Caratheodory's Theorem Research Articles

Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders

<abstract><p>The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators.</p></abstract>

Fast and Accurate Least-Mean-Squares Solvers for High Dimensional Data.

Least-mean-squares (LMS) solvers such as Linear / Ridge-Regression and SVD not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as matrix factorizations. We suggest an algorithm that gets a finite set of n d-dimensional real vectors and returns a subset of d+1 vectors with positive weights whose weighted sum is exactly the same. The constructive proof in Caratheodory's Theorem computes such a subset in O(n2d2) time and thus not used in practice. Our algorithm computes this subset in O(nd+d4logn) time, using O(logn) calls to Caratheodory's construction on small but "smart" subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets. For large values of d, we suggest a faster construction that takes O(nd) time and returns a weighted subset of O(d) sparsified input points. Here, a sparsified point means that some of its entries were set to zero. As an application, we show how to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to x100. Generalization for streaming and distributed data is trivial. Extensive experimental results and open source code are provided.

Escaping Braess's paradox through approximate Caratheodory's theorem

In selfish routing, Braess's paradox demonstrates the counterintuitive fact that removing a part of a network may improve the players' cost at equilibrium. In this work, we use the approximate version of Caratheodory's theorem (Barman, SICOMP 2018) and show how to efficiently approximate the best subnetwork on selfish routing instances with relatively short paths and Lipschitz continuous cost functions. Restriction to networks with short paths is necessary to escape the strong inapproximability result of (Roughgarden, JCSS 2006). We present the first polynomial-time algorithm that approximates (in a certain relaxed sense) the best subnetwork problem on such instances. Specifically, for any constant ε>0, our algorithm computes an ε-Nash flow with maximum latency at most (1+ε)L⁎+ε, where L⁎ is the equilibrium latency of the best subnetwork. Moreover, our algorithm runs in quasipolynomial time on networks with polylogarithmically long paths. As a corollary, we obtain the first polynomial-time approximation scheme for the best subnetwork in the class of random Gn,p instances proven prone to Braess's paradox by (Roughgarden and Valiant, RSA 2010) and (Chung Graham et al., RSA 2012).

Transversal factors and spanning trees

Since the proof of a "colorful" version of [Caratheodory's theorem](https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_%28convex_hull%29) by Bárány in 1982, it has been an important problem to obtain colorful extensions of other classical results in discrete geometry (for instance Tverberg's theorem). The present paper continues this line of research, but in the context of extremal graph theory rather than discrete geometry. Mantel's classical theorem from 1907 states that every $n$-vertex graph on more than $n^2/4$ edges contains a triangle. In [Ron Aharoni, Matt DeVos, Sebastián González Hermosillo de la Maza, Amanda Montejano, and Robert Šámal, A rainbow version of Mantel’s Theorem, Advances in Combinatorics 2020:2, 12 pp](https://arxiv.org/abs/1812.11872v2), a "rainbow", "colored", or "colorful" variant of this problem was considered : given three graphs $G_1,G_2,G_3$ on the same vertex set of size $n$, what average degree conditions on $G_1,G_2,G_3$ force the existence of a "rainbow triangle" (a triangle $\{e_1,e_2,e_3\}$ such that each edge $e_i$ belongs to $G_i$)? By taking three copies of the same graph $G$ we see that the colored version is at least as hard as the original problem, and the paper cited above provided a construction showing that in this case the colorful variant is strictly harder than Mantel's problem. It was suggested to study average degree or minimum degree thresholds for colorful variants of classical problems in extremal combinatorics, such as Dirac's theorem (every $n$-vertex graph of minimum degree at least $n/2$ has a Hamiltonian cycle). In particular, Joos and Kim proved in 2020 that the same minimum degree condition as in Dirac's theorem guarantees a rainbow $n$-cycle: namely if we are given $n$ graphs of minimum degree at least $n/2$ on the same set of $n$ vertices, then there is an $n$-cycle comprising one edge of each graph. The results in the present paper follow the same line of research. The two major results that are extended to the colorful setting here are a theorem of Kühn and Osthus (a sharp minimum degree condition to obtain a perfect packing of copies of any given graph $F$, generalizing the Hajnal-Szemerédi theorem), and a theorem of Komlós, Sárközy and Szemerédi (a sharp degree condition to contain any given spanning tree without large degree vertices). Amazingly, the minimum degree conditions in the (stronger) colorful versions are the same as the original minimum degree conditions.

Caratheodory theorem about prime ends on Riemann surfaces

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio--Ryazanov--Srebro--Yakubov at 2004 and then included in the known monograph of these authors in the modern mapping theory at 2009. As it was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.

