Showing papers on "Extreme point published in 2023"

A Class of Janowski-Type (p,q)-Convex Harmonic Functions Involving a Generalized q-Mittag-Leffler Function

Abstract: This research aims to present a linear operator Lp,qρ,σ,μf utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic (p,q)-convex functions HTp,q(ϑ,W,V) related to the Janowski function. For the harmonic p-valent functions f class, we investigate the harmonic geometric properties, such as coefficient estimates, convex linear combination, extreme points, and Hadamard product. Finally, the closure property is derived using the subclass HTp,q(ϑ,W,V) under the q-Bernardi integral operator.

Off-the-Grid Curve Reconstruction through Divergence Regularization: An Extreme Point Result

Abstract: . We propose a new strategy for curve reconstruction in an image through an off-the-grid variational 5 framework, inspired by spike reconstruction in the literature. We introduce a new functional CROC 6 on the space of 2-dimensional Radon measures with finite divergence denoted (cid:86) , and we estab-7 lish several theoretical tools through the definition of a certificate. Our main contribution lies in 8 the sharp characterisation of the extreme points of the unit ball of the (cid:86) -norm: there are exactly 9 measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise charac-10 terisation of our functional minimisers and further opening a promising avenue for the algorithmic 11 implementation.

Diametral notions for elements of the unit ball of a Banach space

Abstract: We introduce extensions of $\Delta$-points and Daugavet points in which slices are replaced by relative weakly open subsets (super $\Delta$-points and super Daugavet points) or by convex combinations of slices (ccs $\Delta$-points and ccs Daugavet points). We first give a general overview on these new concepts and provide some isometric consequences on the spaces. As examples: if a Banach space contains a super $\Delta$-point, then it does not admit an unconditional FDD with suppression constant smaller than two; if a real Banach space contains a ccs $\Delta$-point, then it does not admit a one-unconditional basis; if a Banach space contains a ccs Daugavet point, then every convex combination of slices of its unit ball has diameter two. We next characterize the notions in some classes of Banach spaces showing, for instance, that all the notions coincide in $L_1$-predual spaces and that all the notions but ccs Daugavet points coincide in $L_1$-spaces. We next remark on some examples which have previously appeared in the literature and provide some new intriguing examples: examples of super $\Delta$-points which are as closed as desired to strongly exposed points (hence failing to be Daugavet points in an extreme way); an example of a super $\Delta$-point which is strongly regular (hence failing to be a ccs $\Delta$-point in the strongest way); a super Daugavet point which fails to be a ccs $\Delta$-point. The extensions of the diametral notions to point in the open unit ball and the consequences on the spaces are also studied. Last, we investigate the Kuratowski measure of relative weakly open subsets and of convex combinations of slices in the presence of super $\Delta$-points or ccs $\Delta$-points, as well as for spaces enjoying diameter 2 properties. We conclude the paper with a section on open problems.

A Novel Neural Network with the Ability to Express the Extreme Points Distribution Features of Higher Derivatives of Physical Processes

Abstract: Higher derivatives are important to interpret the physical process. However, higher derivatives calculated from measured data often deviate from the real ones because of measurement errors. A novel method for data fitting without higher derivatives violating the real physical process is developed in this paper. Firstly, the research on errors’ influence on higher derivatives and the typical functions’ extreme points distribution were conducted, which demonstrates the necessity and feasibility of adopting extreme points distribution features in neural networks. Then, we proposed a new neural network considering the extreme points distribution features, namely, the extreme-points-distribution-based neural network (EDNN), which contains a sample error calculator (SEC) and extreme points distribution error calculator (EDEC). With recursive automatic differentiation, a model calculating the higher derivatives of the EDNN was established. Additionally, a loss function, embedded with the extreme points distribution features, was introduced. Finally, the EDNN was applied to two specific cases to reduce the noise in a second-order damped free oscillation signal and an internal combustion engine cylinder pressure trace signal. It was found that the EDNN could obtain higher derivatives that are more compatible with physical trends without detailed differentiation equations. The standard deviation of derivatives’ error of the EDNN is less than 62.5 percent of that of traditional neural networks. The EDNN provides a novel method for the analysis of physical processes with higher derivatives compatible with real physical trends.

