scholarly journals Star Bodies with Completely Symmetric Sections

2017 ◽  
Vol 2019 (10) ◽  
pp. 3015-3031 ◽  
Author(s):  
Sergii Myroshnychenko ◽  
Dmitry Ryabogin ◽  
Christos Saroglou
Keyword(s):  
Symmetry Group ◽  
General Theorem ◽  
Short Note ◽  
Euclidean Ball ◽  
Star Body ◽  
Open Problems ◽  
Open Questions ◽  
Almost All ◽  
Isotropic Functions ◽  
Special Case

Abstract We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from [12]. Our result is a consequence of a general theorem that we establish, stating that if the restrictions to almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements, and related open problems are discussed, and two additional open questions from [11] and [12] are answered.

Idempotent lifting and ring extensions

2016 ◽  
Vol 15 (06) ◽  
pp. 1650112 ◽  
Author(s):  
Alexander J. Diesl ◽  
Samuel J. Dittmer ◽  
Pace P. Nielsen
Keyword(s):  
Jacobson Radical ◽  
General Theorem ◽  
Positive Side ◽  
Open Questions ◽  
Morita Invariant ◽  
Ring Extensions ◽  
Special Case

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring [Formula: see text] for which idempotents lift modulo the Jacobson radical [Formula: see text], but idempotents do not lift modulo [Formula: see text]. Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if [Formula: see text] and [Formula: see text] are ideals of [Formula: see text] for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over [Formula: see text]. On the positive side, if [Formula: see text] and [Formula: see text] are enabling ideals in [Formula: see text], then [Formula: see text] is also an enabling ideal. We show that if [Formula: see text] is (weakly) enabling in [Formula: see text], then [Formula: see text] is not necessarily (weakly) enabling in [Formula: see text] while [Formula: see text] is (weakly) enabling in [Formula: see text]. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

THE RELATIONSHIP BETWEEN MAXIMUM ENTROPY SPECTRA AND MAXIMUM LIKELIHOOD SPECTRA

Geophysics ◽  
1972 ◽  
Vol 37 (2) ◽  
pp. 375-376 ◽  
Author(s):  
John Parker Burg
Keyword(s):  
Spectral Analysis ◽  
Maximum Likelihood ◽  
Maximum Entropy ◽  
Short Note ◽  
Entropy Method ◽  
Likelihood Method ◽  
Almost All ◽  
The Relationship ◽  
Exact Relationship ◽  
Special Case

In a long needed paper, R. T. Lacoss (1971) has presented many examples of spectra obtained by the maximum likelihood method and by the maximum entropy method and has shown that these newer techniques are in general superior to the more conventional spectral analysis methods. This short note shows that there exists a simple, exact relationship between maximum entropy spectra and maximum likelihood spectra when the correlation function is known at uniform intervals of lag. The data are of this form in almost all practical cases of time series analysis as well as in the special case of wavenumber spectral analysis of wave propagation as seen by a linear array of equally spaced sensors. The wavenumber case will be explicitly considered in this note since it requires the complex variable form of the theory.

Nonwovens: Global Trends in World Economy, European Foreign Trade, and Selected Case Studies from Poland and Asian Brics Countries

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Zofia Wysokińska ◽  
Tomasz Czajkowski ◽  
Katarzyna Grabowska
Keyword(s):  
Case Studies ◽  
World Economy ◽  
World Market ◽  
Global Trends ◽  
Textile Materials ◽  
Brics Countries ◽  
Recent Trends ◽  
Almost All ◽  
Special Case

AbstractNonwovens are one of the most versatile textile materials and have become increasingly popular in almost all sectors of the economy due to their low manufacturing costs and unique properties. In the next few years, the world market of nonwovens is predicted to grow by 7%–8% annually (International Nonwovens & Disposables Association [INDA], European Disposables and Nonwovens Association [EDANA], and Markets and Markets). This article aims to analyze the most recent trends in the global export and import of nonwovens, to present two case studies of Polish companies that produce them, and to present one special case study of the market of nonwoven geotextiles in China and India, which are the Asian transition economies among the BRICS countries (Brazil, Russia, India, China, and South Africa).

scholarly journals DASH-type cryptochromes – solved and open questions

2020 ◽  
Vol 401 (12) ◽  
pp. 1487-1493
Author(s):  
Stephan Kiontke ◽  
Tanja Göbel ◽  
Annika Brych ◽  
Alfred Batschauer
Keyword(s):  
Dna Repair ◽  
Circadian Clock ◽  
Blue Light ◽  
Dna Repair Enzymes ◽  
Repair Enzymes ◽  
Open Questions ◽  
Essential Components ◽  
Repair Activity ◽  
Almost All ◽  
Uv Lesions

AbstractDrosophila, Arabidopsis, Synechocystis, human (DASH)-type cryptochromes (cry-DASHs) form one subclade of the cryptochrome/photolyase family (CPF). CPF members are flavoproteins that act as DNA-repair enzymes (DNA-photolyases), or as ultraviolet(UV)-A/blue light photoreceptors (cryptochromes). In mammals, cryptochromes are essential components of the circadian clock feed-back loop. Cry-DASHs are present in almost all major taxa and were initially considered as photoreceptors. Later studies demonstrated DNA-repair activity that was, however, restricted to UV-lesions in single-stranded DNA. Very recent studies, particularly on microbial organisms, substantiated photoreceptor functions of cry-DASHs suggesting that they could be transitions between photolyases and cryptochromes.

scholarly journals PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS

2015 ◽  
Vol 100 (1) ◽  
pp. 33-41 ◽  
Author(s):  
FRANÇOIS BRUNAULT
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Recent Work ◽  
Short Note ◽  
Torsion Subgroup ◽  
Modular Functions ◽  
Open Questions ◽  
Modular Units

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

AN APPLICATION OF THE TOPOLOGICAL DEGREE TO GRAVITATIONAL LENSES

1998 ◽  
Vol 13 (02) ◽  
pp. 83-86 ◽  
Author(s):  
MARCO LOMBARDI
Keyword(s):  
Point Source ◽  
Mass Distribution ◽  
Topological Degree ◽  
General Theorem ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
General Proof ◽  
Special Case

In this letter we provide a new proof of a general theorem on gravitational lenses, first proven by Burke (1981) for the special case of thin lenses. The theorem states that a transparent gravitational lens with non-singular mass distribution produces an odd number of images of a point source. Our general proof shows that the topological degree finds natural and interesting applications in the theory of gravitational lenses.

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

Radiocarbon Dating and Egyptian Chronology—From the “Curve of Knowns” to Bayesian Modeling

2016 ◽  
Author(s):  
Felix Höflmayer
Keyword(s):  
Bayesian Modeling ◽  
Radiocarbon Dating ◽  
State Of The Art ◽  
Near Eastern ◽  
Open Questions ◽  
The World ◽  
Dating Method ◽  
To Come ◽  
Almost All ◽  
Egyptian Archaeology

Radiocarbon dating has become a standard dating method in archaeology almost all over the world. However, in the field of Egyptology and Near Eastern archaeology, the method is still not fully appreciated. Recent years have seen several major radiocarbon projects addressing Egyptian archaeology and chronology that have led to an intensified discussion regarding the application of radiocarbon dating within the field of Egyptology. This chapter reviews the contribution of radiocarbon dating to the discipline of Egyptology, discusses state-of-the-art applications and their impact on archaeological as well as chronological questions, and presents open questions that will be addressed in the years to come.

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