scholarly journals Reflexive Polytopes of Higher Index and the Number 12

10.37236/2366 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Alexander M. Kasprzyk ◽  
Benjamin Nill
Keyword(s):  
Infinite Class ◽  
Dual Pairs ◽  
Isomorphism Classes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Reflexive Polytopes ◽  
Natural Generalisation ◽  
Lattice Polygons ◽  
Reflexive Polytope

We introduce reflexive polytopes of index $l$ as a natural generalisation of the notion of a reflexive polytope of index $1$. These $l$-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of $l$-reflexive polygons up to index $200$. As another application, we show that any reflexive polygon of arbitrary index satisfies the famous "number $12$" property. This is a new, infinite class of lattice polygons possessing this property, and extends the previously known sixteen instances. The number $12$ property also holds more generally for $l$-reflexive non-convex or self-intersecting polygonal loops. We conclude by discussing higher-dimensional examples and open questions.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

STRONG COUPLING FROM TAU LEPTON DECAYS

2013 ◽  
Vol 28 (24) ◽  
pp. 1360006 ◽  
Author(s):  
MATTHIAS JAMIN
Keyword(s):  
Perturbative Qcd ◽  
Strong Coupling ◽  
Theoretical Description ◽  
Operator Product ◽  
Hadronic Decays ◽  
Open Questions ◽  
Higher Dimensional ◽  
Low Energies ◽  
Consistent Manner ◽  
Lepton Decays

A determination of the strong coupling αs at rather low energies is possible through the analysis of hadronic decays of the τ lepton. In turn, the low energy necessitates sufficient control over perturbative QCD corrections, the nonperturbative condensate contributions in the framework of the operator product expansion (OPE), as well as corrections going beyond the OPE, the duality violations (DVs). Perturbative QCD uncertainties arise from open questions regarding the renormalization group resummation of the series. The fit quantities are moment integrals of the τ spectral function data in a certain energy window and care should be taken to have good perturbative behavior of those moments as well as control over higher-dimensional operator corrections. Furthermore, all parameters occurring in the theoretical description should be extracted from fits to the data in a self-consistent manner.

scholarly journals Directed Rooted Forests in Higher Dimension

10.37236/5819 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Olivier Bernardi ◽  
Caroline J. Klivans
Keyword(s):  
Generating Function ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Graph Laplacian ◽  
Connected Components ◽  
Higher Dimension ◽  
Cell Complexes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Rooted Forest

For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.

scholarly journals STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MARCUS DE CHIFFRE ◽  
LEV GLEBSKY ◽  
ALEXANDER LUBOTZKY ◽  
ANDREAS THOM
Keyword(s):  
Symmetric Groups ◽  
Unitary Groups ◽  
Frobenius Norm ◽  
Central Extensions ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Open Questions ◽  
Higher Dimensional ◽  
First Time ◽  
Finitely Presented

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$ , the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$ . Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $p$ -adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

scholarly journals Ehrhart Polynomial Roots of Reflexive Polytopes

10.37236/7780 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Gábor Hegedüs ◽  
Akihiro Higashitani ◽  
Alexander Kasprzyk
Keyword(s):  
Recent Work ◽  
Ehrhart Polynomial ◽  
Polynomial Roots ◽  
Lattice Polytope ◽  
Reflexive Polytopes ◽  
Half Strip ◽  
Reflexive Polytope

Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension $34$.

Non-periodic Space Structures

1987 ◽  
Vol 2 (2) ◽  
pp. 93-108 ◽  
Author(s):  
Haresh Lalvani
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Structures ◽  
Infinite Class ◽  
Space Filling ◽  
Higher Dimensional ◽  
Centrally Symmetric ◽  
Interesting Class ◽  
Periodic Space ◽  
Three Dimensional Space

An interesting class of two- and three-dimensional space structures can be derived from projections of higher-dimensional structures. Regular polygons and regular-faced polyhedra provide the geometry of families of n-stars from which two- and three-dimensional projections of n-dimensional grids can be derived. These projections are rhombic space grids composed of all-space filling rhombi and rhombohedra with edges parallel to n directions. An infinite class of single-, double- and multi-layered grids can be derived from n-sided polygons and prisms, and a finite class of multi-directional grids from the polyhedral symmetry groups. The grids can be periodic, centrally symmetric or non-periodic, and act as skeletons to generate corresponding classes of space-filling, packings and labyrinths.

scholarly journals Bifurcation currents and equidistribution in parameter space

2014 ◽  
Author(s):  
Romain Dujardin
Keyword(s):  
Dynamical Systems ◽  
Parameter Space ◽  
Rational Maps ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Open Questions ◽  
Higher Dimensional ◽  
Möbius Group ◽  
Positive Currents

This chapter reviews the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P¹ or subgroups of the Möbius group PSL(2,C)). The topics covered include the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties of the parameter space. Emphasis will be placed as much as possible on the similarities between methods of higher-dimensional dynamics, of the study of families of rational maps, and that of Möbius subgroups. The chapter also states a number of open questions to foster further developments of this theory.

ON ISOMORPHISM CLASSES OF COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

2019 ◽  
Vol 85 (1) ◽  
pp. 61-86 ◽  
Author(s):  
URI ANDREWS ◽  
SERIKZHAN A. BADAEV
Keyword(s):  
Explicit Construction ◽  
The Other ◽  
Equivalence Relations ◽  
Index Set ◽  
Isomorphism Classes ◽  
Closed Sets ◽  
Open Questions ◽  
Index Sets ◽  
Image Position ◽  
Computably Enumerable

AbstractWe examine how degrees of computably enumerable equivalence relations (ceers) under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots ,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left( {{E_i}} \right)$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers.

What’s Next?

2011 ◽  
Vol 23 (1) ◽  
pp. 60-63 ◽  
Author(s):  
Peter Vorderer
Keyword(s):  
Cultural Differences ◽  
Factor Model ◽  
New Developments ◽  
Theoretical Concepts ◽  
Open Questions ◽  
Entertainment Products ◽  
Entertainment Theory

This paper points to new developments in the context of entertainment theory. Starting from a background of well-established theories that have been proposed and elaborated mainly by Zillmann and his collaborators since the 1980s, a new two-factor model of entertainment is introduced. This model encompasses “enjoyment” and “appreciation” as two independent factors. In addition, several open questions regarding cultural differences in humans’ responses to entertainment products or the usefulness of various theoretical concepts like “presence,” “identification,” or “transportation” are also discussed. Finally, the question of why media users are seeking entertainment is brought to the forefront, and a possibly relevant need such as the “search for meaningfulness” is mentioned as a possible major candidate for such an explanation.

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