Extreme points in Orlicz spaces equipped with s$s$‐norms and closedness

Abstract: Let Φ be an Orlicz function and L Φ ( X , Σ , μ ) $L^\Phi (X, \Sigma , \mu )$ be the corresponding Orlicz space on a non-atomic, σ-finite, complete measure space ( X , Σ , μ ) $(X,\Sigma ,\mu )$ . We describe the extreme points of unit ball of Orlicz spaces equipped with the s-norm. We also investigate the closedness of the set of extreme points of unit ball. Our study generalizes and unifies the results that have been obtained for the Orlicz norm, the Luxemburg norm and the p-Amemiya norm separately. We further classify the outer functions that generate s-norms with respect to a constant.

On the Extremality of the Tensor Product of Quantum Channels

Abstract: Completely positive and trace preserving (CPT) maps are important for Quantum Information Theory, because they describe a broad class of of transformations of quantum states. There are also two other related classes of maps, the unital completely positive (UCP) maps and the unital completely positive and trace preserving (UCPT) maps. For these three classes, the maps from a finite dimensional Hilbert space $X$ to another one $Y$ is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.

Extreme Points and First-Order Stochastic Dominance: Theory and Applications

Abstract: We characterize the extreme points of first-order stochastic dominance (FOSD) intervals and show how these intervals are at the heart of many topics in economics. An FOSD interval is a set of distributions that dominate a distribution and are simultaneously dominated by another distribution, in the sense of FOSD. The convexity of FOSD intervals means that their extreme points are fundamental to understanding their properties. We show that a distribution is an extreme point of an FOSD interval if and only if the distribution either coincides with one of the FOSD bounds or is flat. Wherever the distribution is flat, at least one end of the flat portion must be attached to one of the FOSD bounds.

Inner functions as strongly extreme points: stability properties

Abstract: Given a Banach space $\mathcal X$, let $x$ be a point in $\text{ball}(\mathcal X)$, the closed unit ball of $\mathcal X$. One says that $x$ is a strongly extreme point of $\text{ball}(\mathcal X)$ if it has the following property: for every $\varepsilon>0$ there is $\delta>0$ such that the inequalities $\|x\pm y\|<1+\delta$ imply, for $y\in\mathcal X$, that $\|y\|<\varepsilon$. We are concerned with certain subspaces of $H^\infty$, the space of bounded holomorphic functions on the disk, that arise upon imposing finitely many linear constraints and can be viewed as finite-dimensional perturbations of $H^\infty$. It is well known that the strongly extreme points of $\text{ball}(H^\infty)$ are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions $f\in H^\infty$ with $\log(1-|f|)$ non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed $H^\infty$-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point.

A subclass of harmonic functions defined by a certain fractional calculus operators

Abstract: In this paper a subclass of p-valent harmonic functions in the open unit disc is introduced by making use of a certain fractional calculus operator and some properties such as coefficient estimates, distortion theorem and extreme points are studied.

Convex Hulls of Random Order Types

Abstract: We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n -point order type, chosen uniformly at random from all such order types, is on average 4+ o (1). For labeled order types, this number has average \(4- \mbox{$\frac{8}{n^2 - n +2}$}\) and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2 d + o (1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős-Szekeres theorem: for any fixed k , as n → ∞, a proportion 1 - O (1/ n ) of the n -point simple order types contain a triangle enclosing a convex k -chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A 4 , S 4 , or A 5 (and each case is possible). These are the finite subgroups of SO (3) and our proof follows the lines of their characterization by Felix Klein.

Some Subclasses of q-Analytic Starlike Functions

Abstract: We define two new subclasses, $\mathcal{S}_{q}^{\ast }(\alpha )$ and $\mathcal{TS}_{q}^{\ast }(\alpha )$, of analytic univalent functions. We obtain a sufficient condition for analytic univalent functions to be in $\mathcal{S}_{q}^{\ast }(\alpha )$ and we prove that this condition is also necessary for the functions in the class $\mathcal{TS}_{q}^{\ast }(\alpha )$. We also obtain extreme points, distortion bounds, covering result, convex combination and convolution properties for the functions in the class $\mathcal{TS}_{q}^{\ast }(\alpha )$.

