A lower bound estimation for the Ricci curvature of a hypersurface in a hyperbolic space

2009 ◽  
Vol 46 (4) ◽  
pp. 539-546
Author(s):  
Mehmet Erdoğan
Keyword(s):  
Maximum Principle ◽  
Lower Bound ◽  
Hyperbolic Space ◽  
Ricci Curvature ◽  
Upper Bound ◽  
The Maximum Principle ◽  
Bound Estimation

In some previous papers the author gave an upper bound estimation for the Ricci curvature of a hypersurface in a hyperbolic space and in a sphere, see [4] and [5]. In the present paper, we give a lower bound estimation for the Ricci curvature of a compact connected embedded hypersurface in a hyperbolic space via the maximum principle given by H. Omori in [11].

scholarly journals Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

2019 ◽  
Author(s):  
Debora Impera ◽  
Michele Rimoldi ◽  
Giona Veronelli
Keyword(s):  
Maximum Principle ◽  
Ricci Curvature ◽  
Sobolev Spaces ◽  
Upper Bound ◽  
Ricci Tensor ◽  
Injectivity Radius ◽  
Quadratic Growth ◽  
Riemann Curvature ◽  
Main Application ◽  
Compactly Supported Functions

Abstract We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

CONSTRUCTIVE ESTIMATION OF APPROXIMATION FOR TRIGONOMETRIC NEURAL NETWORKS

2012 ◽  
Vol 10 (03) ◽  
pp. 1250021
Author(s):  
JIANJUN WANG ◽  
WEIHUA XU ◽  
BIN ZOU
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Lower Bound ◽  
Upper Bound ◽  
Intrinsic Property ◽  
Order Of Approximation ◽  
Integrable Functions ◽  
Hidden Layer ◽  
Hidden Neurons ◽  
Bound Estimation

For the three-layer artificial neural networks with trigonometric weights coefficients, the upper bound and lower bound of approximating 2π-periodic pth-order Lebesgue integrable functions [Formula: see text] are obtained in this paper. Theorems we obtained provide explicit equational representations of these approximating networks, the specification for their numbers of hidden-layer units, the lower bound estimation of approximation, and the essential order of approximation. The obtained results not only characterize the intrinsic property of approximation of neural networks, but also uncover the implicit relationship between the precision (speed) and the number of hidden neurons of neural networks.

Pontryagin's Maximum Principle Applied to the Profile of a Beam

1967 ◽  
Vol 71 (679) ◽  
pp. 513-515 ◽  
Author(s):  
L. C. W. Dixon
Keyword(s):  
Maximum Principle ◽  
Lower Bound ◽  
Modulus Of Elasticity ◽  
Variable Density ◽  
Structural Member ◽  
Constant Density ◽  
Control Vector ◽  
The Maximum Principle ◽  
Centre Line ◽  
Independent Variable

Every structural member deforms under the forces acting upon it. In this paper the Maximum Principle, recently developed by Pontryagin is applied to the problem of determining that profile of a beam that deforms least under its own weight. The beam considered is a solid horizontal cantilever having constant density, modulus of elasticity and width. The height is taken as an independent variable subject to the restrictions that it lies between an upper and lower bound and is symmetrically distributed about the centre-line. The method is attractive in that it can readily be extended to take account of variable density, width and/or hollow members by extending the control vector u defined in the analysis.

scholarly journals An estimate of sectional curvatures of hypersurfaces with positive Ricci curvatures

1995 ◽  
Vol 38 (1) ◽  
pp. 167-170
Author(s):  
Ju Seon Kim ◽  
Sang Og Kim
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Upper Bound ◽  
Mean Curvature ◽  
Nonnegative Constant ◽  
Sectional Curvatures ◽  
Bounded Below

Let M be a hypersurface in Euclidean space and let the Ricci curvature of M be bounded below by some nonnegative constant. In this paper, we estimate the sectional curvature of M in terms of the lower bound of Ricci curvature and the upper bound of mean curvature.

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

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