scholarly journals Dendrites and measures with discrete spectrum

2021 ◽  
pp. 1-11
Author(s):  
MAGDALENA FORYŚ-KRAWIEC ◽  
JANA HANTÁKOVÁ ◽  
JIŘÍ KUPKA ◽  
PIOTR OPROCHA ◽  
SAMUEL ROTH
Keyword(s):  
Discrete Spectrum ◽  
Invariant Measures ◽  
Zero Entropy ◽  
Open Question

Abstract We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrites is countable. This solves an open question which has been around for awhile, and almost completes the characterization of dendrites with this property.

scholarly journals Mean equicontinuity and mean sensitivity

2014 ◽  
Vol 35 (8) ◽  
pp. 2587-2612 ◽  
Author(s):  
JIAN LI ◽  
SIMING TU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Discrete Spectrum ◽  
Stable System ◽  
Positive Entropy ◽  
Zero Entropy ◽  
Open Question ◽  
Mean Sensitivity ◽  
Spectrum Dichotomy

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

scholarly journals On the similarity to nonnegative and Metzler Hessenberg forms

2021 ◽  
Vol 10 (1) ◽  
pp. 1-8
Author(s):  
Christian Grussler ◽  
Anders Rantzer
Keyword(s):  
Standard Form ◽  
Complete Characterization ◽  
Nonnegative Matrices ◽  
Positive Systems ◽  
Third Order ◽  
Hessenberg Form ◽  
Hessenberg Matrices ◽  
Open Question ◽  
Order Continuous

Abstract We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n 4. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n 5 remain similar to Metzler Hessenberg matrices.

The randomized communication complexity of revenue maximization

2021 ◽  
Vol 19 (2) ◽  
pp. 75-83
Author(s):  
Aviad Rubinstein ◽  
Junyao Zhao
Keyword(s):  
Lower Bounds ◽  
Communication Complexity ◽  
Social Choice Rule ◽  
Valuation Function ◽  
Optimal Auctions ◽  
Incentive Compatible ◽  
First Approximation ◽  
Exponential Separation ◽  
Open Question

We study the communication complexity of incentive compatible auction-protocols between a monopolist seller and a single buyer with a combinatorial valuation function over n items [Rubinstein and Zhao 2021]. Motivated by the fact that revenue-optimal auctions are randomized [Thanassoulis 2004; Manelli and Vincent 2010; Briest et al. 2010; Pavlov 2011; Hart and Reny 2015] (as well as by an open problem of Babaioff, Gonczarowski, and Nisan [Babaioff et al. 2017]), we focus on the randomized communication complexity of this problem (in contrast to most prior work on deterministic communication). We design simple, incentive compatible, and revenue-optimal auction-protocols whose expected communication complexity is much (in fact infinitely) more efficient than their deterministic counterparts. We also give nearly matching lower bounds on the expected communication complexity of approximately-revenue-optimal auctions. These results follow from a simple characterization of incentive compatible auction-protocols that allows us to prove lower bounds against randomized auction-protocols. In particular, our lower bounds give the first approximation-resistant, exponential separation between communication complexity of incentivizing vs implementing a Bayesian incentive compatible social choice rule, settling an open question of Fadel and Segal [Fadel and Segal 2009].

scholarly journals New Results on Degree Sequences of Uniform Hypergraphs

10.37236/3414 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sarah Behrens ◽  
Catherine Erbes ◽  
Michael Ferrara ◽  
Stephen G. Hartke ◽  
Benjamin Reiniger ◽  
...  
Keyword(s):  
Efficient Algorithm ◽  
Necessary Conditions ◽  
Degree Sequence ◽  
Sufficient Conditions ◽  
Degree Sequences ◽  
Uniform Hypergraph ◽  
Graphic Sequence ◽  
Uniform Hypergraphs ◽  
Open Question

A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

Entropy of a Convolution Operator

2004 ◽  
Vol 11 (01) ◽  
pp. 79-85 ◽  
Author(s):  
Aleksander Urbański
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Absolutely Continuous ◽  
Doubly Stochastic ◽  
Doubly Stochastic Operator ◽  
Zero Entropy

The concept of the entropy of a doubly stochastic operator was introduced in 1999 by Ghys, Langevin, and Walczak. The idea was developed further by Kamiński and de Sam Lazaro, who also conjectured that the entropy of a convolution operator determined by a probability measure on a compact abelian group is equal to zero. We prove that this is true when the group is connected and the convolution operator is determined by a measure absolutely continuous with respect to the normalized Haar measure. Our result provides also a characterization of the set of doubly stochastic operators with non-zero entropy.

scholarly journals ON THE BEGINNING OF THE EARLY IRON AGE IN THE TERRITORY OF MIDDLE DON BASIN

2019 ◽  
Vol 31 (2) ◽  
pp. 92-101
Author(s):  
A. P. Medvedev ◽  
R. S. Berestnev
Keyword(s):  
Iron Age ◽  
Early Iron Age ◽  
Forest Steppe ◽  
Funerary Monuments ◽  
Metal Plates ◽  
Regional Specificity ◽  
Open Question ◽  
Cultural Genesis ◽  
The Relationship

The article is devoted to the characterization of pre-Scythian time monuments in the forest-steppe course of the Don. The authors come to the conclusion about the regional specificity of the process of cultural genesis in this territory at the beginning of the early Iron Age. The authors analyze the new treasure of Novocherkassk type. It was opened in 2016 in the Podgorensky district of the Voronezh region. This treasure includes psalms, hatchet, metal plates, bracelet-like rings, spearheads. In inventory, it is close to the pre-Scythian burials in the forest-steppe Ukraine (Butenki, Kvitki). Obviously, the population that left the treasure penetrated into the territory of the Middle Don region from the steppes between the Dnieper and Ciscaucasia — the place where the Cimmerian culture was formed in the 9th century. Objects close to the Proto-Meotian, Novocherkassk complexes, their diversity show this process. It remains an open question about the relationship in the studied region of the funerary monuments of Novocherkassk type and Middle-Don mounds of the Scythian time.

scholarly journals Discrete spectrum for amenable group actions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tao Yu ◽  
Guohua Zhang ◽  
Ruifeng Zhang
Keyword(s):  
Invariant Measure ◽  
Discrete Spectrum ◽  
Invariant Measures ◽  
Hamming Distance ◽  
Amenable Group ◽  
Group Actions ◽  
The Mean

<p style='text-indent:20px;'>In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions.</p><p style='text-indent:20px;'>We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.</p>

scholarly journals Upper paired domination versus upper domination

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Graph Classes ◽  
Paired Domination Number ◽  
Upper Domination Number ◽  
Paired Domination ◽  
Cactus Graphs ◽  
Open Question

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

Sequence Entropy and Mild Mixing

1992 ◽  
Vol 44 (1) ◽  
pp. 215-224 ◽  
Author(s):  
Qing Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Infinite Subset ◽  
Mixing Properties ◽  
Sequence Entropy ◽  
Number Of Authors

Entropy characterizations of different spectral and mixing properties of dynamical systems were dealt with by a number of authors (see [5], [6] and [8]).Given an infinite subset Γ = {tn}of N and a dynamical system (X, β,μ, T) one can define sequence entropy along for any finite Petition ξ, and hΓ(T) —supξ hΓ(T,ξ). In [6] Kushnirenko used sequence entropy to give a characterization of systems with discrete spectrum.

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