characteristics and differential conductance, have been numerically analysed. Numerical results show a systematic variation in the number of transmission spectrum peaks as the number of quantum dots side-coupled to the active dot increases. It is found that the number of the transmission probability peaks is directly related to the number of quantum dots and the peak spacings are controlled by the inter-dot coupling strength. However, the magnitude of the current is hardly affected by the increase in the number of side-coupled dots in T-shaped quantum dot systems. [ABSTRACT FROM AUTHOR]

Abraham A. Ungar investigated in 1988 the relativistic velocity addition HT ht on HT ht , which is neither commutative nor associative. Given a nearfield I F i the group HT ht Graph of all affine translations operates sharply 2-transitively on I F i . Conversely, if I T i is a sharply 2-transitive permutation group on a finite set I F i , then, according to Robert D. Carmichael, two operations can be defined on I F i such that I F i is made into a nearfield and I T i is isomorphic to the group HT ht . [Extracted from the article]

In Lin (Calc. Var. Partial Differential Equations 60:196, 2021), the first author proved the Steiner's projection inequality and the Petty projection inequality for sets of finite perimeter but didn't characterize their cases of equality. In this paper, we characterize the case of equality in the Steiner's projection inequality. Moreover, we give a new proof of the equality case in the Petty projection inequality, which was characterized by Wang (Adv. Math. 230:2457–2473, 2012). [ABSTRACT FROM AUTHOR]

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge–Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained. [ABSTRACT FROM AUTHOR]

Let X be a simplicial complex on vertex set V. We say that X is d-representable if it is isomorphic to the nerve of a family of convex sets in R d . We define the d-boxicity of X as the minimal k such that X can be written as the intersection of kd-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of X is a set τ ⊂ V such that τ ∉ X but σ ∈ X for any σ ⊊ τ . We prove that the d-boxicity of a simplicial complex on n vertices without missing faces of dimension larger than d is at most ⌊ n d / (d + 1) ⌋ . The bound is sharp: the d-boxicity of a simplicial complex whose set of missing faces form a Steiner (d , d + 1 , n) -system is exactly n d / (d + 1) . One of the main ingredients in the proof is the following bound on the representability of a complex: let V 1 , ... , V k be subsets of V such that V i ∉ X for all 1 ≤ i ≤ k , and for any missing face τ of X there is some 1 ≤ i ≤ k satisfying | τ \ V i | ≤ 1 . Then, X can be written as an intersection X = ⋂ i = 1 k X i , where, for all 1 ≤ i ≤ k , X i is a (| V i | - 1) -representable complex. In particular, X is (∑ i = 1 k (| V i | - 1)) -representable. [ABSTRACT FROM AUTHOR]

A 3-way (v , k , t) trade T of volume m consists of three pairwise disjoint collections T 1 , T 2 and T 3 , each of m blocks of size k , such that for every t -subset of v -set V , the number of blocks containing this t -subset is the same in each T i for 1 ≤ i ≤ 3 . If any t -subset of found(T ) occurs at most once in each T i for 1 ≤ i ≤ 3 , then T is called 3-way (v , k , t) Steiner trade. We attempt to complete the spectrum S 3 s (v , k) , the set of all possible volume sizes, for 3-way (v , k , 2) Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and r × s packing grid blocks. Previously, we obtained some results about the existence some 3-way (v , k , 2) Steiner trades. In particular, we proved that there exists a 3-way (v , k , 2) Steiner trade of volume m when 12 (k − 1) ≤ m for 15 ≤ k (Rashidi and Soltankhah in Discrete Math. 339(12): 2955–2963, 2016). Now, we show that the claim is correct also for k ≤ 14 . [ABSTRACT FROM AUTHOR]

