scholarly journals Extensions of the Heisenberg-Weyl inequality

1986 ◽  
Vol 9 (1) ◽  
pp. 185-192 ◽  
Author(s):  
H. P. Heinig ◽  
M. Smith
Keyword(s):  
Entropy Inequality ◽  
Optimal Form ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Weyl Inequality ◽  
Uncertainty Inequality ◽  
Weighted Entropy ◽  
Inequality Theorem

In this paper a number of generalizations of the classical Heisenberg-Weyl uncertainty inequality are given. We prove then-dimensional Hirschman entropy inequality (Theorem 2.1) from the optimal form of the Hausdorff-Young theorem and deduce a higher dimensional uncertainty inequality (Theorem 2.2). From a general weighted form of the Hausdorff-Young theorem, a one-dimensional weighted entropy inequality is proved and some weighted forms of the Heisenberg-Weyl inequalities are given.

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals A Fast Elitism Gaussian Estimation of Distribution Algorithm and Application for PID Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qingyang Xu ◽  
Chengjin Zhang ◽  
Li Zhang
Keyword(s):  
Probability Model ◽  
Learning Rule ◽  
Estimation Of Distribution Algorithm ◽  
Model Learning ◽  
One Dimensional ◽  
Fast Learning ◽  
Higher Dimensional ◽  
Intelligent Optimization Algorithm ◽  
Estimation Of Distribution ◽  
Gaussian Estimation

Estimation of distribution algorithm (EDA) is an intelligent optimization algorithm based on the probability statistics theory. A fast elitism Gaussian estimation of distribution algorithm (FEGEDA) is proposed in this paper. The Gaussian probability model is used to model the solution distribution. The parameters of Gaussian come from the statistical information of the best individuals by fast learning rule. A fast learning rule is used to enhance the efficiency of the algorithm, and an elitism strategy is used to maintain the convergent performance. The performances of the algorithm are examined based upon several benchmarks. In the simulations, a one-dimensional benchmark is used to visualize the optimization process and probability model learning process during the evolution, and several two-dimensional and higher dimensional benchmarks are used to testify the performance of FEGEDA. The experimental results indicate the capability of FEGEDA, especially in the higher dimensional problems, and the FEGEDA exhibits a better performance than some other algorithms and EDAs. Finally, FEGEDA is used in PID controller optimization of PMSM and compared with the classical-PID and GA.

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

scholarly journals Quasi-Periodic Bifurcations of Higher-Dimensional Tori

2016 ◽  
Vol 26 (07) ◽  
pp. 1630016 ◽  
Author(s):  
Motomasa Komuro ◽  
Kyohei Kamiyama ◽  
Tetsuro Endo ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Solutions ◽  
Phase Locking ◽  
Period Doubling ◽  
Double Covering ◽  
Heteroclinic Cycle ◽  
One Dimensional ◽  
Local Bifurcations ◽  
Saddle Node ◽  
Higher Dimensional ◽  
The Relationship

We classify the local bifurcations of quasi-periodic [Formula: see text]-dimensional tori in maps (abbr. MT[Formula: see text]) and in flows (abbr. FT[Formula: see text]) for [Formula: see text]. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT[Formula: see text] can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FT[Formula: see text] can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT[Formula: see text] and FT[Formula: see text]: namely, saddle-node cycle and heteroclinic cycle bifurcations of the [Formula: see text]-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST[Formula: see text]) and zero-dimensional tori in sections (abbr. ST[Formula: see text]). The bifurcations of ST[Formula: see text] can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST[Formula: see text] can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST[Formula: see text]/ST[Formula: see text] and the bifurcations of MT[Formula: see text]/FT[Formula: see text]. We present examples of all of these bifurcations.

FAULT TOLERANCE AND DETECTION IN CHAOTIC COMPUTERS

2007 ◽  
Vol 17 (06) ◽  
pp. 1955-1968 ◽  
Author(s):  
MOHAMMAD R. JAHED-MOTLAGH ◽  
BEHNAM KIA ◽  
WILLIAM L. DITTO ◽  
SUDESHNA SINHA
Keyword(s):  
Chaotic Systems ◽  
Chaotic Maps ◽  
Structural Testing ◽  
Testing Time ◽  
Computing Device ◽  
One Dimensional ◽  
Testing Method ◽  
Higher Dimensional ◽  
Parameter Variations ◽  
Logic Blocks

