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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]