In this paper, a study on modelling energy harvesting efficiency of a quasi-zero stiffness system is presented. Mechanical characteristics of the system are identified, and the effect of its stiffness and geometry on the function describing energy potential barrier is determined. It has been shown numerically that an increase in equivalent stiffness of the quasi-zero stiffness system limits the potential barrier width. On the other hand, increased the spacing between compensating springs results in increased barrier width. Simulation results of the quasi-zero stiffness system are compared with those obtained for a triple-well system with permanent magnets. Based on mathematical models, multi-color diagrams depicting the largest Lyapunov exponent are plotted. The effect of selected values of external excitation frequency and amplitude on the efficiency of energy harvesting is determined. The rms value of time sequence is taken as a measure of the energy harvesting efficiency. Obtained numerical results are plotted as phase trajectories. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the paper "Bifurcations of Steiner tree topologies in the plane," where topologies of Steiner networks of a fixed binary type are considered. Here bifurcation diagrams of binary types of Steiner minimal networks for four boundary points are constructed and some properties of such diagrams for any number of points are discussed. [ABSTRACT FROM AUTHOR]

Given a bipartite graph with parts A and B having maximum degrees at most Δ A and Δ B , respectively, consider a list assignment such that every vertex in A or B is given a list of colours of size k A or k B , respectively. We prove some general sufficient conditions in terms of Δ A , Δ B , k A , k B to be guaranteed a proper colouring such that each vertex is coloured using only a colour from its list. These are asymptotically nearly sharp in the very asymmetric cases. We establish one sufficient condition in particular, where Δ A = Δ B = Δ , k A = log Δ and k B = (1 + o (1)) Δ / log Δ as Δ → ∞ . This amounts to partial progress towards a conjecture from 1998 of Krivelevich and the first author. We also derive some necessary conditions through an intriguing connection between the complete case and the extremal size of approximate Steiner systems. We show that for complete bipartite graphs these conditions are asymptotically nearly sharp in a large part of the parameter space. This has provoked the following. In the setup above, we conjecture that a proper list colouring is always guaranteed if k A ≥ Δ A ε and k B ≥ Δ B ε for any ε > 0 provided Δ A and Δ B are large enough; if k A ≥ C log Δ B and k B ≥ C log Δ A for some absolute constant C > 1 ; or if Δ A = Δ B = Δ and k B ≥ C (Δ / log Δ) 1 / k A log Δ for some absolute constant C > 0 . These are asymmetric generalisations of the above-mentioned conjecture of Krivelevich and the first author, and if true are close to best possible. Our general sufficient conditions provide partial progress towards these conjectures. [ABSTRACT FROM AUTHOR]

We introduce the concept of local spanners for planar point sets with respect to a family of regions, and prove the existence of local spanners of small size for some families. For a geometric graph G on a point set P and a region R belonging to a family R , we define G ∩ R to be the part of the graph G that is inside R (or is induced by R). A local t-spanner w.r.t R is a geometric graph G on P such that for any region R ∈ R , the graph G ∩ R is a t-spanner for K (P) ∩ R , where K (P) is the complete geometric graph on P. For any set P of n points and any constant ε > 0 , we prove that P admits local (1 + ε) -spanners of sizes O (n log 6 n) and O (n log n) w.r.t axis-parallel squares and vertical slabs, respectively. If adding Steiner points is allowed, then local (1 + ε) -spanners with O(n) edges and O (n log 2 n) edges can be obtained for axis-parallel squares and disks using O(n) Steiner points, respectively. [ABSTRACT FROM AUTHOR]

We study the generalized minimum Manhattan network (GMMN) problem: given a set P of pairs of points in the Euclidean plane R 2 , we are required to find a minimum-length geometric network which consists of axis-aligned segments and contains a shortest path in the L 1 metric (a so-called Manhattan path) for each pair in P . This problem commonly generalizes several NP-hard network design problems that admit constant-factor approximation algorithms, such as the rectilinear Steiner arborescence (RSA) problem, and it is open whether so does the GMMN problem. As a bottom-up exploration, Schnizler (2015) focused on the intersection graphs of the rectangles defined by the pairs in P , and gave a polynomial-time dynamic programming algorithm for the GMMN problem whose input is restricted so that both the treewidth and the maximum degree of its intersection graph are bounded by constants. In this paper, as the first attempt to remove the degree bound, we provide a polynomial-time algorithm for the star case, and extend it to the general tree case based on an improved dynamic programming approach. [ABSTRACT FROM AUTHOR]

