Smooth Mappings with Higher Dimensional Critical Sets

2010 ◽  
Vol 53 (3) ◽  
pp. 542-549 ◽  
Author(s):  
Cornel Pintea
Keyword(s):  
Lower Bounds ◽  
High Dimensional ◽  
Higher Dimensional ◽  
Critical Sets

AbstractIn this paper we provide lower bounds for the dimension of various critical sets, and we point out some differential maps with high dimensional critical sets.

EXPERIMENTAL MODIFICATIONS OF VOA-BASED AUTONOMOUS AND NONAUTONOMOUS CHUA'S CIRCUITS FOR HIGHER DIMENSIONAL OPERATION

2006 ◽  
Vol 16 (09) ◽  
pp. 2649-2658
Author(s):  
RECAI KILIÇ
Keyword(s):  
Experimental Method ◽  
Operational Amplifier ◽  
Experimental Results ◽  
High Dimensional ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Voltage Mode ◽  
Pspice Simulation ◽  
Chua Circuit ◽  
Higher Dimensional

In order to operate in higher dimensional form of autonomous and nonautonomous Chua's circuits keeping their original chaotic behaviors, we have experimentally modified VOA (Voltage Mode Operational Amplifier)-based autonomous Chua's circuit and nonautonomous MLC [Murali–Lakshmanan–Chua] circuit by using a simple experimental method. After introducing this experimental method, we will present PSpice simulation and experimental results of modified high dimensional autonomous and nonautonomous Chua's circuits.

TIME LOWER BOUNDS FOR PERMUTATION ROUTING ON MULTI-DIMENSIONAL BUSED MESHES

1993 ◽  
Vol 03 (02) ◽  
pp. 129-138
Author(s):  
STEVEN CHEUNG ◽  
FRANCIS C.M. LAU
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Routing Problem ◽  
Higher Dimensional ◽  
Low Dimensional ◽  
Permutation Routing

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.

scholarly journals On the High Dimentional Information Processing in Quaternionic Domain and its Applications

2018 ◽  
Vol 7 (2) ◽  
pp. 177
Author(s):  
Sushil Kumar ◽  
Bipin Kumar Tripathi
Keyword(s):  
Wide Spectrum ◽  
Neural System ◽  
High Dimensional ◽  
Generalization Capability ◽  
Phase Information ◽  
High Dimension Data ◽  
Higher Dimensional ◽  
Dimensional Parameters ◽  
Learning Machine ◽  
Communication Control

<p>There are various high dimensional engineering and scientific applications in communication, control, robotics, computer vision, biometrics, etc.; where researchers are facing problem to design an intelligent and robust neural system which can process higher dimensional information efficiently. The conventional real-valued neural networks are tried to solve the problem associated with high dimensional parameters, but the required network structure possesses high complexity and are very time consuming and weak to noise. These networks are also not able to learn magnitude and phase values simultaneously in space.<strong> </strong> The quaternion is the number, which possesses the magnitude in all four directions and phase information is embedded within it. This paper presents a well generalized learning machine with a quaternionic domain neural network that can finely process magnitude and phase information of high dimension data without any hassle. The learning and generalization capability of the proposed learning machine is presented through a wide spectrum of simulations which demonstrate the significance of the work.</p>

scholarly journals LS-category of moment-angle manifolds and higher order Massey products

2021 ◽  
Vol 33 (5) ◽  
pp. 1179-1205
Author(s):  
Piotr Beben ◽  
Jelena Grbić
Keyword(s):  
Simplicial Complex ◽  
Structural Properties ◽  
Lower Bounds ◽  
Higher Order ◽  
Hyperbolic Manifolds ◽  
Upper And Lower Bounds ◽  
Connected Sum ◽  
Massey Products ◽  
Higher Dimensional ◽  
Lusternik Schnirelmann

Abstract Using the combinatorics of the underlying simplicial complex K, we give various upper and lower bounds for the Lusternik–Schnirelmann (LS) category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} . We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS-category. In particular, we characterize the LS-category of moment-angle complexes 𝒵 K {\mathcal{Z}_{K}} over triangulated d-manifolds K for d ≤ 2 {d\leq 2} , as well as higher-dimensional spheres built up via connected sum, join, and vertex doubling operations. We show that the LS-category closely relates to vanishing of Massey products in H * ⁢ ( 𝒵 K ) {H^{*}(\mathcal{Z}_{K})} , and through this connection we describe first structural properties of Massey products in moment-angle manifolds. Some of the further applications include calculations of the LS-category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and k-neighborly complexes, which double as important examples of hyperbolic manifolds.

scholarly journals Spaces of positive intermediate curvature metrics

2021 ◽  
Author(s):  
Georg Frenck ◽  
Jan-Bernhard Kordaß
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Riemannian Metrics ◽  
High Dimensional ◽  
Homotopy Groups ◽  
Positive Ricci Curvature ◽  
Spaces Of Metrics

AbstractIn this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional $$\mathrm {Spin}$$ Spin -manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

Gradient-like flows on high-dimensional manifolds

1999 ◽  
Vol 19 (2) ◽  
pp. 339-362 ◽  
Author(s):  
R. N. CRUZ ◽  
K. A. DE REZENDE
Keyword(s):  
Level Sets ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
High Dimensional ◽  
Topological Information ◽  
Critical Set ◽  
Critical Sets ◽  
Lyapunov Graph

The main purpose of this paper is to study the implications that the homology index of critical sets of smooth flows on closed manifolds M have on both the homology of level sets of M and the homology of M itself. The bookkeeping of the data containing the critical set information of the flow and topological information of M is done through the use of Lyapunov graphs. Our main result characterizes the necessary conditions that a Lyapunov graph must possess in order to be associated to a Morse–Smale flow. With additional restrictions on an abstract Lyapunov graph L we determine sufficient conditions for L to be associated to a flow on M.

scholarly journals Grid Minors in Damaged Grids

10.37236/3872 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
David Eppstein
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Grid Graph ◽  
Square Grid ◽  
Higher Dimensional

We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.

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