This paper deals with the joint and conditional distributions concerning the maximum of random walk paths and the number of times this maximum is achieved. This joint distribution was studied first by Dwass [1967]. Based on his result, the correlation and some conditional moments are derived. The main contributions are however asymptotic expansions concerning the conditional distribution and conditional moments. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

La sécurité de la cryptographie à clés publiques repose sur des problèmes mathématiques difficiles, notamment en théorie des nombres, tels que la factorisation pour RSA ou le logarithme discret pour ElGamal. Cependant les progrès des algorithmes rendent les protocoles basés sur des problèmes de théorie des nombres de moins en moins efficaces. De plus, l'arrivée de l'ordinateur quantique rendrait ces cryptosystèmes inutilisables. La cryptographie basée sur les codes en métrique rang est une alternative crédible pour concevoir des cryptosystèmes post-quantiques en raison de sa rapidité et de la faible taille de ses clés. Le but de cette thèse est d'étudier les problèmes difficiles en métrique rang et les algorithmes permettant de les résoudre, ainsi que de chercher de nouvelles attaques et de nouvelles primitives basées sur ces problèmes. / Security of public keys cryptography is based on difficult mathematic problems, especially in number field theory, such as the factorization for RSA or the discrete logarithm for ElGamal. However, algorithms are more and more efficient to solve these problems. Furthermore, quantum computers would be able to easily break these cryptosystems. Code-based cryptography in rank metric is a solid candidate to design new postquatum cryptosystems since it is fast and has low weight keysize. The goals of this thesis are to study hard problems in rank metric and algorithms which solve them, also to search for new attacks and new primitives based on these problems.

Development of Cybersecurity processes and strategies should take two main approaches. One is to develop an efficient and effective set of methodologies to identify software vulnerabilities and patch them before being exploited. Second is to develop a set of methodologies to predict the behavior of attackers and execute defending techniques based on attacking behavior. Managing of Vulnerabilities and analyzing them is directly related to the first approach. Developing of methodologies and models to predict the behavior of attackers is related to the second approach. Both these approaches are inseparably interconnected. Our effort in this study mainly focuses on developing useful statistical models that can give us signals about the behavior of cyber attackers. Analytically understanding of vulnerabilities in statistical point of view helps to develop a set of statistical models that works as a bridge between Cybersecurity and Abstract Statistical and Mathematical knowledge. Any such effort should begin with properly understanding the nature of Vulnerabilities in a computer network system. We start this study with analyzing "Vulnerability" based on inferences that can be taken from National Vulnerability Database (NVD). In Cybersecurity context, we apply Markov approach to develop suitable predictive models to successfully estimate the minimum number of steps to compromise a security goal that an attacker would take using the concept of Expected Path Length (EPL). We have further developed Non-Homogeneous Stochastic model by improving EPL estimates in to a time dependent variable. This approach analytically applied in a simple model of computer network with discovered vulnerabilities resulted in several useful observations exemplifying the applicability in real world computer systems. The methodology indicated a measure of the "Risk" associated with the model network as a function of time indicating defending professionals on the threats they are facing and should anticipate to face. Furthermore, using a similar approach taken in well-known Google page rank algorithm, a new ranking algorithm of vulnerability ranks with respect to time for computer network system is also presented in this study. With better IT resources analytical models and methodologies presented in this study can be developed into more generalized versions and apply in real world computer network environments.

COOPERATIVE WIDEBAND SPECTRUM SENSING BASED ON JOINT SPARSITY By Ghazaleh Jowkar, Master of Science A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University Virginia Commonwealth University 2017 Major Director: Dr. Ruixin Niu, Associate Professor of Department of Electrical and Computer Engineering In this thesis, the problem of wideband spectrum sensing in cognitive radio (CR) networks using sub-Nyquist sampling and sparse signal processing techniques is investigated. To mitigate multi-path fading, it is assumed that a group of spatially dispersed SUs collaborate for wideband spectrum sensing, to determine whether or not a channel is occupied by a primary user (PU). Due to the underutilization of the spectrum by the PUs, the spectrum matrix has only a small number of non-zero rows. In existing state-of-the-art approaches, the spectrum sensing problem was solved using the low-rank matrix completion technique involving matrix nuclear-norm minimization. Motivated by the fact that the spectrum matrix is not only low-rank, but also sparse, a spectrum sensing approach is proposed based on minimizing a mixed-norm of the spectrum matrix instead of low-rank matrix completion to promote the joint sparsity among the column vectors of the spectrum matrix. Simulation results are obtained, which demonstrate that the proposed mixed-norm minimization approach outperforms the low-rank matrix completion based approach, in terms of the PU detection performance. Further we used mixed-norm minimization model in multi time frame detection. Simulation results shows that increasing the number of time frames will increase the detection performance, however, by increasing the number of time frames after a number of times the performance decrease dramatically.

