@phdthesis{Ziese2014,
  author    = {Ziese, Ramona},
  title     = {Geometric electroelasticity},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72504},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vi, 113},
  year      = {2014},
  abstract  = {In this work a diffential geometric formulation of the theory of electroelasticity is developed which also includes thermal and magnetic influences. We study the motion of bodies consisting of an elastic material that are deformed by the influence of mechanical forces, heat and an external electromagnetic field. To this end physical balance laws (conservation of mass, balance of momentum, angular momentum and energy) are established. These provide an equation that describes the motion of the body during the deformation. Here the body and the surrounding space are modeled as Riemannian manifolds, and we allow that the body has a lower dimension than the surrounding space. In this way one is not (as usual) restricted to the description of the deformation of three-dimensional bodies in a three-dimensional space, but one can also describe the deformation of membranes and the deformation in a curved space. Moreover, we formulate so-called constitutive relations that encode the properties of the used material. Balance of energy as a scalar law can easily be formulated on a Riemannian manifold. The remaining balance laws are then obtained by demanding that balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space. This generalizes a result by Marsden and Hughes that pertains to bodies that have the same dimension as the surrounding space and does not allow the presence of electromagnetic fields. Usually, in works on electroelasticity the entropy inequality is used to decide which otherwise allowed deformations are physically admissible and which are not. It is alsoemployed to derive restrictions to the possible forms of constitutive relations describing the material. Unfortunately, the opinions on the physically correct statement of the entropy inequality diverge when electromagnetic fields are present. Moreover, it is unclear how to formulate the entropy inequality in the case of a membrane that is subjected to an electromagnetic field. Thus, we show that one can replace the use of the entropy inequality by the demand that for a given process balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space and under linear rescalings of the temperature. On the one hand, this demand also yields the desired restrictions to the form of the constitutive relations. On the other hand, it needs much weaker assumptions than the arguments in physics literature that are employing the entropy inequality. Again, our result generalizes a theorem of Marsden and Hughes. This time, our result is, like theirs, only valid for bodies that have the same dimension as the surrounding space.},
  language  = {en}
}
@phdthesis{Zass2021,
  author    = {Zass, Alexander},
  title     = {A multifaceted study of marked Gibbs point processes},
  doi       = {10.25932/publishup-51277},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-512775},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 104},
  year      = {2021},
  abstract  = {This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry.},
  language  = {en}
}
@phdthesis{Zadorozhnyi2021,
  author    = {Zadorozhnyi, Oleksandr},
  title     = {Contributions to the theoretical analysis of the algorithms with adversarial and dependent data},
  school      = {Universit{\"a}t Potsdam},
  pages     = {144},
  year      = {2021},
  abstract  = {In this work I present the concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under a general functional weak-dependency assumption (the so-called \$\cC-\$mixing). The latter is then used to prove, in the asymptotic framework, excess risk upper bounds of the regularised Hilbert valued statistical learning rules under the τ-mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here, in particular, a question of robustness of the kernel-based forecaster is investigated. Afterwards, in the framework of sequential learning, the multi-armed bandit problem under \$\cC-\$mixing assumption on the arm's outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly, probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma-Hoeffding's and of Burkholder's type) to the partial sums of real-valued weakly dependent random fields (under the type of projective dependence condition).},
  language  = {en}
}
@phdthesis{Witt2003,
  author    = {Witt, Ingo},
  title     = {Beitr{\"a}ge zur Theorie singul{\"a}rer hyperbolischer Differentialoperatoren},
  address   = {Potsdam},
  pages     = {121 S.},
  year      = {2003},
  language  = {de}
}
@phdthesis{Wichitsanguan2016,
  author    = {Wichitsa-nguan, Korakot},
