638 research outputs found
Equilibrium problems on Riemannian manifolds with applications
We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent
for the equilibrium problem on Hadamard manifolds.National Natural Science Foundation of ChinaNatural Science Foundation of Guizhou Province (China)Dirección General de Enseñanza SuperiorJunta de AndalucíaNational Science Council of Taiwa
Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions
In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results
Geometric quantum computation using fictitious spin- 1/2 subspaces of strongly dipolar coupled nuclear spins
Geometric phases have been used in NMR, to implement controlled phase shift
gates for quantum information processing, only in weakly coupled systems in
which the individual spins can be identified as qubits. In this work, we
implement controlled phase shift gates in strongly coupled systems, by using
non-adiabatic geometric phases, obtained by evolving the magnetization of
fictitious spin-1/2 subspaces, over a closed loop on the Bloch sphere. The
dynamical phase accumulated during the evolution of the subspaces, is refocused
by a spin echo pulse sequence and by setting the delay of transition selective
pulses such that the evolution under the homonuclear coupling makes a complete
rotation. A detailed theoretical explanation of non-adiabatic geometric
phases in NMR is given, by using single transition operators. Controlled phase
shift gates, two qubit Deutsch-Jozsa algorithm and parity algorithm in a
qubit-qutrit system have been implemented in various strongly dipolar coupled
systems obtained by orienting the molecules in liquid crystal media.Comment: 37 pages, 17 figure
Approximate Efficient Solutions of the Vector Optimization Problem on Hadamard Manifolds via Vector Variational Inequalities
This article has two objectives. Firstly, we use the vector variational-like inequalities
problems to achieve local approximate (weakly) efficient solutions of the vector optimization problem
within the novel field of the Hadamard manifolds. Previously, we introduced the concepts of
generalized approximate geodesic convex functions and illustrated them with examples. We see the
minimum requirements under which critical points, solutions of Stampacchia, and Minty weak
variational-like inequalities and local approximate weakly efficient solutions can be identified,
extending previous results from the literature for linear Euclidean spaces. Secondly, we show
an economical application, again using solutions of the variational problems to identify Stackelberg
equilibrium points on Hadamard manifolds and under geodesic convexity assumptions
Proximal Point Methods for Solving Mixed Variational Inequalities on the Hadamard Manifolds
We use the auxiliary principle technique to suggest and analyze a proximal point method for
solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results
Application of Microlocal Analysis to the Theory of Quantum Fields Interacting with a Gravitational Field
It is explained how techniques from microlocal analysis can be used to settle
some long-standing questions that arise in the study of the interaction of
quantum matter fields with a classical gravitational background field.Comment: 7 pages, Latex, talk presented at the Conference on Partial
Differential Equations, Potsdam 199
On Mixed Equilibrium Problems in Hadamard Spaces
The main purpose of this paper is to study mixed equilibrium problems in Hadamard spaces. First, we establish the existence of solution of the mixed equilibrium problem and the unique existence of the resolvent operator for the problem. We then prove a strong convergence of the resolvent and a ?-convergence of the proximal point algorithm to a solution of the mixed equilibrium problem under some suitable conditions. Furthermore, we study the asymptotic behavior of the sequence generated by a Halpern-type PPA. Finally, we give a numerical example in a nonlinear space setting to illustrate the applicability of our results. Our results extend and unify some related results in the literature. - 2019 Chinedu Izuchukwu et al.ledgments
.e publication of this article was funded by the Qatar
National Library. .e first and third authors acknowledge
the bursary and financial support from Department of
Science and Technology and National Research Foundation,
Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS)
Doctoral Bursary. .e fourth author is supported in part by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant NumberScopu
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