Given an n-sided polygon P on the plane with n ≥ 4 , a quadrangulation of P is a geometric plane graph such that the boundary of the outer face is P and that each finite face is quadrilateral. Clearly, P is quadrangulatable (i.e., admits a quadrangulation) only if n is even, but there is a non-quadrangulatable even-sided polygon. Ramaswami et al. [Comp Geom 9:257–276, (1998)] proved that every n-sided polygon P with n ≥ 4 even admits a quadrangulation with at most ⌊ n - 2 4 ⌋ Steiner points, where a Steiner point for P is an auxiliary point which can be put in any position in the interior of P. In this paper, introducing the notion of the spirality of P to control a structure of P (independent of n), we estimate the number of Steiner points to quadrangulate P. [ABSTRACT FROM AUTHOR]

The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov (On growth rate in S L 2 (F p) , the affine group and sum-product type implications, 2018. arXiv:1812.01671). We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds in Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound from Rudnev and Shkredov comes from possible applications to sum-product problems. For example, we prove that, for any finite A ⊂ R the following superquadratic bound holds: This improves on a bound with exponent 2 that was given in Rudnev and Shkredov. We also give a threshold-beating asymmetric sum-product estimate for sets with small sum set by proving that there exists a positive constant c such that for all finite A , B ⊂ R , [ABSTRACT FROM AUTHOR]

The studies of the topological properties of systems have recently been extended due to a new concept of higher-order topological insulators and superconductors. Many models were proposed for two-dimensional systems on a square lattice, where corner excitations can appear; however, the problem of existence of such excitations in superconducting systems with a triangular crystal lattice is still poorly understood. Using a topological insulator in the form of a triangle with a chiral superconducting order parameter as an example, we shows that corner excitations can exist in C3-symmetric systems. In spite of a nontopological character, these excitations have energies inside the gap of the first-order edge excitation spectrum over a wide parameter range and are well localized at the corners of the system. Gapless corner excitations are shown to exist in the system at certain parameters. The application of a magnetic field in the system plane removes the triple degeneracy of the corner excitation energy and makes it possible to control the position of the minimum-energy corner excitation using a magnetic field. At the same time the fine adjustment to achieve the gapless excitations at the chosen corner can be made with changing of the magnetic field value. [ABSTRACT FROM AUTHOR]

Mathematical models and the rank optimization method for current Steiner networks are presented in the work. This method is based on the nonlocal minimum condition and on the dynamical decomposition of the current Steiner network in optimized intersecting subnetworks of an equal dimension that follows from it. A computer simulation is presented that shows a sufficient effectiveness of the method in applying it in designing distributive piping networks of water- and gas-supply. [ABSTRACT FROM AUTHOR]

After the departure of Kotzig and Rosa, the leadership of the seminar was assumed by Juraj Bosák, originally a student of Kotzig, later his long-time collaborator, and an active member of the seminar from its very beginnings. Kotzig and Bosák were among the first to present lectures at a graph theory conference (with Kotzig delivering the first talk in 1963, in Smolenice, Slovakia). Many young Slovak mathematicians benefited from these contacts, and had Kotizig and Rosa not left, the development of graph theory in Slovakia at the international level may have been delayed by many years. The Beginning of the Seminar (Early 1960s) The Bratislava graph theory seminar was established through the initiative of the Slovak mathematician Anton Kotzig, who prior to founding the seminar was (among his other activities) the rector of the University of Economics in Bratislava from 1952 to 1958. [Extracted from the article]

The Steiner forest problem asks for a minimum weight forest that spans a given number of terminal sets. We propose new cut- and flow-based integer linear programming formulations for the problem which yield stronger linear programming bounds than the two previous strongest formulations: The directed cut formulation (Balakrishnan et al. in Oper Res 37(5):716–740, 1989; Chopra and Rao in Math Prog 64(1):209–229, 1994) and the advanced flow formulation by Magnanti and Raghavan (Networks 45:61–79, 2005). We further introduce strengthening constraints and provide an example where the integrality gap of our models is 1.5. In an experimental evaluation, we show that the linear programming bounds of the new formulations are indeed strong on practical instances and that the related branch-and-cut algorithm outperforms algorithms based on the previous formulations. [ABSTRACT FROM AUTHOR]