Une classe générale de statistiques de rangs basées sur la fonction caractéristique est introduite afin de tester l'hypothèse composite d'appartenance à une famille de copules multidimensionnelles. Ces statistiques d'adéquation sont définies comme des distances fonctionnelles de type L_2 pondérées entre une version non paramétrique et une version semi-paramétrique de la fonction caractéristique que l'on peut associer à une copule. Il est démontré que ces statistiques de test se comportent asymptotiquement comme des V-statistiques dégénérées d'ordre quatre et que leurs lois limites s'expriment en termes de sommes pondérées de variables khi-deux indépendantes. La convergence des tests sous des alternatives générales est établie, de même que la validité du bootstrap paramétrique pour le calcul de valeurs critiques. Le comportement des nouveaux tests sous des tailles d'échantillons faibles et modérées est étudié à l'aide de simulations et est comparé à celui d'un test concurrent fondé sur la copule empirique. La méthodologie est finalement illustrée sur un jeu de données à plusieurs dimensions.

We consider a quantum system whose unperturbed form consists of a self-adjoint Δ-operator on a 2-dimensional compact Riemannian manifold, which may or may not have a boundary. Then as a perturbation, we add a delta potential/point scatterer at some select point ρ. The perturbed self-adjoint operator is constructed rigorously by means of self-adjoint extension theory. We also consider a corresponding classical dynamical system on the cotangent/cosphere bundle, consisting of geodesic flow on the manifold, with specular reflection if there is a boundary. Chapter 2 describes the mathematics of the unperturbed and perturbed quantum systems, as well as outlining the classical dynamical system. Included in the discussion on the delta-perturbed quantum system is consideration concerning the strength of the delta potential. It is reckoned that the delta potential effectively has negative infinitesimal strength. Chapter 3 continues on with investigations from [KMW10], concerned with perturbed eigenfunctions that approximate to a linear combination of only two "surrounding" unperturbed eigenfunctions. In Thm. 4.4 of [KMW10], conditions are derived under which a sequence of perturbed eigenfunctions exhibits this behaviour in the limit. The approximating pair linear combinations belong to a class of quasimodes constructed within [KMW10]. The aim of Chapter 3 in this thesis is to improve on the result in [KMW10]. In Chapter 3, preliminary results are first derived constituting a broad consideration of the question of when a perturbed eigenfunction subsequence approaches linear combinations of only two surrounding unperturbed eigenfunctions. Afterwards, the central result of this Chapter, namely Thm. 3.4.1, is derived, which serves as an improved version of Thm. 4.4 in [KMW10]. The conditions of this theorem are shown to be weaker than those in [KMW10]. At the same time though, the conclusion does not require the approximating pair linear combinations to be quasimodes contained in the domain of the perturbed operator. Cor. 3.5.2 allows for a transparent comparison between the results of this Chapter and [KMW10]. Chapter 4 deals with the construction of non-singular rank-one perturbations for which the eigenvalues and eigenfunctions approximate those of the delta-perturbed operator. This is approached by means of direct analysis of the construction and formulae for the rank-one-perturbed eigenvalues and eigenfunctions, by comparison that of the delta-perturbed eigenvalues and eigenfunctions. Successful results are derived to this end, the central result being Thm. 4.4.19. This provides conditions on a sequence of non-singular rank-one perturbations, under which all eigenvalues and eigenbasis members within an interval converge to those of the delta-perturbed operator. Comparisons have also been drawn with previous literature such as [Zor80], [AK00] and [GN12]. These deal with rank-one perturbations approaching the delta potential within the setting of a whole Euclidean space Rⁿ, for example by strong resolvent convergence, and by limiting behaviour of generalised eigenfunctions associated with energies at every Eℓ(0,∞). Furthermore in Chapter 4, the suggestion from Chapter 2 that the delta potential has negative infinitessimal strength is further supported, due to the coefficients of the approximating rank-one perturbations being negative and tending to zero. This phenomenon is also in agreement with formulae from [Zor80], [AK00] and [GN12]. Chapter 5 first reviews the correspondence between certain classical dynamics and equidistribution in position space of almost all unperturbed quantum eigenfunctions, as demonstrated for example in [MR12]. Equidistribution in position space of almost all perturbed eigenfunctions, in the case of the 2D rectangular flat torus, is also reviewed. This result comes from [RU12], which is only stated in terms of the "new" perturbed eigenfunctions, which would only be a subset of the full perturbed eigenbasis. Nevertheless, in this Chapter it is explained how it follows that this position space equidistribution result also applies to a full-density subsequence of the full perturbed eigenbasis. Finally three methods of approach are discussed for attempting to derive this position space equidistribution result in the case of a more general delta-perturbed system whose classical dynamics satisfies the particular key property.