  title     = {Modifications and extensions of the logistic regression and Cox model},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-90033},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 131},
  year      = {2016},
  abstract  = {In many statistical applications, the aim is to model the relationship between covariates and some outcomes. A choice of the appropriate model depends on the outcome and the research objectives, such as linear models for continuous outcomes, logistic models for binary outcomes and the Cox model for time-to-event data. In epidemiological, medical, biological, societal and economic studies, the logistic regression is widely used to describe the relationship between a response variable as binary outcome and explanatory variables as a set of covariates. However, epidemiologic cohort studies are quite expensive regarding data management since following up a large number of individuals takes long time. Therefore, the case-cohort design is applied to reduce cost and time for data collection. The case-cohort sampling collects a small random sample from the entire cohort, which is called subcohort. The advantage of this design is that the covariate and follow-up data are recorded only on the subcohort and all cases (all members of the cohort who develop the event of interest during the follow-up process). In this thesis, we investigate the estimation in the logistic model for case-cohort design. First, a model with a binary response and a binary covariate is considered. The maximum likelihood estimator (MLE) is described and its asymptotic properties are established. An estimator for the asymptotic variance of the estimator based on the maximum likelihood approach is proposed; this estimator differs slightly from the estimator introduced by Prentice (1986). Simulation results for several proportions of the subcohort show that the proposed estimator gives lower empirical bias and empirical variance than Prentice's estimator. Then the MLE in the logistic regression with discrete covariate under case-cohort design is studied. Here the approach of the binary covariate model is extended. Proving asymptotic normality of estimators, standard errors for the estimators can be derived. The simulation study demonstrates the estimation procedure of the logistic regression model with a one-dimensional discrete covariate. Simulation results for several proportions of the subcohort and different choices of the underlying parameters indicate that the estimator developed here performs reasonably well. Moreover, the comparison between theoretical values and simulation results of the asymptotic variance of estimator is presented. Clearly, the logistic regression is sufficient for the binary outcome refers to be available for all subjects and for a fixed time interval. Nevertheless, in practice, the observations in clinical trials are frequently collected for different time periods and subjects may drop out or relapse from other causes during follow-up. Hence, the logistic regression is not appropriate for incomplete follow-up data; for example, an individual drops out of the study before the end of data collection or an individual has not occurred the event of interest for the duration of the study. These observations are called censored observations. The survival analysis is necessary to solve these problems. Moreover, the time to the occurence of the event of interest is taken into account. The Cox model has been widely used in survival analysis, which can effectively handle the censored data. Cox (1972) proposed the model which is focused on the hazard function. The Cox model is assumed to be λ(t|x) = λ0(t) exp(β^Tx) where λ0(t) is an unspecified baseline hazard at time t and X is the vector of covariates, β is a p-dimensional vector of coefficient. In this thesis, the Cox model is considered under the view point of experimental design. The estimability of the parameter β0 in the Cox model, where β0 denotes the true value of β, and the choice of optimal covariates are investigated. We give new representations of the observed information matrix In(β) and extend results for the Cox model of Andersen and Gill (1982). In this way conditions for the estimability of β0 are formulated. Under some regularity conditions, ∑ is the inverse of the asymptotic variance matrix of the MPLE of β0 in the Cox model and then some properties of the asymptotic variance matrix of the MPLE are highlighted. Based on the results of asymptotic estimability, the calculation of local optimal covariates is considered and shown in examples. In a sensitivity analysis, the efficiency of given covariates is calculated. For neighborhoods of the exponential models, the efficiencies have then been found. It is appeared that for fixed parameters β0, the efficiencies do not change very much for different baseline hazard functions. Some proposals for applicable