Narrow Results

We consider functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups.

The classical decomposition theory for states on a C*-algebra that are invariant under a group action has been studied by using the theory of orthogonal measures on the state space.⁴

In Ref. 3, we introduced the notion of generalized orthogonal measures on the space of unital completely positive (UCP) maps from a C*-algebra $A$ into $B(H)$. In this paper, we consider a group $G$ that acts on a C*-algebra $A$, and the collection of $G$-invariant UCP maps from $A$ into $B(H)$. This paper examines a $G$-invariant decomposition of UCP maps by using the theory of generalized orthogonal measures on the space of UCP maps, developed in Ref. 3. Further, the set of all $G$-invariant UCP maps is a compact and convex subset of a topological vector space. Hence, by characterizing the extreme points of this set, we complete the picture of barycentric decomposition in the space of $G$-invariant UCP maps. We establish this theory in Stinespring and Paschke dilations of completely positive maps. We end this note by mentioning some examples of UCP maps admitting a decomposition into $G$-invariant UCP maps.

The primary objective of this paper is to explore a novel subclass ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ of $p$-harmonic mappings, along with the associated subclass ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$. We demonstrate that the mapping ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ is both univalent and sense-preserving within the unit disk $𝕌$. Furthermore, we determine the extreme points of ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$ and establish that ${{\mathscr{H}}{𝒮}}_p(q,\alpha )\cap {{𝒯}}_p$ is defined. Additional results include the derivation of distortion bounds, the convolution condition, and the convex combination for this subclass. Finally, we examine the class-preserving integral operator and introduce a $q$-Jackson type integral operator.

This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑_kR_kTrF_kρ where each R_k is a density matrix and F_k>0. If, in addition, Φ is trace-preserving, the {F_k} must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given.

Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.

This paper continues the study of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable in the case of qubits. We give a detailed description of entanglement-breaking qubit channels, and show that such maps are precisely the convex hull of those known as classical-quantum channels. We also review the complete positivity conditions in a canonical parameterization and show how they lead to entanglement-breaking conditions.

Multilevel programming appears in many decision-making situations. Investigation of the main properties of quasiconcave multilevel programming (QCMP) problems, to date, is limited to bilevel programming (only two levels). In this paper, first, we present an extension of the properties of quasiconcave bilevel programming (QCBP) problems for the case when three levels exist. Then, by induction on n (the number of levels), we prove the existence of an extreme point of the polyhedral constraint region that solves the QCMP problem under given conditions. Ultimately, a number of numerical examples are illustrated to verify the results.

A new approach for ensemble construction based on restricting a set of weights of examples in training data to avoid overfitting is proposed in the paper. The algorithm called EPIBoost (Extreme Points Imprecise Boost) applies imprecise statistical models to restrict the set of weights. The updating of the weights within the restricted set is carried out by using its extreme points. The approach allows us to construct various algorithms by applying different imprecise statistical models for producing the restricted set. It is shown by various numerical experiments with real data sets that the EPIBoost algorithm may outperform the standard AdaBoost for some parameters of imprecise statistical models.

The number of inner points excluded in an initial convex hull (ICH) is vital to the efficiency getting the convex hull (CH) in a planar point set. The maximum inscribed circle method proposed recently is effective to remove inner points in ICH. However, limited by density distribution of a planar point set, it does not always work well. Although the affine transformation method can be used, it is still hard to have a better performance. Furthermore, the algorithm mentioned above fails to deal with the exceptional distribution: the gravity centroid (GC) of a planar point set is outside or on the edge formed by the extreme points in ICH. This paper considers how to remove more inner points in ICH when GC is inside of ICH and completely process the case which mentioned above. Further, we presented a complete algorithm architecture: (1) using the ellipse and elasticity ellipse methods (EM and EEM) to remove more inner points in ICH and process the cases: GC is inside or outside of ICH. (2) Using the traditional methods to process the situation: the initial centroid is on the edge in ICH. It is adaptive to more data sets than other algorithms. The experiments under seven distributions show that the proposed method performs better than other traditional algorithms in saving time and space.

A framework for constructing robust classification models is proposed in the paper. An assumption about importance of one of the classes in comparison with other classes is incorporated into the models. It often takes place in the real application, for example, in reliability, in medical diagnostic, etc. A main idea underlying the models is to consider a set of probability distributions on training examples produced by the imprecise probability models such as linear-vacuous mixture and constant odd-ratio contaminated models. Extreme points of the sets of probability distributions are a main tool for constructing the robust classifiers. It is shown that algorithms for computing optimal classification parameters are reduced to a finite number of weighted support vector machines with weights determined by the extreme points. Experimental results with synthetic and real data illustrate the proposed models.

We give a characterization of the convex hull of self-similar sets in ℝ³ which extends the results of Panzone¹ in ℝ². As an application, we show a convex set which cannot correspond to the convex hull of a self-similar set.

Under an epistemic interpretation, an upper probability can be regarded as equivalent to the set of probability measures it dominates, sometimes referred to as its core. In this paper, we study the properties of the number of extreme points of the core of a possibility measure, and investigate in detail those associated with (uni- and bi-)variate p-boxes, that model the imprecise information about a cumulative distribution function.

In this paper, different types of saddle pairs of vector-valued functions are investigated. Main properties of these pairs are examined. Structure of images of saddle pair sets is found and these images are constructed by means of cone extreme points in an explicit way as well. The obtained results provide possibility to construct dual problems in general cases of multiple objective problems and to investigate how to solve them using saddle pairs approach.

The bilevel programming problem is a leader–follower game in which two players try to maximize their own objective functions over a common constraint region. This paper discusses an integer nonlinear bilevel programming problem with box constraints by exploiting the quasimonotone character of the indefinite quadratic fractional function, considered as leader's objective. By making use of the duality theory, given nonlinear bilevel programming problem is transformed into single level programming problem. Various cuts have been discussed in this paper which successively rank and scan all integer feasible points of the single level programming problem in the decreasing value of objective function. An iterative algorithm is proposed, which by making use of these cuts repeatedly, solves the problem.

By virtue of coefficient inequalities, certain generalized subclasses of normalized analytic functions are introduced. Several integral means inequalities are obtained for higher-order and fractional calculus of functions belonging to the generalized suclasses. In these results, certain hypotheses for the generalized subclasses is weakened than those of the earlier results by T. Sekine, K. Tsurumi and H. M. Srivastava.