Autonomous Toy Drone via Coresets for Pose Estimation

A coreset of a dataset is a small weighted set, such that querying the coreset provably yields a ()-factor approximation to the original (full) dataset, for a given family of queries. This paper suggests accurate coresets () that are subsets of the input for fundamental optimization problems. These coresets enabled us to implement a “Guardian Angel” system that computes pose-estimation in a rate frames per second. It tracks a toy quadcopter which guides guests in a supermarket, hospital, mall, airport, and so on. We prove that any set of n matrices in whose sum is a matrix S of rank r, has a coreset whose sum has the same left and right singular vectors as S, and consists of matrices, independent of n. This implies the first (exact, weighted subset) coreset of points to problems such as linear regression, PCA/SVD, and Wahba’s problem, with corresponding streaming, dynamic, and distributed versions. Our main tool is a novel usage of the Caratheodory Theorem for coresets, an algorithm that computes its set in time that is linear in its cardinality. Extensive experimental results on both synthetic and real data, companion video of our system, and open code are provided.

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

We prove a common strengthening of Barany’s colorful Caratheodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a colorful KKMS theorem due to Shih and Lee [Math. Ann. 296 (1993), no. 1, 35–61] as well as a polytopal KKMS theorem due to Komiya [Econ. Theory 4 (1994), no. 3, 463–466]. The (seemingly unrelated) colorful Caratheodory theorem is a special case as well. We apply our theorem to establish an upper bound on the piercing number of colorful d-interval hypergraphs, extending earlier results of Tardos [Combinatorica 15 (1995), no. 1, 123–134] and Kaiser [Discrete Comput. Geom. 18 (1997), no. 2, 195–203].

Unified halo-independent formalism from convex hulls for direct dark matter searches

Using the Fenchel-Eggleston theorem for convex hulls (an extension of the Caratheodory theorem), we prove that any likelihood can be maximized by either a dark matter 1- speed distribution F(v) in Earth's frame or 2- Galactic velocity distribution fgal(u⃗), consisting of a sum of delta functions. The former case applies only to time-averaged rate measurements and the maximum number of delta functions is (\U0001d4a9−1), where \U0001d4a9 is the total number of data entries. The second case applies to any harmonic expansion coefficient of the time-dependent rate and the maximum number of terms is \U0001d4a9. Using time-averaged rates, the aforementioned form of F(v) results in a piecewise constant unmodulated halo function η̃0BF(vmin) (which is an integral of the speed distribution) with at most (\U0001d4a9-1) downward steps. The authors had previously proven this result for likelihoods comprised of at least one extended likelihood, and found the best-fit halo function to be unique. This uniqueness, however, cannot be guaranteed in the more general analysis applied to arbitrary likelihoods. Thus we introduce a method for determining whether there exists a unique best-fit halo function, and provide a procedure for constructing either a pointwise confidence band, if the best-fit halo function is unique, or a degeneracy band, if it is not. Using measurements of modulation amplitudes, the aforementioned form of fgal(u⃗), which is a sum of Galactic streams, yields a periodic time-dependent halo function η̃BF(vmin, t) which at any fixed time is a piecewise constant function of vmin with at most \U0001d4a9 downward steps. In this case, we explain how to construct pointwise confidence and degeneracy bands from the time-averaged halo function. Finally, we show that requiring an isotropic Galactic velocity distribution leads to a Galactic speed distribution F(u) that is once again a sum of delta functions, and produces a time-dependent η̃BF(vmin, t) function (and a time-averaged η̃0BF(vmin)) that is piecewise linear, differing significantly from best-fit halo functions obtained without the assumption of isotropy.

Quasiconformality of the injective mappings transforming spheres to quasispheres

We prove that every injective mapping of a domain \(D \subset \overline {{\mathbb{R}^n}} \) transforming spheres Σ ⊂ D to K-quasispheres (the images of spheres under K-quasiconformal automorphisms of \(\overline {{\mathbb{R}^n}} \)) is K′-quasiconformal with K′ depending only on K and tending to 1 as K → 1. This is a quasiconformal analog of the classical Caratheodory Theorem on the Mobius property of an injective mapping of a domain D ⊂ Rn which sends spheres to spheres.

Caratheodory's theorem for \mathbb{B}^{-1}-convex sets

In this article, our main concept is $\mathbb{B}^{-1}$-convexity that is a new abstract convexity type. For the $\mathbb{B}^{-1}$-convex sets, Caratheodory's Theorem which is one of the most important results in convexity theory is proved and its corollary is given.

Costly Circuits, Submodular Schedules and Approximate Carathéodory Theorems

Hybrid switching -- in which a high bandwidth circuit switch (optical or wireless) is used in conjunction with a low bandwidth packet switch -- is a promising alternative to interconnect servers in today's large scale data centers. Circuit switches offer a very high link rate, but incur a non-trivial reconfiguration delay which makes their scheduling challenging. In this paper, we demonstrate a lightweight, simple and nearly-optimal scheduling algorithm that trades-off reconfiguration costs with the benefits of reconfiguration that match the traffic demands. Seen alternatively, the algorithm provides a fast and approximate solution towards a constructive version of Caratheodory's Theorem for the Birkhoff polytope. The algorithm also has strong connections to submodular optimization, achieves a performance at least half that of the optimal schedule and strictly outperforms state of the art in a variety of traffic demand settings. These ideas naturally generalize: we see that indirect routing leads to exponential connectivity; this is another phenomenon of the power of multi-hop routing, distinct from the well-known load balancing effects.

