The Identification of Convex Function on Riemannian Manifold

The necessary and sufficient condition of convex function is significant in nonlinear convex programming. This paper presents the identification of convex function on Riemannian manifold by use of Penot generalized directional derivative and the Clarke generalized gradient. This paper also presents a method for judging whether a point is the global minimum point in the inequality constraints. Our objective here is to extend the content and proof the necessary and sufficient condition of convex function to Riemannian manifolds.

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Existence of solutions for a class of variational-hemivariational-like inequalities in Banach spaces

Filomat ◽

10.2298/fil1810609c ◽

2018 ◽

Vol 32 (10) ◽

pp. 3609-3622

Author(s):

Lu-Chuan Ceng ◽

Jen-Chih Yao ◽

Yonghong Yao

Keyword(s):

Banach Spaces ◽

Existence Of Solutions ◽

Directional Derivative ◽

Existence Results ◽

Solution Set ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Generalized Gradient ◽

Reflexive Banach Spaces ◽

Necessary And Sufficient

This paper is devoted to study the existence of solutions for a class of variational-hemivariationallike inequalities in reflexive Banach spaces. Using the notion of the stable (?,?)-quasimonotonicity, the properties of Clarke?s generalized directional derivative and Clarke?s generalized gradient, we establish some existence results of solutions when the constrained set is nonempty, bounded (or unbounded), closed and convex. Moreover, a sufficient condition to the boundedness of the solution set and a necessary and sufficient condition to the existence of solutions are also derived.

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On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽

10.3390/sym12121941 ◽

2020 ◽

Vol 12 (12) ◽

pp. 1941

Author(s):

Sharief Deshmukh ◽

Nasser Bin Turki ◽

Haila Alodan

Keyword(s):

Riemannian Manifold ◽

Vector Field ◽

Euclidean Space ◽

Riemannian Manifolds ◽

Vector Fields ◽

Compact Riemannian Manifold ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Rigidity Results ◽

Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

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A Characterization of Invariant Affine Connections

Nagoya Mathematical Journal ◽

10.1017/s0027763000007534 ◽

1960 ◽

Vol 16 ◽

pp. 35-50 ◽

Cited By ~ 12

Author(s):

Bertram Kostant

Keyword(s):

Riemannian Manifold ◽

Simple Proof ◽

General Theorem ◽

Transitive Group ◽

Complete Riemannian Manifold ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

Simply Connected

1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the Tx of [1] is just the ax of [6] when X is restricted to p0, see [6], p. 539).

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TOTALLY GEODESIC SUBMANIFOLDS OF GL-MANIFOLDS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500028 ◽

2012 ◽

Vol 09 (01) ◽

pp. 1250002

Author(s):

ABOLGHASEM LALEH ◽

MORTEZA M. REZAII ◽

ATAABAK BAAGHERZADEH HUSHMANDI

Keyword(s):

Riemannian Manifold ◽

Finsler Manifold ◽

Finsler Metric ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Totally Geodesic ◽

Geodesic Submanifolds ◽

Totally Geodesic Submanifolds ◽

Necessary And Sufficient ◽

Sasakian Metric

In this paper, for a Finsler manifold (M, F) with a Finsler metric gij(x, y) we shall consider a generalized Lagrange metrics (FGL-metrics) as the form *gij(x, y) = gij(x, y) + σ(x, y)Bi(x, y)Bj(x, y) on TM. Then we shall consider a Riemannian manifold (TM, *G) in which *G is a generalized Sasakian metric of *g on [Formula: see text]. Then we restrict the above FGL-metrics to a submanifold of [Formula: see text], and show that it admits a GL-metric structure. Then we shall find a necessary and sufficient condition for this submanifold to be totally geodesic.

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Dispersive ordering by dilation

Journal of Applied Probability ◽

10.1017/s0021900200038900 ◽

1990 ◽

Vol 27 (02) ◽

pp. 440-444 ◽

Cited By ~ 1

Author(s):

J. Muñoz-Perez ◽

A. Sanchez-Gomez

Keyword(s):

Distribution Function ◽

Convex Function ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Dispersive Ordering ◽

Necessary And Sufficient

In this paper a necessary and sufficient condition for the dispersive ordering in dilation sense is given by a convex function which is called the dispersive function and characterizes the distribution function. Some interesting properties of the ordering follow from this result.

