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- ItemAssessment of Stability in Partitional Clustering Using Resampling Techniques(Karlsruhe : KIT Scientific Publishing, 2016) Mucha, Hans-JoachimThe assessment of stability in cluster analysis is strongly related to the main difficult problem of determining the number of clusters present in the data. The latter is subject of many investigations and papers considering different resampling techniques as practical tools. In this paper, we consider non-parametric resampling from the empirical distribution of a given dataset in order to investigate the stability of results of partitional clustering. In detail, we investigate here only the very popular K-means method. The estimation of the sampling distribution of the adjusted Rand index (ARI) and the averaged Jaccard index seems to be the most general way to do this. In addition, we compare bootstrapping with different subsampling schemes (i.e., with different cardinality of the drawn samples) with respect to their performance in finding the true number of clusters for both synthetic and real data.
- ItemA boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions(Berlin ; Boston, Mass. : de Gruyter, 2015) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenA boundary control problem for the pure Cahn–Hilliard equations with possibly singular potentialsand dynamic boundary conditions is studied and rst-order necessary conditions for optimality are proved.
- ItemDistributed optimal control of a nonstandard nonlocal phase field system(Springfield, MO : AIMS Press, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenWe investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
- ItemWeak-strong uniqueness for the general Ericksen-Leslie system in three dimensions(Springfield, Mo. : American Institute of Mathematical Sciences, 2018) Emmrich, Etienne; Lasarzik, RobertWe study the Ericksen-Leslie system equipped with a quadratic free energy functional. The norm restriction of the director is incorporated by a standard relaxation technique using a double-well potential. We use the relative energy concept, often applied in the context of compressible Euler- or related systems of fluid dynamics, to prove weak-strong uniqueness of solutions. A main novelty, not only in the context of the Ericksen-Leslie model, is that the relative energy inequality is proved for a system with a nonconvex energy.
- ItemPrevention and trust evaluation scheme based on interpersonal relationships for large-scale peer-to-peer networks(New York, NY : Hindawi Publishing Corporation, 2014) Li, L.; Kurths, J.; Yang, Y.; Liu, G.In recent years, the complex network as the frontier of complex system has received more and more attention. Peer-to-peer (P2P) networks with openness, anonymity, and dynamic nature are vulnerable and are easily attacked by peers with malicious behaviors. Building trusted relationships among peers in a large-scale distributed P2P system is a fundamental and challenging research topic. Based on interpersonal relationships among peers of large-scale P2P networks, we present prevention and trust evaluation scheme, called IRTrust. The framework incorporates a strategy of identity authentication and a global trust of peers to improve the ability of resisting the malicious behaviors. It uses the quality of service (QoS), quality of recommendation (QoR), and comprehensive risk factor to evaluate the trustworthiness of a peer, which is applicable for large-scale unstructured P2P networks. The proposed IRTrust can defend against several kinds of malicious attacks, such as simple malicious attacks, collusive attacks, strategic attacks, and sybil attacks. Our simulation results show that the proposed scheme provides greater accuracy and stronger resistance compared with existing global trust schemes. The proposed scheme has potential application in secure P2P network coding.
- ItemSynchronization of a Class of Memristive Stochastic Bidirectional Associative Memory Neural Networks with Mixed Time-Varying Delays via Sampled-Data Control(London : Hindawi Limited, 2018) Yuan, M.; Wang, W.; Luo, X.; Ge, C.; Li, L.; Kurths, J.; Zhao, W.The paper addresses the issue of synchronization of memristive bidirectional associative memory neural networks (MBAMNNs) with mixed time-varying delays and stochastic perturbation via a sampled-data controller. First, we propose a new model of MBAMNNs with mixed time-varying delays. In the proposed approach, the mixed delays include time-varying distributed delays and discrete delays. Second, we design a new method of sampled-data control for the stochastic MBAMNNs. Traditional control methods lack the capability of reflecting variable synaptic weights. In this paper, the methods are carefully designed to confirm the synchronization processes are suitable for the feather of the memristor. Third, sufficient criteria guaranteeing the synchronization of the systems are derived based on the derive-response concept. Finally, the effectiveness of the proposed mechanism is validated with numerical experiments.
- ItemLarge Deviations of Continuous Regular Conditional Probabilities(New York, NY [u.a.] : Springer Science + Business Media B.V., 2016) van Zuijlen, W.We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate.
- ItemSequential decision problems, dependent types and generic solutions(Braunschweig : Department of Theoretical Computer Science, Technical University of Braunschweig, 2017) Botta, N.; Jansson, P.; Ionescu, C.; Christiansen, D.R.; Brady, E.We present a computer-checked generic implementation for solving finite-horizon sequential decision problems. This is a wide class of problems, including inter-temporal optimizations, knapsack, optimal bracketing, scheduling, etc. The implementation can handle time-step dependent control and state spaces, and monadic representations of uncertainty (such as stochastic, non-deterministic, fuzzy, or combinations thereof). This level of genericity is achievable in a programming language with dependent types (we have used both Idris and Agda). Dependent types are also the means that allow us to obtain a formalization and computer-checked proof of the central component of our implementation: Bellman’s principle of optimality and the associated backwards induction algorithm. The formalization clarifies certain aspects of backwards induction and, by making explicit notions such as viability and reachability, can serve as a starting point for a theory of controllability of monadic dynamical systems, commonly encountered in, e.g., climate impact research.
- ItemCanonical sets of best L1-approximation(New York, NY : Hindawi, 2012) Dryanov, D.; Petrov, P.In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L 1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities.
- ItemA rough path perspective on renormalization(Amsterdam [u.a.] : Elsevier, 2019) Bruned, Y.; Chevyrev, I.; Friz, P.K.; Preiß, R.We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned–Hairer–Zambotti (2016), the links with which are discussed in detail. © 2019 The Author(s)