optimal covariates and a calculation procedure for finding optimal covariates are discussed. Furthermore, the extension of the Cox model where time-dependent coefficient are allowed, is investigated. In this situation, the maximum local partial likelihood estimator for estimating the coefficient function β(·) is described. Based on this estimator, we formulate a new test procedure for testing, whether a one-dimensional coefficient function β(·) has a prespecified parametric form, say β(·; ϑ). The score function derived from the local constant partial likelihood function at d distinct grid points is considered. It is shown that the distribution of the properly standardized quadratic form of this d-dimensional vector under the null hypothesis tends to a Chi-squared distribution. Moreover, the limit statement remains true when replacing the unknown ϑ0 by the MPLE in the hypothetical model and an asymptotic α-test is given by the quantiles or p-values of the limiting Chi-squared distribution. Finally, we propose a bootstrap version of this test. The bootstrap test is only defined for the special case of testing whether the coefficient function is constant. A simulation study illustrates the behavior of the bootstrap test under the null hypothesis and a special alternative. It gives quite good results for the chosen underlying model. References P. K. Andersen and R. D. Gill. Cox's regression model for counting processes: a large samplestudy. Ann. Statist., 10(4):1100{1120, 1982. D. R. Cox. Regression models and life-tables. J. Roy. Statist. Soc. Ser. B, 34:187{220, 1972. R. L. Prentice. A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika, 73(1):1{11, 1986.},
  language  = {en}
}
@phdthesis{Welke1996,
  author    = {Welke, Dirk},
  title     = {Hyperidentit{\"a}ten partieller Algebren},
  pages     = {104 S.},
  year      = {1996},
  language  = {de}
}
@phdthesis{Wei2008,
  author    = {Wei, Yawei},
  title     = {Operators on manifolds with edge and boundary with singular trace conditions},
  address   = {Potsdam},
  pages     = {128 S.: graph. Darst.},
  year      = {2008},
  language  = {en}
}
@phdthesis{Wei2008,
  author    = {Wei, Yawei},
  title     = {Operators on Manifolds with Edge and Boundary with Singular Trace Conditions},
  address   = {Potsdam},
  pages     = {127 S., : graph. Darst.},
  year      = {2008},
  language  = {en}
}
@phdthesis{Waltz1994,
  author    = {Waltz, Christine},
  title     = {Adjungierte Polynomideale und duale Typen auf Banachr{\"a}umen und ihre Anwendung},
  pages     = {109 Bl.},
  year      = {1994},
  language  = {de}
}
@phdthesis{Wallenta2015,
  author    = {Wallenta, Daniel},
  title     = {Sequences of compact curvature},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-87489},
  school      = {Universit{\"a}t Potsdam},
  pages     = {viii, 73},
  year      = {2015},
  abstract  = {By perturbing the differential of a (cochain-)complex by "small" operators, one obtains what is referred to as quasicomplexes, i.e. a sequence whose curvature is not equal to zero in general. In this situation the cohomology is no longer defined. Note that it depends on the structure of the underlying spaces whether or not an operator is "small." This leads to a magical mix of perturbation and regularisation theory. In the general setting of Hilbert spaces compact operators are "small." In order to develop this theory, many elements of diverse mathematical disciplines, such as functional analysis, differential geometry, partial differential equation, homological algebra and topology have to be combined. All essential basics are summarised in the first chapter of this thesis. This contains classical elements of index theory, such as Fredholm operators, elliptic pseudodifferential operators and characteristic classes. Moreover we study the de Rham complex and introduce Sobolev spaces of arbitrary order as well as the concept of operator ideals. In the second chapter, the abstract theory of (Fredholm) quasicomplexes of Hilbert spaces will be developed. From the very beginning we will consider quasicomplexes with curvature in an ideal class. We introduce the Euler characteristic, the cone of a quasiendomorphism and the Lefschetz number. In particular, we generalise Euler's identity, which will allow us to develop the Lefschetz theory on nonseparable Hilbert spaces. Finally, in the third chapter the abstract theory will be applied to elliptic quasicomplexes with pseudodifferential operators of arbitrary order. We will show that the Atiyah-Singer index formula holds true for those objects and, as an example, we will compute the Euler characteristic of the connection quasicomplex. In addition to this we introduce geometric quasiendomorphisms and prove a generalisation of the Lefschetz fixed point theorem of Atiyah and Bott.},
  language  = {en}
}