Wide- and small-angle X-ray scattering is used to study the structure of the single-phase glass of the Bi2O3–GeO2 system and three-component Bi2O3–GeO2–SiO2 system in three sections with the GeO2/SiO2 molar ratios of 3, 1, and 1/3. The Bi2O3 content is changed from 20 to 50 mol % in the first two sections; and from 10 to 50 mol %, in the third one. Based on the data of wide-angle X-ray scattering, it can be concluded that structural groupings containing BiO3 triangles and SiO2 and GeO4 tetrahedra connected with them are formed in the glass of the studied triple system due to the chemical reactions of Bi2O3, SiO2, and GeO2. When the Bi2O3 content is increased the structural groupings are combined and local areas with three-dimensional structure are formed. As a result, each BiO3 triangle is connected with three tetrahedra, and each tetrahedron is connected with four triangles. It is established that the initial stages of the appearance of crystalline phases during the cooling of the melts are observed in the glass containing not less than 40 mol % of Bi2O3. [ABSTRACT FROM AUTHOR]

A loop X is said to satisfy Moufang's theorem if for every x , y , z ∈ X such that x (y z) = (x y) z the subloop generated by x, y, z is a group. We prove that the variety V of Steiner loops satisfying the identity (x z) (((x y) z) (y z)) = ((x z) ((x y) z)) (y z) is not contained in the variety of Moufang loops, yet every loop in V satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah. [ABSTRACT FROM AUTHOR]

A Steiner triple system (STS) contains a transversal subdesign TD(3, w) if its point set has three pairwise disjoint subsets A, B, C of size w and w2 blocks of the STS intersect with each of A, B, C (those w2 blocks form a TD(3,w)). We prove several structural properties of Steiner triple systems of order 3w + 3 that contain one or more transversal subdesigns TD(3, w). Using exhaustive search, we find that there are 2 004 720 isomorphism classes of STS(21) containing a subdesign TD(3, 6) (or, equivalently, a 6 × 6 Latin square). [ABSTRACT FROM AUTHOR]

We introduce a new model of random d-dimensional simplicial complexes, for d ≥ 2 , whose (d - 1) -cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash's recent result on designs (The existence of designs; arXiv:1401.3665, 2014), and the proof of the expansion uses techniques developed by Evra and Kaufman in (Bounded degree cosystolic expanders of every dimension; arXiv:1510.00839, 2015). This gives a full solution to a question raised in Dotterrer and Kahle (J Topol Anal 4(4): 499–514, 2012), which was solved in the two-dimensional case by Lubotzky and Meshulam (Adv Math 272: 743–760, 2015). [ABSTRACT FROM AUTHOR]

A graph is n-existentially closed (n-e.c.) if for any disjoint subsets A, B of vertices with A ∪ B = n , there is a vertex z ∉ A ∪ B adjacent to every vertex of A and no vertex of B. For a block design with block set B , its block intersection graph is the graph whose vertex set is B and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be 2-e.c. In particular, we study the λ -fold triple systems with λ ≥ 2 and determine for which parameters their block intersection graphs are 1- or 2-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called { 1 } -block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be 2-e.c. are established. [ABSTRACT FROM AUTHOR]

The paper is devoted to a semi-analytical method, to develop analytical expressions for the wave field and the scattering parameters—the reflection and transmission coefficients, when a longitudinal plane wave falls on the system of a finite number of identical doubly periodic gratings parallel to each other, each of them consisting of a periodic array of rectangular cracks in the elastic isotropic medium. In the one-mode frequency range, the problem is reduced to a system of hypersingular integral equations whose solution gives the physical parameters as well as an explicit representation of the wave field inside the structure. The present work continues to study the 3-D problem for arbitrary finite number of such plane arrays which form a triple-periodic system of cracks. The principal aim is to generalize the results obtained previously for elastic 2-D and 3-D problems. Some presented diagrams emphasize new physical properties of the cascading structure. [ABSTRACT FROM AUTHOR]