We introduce a structural testing method for a dynamics based computing device. Our scheme detects different physical defects, manifesting themselves as parameter variations in the chaotic system at the core of the logic blocks. Since this testing method exploits the dynamical properties of chaotic systems to detect damaged logic blocks, the damaged elements can be detected by very few testing inputs, leading to very low testing time. Further the method does not entail dedicated or extra hardware for testing. Specifically, we demonstrate the method on one-dimensional unimodal chaotic maps. Some ideas for testing higher dimensional maps and flows are also presented.

scholarly journals Dimensional Enhancement via Supersymmetry

2011 ◽  
Vol 2011 ◽  
pp. 1-45 ◽  
Author(s):  
M. G. Faux ◽  
K. M. Iga ◽  
G. D. Landweber
Keyword(s):  
Quantum Mechanics ◽  
Representation Theory ◽  
Gauge Invariance ◽  
Extra Dimensions ◽  
Field Theories ◽  
Consistency Test ◽  
One Dimensional ◽  
Supersymmetric Field Theories ◽  
Higher Dimensional ◽  
Supersymmetric Models

We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited, and utilized in this paper, which indicate which subset of one-dimensional supersymmetric models describes “shadows” of higher-dimensional models. This formalism delineates that minority of one-dimensional supersymmetric models which can “enhance” to accommodate extra dimensions. As a consistency test, we use our formalism to reproduce well-known conclusions about supersymmetric field theories using one-dimensional reasoning exclusively. And we introduce the notion of “phantoms” which usefully accommodate higher-dimensional gauge invariance in the context of shadow multiplets in supersymmetric quantum mechanics.

Coordination Polymers with a Bulky Perylene-Based Tetracarboxylate Ligand: Syntheses, Crystal Structures, and Luminescent Properties

2010 ◽  
Vol 63 (3) ◽  
pp. 463 ◽  
Author(s):  
Chun-Sen Liu ◽  
Min Hu ◽  
Song-Tao Ma ◽  
Qiang Zhang ◽  
Li-Ming Zhou ◽  
...  
Keyword(s):  
Coordination Polymers ◽  
Three Dimensional ◽  
Luminescent Properties ◽  
Supramolecular Interactions ◽  
Polymeric Chain ◽  
Stacking Interactions ◽  
One Dimensional ◽  
Π Stacking ◽  
Higher Dimensional ◽  
Interchain Interactions

To explore the coordination possibilities of perylene-based ligands with a larger conjugated π-system, four ZnII, MnII, and CoII coordination polymers with perylene-3,4,9,10-tetracarboxylate (ptc) and the chelating 1,10-phenanthroline (phen) ligands were synthesized and characterized: {[Zn2(ptc)(phen)2](H2O)10}∞ (1), {[Zn3(ptc)(OH)2(phen)2](H2O)3}∞ (2), {[Mn(ptc)0.5(phen)(H2O)2](H2O)1.5}∞ (3), and {[Co(ptc)0.5(phen)(H2O)2](H2O)2.5}∞ (4). Structural analysis reveals that complexes 1 and 2 both take one-dimensional polymeric chain structures with dinuclear and trinuclear units as nodes, respectively, which are further extended via the accessorial secondary interchain interactions, such as C–H···O H-bonding or aromatic π···π stacking interactions, to give rise to the relevant higher-dimensional frameworks. Compound 3 has a two-dimensional sheet structure that is further assembled to form a three-dimensional framework by interlayer π···π stacking interactions. Complex 4 is a one-dimensional ribbon-like array structure that is interlinked by the co-effects of intermolecular π···π stacking and C–H···π supramolecular interactions, resulting in a higher-dimensional framework from the different crystallographic directions. Moreover, complexes 1–4 exhibit strong solid-state luminescence emissions at room temperature, which mainly originate from intraligand π→π* transitions of ptc.

AN INTRODUCTION TO QUANTITATIVE POINCARÉ RECURRENCE IN DYNAMICAL SYSTEMS

2009 ◽  
Vol 21 (08) ◽  
pp. 949-979 ◽  
Author(s):  
BENOIT SAUSSOL
Keyword(s):  
Dynamical Systems ◽  
Thermodynamic Formalism ◽  
Dimension Theory ◽  
Recurrence Rates ◽  
One Dimensional ◽  
Expanding Map ◽  
System A ◽  
Higher Dimensional ◽  
Technical Details ◽  
Hyperbolic Behavior

We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

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