Many trace inequalities can be expressed either as concavity/convexity theorems or monotonicity theorems. A classic example is the joint convexity of quantum relative entropy which equivalent to Data Processing Inequality. The latter says that operations never increase entropy. versions often have many advantages, and direct physical application, in just mentioned. Moreover, results are valid f...

Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity assumptions. The bangbang property is proved for a new class of “concave” multifunctions, characterized by the existence of suitable linear selections. The proofs rely ...

Abstract We discuss some frequency-domain criteria for the exponential stability of nonlinear feedback systems based on Popov’s hyperstability theorem. The main results are new bounds convergence rates perturbations damped harmonic oscillator, including Hessian-damped case, when satisfy certain sectoriality or convexity hypotheses. also include a simplified proof theorem in its general form, wh...

The existence and regularity of Young measure-valued solutions and weak solutions to non-Newtonian flows are considered. Galerkin approximation and an L2 compactness theorem are main ingredients for the proof of the existence of Young measurevalued solutions. Under a certain convexity condition for the energy, we prove that Young measure-valued solutions are weak solutions. Also, for the limite...

In this paper, we study φ-Laplacian problems for differential inclusions with Dirichlet boundary conditions. We prove the existence of solutions under both convexity and nonconvexity conditions on the multivalued right-hand side. The nonlinearity satisfies either a Nagumotype growth condition or an integrably boundedness one. The proofs rely on the Bonhnenblust-Karlin fixed point theorem and th...

There are many subclasses of analytic and univalent functions. The object of this paper is to introduce new classes and we have attempted to obtain coefficient estimate, distortion theorem, radius of starlikeness, convexity and closure theorem for the classes S∗(α, β, ξ, γ) and K∗(α, β, ξ, γ) on the lines of [1] and [2]. Results obtained by [1] and [2] which are particular cases of the paramete...

Vogt’s theorem, concerning boundary angles of a convex arc with monotonic curvature (spiral arc), is taken as a starting point to establish basic properties of spirals. The theorem is expanded by removing requirements of convexity and curvature continuity; the cases of inflection and multiple windings are considered. Positional restrictions for a spiral arc with two given curvature elements at ...

Helly’s Theorem and its relatives, the theorems of Radon and Caratheodory have played an important role in Discrete and Convex Geometry, there are numerous generalizations and variations of them that have been studied from different perspectives and points of view. The classical lemma of Radon for instance states that any d + 2 points in R can be partitioned into two classes with intersecting c...

This is a survey of the results in [1, 2, 3, 4, 5, 6] on convexity numbers of closed sets in Rn, homogeneity numbers of continuous colorings on Polish spaces and families of functions covering Rn. 1. Convexity numbers of closed sets in R A natural way to measure the degree of non-convexity of a subset S of a real vector space is the convexity number γ(S), the least size of a family C of convex ...

The seminal work by Edmonds [9] and Lovász [39] shows the strong connection between submodular functions and convex functions. Submodular functions have tight modular lower bounds, and a subdifferential structure [16] in a manner akin to convex functions. They also admit polynomial time algorithms for minimization and satisfy the Fenchel duality theorem [18] and the Discrete Seperation Theorem ...