Let v ≡ 4 (mod 6) and v ≥ 4 . A canonical Kirkman packing design of order v, denoted by CKPD (v) , is a resolvable packing with r = (v - 4) / 2 parallel classes such that (i) each parallel class consists of a size 4 block and (v - 4) / 3 triples; (ii) the leave consists of the union of (v - 4) / 2 vertex-disjoint edges and a K 4 with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable CKPD (v) s with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained. [ABSTRACT FROM AUTHOR]

. [Extracted from the article]

A μ-way (v, k, t) trade is a pair T=(X,{T1,T2,...,Tμ}) such that for eacht-subset of v-set X the number of blocks containing this t-subset is the same in each Ti(1≤i≤μ). In the other words for each 1≤i

We introduce m-near-resolvable block designs. We establish a correspondence between such block designs and a subclass of (optimal equidistant) q-ary constant-weight codes meeting the Johnson bound. We present constructions of m-near-resolvable block designs, in particular based on Steiner systems and super-simple t-designs. [ABSTRACT FROM AUTHOR]

In this paper, we give some properties of the generalized derivation algebra GDer(T) of a Hom-Lie triple systems T. In particular, we prove that GDer(T)=QDer(T)+QC(T), the sum of the quasiderivation algebra and the quasicentroid. We also prove that QDer(T) can be embedded as derivations in a larger Hom-Lie triple system. General results on centroids of Hom-Lie triple systems are also developed in this paper. [ABSTRACT FROM AUTHOR]

In this paper we prove some bounds for Steiner distance in Cartesian product. We investigate properties of connected subgraphs that are not Steiner convex. Those results are the key in the characterization of Steiner convex sets of grids and also in the characterization of 3-Steiner convex sets of Cartesian product graphs. [ABSTRACT FROM AUTHOR]

In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk et al. (55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, 2014) gave a polynomial kernel for Steiner Tree in planar graphs and graphs of bounded genus, when parameterized by $$|T|+k$$ , the total number of vertices in the constructed subgraph. In this paper we present several polynomial time applicable reduction rules for Steiner Tree in graphs of bounded genus. In an instance reduced with respect to the presented reduction rules, the number of terminals | T| is at most cubic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in graphs of bounded genus for the parameterization by the number k of Steiner vertices in the solution. We give better bounds for Steiner Tree in planar graphs. [ABSTRACT FROM AUTHOR]

Erdős [1] conjectured that for all $${k \geq 2}$$ , $${s \geq 1}$$ and $${n \geq {k(s+1)}}$$ , an n-vertex k-uniform hypergraph $${\mathcal{F}}$$ with $${\nu(\mathcal{F})=s}$$ cannot have more than $${\max\{\binom{sk+k-1}k,\binom nk-\binom{n-s}k\}}$$ edges. It took almost fifty years to prove it for triple systems. In [5] we proved the conjecture for all s and all $${n \geq 4(s+1)}$$ . Then Łuczak and Mieczkowska [6] proved the conjecture for sufficiently large s and all n. Soon after, Frankl proved it for all s. Here we present a simpler version of that proof which yields Erdős' conjecture for $${s \geq 33}$$ . Our motivation is to lay down foundations for a possible proof in the much harder case k = 4, at least for large s. [ABSTRACT FROM AUTHOR]

The article discusses variations on Euclidean geometry theorem, Steiner's porism, which is named after geometer Jacob Steiner. Topics include the properties of Steiner configuration, the radical line or radical axis as the locus of points of equal power, the importance of Steiner porism for understanding the elementary properties of pencils of coaxial circles.

The bottleneck of the currently best $$(\ln (4)+{\varepsilon })$$ -approximation algorithm for the NP-hard Steiner tree problem is the solution of its large, so called hypergraphic, linear programming relaxation (HYP). Hypergraphic LPs are strongly NP-hard to solve exactly, and it is a formidable computational task to even approximate them sufficiently well. We focus on another well-studied but poorly understood LP relaxation of the problem: the bidirected cut relaxation (BCR). This LP is compact, and can therefore be solved efficiently. Its integrality gap is known to be greater than 1.16, and while this is widely conjectured to be close to the real answer, only a (trivial) upper bound of 2 is known. In this article, we give an efficient constructive proof that BCR and HYP are polyhedrally equivalent in instances that do not have an (edge-induced) claw on Steiner vertices, i.e., they do not contain a Steiner vertex with three Steiner neighbours. This implies faster $$\ln (4)$$ -approximations for these graphs, and is a significant step forward from the previously known equivalence for (so called quasi-bipartite) instances in which Steiner vertices form an independent set. We complement our results by showing that even restricting to instances where Steiner vertices induce one single star, determining whether the two relaxations are equivalent is NP-hard. [ABSTRACT FROM AUTHOR]

