scholarly journals On a Separation Theorem for Delta-Convex Functions

2020 ◽  
Vol 34 (1) ◽  
pp. 133-141
Author(s):  
Andrzej Olbryś
Keyword(s):  
Convex Function ◽  
Partial Order ◽  
Convex Functions ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Separation Theorem ◽  
Necessary And Sufficient Conditions ◽  
Lorentz Cone ◽  
Necessary And Sufficient

AbstractIn the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.

scholarly journals On separation by h-convex functions

2015 ◽  
Vol 62 (1) ◽  
pp. 105-111 ◽  
Author(s):  
Andrzej Olbryś
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Abstract In the present paper, we establish the necessary and sufficient conditions under which two functions can be separated by h-convex function, in the case, when the function h is multiplicative. This result is related to the theorem on separation by convex functions presented in Baron, K. Matkowski, J. Nikodem, K. [A sandwich with convexity, Math. Pannon. 5 (1994), 139 144].

scholarly journals Jensen-type geometric shapes

2020 ◽  
Vol 19 (1) ◽  
pp. 27-33 ◽  
Author(s):  
Paweł Pasteczka
Keyword(s):  
Convex Function ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Centre Of Mass ◽  
Geometric Shapes ◽  
Necessary And Sufficient

AbstractWe present both necessary and sufficient conditions for a convex closed shape such that for every convex function the average integral over the shape does not exceed the average integral over its boundary.It is proved that this inequality holds for n-dimensional parallelotopes, n-dimensional balls, and convex polytopes having the inscribed sphere (tangent to all its facets) with the centre in the centre of mass of its boundary.

scholarly journals Extensions of strict partial orders in N-Groups

1978 ◽  
Vol 25 (2) ◽  
pp. 241-249 ◽  
Author(s):  
K. B. Prabhakara Rao
Keyword(s):  
Partial Order ◽  
Sufficient Conditions ◽  
Partial Orders ◽  
Necessary And Sufficient Conditions ◽  
Full Extension ◽  
Partially Ordered ◽  
Strict Partial Order ◽  
Positive Elements ◽  
Necessary And Sufficient

AbstractAn attempt is made to extend the theory of extensions of partial orders in groups to strict partially ordered N-groups. Necessary and sufficient conditions, for a strict partial order of an N-group to have a strict full extension, and for a strict partial order of an N-group to be an intersection of strict full orders, are obtained when the partially ordered near-ring N and the N-group G satisfy the condition (− x) n = − xn for all elements x in G and positive elements n in N.

The displacement problem for elastic crystals

1989 ◽  
Vol 113 (1-2) ◽  
pp. 159-180 ◽  
Author(s):  
I. Fonseca ◽  
L. Tartar
Keyword(s):  
Boundary Conditions ◽  
Convex Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Displacement Boundary ◽  
Displacement Problem ◽  
Body Forces ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Elastic Crystals

SynopsisIn this paper we obtain necessary and sufficient conditions for the existence of Lipschitz minimisers of a functional of the typewhere h is a convex function converging to infinity at zero and u is subjected to displacement boundary conditions. We provide examples of body forces f for which the infimum of J(.) is not attained.

Almost sure stability of maxima

1973 ◽  
Vol 10 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

scholarly journals Convex functions and the rolling circle criterion

1995 ◽  
Vol 18 (4) ◽  
pp. 799-811
Author(s):  
V. Srinivas ◽  
O. P. Juneja ◽  
G. P. Kapoor
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rolling Circle ◽  
Circle Criterion ◽  
Rotation Theorem ◽  
Distortion Theorems ◽  
Necessary And Sufficient ◽  
Growth And Distortion Theorems

Given0≤R1≤R2≤∞,CVG(R1,R2)denotes the class of normalized convex functionsfin the unit discU, for which∂f(U)satisfies a Blaschke Rolling Circles Criterion with radiiR1andR2. Necessary and sufficient conditions forR1=R2, growth and distortion theorems forCVG(R1,R2)and rotation theorem for the class of convex functions of bounded type, are found.

scholarly journals A metrizable semitopological semilattice with non-closed partial order

2020 ◽  
Vol 8 (1) ◽  
pp. 67-75
Author(s):  
Taras Banakh ◽  
Serhii Bardyla ◽  
Alex Ravsky
Keyword(s):  
Partial Order ◽  
Dense Subset ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Countable Family ◽  
Hausdorff Topology ◽  
Necessary And Sufficient ◽  
Convergent Sequences

AbstractWe construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.

scholarly journals UPPER BOUNDS IN THE OHTSUKI–RILEY–SAKUMA PARTIAL ORDER ON 2-BRIDGE KNOTS

2012 ◽  
Vol 21 (09) ◽  
pp. 1250084 ◽  
Author(s):  
SCOTT M. GARRABRANT ◽  
JIM HOSTE ◽  
PATRICK D. SHANAHAN
Keyword(s):  
Partial Order ◽  
Upper Bound ◽  
Continued Fractions ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K1 we characterize all other 2-bridge knots K2 such that {K1, K2} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.

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