On connection between properties of a compact set in ¢n and its conjugate set

1980 ◽  
pp. 465-476
Author(s):  
Juri Borisovič Zelinski
Keyword(s):  
Compact Set

scholarly journals A Robust Observer—Based Adaptive Control of Second—Order Systems with Input Saturation via Dead-Zone Lyapunov Functions

Computation ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 82
Author(s):  
Alejandro Rincón ◽  
Gloria M. Restrepo ◽  
Fredy E. Hoyos
Keyword(s):  
Lyapunov Functions ◽  
Dead Zone ◽  
Tracking Error ◽  
Input Saturation ◽  
Chemical Reactor ◽  
Second Order ◽  
Control Law ◽  
Compact Set ◽  
State Variables ◽  
Robust Observer

In this study, a novel robust observer-based adaptive controller was formulated for systems represented by second-order input–output dynamics with unknown second state, and it was applied to concentration tracking in a chemical reactor. By using dead-zone Lyapunov functions and adaptive backstepping method, an improved control law was derived, exhibiting faster response to changes in the output tracking error while avoiding input chattering and providing robustness to uncertain model terms. Moreover, a state observer was formulated for estimating the unknown state. The main contributions with respect to closely related designs are (i) the control law, the update law and the observer equations involve no discontinuous signals; (ii) it is guaranteed that the developed controller leads to the convergence of the tracking error to a compact set whose width is user-defined, and it does not depend on upper bounds of model terms, state variables or disturbances; and (iii) the control law exhibits a fast response to changes in the tracking error, whereas the control effort can be reduced through the controller parameters. Finally, the effectiveness of the developed controller is illustrated by the simulation of concentration tracking in a stirred chemical reactor.

scholarly journals Application of Psychometric Approach for ASD Evaluation in Russian 3–4-Year-Olds

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1608
Author(s):  
Andrey Nasledov ◽  
Sergey Miroshnikov ◽  
Liubov Tkacheva ◽  
Kirill Miroshnik ◽  
Meriam Uld Semeta
Keyword(s):  
Autistic Spectrum Disorder ◽  
Online Survey ◽  
Total Sample ◽  
Compact Set ◽  
Healthy Controls ◽  
Screening System ◽  
Children With Asd ◽  
Discriminant Analyses ◽  
Biological Problem ◽  
Choice Tasks

Background: Autistic spectrum disorder (ASD) is a significant socio-biological problem due to its wide prevalence and negative outcomes. In the current study, we aimed to develop an autism scale for early and accurate differentiation of 3- to 4-year-olds at risk for ASD since there is no systematic monitoring of young children in Russia yet. Methods: The total sample (N = 324) included 116 children with ASD, 131 children without ASD (healthy controls), and 77 children with developmental delay (DD). An online survey of specialists working with children was conducted based on a specially designed autism questionnaire consisting of 85 multiple-choice tasks distributed across 12 domains. Initially, each child was assessed by 434 items using a dichotomous scale (0 = no, 1 = yes). Factor and discriminant analyses were performed to identify a compact set of subscales that most accurately and with sufficient reliability predicted whether a child belongs to the ASD group. Results: As a result, four subscales were obtained: Sensorics, Emotions, Hyperactivity, and Communication. The high discriminability of the subscales in distinguishing the ASD group from the non-ASD group was revealed (accuracy 85.5–87.0%). Overall, the obtained subscales meet psychometric requirements and allow for creating an online screening system for wide application.

scholarly journals Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

4OR ◽  
2020 ◽  
Author(s):  
Michele Conforti ◽  
Marianna De Santis ◽  
Marco Di Summa ◽  
Francesco Rinaldi
Keyword(s):  
Integer Point ◽  
Cutting Plane ◽  
Integer Program ◽  
Compact Set ◽  
Cutting Plane Method ◽  
Integer Points ◽  
Gomory Cuts ◽  
The Family ◽  
Split Cuts ◽  
Number Of Iterations

AbstractWe consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

Some properties of the space of fuzzy-valued continuous functions on a compact set

2009 ◽  
Vol 160 (11) ◽  
pp. 1620-1631 ◽  
Author(s):  
Jin-Xuan Fang ◽  
Qiong-Yu Xue
Keyword(s):  
Continuous Functions ◽  
Compact Set

On the convex kernel of a compact set

1967 ◽  
Vol 63 (2) ◽  
pp. 311-313 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Linear Space ◽  
Compact Subset ◽  
Line Segment ◽  
Linear Dimension ◽  
Single Point ◽  
Compact Set ◽  
Topological Linear Space ◽  
Topological Linear

Suppose that E is a compact subset of a topological linear space ℒ. Then the convex kernel K, of E, is such that a point k belongs to K if every point of E can be seen, via E, from k. Valentine (l) has asked for conditions on E which ensure that the convex kernel K, of E, consists of exactly one point, and in this note we give such a condition. If A, B, C are three subsets of E, we use (A, B, C) to denote the set of those points of E, which can be seen, via E, from a triad of points a, b, c, with a ∈ A, b ∈ B, c ∈ C. We shall say that E has the property if, whenever A is a line segment and B, C are points of E which are not collinear with any point of A, the set (A, B, C) has linear dimension of at most one, and degenerates to a single point whenever A is a point.

scholarly journals A New Method for Solving Multiobjective Bilevel Programs

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Ying Ji ◽  
Shaojian Qu ◽  
Zhensheng Yu
Keyword(s):  
Multiobjective Optimization ◽  
Demand Uncertainty ◽  
Pareto Optimum ◽  
Compact Set ◽  
Supply Chain Risk Management ◽  
Equilibrium Constraints ◽  
Supply Chain Risk ◽  
Worst Case ◽  
Bilevel Programs ◽  
Real World Problems

We study a class of multiobjective bilevel programs with the weights of objectives being uncertain and assumed to belong to convex and compact set. To the best of our knowledge, there is no study about this class of problems. We use a worst-case weighted approach to solve this class of problems. Our “worst-case weighted multiobjective bilevel programs” model supposes that each player (leader or follower) has a set of weights to their objectives and wishes to minimize their maximum weighted sum objective where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto optimum concept, which we call “robust-weighted Pareto optimum”; for the worst-case weighted multiobjective optimization with the weight set of each player given as a polytope, we show that a robust-weighted Pareto optimum can be obtained by solving mathematical programing with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multiobjective optimization to a supply chain risk management under demand uncertainty. By the comparison with the existing weighted approach, we show that our method is more robust and can be more efficiently applied to real-world problems.

The dimension of self-affine fractals II

1992 ◽  
Vol 111 (1) ◽  
pp. 169-179 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Exact Expression ◽  
Singular Value ◽  
Compact Set ◽  
Value Functions ◽  
Affine Transformations ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Almost All

AbstractA family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

scholarly journals TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

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