Cohomology of fixed point sets and deformation of algebras

Under certain conditions the cohomology algebra (with appropriate coefficients) of the fixed point set of a G-space (G=Z/pZ=Zp, p prime or G=S¹) can be considered a deformation - in a purely algebraic sense - of the cohomology algebra of the space itself. On one hand this gives restrictions on the isomorphism type of algebras that can occur as cohomology algebra of the fixed point set of G-spaces if the cohomology algebra of the space itself is given. On the other hand it leads to the definition of obstructions for the cohomology algebra of the fixed point set to be isomorphic (as an ungraded algebra) to the cohomology algebra of the space itself.     

Soliton Theory, Symmetric Functions and Matrix Integrals

We consider a certain scalar product of symmetric functions which is parameterized by a function r and an integer n. On the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of this product with the help of multi-integrals. This gives links between a theory of symmetric functions, soliton theory and models of random matrices (such as a model of normal matrices).     

Thermomechanical behavior of pseudoelastic NiTi shape memory alloys

The thermomechanical behavior of polycrystalline Ni-rich pseudoelastic NiTi shape memory alloys is analyzed. Special focus is on regions within the stress strain diagram which are regarded as linear elastic in common phenomenological material models, i.e. the region between zero stress and the beginning of the pseudoelastic plateau as well as the region within the hysteresis. In both cases, severe temperature changes can be observed. A possible explanation for this effect is twofold: On the one hand, it might be explained by the presence of an R-phase transformation. On the other hand, unstructured martensite of the B19' phase may form. However, the assumption of a purely thermo-elastic material behavior in those regions does not seem to hold true in general. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantum mechanics as the quadratic Taylor approximation of classical mechanics: The finite-dimensional case

We show that in contrast to a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. We consider an approximation based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the second-order term. To escape technical difficulties related to the infinite dimensionality of the phase space for quantum mechanics, we consider finite-dimensional quantum mechanics. On one hand, this is a simple example with high pedagogical value. On the other hand, quantum information operates in a finite-dimensional state space. Therefore, our investigation can be considered a construction of a classical statistical model for quantum information. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 278–291, August, 2007.     

Paid-incurred chain claims reserving method

We present a novel stochastic model for claims reserving that allows us to combine claims payments and incurred losses information. The main idea is to combine two claims reserving models (Hertig’s (1985) model and Gogol’s (1993) model ) leading to a log-normal paid-incurred chain (PIC) model. Using a Bayesian point of view for the parameter modelling we derive in this Bayesian PIC model the full predictive distribution of the outstanding loss liabilities. On the one hand, this allows for an analytical calculation of the claims reserves and the corresponding conditional mean square error of prediction. On the other hand, simulation algorithms provide any other statistics and risk measure on these claims reserves.     

Measure-free viewpoint on p-adic and adelic wavelets

Recently there were presented new examples of MRAs based on p-adic and adelic wavelets. This is a note of the methodological nature. Its aim is embed all known examples MRAs into the measure-free scheme based on abstract Hilbert space formalism. On the one hand, such an embedding unifies the known examples of MRAs. On the other hand, the presented measure free scheme can be used for construction of new examples of MRAs, especially in the infinite dimensional case. The last section of this paper is devoted to another abstract MRAs scheme based on Riesz spaces. We start paper with a review on p-adic and adelic wavelets.     

Nonintersecting Paths, Pfaffians, and ?-Matroids

In this paper, we study -matroids induced by nonintersecting paths in a directed graph. The association between nonintersecting paths and -matroids is derived from Pfaffians. On the one hand, certain numbers ofk-tuples of nonintersecting paths may (often) be expressed as a Pfaffian, while, on the other hand, representability problems for -matroids may be studied in terms of Pfaffians. It is shown that -matroids induced by nonintersecting paths are representable over fields of any characteristic and that weightings defined on the edge set and with values in some linearly ordered abelian group give rise to valuated -matroids.     

