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In this paper we give sufficient conditions for a compactum in ? n to have Carathéodory number less than n+1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem a...
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In this paper we give sufficient conditions for a compactum in ? n to have Carathéodory number less than n+1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg-type theorem for families of convex compacta.
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It is known that the existence of a convex (resp., concave) separator between two given functions can be characterized via a simple inequality. The notion of convexity can be generalized applying regular pairs (in other words, two...
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It is known that the existence of a convex (resp., concave) separator between two given functions can be characterized via a simple inequality. The notion of convexity can be generalized applying regular pairs (in other words, two dimensional Chebyshev systems). The aim of the present note is to extend the above mentioned result to this setting. In the proof, a modified version of the classical Carathéodory's theorem and the characterization of convex functions play the key role.
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In this paper, a boundary version of Carath′ eodory’s inequal-ity on the right half plane for p-valent is investigated. Let Z(s) = 1+c p (s ? 1) p +c p+1 (s ? 1) p+1 +··· be an analytic function in the right half plane with 1...
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In this paper, a boundary version of Carath′ eodory’s inequal-ity on the right half plane for p-valent is investigated. Let Z(s) = 1+c p (s ? 1) p +c p+1 (s ? 1) p+1 +··· be an analytic function in the right half plane with 1) for 收起
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We consider the problem of uniform long-time behavior of all globally defined weak solutions of a non-autonomous reaction-diffusion system with Carathéodory's nonlinearity satisfying standard sign and polynomial growth assumption...
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We consider the problem of uniform long-time behavior of all globally defined weak solutions of a non-autonomous reaction-diffusion system with Carathéodory's nonlinearity satisfying standard sign and polynomial growth assumptions. The main contributions of this paper are: (i) the existence of a uniform trajectory attractor for all globally defined weak solutions of non-autonomous reaction-diffusion equations with Carathéodory's nonlinearity, (ii) sufficient conditions for the existence of a uniform trajectory attractor in strongest topologies, and (iii) new topological properties of weak solutions.
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Knowing an upper bound on the number of optimal design points greatly simplifies the search for an optimal design. Carathéodory's Theorem is commonly used to identify an upper bound. However, the upper bound from Carathéodory's ...
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Knowing an upper bound on the number of optimal design points greatly simplifies the search for an optimal design. Carathéodory's Theorem is commonly used to identify an upper bound. However, the upper bound from Carathéodory's Theorem is relatively loose when there are three or more parameters in the model. In this paper, an alternative approach of finding a sharper upper bound for classical optimality criteria is proposed. Examples are given to demonstrate how to use the new approach.
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We consider conformal homeomorphisms ? of generalized Jordan domains U onto planar domains Ω that satisfy both of the next two conditions: (1) at most countably many boundary components of Ω are non-degenerate and their diameter...
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We consider conformal homeomorphisms ? of generalized Jordan domains U onto planar domains Ω that satisfy both of the next two conditions: (1) at most countably many boundary components of Ω are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of Ω or those of U form a set of sigma-finite linear measure. We prove that ? continuously extends to the closure of U if and only if every boundary component of Ω is locally connected. This generalizes the Carathéodory's Continuity Theorem and leads us to a new generalization of the well known Osgood-Taylor-Carathéodory Theorem. There are three issues that are noteworthy. Firstly, none of the above conditions (1) and (2) can be removed. Secondly, our results remain valid for non-cofat domains and do not follow from the extension results, of a similar nature, that are obtained in very recent studies on the conformal rigidity of circle domains. Finally, when ? does extend continuously to the closure of U, the boundary of Ω is a Peano compactum. Therefore, we also show that the following properties are equivalent for any planar domain Ω: (1) The boundary of Ω is a Peano compactum. (2) Ω has Property S. (3) Every point on the boundary of Ω is locally accessible. (4) Every point on the boundary of Ω is locally sequentially accessible. (5) Ω is finitely connected at the boundary. (6) The completion of Ω under the Mazurkiewicz distance is compact.
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In this work, we establish the existence of multiple positive solutions for a general higher order fractional differential equation with derivatives and a sign-changing Carathèodory perturbed term-D ~αx(t)=p(t)ft, x(t),D ~(μ1)x...
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In this work, we establish the existence of multiple positive solutions for a general higher order fractional differential equation with derivatives and a sign-changing Carathèodory perturbed term-D ~αx(t)=p(t)ft, x(t),D ~(μ1)x(t),D ~(μ2)x(t),?,Dμn- _1x(t)-gt,x(t),D ~(μ1)x(t),D ~(μ2)x(t),?, Dμn- _1x(t),D ~(μi)x(0)=0,1≤i≤n-1,Dμn- _1+1x(0)=0,Dμn- _1x(1)=∑j=1m-2 ~(aj)Dμn- 1x(~(ξj)),where n-_1<α≤n,n∈N and n≥3 with 0< ~(μ1)< ~(μ2)收起
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Let C ? ?~n be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ?, or ?~n. The C-ball convex hull of two points is called ...
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Let C ? ?~n be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ?, or ?~n. The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.
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The main goal of this article is to generalize Hadamard's real part theorem and invariant forms of Borel-Carathéodory's theorem from complex analysis to solutions of the Riesz system in the three-dimensional Euclidean space in th...
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The main goal of this article is to generalize Hadamard's real part theorem and invariant forms of Borel-Carathéodory's theorem from complex analysis to solutions of the Riesz system in the three-dimensional Euclidean space in the framework of quaternionic analysis.
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A Wasserstein space is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be made precise. In...
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A Wasserstein space is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be made precise. In the first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a natural set of spaces generalizing the usual Hilbert cube. These invariants are very similar to concepts initiated by Rogers, but our variant is specifically suited to tackle Lipschitz comparison. In the second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide uniform bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d. These arguments are very easily adapted to study the space of closed subsets of a compact metric space, partly generalizing results of Boardman, Goodey and McClure.
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