On Eigenvalues of the Transition Matrix of Some Count-Data Markov Chains

We analyze the eigenstructure of count-data Markov chains. Our main focus is on so-called CLAR(1) models, which are characterized by having a linear conditional mean, and also on the case of a finite range, where the second largest eigenvalue determines the speed of convergence of the forecasting distributions. We derive a lower bound for the second largest eigenvalue, which often (but not always) even equals this eigenvalue. This becomes clear by deriving the complete set of eigenvalues for several specific cases of CLAR(1) models.     

Roots of Unity in Orders

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite rings. Our techniques, which are taken from commutative algebra, may have further potential in the context of cryptology and computer algebra.     

Micromorphic homogenisation and its application to a model of ductile damage

Heuristic gradient extensions of classical damage models are established to overcome their spurious mesh dependency. In contrast, an extension of Gurson's model of ductile damage towards the microdilatational theory is presented through a straight-forward homogenization. FEM simulations prove the regularization capabilities of the model. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of vortex roll-up processes using different wingtip shapes

Vortex roll-up processes of different generic wingtip shapes are investigated using the DLR TAU flow solver in combination with a Reynolds Stress Model. In contrast to a non-rounded layout an elliptical rounding of the wingtip planform leads to multiple vortices at the wingtip and results in an axial velocity deficit in the vortex core further downstream. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electrothermal Description of Organic Semiconductor Devices by p(x)-Laplace Thermistor Models

We consider a p(x)-Laplace thermistor model for electrothermal effects in organic semiconductor devices and propose a hybrid finite-volume/finite-element scheme which is combined with path-following methods to find a numerical solution. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any $k\ge 2$, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.     

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

In this paper we develop a relative Gröbner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and difference–differential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchberger-type algorithm for constructing relative Gröbner bases of filtered free modules.     

Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.