scholarly journals The Frobenius and Factor Universality Problems of the Kleene Star of a Finite Set of Words

2021 ◽  
Vol 68 (3) ◽  
pp. 1-22
Author(s):  
Maksymilian Mika ◽  
Marek Szykuła
Keyword(s):  
Infinite Family ◽  
Upper Bounds ◽  
Open Problems ◽  
Rewriting Systems ◽  
Finite Set ◽  
General Tool

We solve open problems concerning the Kleene star of a finite set of words over an alphabet . The Frobenius monoid problem is the question for a given finite set of words , whether the language is cofinite. We show that it is PSPACE-complete. We also exhibit an infinite family of sets such that the length of the longest words not in (when is cofinite) is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . The factor universality problem is the question for a given finite set of words , whether every word over is a factor (substring) of some word from . We show that it is also PSPACE-complete. Besides that, we exhibit an infinite family of sets such that the length of the shortest words not being a factor of any word in is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . This essentially settles in the negative the longstanding Restivo’s conjecture (1981) and its weak variations. All our solutions are based on one shared construction, and as an auxiliary general tool, we introduce the concept of set rewriting systems . Finally, we complement the results with upper bounds.

scholarly journals Quantifying Noninvertibility in Discrete Dynamical Systems

10.37236/9475 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Colin Defant ◽  
James Propp
Keyword(s):  
Dynamical Systems ◽  
Real Number ◽  
Arbitrary Function ◽  
Discrete Dynamical Systems ◽  
Open Problems ◽  
Natural Measure ◽  
Stack Sorting ◽  
Finite Set

Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\deg(f^k)}{\deg(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})\] for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.  

scholarly journals Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

scholarly journals A hybrid VNS matheuristic for a bin packing problem with a color constraint

2021 ◽  
pp. 9-9
Author(s):  
Yury Kochetov ◽  
Arteam Kondakov
Keyword(s):  
Column Generation ◽  
Bin Packing ◽  
Packing Problem ◽  
Upper Bounds ◽  
Hard Problem ◽  
Generation Technique ◽  
Bin Packing Problem ◽  
New Variant ◽  
Finite Set ◽  
Np Hard Problem

We study a new variant of the bin packing problem with a color constraint. Given a finite set of items, each item has a set of colors. Each bin has a color capacity, the total number of colors for a bin is the unification of colors for its items and cannot exceed the bin capacity. We need to pack all items into the minimal number of bins. For this NP-hard problem we present approximability results and design a hybrid matheuristic based on the column generation technique. A hybrid VNS heuristic is applied to the pricing problem. The column generation method provides a lower bound and a core subset of the most promising bin patterns. Fast heuristics and exact solution for this core produce upper bounds. Computational experiments for test instances with number of items up to 500 illustrate the efficiency of the approach.

From the Outside In

2015 ◽  
Author(s):  
Derek Smith
Keyword(s):  
Positive Integer ◽  
Infinite Family ◽  
Open Problems ◽  
The Family

This chapter discusses Slothouber–Graatsma–Conway puzzle, which asks one to assemble six 1 × 2 × 2 pieces and three 1 × 1 × 1 pieces into the shape of a 3 × 3 × 3 cube. The puzzle has been generalized to larger cubes, and there is an infinite family of such puzzles. The chapter's primary argument is that, for any odd positive integer n = 2k + 1, there is exactly one way, up to symmetry, to make an n × n × n cube out of n tiny 1 × 1 × 1 cubes and six of each of a set of rectangular blocks. The chapter describes a way to solve each puzzle in the family and explains why there are no other solutions. It then presents several related open problems.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals Computational aspects of relaxation complexity: possibilities and limitations

2021 ◽  
Author(s):  
Gennadiy Averkov ◽  
Christopher Hojny ◽  
Matthias Schymura
Keyword(s):  
Explicit Formula ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Upper Bounds ◽  
Negative Answer ◽  
Integer Points ◽  
Finite Set ◽  
Computational Aspects

AbstractThe relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) of the set of integer points X contained in a polyhedron is the smallest number of facets of any polyhedron P such that the integer points in P coincide with X. It is a useful tool to investigate the existence of compact linear descriptions of X. In this article, we derive tight and computable upper bounds on $${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$ rc Q ( X ) , a variant of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) in which the polyhedra P are required to be rational, and we show that $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) can be computed in polynomial time if X is 2-dimensional. Further, we investigate computable lower bounds on $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) with the particular focus on the existence of a finite set $$Y \subseteq \mathbb {Z}^d$$ Y ⊆ Z d such that separating X and $$Y \setminus X$$ Y \ X allows us to deduce $${{\,\mathrm{rc}\,}}(X) \ge k$$ rc ( X ) ≥ k . In particular, we show for some choices of X that no such finite set Y exists to certify the value of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) , providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for specific classes of sets X and present the first practically applicable approach to compute $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for sets X that admit a finite certificate.