We propose a new approach to studies on partial Steiner triple systems consisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs. [ABSTRACT FROM AUTHOR]

We study the symmetry group of a binary perfect Mollard code M( C,D) of length tm + t + m containing as its subcodes the codes C and D formed from perfect codes C and D of lengths t and m, respectively, by adding an appropriate number of zeros. For the Mollard codes, we generalize the result obtained in [1] for the symmetry group of Vasil'ev codes; namely, we describe the stabilizer Sym( M( C,D)) of the subcode D in the symmetry group of the code M( C,D) (with the trivial function). Thus we obtain a new lower bound on the order of the symmetry group of the Mollard code. A similar result is established for the automorphism group of Steiner triple systems obtained by the Mollard construction but not necessarily associated with perfect codes. To obtain this result, we essentially use the notions of 'linearity' of coordinate positions (points) of a nonlinear perfect code and a nonprojective Steiner triple system. [ABSTRACT FROM AUTHOR]

A pure Mendelsohn triple system of order v, denoted by PMTS( v), is a pair $$(X,\mathcal {B})$$ where X is a v-set and $$\mathcal {B}$$ is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of $$\mathcal {B}$$ and if $$\langle a,b,c\rangle \in \mathcal {B}$$ implies $$\langle c,b,a\rangle \notin \mathcal {B}$$ . An overlarge set of PMTS( v), denoted by OLPMTS( v), is a collection $$\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i$$ , where Y is a $$(v+1)$$ -set, $$y_i\in Y$$ , each $$(Y{\setminus }\{y_i\},{\mathcal {A}}_i)$$ is a PMTS( v) and these $${\mathcal {A}}_i$$ s form a partition of all cyclic triples on Y. It is shown in [] that there exists an OLPMTS( v) for $$v\equiv 1,3$$ (mod 6), $$v>3$$ , or $$v \equiv 0,4$$ (mod 12). In this paper, we shall discuss the existence problem of OLPMTS( v) s for $$v\equiv 6,10$$ (mod 12) and get the following conclusion: there exists an OLPMTS( v) if and only if $$v\equiv 0,1$$ (mod 3), $$v>3$$ and $$v\ne 6$$ . [ABSTRACT FROM AUTHOR]

We introduce a class of non-Moufang loops satisfying Moufang's theorem. [ABSTRACT FROM AUTHOR]

We consider generalized Preparata codes with a noncommutative group operation. These codes are shown to induce new partitions of Hamming codes into cosets of these Preparata codes. The constructed partitions induce 2-resolvable Steiner quadruple systems S( n, 4, 3) (i.e., systems S( n, 4, 3) that can be partitioned into disjoint Steiner systems S( n, 4, 2)). The obtained partitions of systems S( n, 4, 3) into systems S( n, 4, 2) are not equivalent to such partitions previously known. [ABSTRACT FROM AUTHOR]

The mathematical analysis of the shape of chemical reactors is studied in this paper through the research of the optimization of its effectiveness $$\eta$$ such as introduced by R. Aris around 1960. Although our main motivation is the consideration of reactors specially designed for the treatment of wastewaters our results are relevant also in more general frameworks. We simplify the modeling by assuming a single chemical reaction with a monotone kinetics leading to a parabolic equation with a non-necessarily differentiable function. In fact we consider here the case of a single, non-reversible catalysis reaction of chemical order $$q, 00$$ ). We assume the chemical reactor of cylindrical shape $$\Omega =G\times (0,H)$$ with G and open regular set of $$\mathbb {R}^{2}$$ not necessarily symmetric. We show that among all the sections G with prescribed area the ball is the set of lowest effectiveness $$\eta (t,G)$$ . The proof uses the notions of Steiner rearrangement. Finally, we show that if the height H is small enough then the effectiveness can be made as close to 1 as desired. [ABSTRACT FROM AUTHOR]

