scholarly journals Tensor Methods for Minimizing Convex Functions with Hölder Continuous Higher-Order Derivatives

2020 ◽  
Vol 30 (4) ◽  
pp. 2750-2779 ◽  
Author(s):  
G. N. Grapiglia ◽  
Yu. Nesterov
Keyword(s):  
Convex Functions ◽  
Higher Order ◽  
Hölder Continuous ◽  
Tensor Methods

scholarly journals Pointwise estimates for higher order convexity preserving polynomial approximation

1994 ◽  
Vol 36 (2) ◽  
pp. 213-233 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Polynomial Approximation ◽  
Convex Functions ◽  
Higher Order ◽  
Second Order ◽  
Convolution Type ◽  
Shape Preservation ◽  
Moduli Of Continuity ◽  
Pointwise Estimates ◽  
Higher Order Convexity ◽  
Boolean Sum

AbstractDeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

scholarly journals Simpson type inequalities and applications

2021 ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Michael Th. Rassias ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Quadrature Formula ◽  
Convex Functions ◽  
Integral Identity ◽  
Higher Order ◽  
Auxiliary Result ◽  
First Order ◽  
New Class ◽  
Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

scholarly journals Higher order strongly general convex functions and variational inequalities

2020 ◽  
Vol 5 (4) ◽  
pp. 3646-3663 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Convex Functions ◽  
Higher Order ◽  
General Convex

scholarly journals Levinson type inequalities for higher order convex functions via Abel–Gontscharoff interpolation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Muhammad Adeel ◽  
Khuram Ali Khan ◽  
Ðilda Pečarić ◽  
Josip Pečarić
Keyword(s):  
Convex Functions ◽  
Higher Order ◽  
Data Points

Abstract In this paper, Levinson type inequalities are studied for the class of higher order convex functions by using Abel–Gontscharoff interpolation. Cebyšev, Grüss, and Ostrowski-type new bounds are also found for the functionals involving data points of two types.

Pathwise convergent higher order numerical schemes for random ordinary differential equations

2007 ◽  
Vol 463 (2087) ◽  
pp. 2929-2944 ◽  
Author(s):  
Peter E Kloeden ◽  
Arnulf Jentzen
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Time Variable ◽  
Numerical Schemes ◽  
Hölder Continuous ◽  
Usual Order

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

Higher Order Strongly m-convex Functions

2021 ◽  
pp. 319-339
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Higher Order

scholarly journals The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point

2008 ◽  
Vol 145 (3) ◽  
pp. 643-667 ◽  
Author(s):  
DANIELA KRAUS ◽  
OLIVER ROTH
Keyword(s):  
Continuous Function ◽  
Liouville Equation ◽  
Gaussian Curvature ◽  
Boundary Point ◽  
Classical Result ◽  
Riemannian Metrics ◽  
Higher Order ◽  
Isolated Singularities ◽  
Hölder Continuous ◽  
Curvature Equation

AbstractA classical result of Nitsche [22] about the behaviour of the solutions to the Liouville equation Δu= 4e2unear isolated singularities is generalized to solutions of the Gaussian curvature equation Δu= −κ(z)e2uwhere κ is a negative Hölder continuous function. As an application a higher–order version of the Yau–Ahlfors–Schwarz lemma for complete conformal Riemannian metrics is obtained.

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