Free Extreme points of free spectrahedrops and generalized free spectrahedra

Abstract: Matrix convexity generalizes convexity to the dimension free setting and has connections to many mathematical and applied pursuits including operator theory, quantum information, noncommutative optimization, and linear control systems. In the setting of classical convex sets, extreme points are central objects which exhibit many important properties. For example, the Minkowski theorem shows that any element of a closed bounded convex set can be expressed as a convex combination of extreme points. Extreme points are also of great interest in the dimension free setting of matrix convex sets; however, here the situation requires more nuance. In the dimension free setting, there are many different types of extreme points. Of particular importance are free extreme points, a highly restricted type of extreme point that is closely connected to the dilation theoretic Arveson boundary. If free extreme points span a matrix convex set through matrix convex combinations, then they satisfy a strong notion of minimality in doing so. However, not all closed bounded matrix convex sets even have free extreme points. Thus, a major goal is to determine which matrix convex sets are spanned by their free extreme points. Building on a recent work of J. W. Helton and the author which shows that free spectrahedra, i.e., dimension free solution sets to linear matrix inequalities, are spanned by their free extreme points, we establish two additional classes of matrix convex sets which are the matrix convex hull of their free extreme points. Namely, we show that closed bounded free spectrahedrops, i.e, closed bounded projections of free spectrahedra, are the span of their free extreme points. Furthermore, we show that if one considers linear operator inequalities that have compact operator defining tuples, then the resulting ``generalized" free spectrahedra are spanned by their free extreme points.

On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

Abstract: In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carath\'eodory.

Analytical Study and Efficient Evaluation of the Josephus Function

Abstract: A new approach to analyzing intrinsic properties of the Josephus function, $J_{_k}$, is presented in this paper. The linear structure between extreme points of $J_{_k}$ is fully revealed, leading to the design of an efficient algorithm for evaluating $J_{_k}(n)$. Algebraic expressions that describe how recursively compute extreme points, including fixed points, are derived. The existence of consecutive extreme and also fixed points for all $k\geq 2$ is proven as a consequence, which generalizes Knuth result for $k=2$. Moreover, an extensive comparative numerical experiment is conducted to illustrate the performance of the proposed algorithm for evaluating the Josephus function compared to established algorithms. The results show that the proposed scheme is highly effective in computing $J_{_k}(n)$ for large inputs.

Rigorous Runtime Analysis of MOEA/D for Solving Multi-Objective Minimum Weight Base Problems

Abstract: We study the multi-objective minimum weight base problem, an abstraction of classical NP-hard combinatorial problems such as the multi-objective minimum spanning tree problem. We prove some important properties of the convex hull of the non-dominated front, such as its approximation quality and an upper bound on the number of extreme points. Using these properties, we give the first run-time analysis of the MOEA/D algorithm for this problem, an evolutionary algorithm that effectively optimizes by decomposing the objectives into single-objective components. We show that the MOEA/D, given an appropriate decomposition setting, finds all extreme points within expected fixed-parameter polynomial time in the oracle model, the parameter being the number of objectives. Experiments are conducted on random bi-objective minimum spanning tree instances, and the results agree with our theoretical findings. Furthermore, compared with a previously studied evolutionary algorithm for the problem GSEMO, MOEA/D finds all extreme points much faster across all instances.

A q-Starlike Class of Harmonic Meromorphic Functions Defined by q-Derivative Operator

Abstract: In this present work, we introduces a subclass of harmonic meromorphic functions associated with q-calculus operator. With that, we study various interesting properties of this class, like, the coefficients bounds, distortion theorems, extreme points, convolution, convex combinations. Further, q-integral operator is also defined and we show that the new class aforementioned is closed under this q-derivative operator.

Diagonals of self-adjoint operators II: Non-compact operators

Abstract: Given a self-adjoint operator $T$ on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set $\mathcal D(T)$ of all possible diagonals of $T$. For operators $T$ with at least two points in their essential spectrum $\sigma_{ess}(T)$, we give a complete characterization of $\mathcal D(T)$ for the class of self-adjoint operators sharing the same spectral measure as $T$ with a possible exception of multiplicities of eigenvalues at the extreme points of $\sigma_{ess}(T)$. We also give a more precise description of $\mathcal D(T)$ for a fixed self-adjoint operator $T$, albeit modulo the kernel problem for special classes of operators. These classes consist of operators $T$ for which an extreme point of the essential spectrum $\sigma_{ess}(T)$ is also an extreme point of the spectrum $\sigma(T)$. Our results generalize a characterization of diagonals of orthogonal projections by Kadison, Blaschke-type results of M\"uller and Tomilov, and Loreaux and Weiss, and a characterization of diagonals of operators with finite spectrum by the authors.

