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The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequences "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means.
The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.

In a bounded domain with smooth boundary in R^3 we consider the stationary Maxwell equations
for a function u with values in R^3 subject to a nonhomogeneous condition
(u,v)_x = u_0 on
the boundary, where v is a given vector field and u_0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.

The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine’s word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.

This essay reads Sam Selvon’s novel The Lonely Londoners (1956) as a milestone in the decolonisation of British fiction. After an introduction to Selvon and the core composition of the novel, it discusses the ways in which the narrative takes on issues of race and racism, how it in the tradition of the Trinidadian carnival confronts audiences with sexual profanation and black masculine swagger, and not least how the novel, especially through its elaborate use of creole Englishes, reimagines London as a West Indian metropolis. The essay then turns more systematically to the ways in which Selvon translates Western literary models and their isolated subject positions into collective modes of narrative performance taken from Caribbean orature and the calypsonian tradition. The Lonely Londoners breathes entirely new life into the ossified conventions of the English novel, and imbues it with unforeseen aesthetic, ethical, political and epistemological possibilities.

This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.

In order to evade detection by network-traffic analysis, a growing proportion of malware uses the encrypted HTTPS protocol. We explore the problem of detecting malware on client computers based on HTTPS traffic analysis. In this setting, malware has to be detected based on the host IP address, ports, timestamp, and data volume information of TCP/IP packets that are sent and received by all the applications on the client. We develop a scalable protocol that allows us to collect network flows of known malicious and benign applications as training data and derive a malware-detection method based on a neural networks and sequence classification. We study the method's ability to detect known and new, unknown malware in a large-scale empirical study.

Convoluted Brownian motion
(2016)

In this paper we analyse semimartingale properties of a class of Gaussian periodic processes, called convoluted Brownian motions, obtained by convolution between a deterministic function and a Brownian motion. A classical
example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are neither
Markovian nor satisfy a time-Markov field property. Nevertheless, by enlargement
of filtration and/or addition of a one-dimensional component, one can in some case recover the Markovianity. We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the higher-dimensional monomial convoluted Brownian motion.

We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set.

We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true
regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.

We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed
problems, and construct an explicit formula for approximate solutions.

When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for
the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on
a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.

Using an algorithm based on a retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical difficulty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.

We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.

Reflections of Lusáni Cissé
(2016)

Postcolonial Justice
(2016)

Kleine Kosmopolitismen
(2016)

Recollecting Bones
(2016)

In the same “guarded, roundabout and reticent way” which Lindsay Barrett invokes for Australian conversations about imperial injustice, Germans, too, must begin to more systematically explore, in Paul Gilroy’s words, “the connections and the differences between anti-semitism and anti-black and other racisms and asses[s] the issues that arise when it can no longer be denied that they interacted over a long time in what might be seen as Fascism’s intellectual, ethical and scientific pre-history” (Gilroy 1996: 26). In the meantime, we need to care for the dead. We need to return them, first, from the status of scientific objects to the status of ancestral human beings, and then progressively, and proactively, as close as possible to the care of those communities from whom they were stolen.