Conditions for the absence of jumps of the solution to the adjoint system of the maximum principle for optimal control problems with state constraints

Properties of Lagrange multipliers from the Pontryagin maximum principle for problems with state constraints are investigated. Sufficient conditions for the continuity of the solution of the adjoint equation depending on how the extremal trajectory approaches the state constraint boundary are obtained. The proof uses the notion of closure with respect to measure of a Lebesgue measurable function and the Caratheodory theorem.

Combinatorial properties of support vectors of separating hyperplanes

In this paper we study the relationship between separating hyperplanes and the Radon partitions of their support vectors. This study is relevant for maximal margin separations, which appear in Support Vector Machines (SVM), as well as for separations that optimize a Chebyshev norm. We propose a new version of the Stiefel exchange algorithm where we exploit the property that each Stiefel exchange is in fact a Radon exchange. Originally, the Stiefel exchange algorithm was developed to find Chebyshev approximations, but we show that it is also suited for finding hyperplane separations. We also show that many important properties in approximation theory are closely related to fundamental results in convex set theory, in particular to Helly's, Radon's and Caratheodory's Theorem. Within this context, we prove a new result that generalizes both Radon's and Caratheodory's Theorem.

On the Zeros of Eigenpolynomials of Hermitian Toeplitz Matrices

This article refines a result of Delsarte, Genin, Kamp (Circuits Syst Signal Process 3:207–223, 1984), and Delsarte and Genin (Springer Lect Notes Control Inf Sci 58:194–213, 1984), regarding the number of zeros on the unit circle of eigenpolynomials of complex Hermitian Toeplitz matrices and generalized Caratheodory representations of such matrices. This is achieved by exploring a key observation of Schur (Uber einen Satz von C. Caratheodory. Sitzungsberichte der Koniglich Preusischen Akademie der Wissenschaften, pp. 4–15, 1912) stated in his proof of a famous theorem of Caratheodory (Rendiconti del Circolo Matematico di Palermo 32:193–217, 1911). In short, Schur observed that companion matrices corresponding to eigenpolynomials of Hermitian Toeplitz matrices H define isometries with respect to (spectrum shifted) submatrices of H. Looking at possible normal forms of these isometries leads directly to the results. This geometric, conceptual approach can be generalized to Hermitian or symmetric Toeplitz matrices over arbitrary fields. Furthermore, as a byproduct, Iohvidov’s law in the jumps of the ranks and the connection between the Iohvidov parameter and the Witt index are established for such Toeplitz matrices.

On Nonsmooth Semi-Infinite Minimax Programming Problem with(Φ,ρ)-Invexity

We are interested in a nonsmooth minimax programming Problem (SIP). Firstly, we establish the necessary optimality conditions theorems for Problem (SIP) when using the well-known Caratheodory's theorem. Under the Lipschitz(Φ,ρ)-invexity assumptions, we derive the sufficiency of the necessary optimality conditions for the same problem. We also formulate dual and establish weak, strong, and strict converse duality theorems for Problem (SIP) and its dual. These results extend several known results to a wider class of problems.

Borel-Carathéodory and Fan Type Inequalities for Hilbert Space Bicontractions

This paper deals with some operator versions of the classical Borel–Caratheodory and Ky Fan theorems for analytic functions of complex variable. Our results refer to the functional calculus with analytic functions on the unit bidisk induced by strict bicontractions on a complex Hilbert space.

Inequalities for angular derivatives and boundary interpolation

The classical Julia–Wolff–Caratheodory theorem asserts that the angular derivative of a holomorphic self-mapping of the open unit disk (Schur function) at its boundary fixed point is a positive number. Cowen and Pommerenke (J Lond Math Soc 26:271–289, 1982) proved that if a Schur function has several boundary regular fixed (or mutual contact) points, then the angular derivatives at these points are subject to certain inequalities. We develop a unified approach to establish relations between angular derivatives of Schur functions with a prescribed (possibly, infinite) collection of either mutual contact points or boundary fixed points. This approach yields diverse inequalities improving both classical and more recent results. We apply them to study the Nevanlinna–Pick interpolation problem with boundary data. Our methods lead to fairly explicit formulas describing the set of solutions.

Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

Lattice‐valued fuzzy measures are lattice‐valued set functions which assign the bottom element of the lattice to the empty set and the top element of the lattice to the entire universe, satisfying the additive properties and the property of monotonicity. In this paper, we use the lattice‐valued fuzzy measures and outer measure definitions and generalize the Caratheodory extension theorem for lattice‐valued fuzzy measures.

Carathéodory-type Theorems à la Bárány

We generalise the famous Helly–Lovasz theorem leading to a generalisation of the Barany–Caratheodory theorem for oriented matroids in dimension ≤3. We also provide a non-metric proof of the latter ...

Carathéodory-type Theorems à la Bárány

We generalise the famous Helly–Lovasz theorem leading to a generalisation of the Barany–Caratheodory theorem for oriented matroids in dimension ≤3. We also provide a non-metric proof of the latter colourful theorem for arbitrary dimensions and explore some generalisations in dimension 2.