On a family of p-valently analytic functions missing initial Taylor coefficients

Abstract: For k ? 0, 0 ? ? ? 1, and some convolution operator 1, the object of this paper is to introduce a generalized family TUnp (1, ?, k, b, ?) of p-valently analytic functions of complex order b ? C \ {0} and type ? ? [0, p). Apart from studying certain coefficient, radii and subordination problems, we prove that TUnp (1, ?, k, b, ?) is convex and derive its extreme points. Moreover, the closedness of this family under the modified Hadamard product is discussed. Several previously established results are obtained as particular cases of our theorems.

Notes on meromorphic functions with positive coefficients defined by Rafid-operator

Abstract: In this paper, we investigate and study a new subclass of meromorphic univalent functions defined by Rafid operator. We obtain coefficient inequalities, growth and distortion theorems, extreme points and closure theorems.

Unit balls of Polyhedral Banach spaces with many extreme points

Abstract: Let $E$ be a $(\mathrm{IV})$-polyhedral Banach space. We show that, for each $\epsilon>0$, $E$ admits an $\epsilon$-equivalent $\mathrm{(V)}$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points. In particular, we obtain that every separable isomorphically polyhedral Banach space, for each $\epsilon>0$, admits an $\epsilon$-equivalent $(\mathrm{V})$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points.

C*-extreme entanglement breaking maps on operator systems

Abstract: Let $\mathcal E$ denote the set of all unital entanglement breaking (UEB) linear maps defined on an operator system $\mathcal S \subset M_d$ and, mapping into $M_n$. As it turns out, the set $\mathcal E$ is not only convex in the classical sense but also in a quantum sense, namely it is $C^*$-convex. The main objective of this article is to describe the $C^*$-extreme points of this set $\mathcal E$. By observing that every EB map defined on the operator system $\mathcal S$ dilates to a positive map with commutative range and also extends to an EB map on $M_d$, We show that the $C^*$-extreme points of the set $\mathcal E$ are precisely the UEB maps that are maximal in the sense of Arveson (\cite{A} and \cite{A69}) and that they are also exactly the linear extreme points of the set $\mathcal E$ with commutative range. We also determine their explicit structure, thereby obtaining operator system generalizations of the analogous structure theorem and the Krein-Milman type theorem given in \cite{BDMS}. As a consequence, we show that $C^*$-extreme (UEB) maps in $\mathcal E$ extend to $C^*$-extreme UEB maps on the full algebra. Finally, we obtain an improved version of the main result in \cite{BDMS}, which contains various characterizations of $C^*$-extreme UEB maps between the algebras $M_d$ and $M_n$.

Determining the Viability of an Unbounded Polyhedron for a Switched System

Abstract: This paper proposes a new method to determine the viability of a switched system on a cone and an unbounded polyhedron. First, we investigate the viability condition on a cone. Then, a sufficient viability criterion for a polyhedron, which is expressed by a convex hull of finite number of extreme points and a nonnegative linear combination of finite extreme directions, is presented by using nonsmooth analysis. Based on this criterion, instead of all boundary points, just several extreme points and extreme directions are needed to be verified whether satisfying some conditions. The advantage of the proposed methods is that determining the viability for a switched system is easy to be implemented. Finally, an example is listed to illustrate the effectiveness of the proposed methods.

Peak Estimation of Hybrid Systems with Convex Optimization

Abstract: Peak estimation of hybrid systems aims to upper bound extreme values of a state function along trajectories, where this state function could be different in each subsystem. This finite-dimensional but nonconvex problem may be lifted into an infinite-dimensional linear program (LP) in occupation measures with an equal objective under mild finiteness/compactness and smoothness assumptions. This LP may in turn be approximated by a convergent sequence of upper bounds attained from solutions of Linear Matrix Inequalities (LMIs) using the Moment-Sum-of-Squares hierarchy. The peak estimation problem is extended to problems with uncertainty and safety settings, such as measuring the distance of closest approach between points along hybrid system trajectories and unsafe sets.

