scholarly journals Conformal scalar curvature equation on Sn: Functions with two close critical points (Twin Pseudo-Peaks)

2018 ◽  
Vol 20 (05) ◽  
pp. 1750051
Author(s):  
Man Chun Leung ◽  
Feng Zhou
Keyword(s):  
Scalar Curvature ◽  
Critical Points ◽  
Reduction Method ◽  
Existence Results ◽  
The Other ◽  
Curvature Equation ◽  
Two Bubbles

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on [Formula: see text] ([Formula: see text]) when the prescribed function (after being projected to [Formula: see text]) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness [Formula: see text]), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.

CONSTRUCTION OF BLOW-UP SEQUENCES FOR THE PRESCRIBED SCALAR CURVATURE EQUATION ON Sn I: UNIFORM CANCELLATION

2012 ◽  
Vol 14 (02) ◽  
pp. 1250008 ◽  
Author(s):  
MAN CHUN LEUNG
Keyword(s):  
Scalar Curvature ◽  
Positive Solutions ◽  
Infinite Number ◽  
Reduction Method ◽  
Blow Up ◽  
Curvature Function ◽  
Cancellation Property ◽  
First Order ◽  
Curvature Equation ◽  
Derivatives Of

For n ≥ 6, using the Lyapunov–Schmidt reduction method, we describe how to construct (scalar curvature) functions on Sn, so that each of them enables the conformal scalar curvature equation to have an infinite number of positive solutions, which form a blow-up sequence. The prescribed scalar curvature function is shown to have Cn - 1,β smoothness. We present the argument in two parts. In this first part, we discuss the uniform cancellation property in the Lyapunov–Schmidt reduction method for the scalar curvature equation. We also explore relation between the Kazdan–Warner condition and the first-order derivatives of the reduced functional, and symmetry in the second-order derivatives of the reduced functional.

scholarly journals Prescribing the Scalar Curvature under Minimal Boundary Conditions on the Half Sphere

2002 ◽  
Vol 2 (2) ◽  
Author(s):  
Mohamed Ben Ayed ◽  
Khalil El Mehdi ◽  
Mohameden Ould Ahmedou
Keyword(s):  
Boundary Conditions ◽  
Variational Problem ◽  
Scalar Curvature ◽  
Critical Points ◽  
Existence Results ◽  
Topological Methods ◽  
Critical Points At Infinity

AbstractThis paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

scholarly journals Existence and Multiplicity Results for the Scalar Curvature Problem on the Half-Sphere 𝕊+3

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ridha Yacoub
Keyword(s):  
Scalar Curvature ◽  
Critical Points ◽  
Mean Curvature ◽  
Existence Results ◽  
Curvature Condition ◽  
Multiplicity Results ◽  
3 Dimensional ◽  
Existence And Multiplicity ◽  
Curvature Problem ◽  
Boundary Mean Curvature

In this paper we deal with the scalar curvature problem under minimal boundary mean curvature condition on the standard 3-dimensional half-sphere. Using tools related to the theory of critical points at infinity, we give existence results under perturbative and nonperturbative hypothesis, and with the help of some “Morse inequalities at infinity”, we provide multiplicity results for our problem.

scholarly journals On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 161-191
Author(s):  
Otis Chodosh ◽  
Michael Eichmair
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Existence Results ◽  
Asymptotically Flat ◽  
Weingarten Surfaces ◽  
Differential Geom

AbstractWe extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.

scholarly journals An efficient stock market prediction model using hybrid feature reduction method based on variational autoencoders and recursive feature elimination

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Hakan Gunduz
Keyword(s):  
Reduction Method ◽  
Feature Reduction ◽  
The Other ◽  
Recursive Feature Elimination ◽  
Accuracy Rate ◽  
Stock Performance ◽  
Attention Model ◽  
Feature Sets ◽  
Accuracy Rates ◽  
F Measure