The thesis deals with a topic of SEO onpage signals, which are important for search engines because of sorting pages in a search engine result page. It focuses on importance of these signals for mobile SERP. Main goals of this study are to describe current recommendations for SEO on-page factors for mobile web and experimentally test real importance of these signals. Based on the results I composed an optimal set of factors with the most benefit for SEO. Theoretical part of the study summarizes basic facts about mobile searching, describes specific mobile users behaviour and describes current recommendations for mobile web onpage optimizing from Google and Seznam.cz. In practical part there is a comparative study of chosen on-page signals. For its needs I had to create few one page static websites. Each one has been optimized for on factor. All websites focused on the same very specific topic to ensure the same initial conditions. By keywords rank tracking in a SERP I was able to determine which signal is more important than others for search engines. The study results contribute to actual evaluation of each on-page signals importance for mobile website. The study could be beneficial for smaller companys websites, which need to get more visible on the net. They are able to optimize their costs by choosing the right set of on-page factors.

abstract: C*-algebras of categories of paths were introduced by Spielberg in 2014 and generalize C*-algebras of higher rank graphs. An approximately finite dimensional (AF) C*-algebra is one which is isomorphic to an inductive limit of finite dimensional C*-algebras. In 2012, D.G. Evans and A. Sims proposed an analogue of a cycle for higher rank graphs and show that the lack of such an object is necessary for the associated C*-algebra to be AF. Here, I give a class of examples of categories of paths whose associated C*-algebras are Morita equivalent to a large number of periodic continued fraction AF algebras, first described by Effros and Shen in 1980. I then provide two examples which show that the analogue of cycles proposed by Evans and Sims is neither a necessary nor a sufficient condition for the C*-algebra of a category of paths to be AF. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2020

In many high dimensional problems, the dependence structure among the variables can be quite complex. An appropriate use of the regularization techniques coupled with other classical statistical methods can often improve estimation and prediction accuracy and facilitate model interpretation, by seeking a parsimonious model representation that involves only the subset of revelent variables. We propose two regularized stochastic regression approaches, for efficiently estimating certain sparse dependence structure in the data. We first consider a multivariate regression setting, in which the large number of responses and predictors may be associated through only a few channels/pathways and each of these associations may only involve a few responses and predictors. We propose a regularized reduced-rank regression approach, in which the model estimation and rank determination are conducted simultaneously and the resulting regularized estimator of the coefficient matrix admits a sparse singular value decomposition (SVD). Secondly, we consider model selection of subset autoregressive moving-average (ARMA) modelling, for which automatic selection methods do not directly apply because the innovation process is latent. We propose to identify the optimal subset ARMA model by fitting a penalized regression, e.g. adaptive Lasso, of the time series on its lags and the lags of the residuals from a long autoregression fitted to the time-series data, where the residuals serve as proxies for the innovations. Computation algorithms and regularization parameter selection methods for both proposed approaches are developed, and their properties are explored both theoretically and by simulation. Under mild regularity conditions, the proposed methods are shown to be selection consistent, asymptotically normal and enjoy the oracle properties. We apply the proposed approaches to several applications across disciplines including cancer genetics, ecology and macroeconomics.

In this era of big data, multivariate time-series (MTS) data are prevalent in diverse domains and often high dimensional. However, there have been limited studies of building a capable classifier with MTS via classical machine learning methods that can deal with the double curse of dimensionality due to high variable dimension and long time series (large sample size). In this thesis, we propose two approaches to address this problem for multiclass classification with high dimensional MTS. Both approaches leverage the dynamics of an MTS captured by non-parametric modeling of its spectral density function. In the first approach, we introduce the reduced-rank spectral classifier (RRSC), which utilizes low-rank estimation and some new discrimination functions. We illustrate the efficacy of the RRSC with both simulations and real applications. For binary classification, we establish the consistency of the RRSC and provide an asymptotic formula for the misclassification error rates, under some regularity conditions. The second approach concerns the development of the random projection ensemble classifier for time series (RPECTS). This method first applies dimension reduction in the time domain via projecting the time-series variables into some low dimensional space, followed by measuring the disparity via some novel base classifier between the data and the candidate generating processes in the projected space. We assess the classification performance of our new approaches by simulations and compare them with some existing methods using real applications. Finally, we elaborate two R packages that implement the aforementioned methods.