SuperHyperGirth Type-Results on extreme SuperHyperGirth theory and (Neutrosophic) SuperHyperGraphs Toward Cancer's extreme Recognition

Abstract: In this research, the extreme SuperHyperNotion, namely, extreme SuperHyperGirth, is up. $E_1$ and $E_3$ are some empty extreme SuperHyperEdges but $E_2$ is a loop extreme SuperHyperEdge and $E_4$ is an extreme SuperHyperEdge. Thus in the terms of extreme SuperHyperNeighbor, there's only one extreme SuperHyperEdge, namely, $E_4.$ The extreme SuperHyperVertex, $V_3$ is extreme isolated means that there's no extreme SuperHyperEdge has it as an extreme endpoint. Thus the extreme SuperHyperVertex, $V_3,$ is excluded in every given extreme SuperHyperGirth. $ \mathcal{C}(NSHG)=\{E_i\}~\text{is an extreme SuperHyperGirth.} \ \ \mathcal{C}(NSHG)=jz^i~\text{is an extreme SuperHyperGirth SuperHyperPolynomial.} \ \ \mathcal{C}(NSHG)=\{V_i\}~\text{is an extreme R-SuperHyperGirth.} \ \ \mathcal{C}(NSHG)=jz^I~{\small\text{is an extreme R-SuperHyperGirth SuperHyperPolynomial.}} $ The following extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices] is the extreme type-SuperHyperSet of the extreme SuperHyperGirth. The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices], is the extreme type-SuperHyperSet of the extreme SuperHyperGirth. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices], is an extreme SuperHyperGirth $\mathcal{C}(ESHG)$ for an extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme type-SuperHyperSet with the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's only one extreme consecutive sequence of the extreme SuperHyperVertices and the extreme SuperHyperEdges form only one extreme SuperHyperCycle. There are not only four extreme SuperHyperVertices inside the intended extreme SuperHyperSet. Thus the non-obvious extreme SuperHyperGirth isn't up. The obvious simple extreme type-SuperHyperSet called the extreme SuperHyperGirth is an extreme SuperHyperSet includes only less than four extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices], doesn't have less than four SuperHyperVertices inside the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the extreme SuperHyperGirth isn't up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices], isn't the non-obvious simple extreme type-SuperHyperSet of the extreme SuperHyperGirth. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices], is an extreme SuperHyperGirth $\mathcal{C}(ESHG)$ for an extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices[SuperHyperEdges] such that there's only one extreme consecutive extreme sequence of extreme SuperHyperVertices and extreme SuperHyperEdges form only one extreme SuperHyperCycle given by that extreme type-SuperHyperSet called the extreme SuperHyperGirth and it's an extreme SuperHyperGirth . Since it 's the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's only one extreme consecutive extreme sequence of extreme SuperHyperVertices and extreme SuperHyperEdges form only one extreme SuperHyperCycle. There are only less than four extreme SuperHyperVertices inside the intended extreme SuperHyperSet, thus the obvious extreme SuperHyperGirth, is up. The obvious simple extreme type-SuperHyperSet of the extreme SuperHyperGirth, is: ,is the extreme SuperHyperSet, is: does includes only less than four SuperHyperVertices in a connected extreme SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only simple extreme type-SuperHyperSet called the extreme SuperHyperGirth amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the neutrosophic SuperHyperGirth , is only and only. A basic familiarity with extreme SuperHyperGirth theory, SuperHyperGraphs, and extreme SuperHyperGraphs theory are proposed.

On a family of holomorphic functions involving linear operator

Abstract: Abstract

Geometry of Feasible Region

Abstract: The feasible region P, defined by Definition 1.4.1 , is of great importance to the LP problem. Theories and methods of LP are closely related to P, without exception. In this chapter, we introduce its special structure in geometry, including P as a polyhedral convex set, interior point, relative interior point, face, vertex, extreme direction, representation of P, optimal face and optimal vertex, graphic approach to LP, heuristic characteristic of an optimal solution, and feasible direction and active constraint.

Extreme problem for a mosaic system of points on the open sets and nonoverlapping domains

Abstract: In the geometric theory of functions of a complex variable, the well-known direction is relatedIn the geometric theory of functions of a complex variable, the well-known direction is related to the estimates of the products of the inner radii of pairwise nonoverlapping domains. This direction is called extreme problems in classes of pairwise nonoverlapping domains. One of the problems of this type is considered in the present work