AbstractIn this study, the hourly directions of eight banking stocks in Borsa Istanbul were predicted using linear-based, deep-learning (LSTM) and ensemble learning (LightGBM) models. These models were trained with four different feature sets and their performances were evaluated in terms of accuracy and F-measure metrics. While the first experiments directly used the own stock features as the model inputs, the second experiments utilized reduced stock features through Variational AutoEncoders (VAE). In the last experiments, in order to grasp the effects of the other banking stocks on individual stock performance, the features belonging to other stocks were also given as inputs to our models. While combining other stock features was done for both own (named as allstock_own) and VAE-reduced (named as allstock_VAE) stock features, the expanded dimensions of the feature sets were reduced by Recursive Feature Elimination. As the highest success rate increased up to 0.685 with allstock_own and LSTM with attention model, the combination of allstock_VAE and LSTM with the attention model obtained an accuracy rate of 0.675. Although the classification results achieved with both feature types was close, allstock_VAE achieved these results using nearly 16.67% less features compared to allstock_own. When all experimental results were examined, it was found out that the models trained with allstock_own and allstock_VAE achieved higher accuracy rates than those using individual stock features. It was also concluded that the results obtained with the VAE-reduced stock features were similar to those obtained by own stock features.

scholarly journals Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping

2020 ◽  
pp. 56-71
Author(s):  
Jenica Cringanu
Keyword(s):  
Banach Space ◽  
Variational Method ◽  
Critical Points ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Existence Results ◽  
Duality Mapping ◽  
Topological Methods ◽  
Carathéodory Function ◽  
Direct Variational Method

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 < q < p∗, p∗ is the Sobolev conjugate exponent.Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.

scholarly journals A basic inequality for submanifolds in a cosymplectic space form

2003 ◽  
Vol 2003 (9) ◽  
pp. 539-547 ◽  
Author(s):  
Jeong-Sik Kim ◽  
Jaedong Choi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
The Other ◽  
Space Forms ◽  
Basic Inequality

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely, its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely, squared mean curvature on the other side. Some applications, including inequalities between the intrinsic invariantδMand the squared mean curvature, are given. The equality cases are also discussed.

scholarly journals Reduction method of residual voltage by connecting lead length of surge protection device

2018 ◽  
Vol 7 (3.3) ◽  
pp. 384
Author(s):  
Young Dal Kim ◽  
Young Chan Kim ◽  
Yun Mi Jeong ◽  
Dae Dong Lee
Keyword(s):  
Reduction Method ◽  
The Other ◽  
Installation Space ◽  
Lead Wire ◽  
Protection Device ◽  
Design Technique ◽  
Wire Length ◽  
Residual Voltage ◽  
Protection Level ◽  
Surge Protection

Background/Objectives: In order to minimize the damage and malfunction of the equipment and system from various surges, we studied the method of reducing the residual voltage according to the lead wire length of the surge protector.Methods/Statistical analysis: In buildings, SPD installation space is insufficient or narrow, resulting in longer lead wire of SPD, and SPD protection performance is decreased due to increase of voltage protection level and residual voltage. In this study, the voltage protection level and the residual voltage of the conventional SPD model and the proposed SPD model are analyzed according to the change of the connecting conductor length from 0.5to 100m.Findings: In the case of the conventional SPD model, the protection level of the SPD is excellent by measuring the voltage protection level at 1,410V even if the lead wire length of the connecting conductor is changed to 10m, but when it exceeds 10m, the protection performance and the protection cooperation are reduced. On the other hand, in the case of the proposed SPD model, the voltage protection level was measured to be 50 V or less even if the lead wire length of the connecting conductor was changed to100 m. Therefore, it is considered that SPD protection performance and protection cooperation are excellent.Improvements/Applications: The design technique of SPD obtained through this study will help to select the optimal installation site and reduce the budget.  

scholarly journals Capacity, quasi-local mass, and singular fill-ins

2020 ◽  
Vol 2020 (768) ◽  
pp. 55-92 ◽  
Author(s):  
Christos Mantoulidis ◽  
Pengzi Miao ◽  
Luen-Fai Tam
Keyword(s):  
Scalar Curvature ◽  
The Other ◽  
Independent Interest ◽  
Asymptotically Flat ◽  
Local Mass ◽  
Singular Metrics ◽  
Comparison Results ◽  
Mean Convex ◽  
New Inequalities

AbstractWe derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown–York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.

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