scholarly journals The last problem of Harald Bohr

1989 ◽  
Vol 47 (1) ◽  
pp. 133-152 ◽  
Author(s):  
Jean-Pierre Kahane
Keyword(s):  
Dirichlet Series ◽  
Negative Answer ◽  
Lindelöf Hypothesis ◽  
Open Question

AbstractThe first and last papers of Harald Bohr deal with ordinary Dirichlet series and their order (or Lindelöf) function μ(σ) (= inf{α;f(σ + it) + 0(|t|α)}). The Lindelöf hypothesis is μ(σ) = inf(0, ½ − t) when an = (−1)n. Are there ordinary Dirichlet series with −l < μ′ (σ) < 0 for some σ? A negative answer would imply Lindelöf's hypothesis. This is the last problem of Harald Bohr. This paper gives (1) a review on Bohr's results on ordinary Dinchlet series; (2) a review on results of the author and of Queffelec on “almost sure” and “quasi sure” properties of series with the solution of a previous problem of Bohr; (3) the following answer to the last problem: μ′(σ) can approach − ½, and necessarily μ(σ + μ(σ) + ½) = 0. The characterization of the order functions of ordinary Dirichlet series remains an open question.

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.

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

scholarly journals IDEAL PROJECTIONS AND FORCING PROJECTIONS

2014 ◽  
Vol 79 (4) ◽  
pp. 1247-1285 ◽  
Author(s):  
SEAN COX ◽  
MARTIN ZEMAN
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Normal Ideal ◽  
List Type ◽  
Type Number ◽  
Inner Model ◽  
Image Position ◽  
Open Question ◽  
Woodin Cardinal

AbstractIt is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1)If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;(2)For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

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 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.

Diametrically Maximal and Constant Width Sets in Banach Spaces

2006 ◽  
Vol 58 (4) ◽  
pp. 820-842 ◽  
Author(s):  
J. P. Moreno ◽  
P. L. Papini ◽  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Hausdorff Space ◽  
Constant Width ◽  
Negative Answer ◽  
Empty Interior ◽  
Finite Dimensional ◽  
Jung Constant

AbstractWe characterize diametrically maximal and constant width sets inC(K), whereKis any compact Hausdorff space. These results are applied to prove that the sum of two diametrically maximal sets needs not be diametrically maximal, thus solving a question raised in a paper by Groemer. A characterization of diametrically maximal sets inis also given, providing a negative answer to Groemer's problem in finite dimensional spaces. We characterize constant width sets inc0(I), for everyI, and then we establish the connections between the Jung constant of a Banach space and the existence of constant width sets with empty interior. Porosity properties of families of sets of constant width and rotundity properties of diametrically maximal sets are also investigated. Finally, we present some results concerning non-reflexive and Hilbert spaces.

scholarly journals When you must forget: Beyond strong persistence when forgetting in answer set programming

2017 ◽  
Vol 17 (5-6) ◽  
pp. 837-854
Author(s):  
RICARDO GONÇALVES ◽  
MATTHIAS KNORR ◽  
JOÃO LEITE ◽  
STEFAN WOLTRAN
Keyword(s):  
Computational Complexity ◽  
Answer Set Programming ◽  
Correct Result ◽  
Open Question ◽  
Shed Light ◽  
Answer Set

AbstractAmong the myriad of desirable properties discussed in the context of forgetting in Answer Set Programming, strong persistence naturally captures its essence. Recently, it has been shown that it is not always possible to forget a set of atoms from a program while obeying this property, and a precise criterion regarding what can be forgotten has been presented, accompanied by a class of forgetting operators that return the correct result when forgetting is possible. However, it is an open question what to do when we have to forget a set of atoms, but cannot without violating this property. In this paper, we address this issue and investigate three natural alternatives to forget when forgetting without violating strong persistence is not possible, which turn out to correspond to the different possible relaxations of the characterization of strong persistence. Additionally, we discuss their preferable usage, shed light on the relation between forgetting and notions of relativized equivalence established earlier in the context of Answer Set Programming, and present a detailed study on their computational complexity.

scholarly journals DENSITY-1-BOUNDING AND QUASIMINIMALITY IN THE GENERIC DEGREES

2017 ◽  
Vol 82 (3) ◽  
pp. 931-957 ◽  
Author(s):  
PETER CHOLAK ◽  
GREGORY IGUSA
Keyword(s):  
Positive Result ◽  
Wide Class ◽  
Minimal Pairs ◽  
Open Question

AbstractWe consider the question “Is every nonzero generic degree a density-1-bounding generic degree?” By previous results [8] either resolution of this question would answer an open question concerning the structure of the generic degrees: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees.We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide class of assumptions is sufficient to prove density-1-bounding.We also consider a historic difficulty in constructing a potential counterexample: By previous results [7] any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal.Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them.Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp [4], this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees.

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