scholarly journals Lower semi-continuity for $\mathcal A$-quasiconvex functionals under convex restrictions

2021 ◽  
Author(s):  
Emil Wiedemann ◽  
Jack Skipper
Keyword(s):  
Convex Constraint ◽  
Lower Semi

We show weak lower semi-continuity of functionals assuming the new notion of a ``convexly constrained''  $\mathcal A$-quasiconvex integrand. We assume $\mathcal A$-quasiconvexity only for functions defined on a set $K$ which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent ``convexly constrained''  $\mathcal A$-free sequences. In a motivating example, the integrand is $-\det^{\frac{1}{d-1}}$ and the convex constraint is positive semi-definiteness of a matrix field.

Deep generative model for non-convex constraint handling

2020 ◽  
Author(s):  
Naoki Sakamoto ◽  
Eiji Semmatsu ◽  
Kazuto Fukuchi ◽  
Jun Sakuma ◽  
Youhei Akimoto
Keyword(s):  
Generative Model ◽  
Constraint Handling ◽  
Convex Constraint

Convex extensions and envelopes of lower semi-continuous functions

2002 ◽  
Vol 93 (2) ◽  
pp. 247-263 ◽  
Author(s):  
Mohit Tawarmalani ◽  
Nikolaos V Sahinidis
Keyword(s):  
Continuous Functions ◽  
Lower Semi

scholarly journals A Fenchel-Rockafellar type duality theorem for maximization

1979 ◽  
Vol 20 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Space ◽  
Duality Theorem ◽  
Locally Convex Space ◽  
Bounded Subset ◽  
Convex Functional ◽  
Lower Semi ◽  
Locally Convex

We prove that sup(f-h)(E) = sup(h*-f*)(E*), where f is a proper lower semi-continuous convex functional on a real locally convex space E, h: E → = [-∞, +∞] is an arbitrary-functional and, f*, h* are their convex conjugates respectively. When h = δG, the indicator of a bounded subset G of E, this yields a formula for sup f(G).

scholarly journals On a new structure of the pantograph inclusion problem in the Caputo conformable setting

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sabri T. M. Thabet ◽  
Sina Etemad ◽  
Shahram Rezapour
Keyword(s):  
Differential Equation ◽  
Inclusion Problem ◽  
Integral Conditions ◽  
Lower Semi ◽  
First Time ◽  
Existence Criteria

Abstract In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann–Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann–Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.

scholarly journals Semi-continuity for total dimension divisors of étale sheaves

2017 ◽  
Vol 28 (01) ◽  
pp. 1750001
Author(s):  
Haoyu Hu ◽  
Enlin Yang
Keyword(s):  
Continuity Property ◽  
Lower Semi ◽  
Total Dimension

In this paper, we extend an inequality that compares the pull-back of the total dimension divisor of an étale sheaf and the total dimension divisor of the pull-back of the sheaf due to Saito. Using this formula, we generalize Deligne and Laumon’s lower semi-continuity property for Swan conductors of étale sheaves on relative curves to higher relative dimensions in a geometric situation.

More on Lower Semi-Continuity

1976 ◽  
Vol 83 (1) ◽  
pp. 39
Author(s):  
Oskar Feichtinger
Keyword(s):  
Lower Semi

scholarly journals On the Continuity of Orthogonal Sets in the Sense of Operator Orthogonality

2018 ◽  
Vol 11 (3) ◽  
pp. 793-802
Author(s):  
Mahdi Iranmanesh ◽  
M. Saeedi Khojasteh ◽  
M. K. Anwary
Keyword(s):  
Numerical Range ◽  
Alternative Definition ◽  
Normed Spaces ◽  
Operator Approach ◽  
Linear Spaces ◽  
Continuity Properties ◽  
Lower Semi

In this paper, we introduce the operator approach for orthogonality in linear spaces. In particular, we represent the concept of orthogonal vectors using an operator associated with them, in normed spaces. Moreover, we investigate some of continuity properties of this kind of orthogonality. More precisely, we show that the set valued function F(x; y) = {μ : μ ∈ C, p(x − μy, y) = 1} is upper and lower semi continuous, where p(x, y) = sup{pz1,...,zn−2 (x, y) : z1, . . . , zn−2 ∈ X} and pz1,...,zn−2 (x, y) = kPx,z1,...,zn−2,yk−1 where Px,z1,...,zn−2,y denotes the projection parallel to y from X to the subspace generated by {x, z1, . . . , zn−2}. This can be considered as an alternative definition for numerical range in linear spaces.

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