scholarly journals Cofinitely and co-countably projective spaces

2002 ◽  
Vol 3 (2) ◽  
pp. 185
Author(s):  
Pablo Mendoza Iturralde ◽  
Vladimir V. Tkachuk
Keyword(s):  
Topological Group ◽  
Projective Space ◽  
Infinite Family ◽  
Projective Spaces ◽  
Finite Union ◽  
Paracompact Space ◽  
Finite Set ◽  
Discrete Spaces

<p>We show that X is cofinitely projective if and only if it is a finite union of Alexandroff compactatifications of discrete spaces. We also prove that X is co-countably projective if and only if X admits no disjoint infinite family of uncountable cozero sets. It is shown that a paracompact space X is co-countably projective if and only if there exists a finite set B C X such that B C U ϵ τ (X) implies │X\U│ ≤ ω. In case of existence of such a B we will say that X is concentrated around B. We prove that there exists a space Y which is co-countably projective while there is no finite set B C Y around which Y is concentrated. We show that any metrizable co-countably projective space is countable. An important corollary is that every co-countably projective topological group is countable.</p>

scholarly journals Residual $q$-Fano Planes and Related Structures

10.37236/7107 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tuvi Etzion ◽  
Niv Hooker
Keyword(s):  
Block Design ◽  
Infinite Family ◽  
Steiner System ◽  
Interesting Case ◽  
Common Belief ◽  
Steiner Systems ◽  
Fano Plane ◽  
Open Problems ◽  
One Step ◽  
Existence Question

One of the most intriguing problems for $q$-analogs of designs, is the existence question of an infinite family of $q$-Steiner systems that are not spreads. In particular the most interesting case is the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We construct different residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.

scholarly journals Searching for gauge theories with the conformal bootstrap

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Zhijin Li ◽  
David Poland
Keyword(s):  
Fixed Points ◽  
Gauge Theories ◽  
Free Field ◽  
Infinite Family ◽  
Preliminary Evidence ◽  
Upper Bounds ◽  
Free Fermion ◽  
Conformal Bootstrap ◽  
General Dimension ◽  
Show Evidence

Abstract Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for SO(N) vector 4-point functions in general dimension D. In the large N limit, upper bounds on the scaling dimensions of the lowest SO(N) singlet and traceless symmetric scalars interpolate between two solutions at ∆ = D/2 − 1 and ∆ = D − 1 via generalized free field theory. In 3D the critical O(N) vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching ∆ = 1/2 at large N. We show that the bootstrap bounds also admit another infinite family of kinks $$ {\mathcal{T}}_D $$ T D , which at large N approach solutions containing free fermion bilinears at ∆ = D − 1 from below. The kinks $$ {\mathcal{T}}_D $$ T D appear in general dimensions with a D-dependent critical N* below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with SO(N) vectors, SU(N) fundamentals, and SU(N) × SU(N) bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of $$ {\mathcal{T}}_D $$ T D are subgroups of SO(N), and we speculate that the kinks $$ {\mathcal{T}}_D $$ T D relate to the fixed points of gauge theories coupled to fermions.

scholarly journals A Survey on Extremal Problems of Eigenvalues

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Ping Yan ◽  
Meirong Zhang
Keyword(s):  
Variational Method ◽  
Lebesgue Space ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Lower And Upper Bounds ◽  
Open Problems ◽  
One Dimensional ◽  
Sturm Liouville ◽  
The One ◽  
Neumann Eigenvalues

Given an integrable potentialq∈L1([0,1],ℝ), the Dirichlet and the Neumann eigenvaluesλnD(q)andλnN(q)of the Sturm-Liouville operator with the potentialqare defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when theL1metric forqis given;∥q∥L1=r. Note that theL1spheres andL1balls are nonsmooth, noncompact domains of the Lebesgue space(L1([0,1],ℝ),∥·∥L1). To solve these extremal problems, we will reveal some deep results on the dependence of eigenvalues on potentials. Moreover, the variational method for the approximating extremal problems on the balls of the spacesLα([0,1],ℝ),1<α<∞will be used. Then theL1problems will be solved by passingα↓1. Corresponding extremal problems for eigenvalues of the one-dimensionalp-Laplacian with integrable potentials have also been solved. The results can yield optimal lower and upper bounds for these eigenvalues. This paper will review the most important ideas and techniques in solving these difficult and interesting extremal problems. Some open problems will also be imposed.

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