scholarly journals Bounds on Antipodal Spherical Designs with Few Angles

10.37236/9891 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Zhiqiang Xu ◽  
Zili Xu ◽  
Wei-Hsuan Yu
Keyword(s):  
Unit Sphere ◽  
Finite Subset ◽  
Upper Bound ◽  
Maximum Size ◽  
Tight Frames ◽  
Previous Estimate ◽  
Spherical Designs ◽  
Equiangular Tight Frames ◽  
Special Cases ◽  
Distance Set

A finite subset $X$ on the unit sphere $\mathbb{S}^d$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X, \mathbf{x}\neq\mathbf{y} \}$ has size $s$, and $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. In this paper, we consider to estimate the maximum size of such antipodal set $X$ for small $s$. Motivated by the method developed by Nozaki and Suda, for each even integer $s\in[\frac{t+5}{2}, t+1]$ with $t\geq 3$, we improve the best known upper bound of Delsarte, Goethals and Seidel. We next focus on two special cases: $s=3,\ t=3$ and $s=4,\ t=5$. Estimating the size of $X$ for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on $|X|$ when $s=3,\ t=3$ and $s=4,\ t=5$, respectively.

scholarly journals Steiner equiangular tight frames

2012 ◽  
Vol 436 (5) ◽  
pp. 1014-1027 ◽  
Author(s):  
Matthew Fickus ◽  
Dustin G. Mixon ◽  
Janet C. Tremain
Keyword(s):  
Tight Frames ◽  
Equiangular Tight Frames

Equiangular tight frames

2009 ◽  
Vol 157 (6) ◽  
pp. 789-815 ◽  
Author(s):  
V. N. Malozemov ◽  
A. B. Pevnyi
Keyword(s):  
Tight Frames ◽  
Equiangular Tight Frames

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

scholarly journals Power-law bounds for critical long-range percolation below the upper-critical dimension

2021 ◽  
Author(s):  
Tom Hutchcroft
Keyword(s):  
Long Range ◽  
Power Law ◽  
Upper Bound ◽  
Mean Field ◽  
Maximum Size ◽  
Finite Graph ◽  
Critical Dimension ◽  
Point Function ◽  
Upper Critical Dimension ◽  
Infinite Cluster

AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$ Z d in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$ 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$ 0 < α < d then there is no infinite cluster at the critical parameter $$\beta _c$$ β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$ P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every $$n\ge 1$$ n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$ ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent $$\delta $$ δ and two-point function exponent $$\eta $$ η .

Quasi-symmetric designs and equiangular tight frames

2015 ◽  
Author(s):  
Matthew Fickus ◽  
John Jasper ◽  
Dustin Mixon ◽  
Jesse Peterson
Keyword(s):  
Tight Frames ◽  
Symmetric Designs ◽  
Equiangular Tight Frames

scholarly journals Inversions Within Restricted Fillings of Young Tableaux

10.37236/1030 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Sarah Iveson
Keyword(s):  
Upper Bound ◽  
Exact Expression ◽  
Flag Varieties ◽  
Young Tableaux ◽  
Geometric Properties ◽  
Number Of Components ◽  
Hessenberg Varieties ◽  
Special Cases

In this paper we study inversions within restricted fillings of Young tableaux. These restricted fillings are of interest because they describe geometric properties of certain subvarieties, called Hessenberg varieties, of flag varieties. We give answers and partial answers to some conjectures posed by Tymoczko. In particular, we find the number of components of these varieties, give an upper bound on the dimensions of the varieties, and give an exact expression for the dimension in some special cases. The proofs given are all combinatorial.

scholarly journals Study of Second Hankel Determinant for Certain Subclasses of Functions Defined by Al-Oboudi Differential Operator

2020 ◽  
Vol 17 (1(Suppl.)) ◽  
pp. 0353
Author(s):  
K. A. Challab et al.
Keyword(s):  
Differential Operator ◽  
Upper Bound ◽  
Unit Disc ◽  
Hankel Determinant ◽  
Special Cases ◽  
Second Hankel Determinant

The concern of this article is the calculation of an upper bound of second Hankel determinant for the subclasses of functions defined by Al-Oboudi differential operator in the unit disc. To study special cases of the results of this article, we give particular values to the parameters A, B and λ

scholarly journals Branch and Bound Method to Solve Multi Objectives Function

2016 ◽  
Vol 12 (3) ◽  
pp. 5964-5974
Author(s):  
Tahani Jabbar Kahribt ◽  
Mohammed Kadhim Al- Zuwaini
Keyword(s):  
Lower Bounds ◽  
Branch And Bound ◽  
Single Machine ◽  
Upper Bound ◽  
Completion Time ◽  
Heuristic Method ◽  
Total Weighted Completion Time ◽  
Branch And Bound Method ◽  
Special Cases ◽  
Weighted Completion Time

This paper  presents  a  branch  and  bound  algorithm  for  sequencing  a  set  of  n independent  jobs  on  a single  machine  to  minimize sum of the discounted total weighted completion time and maximum lateness,  this problems is NP-hard. Two lower bounds were proposed and heuristic method to get an upper bound. Some special cases were  proved and some dominance rules were suggested and proved, the problem solved with up to 50 jobs.

Inductive graph invariants and approximation algorithms

2021 ◽  
Author(s):  
C. R. Subramanian
Keyword(s):  
Optimization Problems ◽  
Independence Number ◽  
Optimal Solution ◽  
Maximum Size ◽  
Chordal Graphs ◽  
Minimum Size ◽  
Optimal Size ◽  
Induced Subgraph ◽  
Special Cases ◽  
Optimal Formula

We introduce and study an inductively defined analogue [Formula: see text] of any increasing graph invariant [Formula: see text]. An invariant [Formula: see text] is increasing if [Formula: see text] whenever [Formula: see text] is an induced subgraph of [Formula: see text]. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing [Formula: see text], this gets us several new invariants and many of which are also increasing. It is also shown that [Formula: see text] is the minimum (over all orderings) of a value associated with each ordering. We also explore the possibility of computing [Formula: see text] (and a corresponding optimal vertex ordering) and identify some pairs [Formula: see text] for which [Formula: see text] can be computed efficiently for members of [Formula: see text]. In particular, it includes graphs of bounded [Formula: see text] values. Some specific examples (like the class of chordal graphs) have already been studied extensively. We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing [Formula: see text] to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of [Formula: see text]-neighborhoods for arbitrary but fixed [Formula: see text]. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms 8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced [Formula: see text]-subgraphs and optimal [Formula: see text]-colorings (for hereditary [Formula: see text]’s) within multiplicative factors of (essentially) [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced [Formula: see text]-subgraph of the input and [Formula: see text] is the minimum size of a forbidden induced subgraph of [Formula: see text]. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal [Formula: see text]-subgraphs and [Formula: see text]-colorings for arbitrary hereditary classes [Formula: see text]. As a corollary, it is also shown that any maximal [Formula: see text]-subgraph approximates an optimal solution within a factor of [Formula: see text] for unweighted graphs, where [Formula: see text] is maximum size of any induced [Formula: see text]-subgraph in any local neighborhood [Formula: see text].

scholarly journals On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1778
Author(s):  
Fangyun Tao ◽  
Ting Jin ◽  
Yiyou Tu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Equitable Partition ◽  
Special Cases ◽  
Vertex Set ◽  
Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

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