scholarly journals Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus

2018 ◽  
Vol 38 (11) ◽  
pp. 5379-5387
Author(s):  
Pedro Duarte ◽  
◽  
Silvius Klein ◽  
Keyword(s):  
Dimensional Torus ◽  
Dominated Splitting ◽  
Higher Dimensional ◽  
Topological Obstructions

scholarly journals Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

2017 ◽  
Vol 2019 (13) ◽  
pp. 4004-4046
Author(s):  
Corey Bregman
Keyword(s):  
Unit Circle ◽  
Dimensional Torus ◽  
Torus Bundle ◽  
Generating Set ◽  
Growth Series ◽  
Torus Bundles ◽  
Almost Convex ◽  
Higher Dimensional ◽  
Finite Index Subgroup ◽  
Almost Convexity

AbstractGiven a matrix $A\in SL(N,\mathbb{Z})$, form the semidirect product $G=\mathbb{Z}^N\rtimes_A \mathbb{Z}$ where the $\mathbb{Z}$-factor acts on $\mathbb{Z}^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this article, we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.

scholarly journals ON THE TOPOLOGY OF T-DUALITY

2005 ◽  
Vol 17 (01) ◽  
pp. 77-112 ◽  
Author(s):  
ULRICH BUNKE ◽  
THOMAS SCHICK
Keyword(s):  
Homotopy Type ◽  
Cohomology Class ◽  
Dimensional Torus ◽  
Duality Transformation ◽  
Classifying Space ◽  
Simple Derivation ◽  
Torus Bundles ◽  
Twisted Cohomology ◽  
Higher Dimensional ◽  
Topological Version

We study a topological version of the T-duality relation between pairs consisting of a principal U(1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T-duality transformation. We give a simple derivation of a T-duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and discuss an example of iterated T-duality for higher-dimensional torus bundles.

scholarly journals Homological Percolation: The Formation of Giant k-Cycles

2020 ◽  
Author(s):  
Omer Bobrowski ◽  
Primoz Skraba
Keyword(s):  
Thermodynamic Limit ◽  
Exponential Decay ◽  
Algebraic Topology ◽  
Percolation Model ◽  
Critical Values ◽  
Giant Component ◽  
Connected Components ◽  
Continuum Percolation ◽  
Dimensional Torus ◽  
Higher Dimensional

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

Emergence of Spinless and Spin-1/2 Particles from Higher-Dimensional Gravity with D-Dimensional Torus as a Compact Manifold

1997 ◽  
Vol 12 (38) ◽  
pp. 2933-2942 ◽  
Author(s):  
S. K. Srivastava ◽  
K. P. Sinha
Keyword(s):  
High Energy ◽  
Physical Aspect ◽  
Dimensional Torus ◽  
Dimensional Theory ◽  
High Energy Level ◽  
Physical Universe ◽  
Higher Dimensional ◽  
Dimensional Gravity ◽  
Fermion Pairs ◽  
Geometrical Field

Using gravitational action for four-dimensional theory of R2-gravity, it has been shown earlier that, at high energy level, Ricci scalar R behaves like a physical field in addition to its usual nature as a geometrical field. The physical aspect of R is represented by spinless particles, called riccions. It is shown here that riccions can also be obtained from multi-dimensional R2-gravity. Further it is shown that these riccions disintegrate into fermion and anti-fermion pairs under certain conditions. Some physical properties of these fermions (here called riccinos) are discussed. On the basis of the results obtained here, one is tempted to speculate that our physical universe might have emerged through decay of riccions and riccinos.

scholarly journals A stability theorem for minimal foliations on a torus

1988 ◽  
Vol 8 (8) ◽  
pp. 251-281 ◽  
Keyword(s):  
Variational Problem ◽  
Minimal Surfaces ◽  
Invariant Tori ◽  
Elliptic Partial Differential Equations ◽  
Stability Theorem ◽  
Small Divisor ◽  
Dimensional Torus ◽  
Higher Dimensional ◽  
Minimal Foliations ◽  
The Stability

AbstractThis paper is concerned with minimal foliations; these are foliations whose leaves are extremals of a prescribed variational problem, as for example foliations consisting of minimal surfaces. Such a minimal foliation is called stable if for any small perturbation of the variational problem there exists a minimal foliation conjugate under a smooth diffeomorphism to the original foliation. In this paper the stability of special foliations of codimension 1 on a higher-dimensional torus is established. This result requires small divisor assumptions similar to those encountered in dynamical systems. This theorem can be viewed as a generalization of the perturbation theory of invariant tori for Hamiltonian systems to elliptic partial differential equations for which one obtains quasi-periodic solutions.

Spectroscopy-Based Mapping with Scanning Microwave Impedance Microscopy

2018 ◽  
Author(s):  
Peter De Wolf ◽  
Zhuangqun Huang ◽  
Bede Pittenger
Keyword(s):  
Single Point ◽  
Electrical Characterization ◽  
Principal Component ◽  
High Sensitivity ◽  
Data Cube ◽  
Nanometer Scale ◽  
Learning Approaches ◽  
Data Set ◽  
3D Data ◽  
Higher Dimensional

Abstract Methods are available to measure conductivity, charge, surface potential, carrier density, piezo-electric and other electrical properties with nanometer scale resolution. One of these methods, scanning microwave impedance microscopy (sMIM), has gained interest due to its capability to measure the full impedance (capacitance and resistive part) with high sensitivity and high spatial resolution. This paper introduces a novel data-cube approach that combines sMIM imaging and sMIM point spectroscopy, producing an integrated and complete 3D data set. This approach replaces the subjective approach of guessing locations of interest (for single point spectroscopy) with a big data approach resulting in higher dimensional data that can be sliced along any axis or plane and is conducive to principal component analysis or other machine learning approaches to data reduction. The data-cube approach is also applicable to other AFM-based electrical characterization modes.

HIGHER DIMENSIONAL ROBERTSON WALKER UNIVERSES INTERACTING WITH BRANS-DICKE FIELD

2020 ◽  
Vol 9 (10) ◽  
pp. 8545-8557
Author(s):  
K. P. Singh ◽  
T. A. Singh ◽  
M. Daimary
Keyword(s):  
Higher Dimensional

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

Sign in / Sign up

Export Citation Format

Share Document