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Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-202453 ◽

2021 ◽

pp. 1-16

Author(s):

Sunae Pak ◽

Huichol Choe ◽

Kinam Sin ◽

Sunghyok Kwon

Keyword(s):

Initial Value Problem ◽

Existence Of Solutions ◽

Sufficient Conditions ◽

Inequality Constraints ◽

Necessary And Sufficient Conditions ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Initial Value ◽

Solution Representation ◽

Necessary And Sufficient

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

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On the ME-manifold inn-*g-UFT and its conformal change

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171294000128 ◽

1994 ◽

Vol 17 (1) ◽

pp. 79-90

Author(s):

Kyung Tae Chung ◽

Gwang Sik Eun

Keyword(s):

Riemannian Manifold ◽

Geometric Structure ◽

Tensor Field ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Unique Existence ◽

Conformal Change ◽

Necessary And Sufficient

An Einstein's connection which takes the form (3.1) is called an ME-connection. A generalizedn-dimensional Riemannian manifoldXnon which the differential geometric structure is imposed by a tensor field*gλνthrough a unique ME-connection subject to the conditions of Agreement (4.1) is called*g-ME-manifold and we denote it by*g-MEXn. The purpose of the present paper is to introduce this new concept of*g-MEXnand investigate its properties. In this paper, we first prove a necessary and sufficient condition for the unique existence of ME-connection inXn, and derive a surveyable tensorial representation of the ME-connection. In the second, we investigate the conformal change of*g-MEXnand present a useful tensorial representation of the conformal change of the ME-connection.

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On generalized concircularly recurrent manifolds

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.46.2009.2.1091 ◽

2009 ◽

Vol 46 (2) ◽

pp. 287-296

Author(s):

U. De ◽

A. Kalam Gazi

Keyword(s):

Riemannian Manifold ◽

Scalar Curvature ◽

Einstein Manifold ◽

Constant Scalar Curvature ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Recurrent Manifold ◽

New Type ◽

Necessary And Sufficient

In this paper we study a new type of Riemannian manifold called generalized concircularly recurrent manifold. We obtain a necessary and sufficient condition for the constant scalar curvature of such a manifold. Next we study Ricci symmetric generalized concircularly recurrent manifold and prove that such a manifold is an Einstein manifold. Finally, we obtain a sufficient condition for a generalized concircularly recurrent manifold to be a special quasi-Einstein manifold.

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Remarks on metallic warped product manifolds

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1802269b ◽

2018 ◽

Vol 33 (2) ◽

pp. 269

Author(s):

Adara-Monica Blaga ◽

Cristina-Elena Hretcanu

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Warped Product ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Metallic Structure ◽

Necessary And Sufficient

We characterize the metallic structure on the product of two metallic manifolds in terms of metallic maps and provide a necessary and sufficient condition for the warped product of two locally metallic Riemannian manifolds to be locally metallic. The particular case of product manifolds is discussed and an example of metallic warped product Riemannian manifold is provided.

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Characterizing approximate global minimizers of the difference of two abstract convex functions with applications

Filomat ◽

10.2298/fil1908431s ◽

2019 ◽

Vol 33 (8) ◽

pp. 2431-2445

Author(s):

A.R. Sattarzadeh ◽

H. Mohebi

Keyword(s):

Mild Condition ◽

Convex Functions ◽

Global Minimum ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Maximal Elements ◽

Global Minimizers ◽

Abstract Subdifferential ◽

The Difference ◽

Necessary And Sufficient

In this paper, we first investigate characterizations of maximal elements of abstract convex functions under a mild condition. Also, we give various characterizations for global "-minimum of the difference of two abstract convex functions and, by using the abstract Rockafellar?s antiderivative, we present the abstract ?-subdifferential of abstract convex functions in terms of their abstract subdifferential. Finally, as an application, a necessary and sufficient condition for global ?-minimum of the difference of two increasing and positively homogeneous (IPH) functions is presented.

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