A complete weighted graph, $$G(X,\varGamma ,W)$$ , is considered. Let $$\tilde{X}\subset X$$ be some subset of vertices and, by definition, a Steiner tree is any tree in the graph G such that the set of the tree vertices includes set $$\tilde{X}$$ . The Steiner tree problem consists of constructing the minimum-length Steiner tree in graph G, for a given subset of vertices $$\tilde{X}$$ The effectively computable estimate of the minimal Steiner tree is obtained in terms of the mean value and the variance over the set of all Steiner trees. It is shown that in the space of the lengths of the graph edges, there exist some regions where the obtained estimate is better than the minimal spanning tree-based one. [ABSTRACT FROM AUTHOR]

We present a greedy algorithm for the directed Steiner tree problem (DST), where any tree rooted at any (uncovered) terminal can be a candidate for greedy choice. It will be shown that the algorithm, running in polynomial time for any constant $$l$$ , outputs a directed Steiner tree of cost no larger than $$2(l-1)(\ln n+1)$$ times the cost of any $$l$$ -restricted Steiner tree, which is such a Steiner tree in which every terminal is at most $$l$$ arcs away from the root or another terminal. We derive from this result that (1) DST for a class of graphs, including quasi-bipartite graphs, in which the length of paths induced by Steiner vertices is bounded by some constant can be approximated within a factor of $$O(\log n)$$ , and (2) the tree cover problem on directed graphs can also be approximated within a factor of $$O(\log n)$$ . [ABSTRACT FROM AUTHOR]

We present an approximation algorithm for the minimum bounded degree Steiner network problem that returns a Steiner network of cost at most two times the optimal and the degree on each vertex $$v$$ is at most $$\min \{b_v + 3r_{\mathrm{max}}, 2b_v+2\}$$ , where $$r_{\mathrm{max}}$$ is the maximum connectivity requirement and $$b_v$$ is the given degree bound on $$v$$ . This unifies, simplifies, and improves the previous results for this problem. [ABSTRACT FROM AUTHOR]

A 6-cycle is said to be squashed if we identify a pair of opposite vertices and name one of them with the other (thereby turning the 6-cycle into a pair of triples with a common vertex). A 6-cycle can be squashed in six different ways. The spectrum for 6-cycle systems that can be squashed into Steiner triple systems has been determined by Lindner et al.(J Comb Des 22:189-195, ). The squashing of maximum packings of K with 6-cycles into maximum packings of K with triples has also been fully dealt with by Lindner et al. (Squashing maximum packings of 6-cycles into maximum packings of triples (submitted)). The object of this paper is to extend these results to minimum coverings of K with 6-cycles into minimum coverings of K with triples. We give a complete solution. [ABSTRACT FROM AUTHOR]

We study the problem of finding small s- t separators that induce graphs having certain properties. It is known that finding a minimum clique s- t separator is polynomial-time solvable (Tarjan in Discrete Math. 55:221-232, ), while for example the problems of finding a minimum s- t separator that induces a connected graph or forms an independent set are fixed-parameter tractable when parameterized by the size of the separator (Marx et al. in ACM Trans. Algorithms 9(4): 30, ). Motivated by these results, we study properties that generalize cliques, independent sets, and connected graphs, and determine the complexity of finding separators satisfying these properties. We investigate these problems also on bounded-degree graphs. Our results are as follows: In order to prove (1), we show that the natural c-connected generalization of the well-known Steiner Tree problem is FPT for c=2 and W[1]-hard for any c≥3. [ABSTRACT FROM AUTHOR]

We consider the Pólya-Szegö type weighted inequality. We prove this inequality for monotone rearrangement and for Steiner's symmetrization. In particular we fill the gap in the paper of Brock (Calc Var PDEs 8:15-25, ) for 1D case. [ABSTRACT FROM AUTHOR]