The sl3 Jones polynomial of the trefoil: A case study of q-holonomic sequences

The ${\mathfrak{sl}}_3$ colored Jones polynomial of the trefoil knot is a q-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.     

Long time asymptotics for 2 by 2 hyperbolic systems

The goal of this paper is to study the behavior of the energy for 2 by 2 strictly hyperbolic systems. On the one hand we are interested in generalized energy conservation which excludes blow-up and decay of the energy for t→∞. On the other hand we present scattering results which take account of terms of order zero.     

From how to why: A quest for the common mathematical meanings behind two different division algorithms

During their education cycle in mathematics, students are exposed to algorithms as early as primary school. Several studies show how students frequently learn to perform these algorithms without controlling the mathematical meanings behind them. On the other hand, several National Standards have highlighted the need to construct meanings in mathematics from the first cycle of education. In this paper we focus on division algorithms, investigating to what extent 6th graders can be guided to understanding the whys behind an algorithm, through the comparison of two different algorithms for integer division. Our results suggest, on the one hand, that “it could work!”, and on the other hand, that the exposure to different algorithms for the same mathematical operation seems particularly significant for bringing out the whys behind such algorithms, as well as for capturing the difference between a mathematical operation and algorithms for calculating the result of such an operation.     

Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.     

Metric aggregation functions revisited

In 1981, J. Borsík and J. Doboš analyzed what properties a function must satisfy in order to merge a collection of metric spaces into a single one. Later on, E. Castiñeira, A. Pradera and E. Trillas studied a variant of the same problem in which each metric of the collection to be merged is defined on the same non-empty set. In this, paper we continue the work in this last aforesaid direction. On the one hand, we provide a new characterization of such functions and a few methods to construct them. On the other hand, we analyze the existence of absorbent, idempotent and neutral elements for such class of functions and, thus, we design techniques that allow to discard those functions that are not useful for merging metrics. Finally, we discuss when the functions under study preserve metrics.     

Richard von Mises and the “problem of two races”: A statistical satire in 1934

The article “Problem of two races,” which appeared in 1934 in French and Russian in the Matematichesky Sbornik (Moscow), authored by the noted applied mathematician and statistician Richard von Mises (1883–1953), is interpreted as having a twofold agenda. On its surface it is merely a detailed mathematical analysis of the statistical problem of comparing the distributions of a quantitative characteristic in two different classes. But on closer examination it serves two auxiliary purposes. On the one hand, von Mises, who as a Jew fled Nazi Germany in 1933, satirically attacked Nazi racial doctrines by resorting to statistical parody. On the other hand, von Mises employed his theoretical results to further his own distinctive program of objective Bayesian statistics.     

Nonlocal second-order geometric equations arising in tomographic reconstruction

In this paper, we study new nonlocal geometric equations which are related to tomographic reconstruction when using the level-set approach. We treat two additional difficulties which make the work original. On one hand, the level lines do not evolve along normal directions and the nonlocal term is not of “convolution type”. On the other hand, the speed is not necessarily bounded compared to the nonlocal term. We prove an existence and uniqueness result for our equation.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

Irreducible Fuchsian system with reducible monodromy representation

We present an example of the reducible representation χ = χ₁ ? χ₂, which, on the one hand, is the monodromy representation of a Fuchsian system. On the other hand, the representation χ₂ is a counterexample to the Riemann-Hilbert problem. Using a meromorphic gauge transformation, one cannot reduce this system to the direct sum of Fuchsian systems corresponding to the subrepresentations.     

Hilbert,Fock and Cantorian spaces in the quantum two-slit gedanken experiment

On the one hand, a rigorous mathematical formulation of quantum mechanics requires the introduction of a Hilbert space and as we move to the second quantization, a Fock space. On the other hand, the Cantorian E-infinity approach to quantum physics was developed largely without any direct reference to the afore mentioned mathematical spaces. In the present work we utilize some novel reinterpretations of basic E^(∞) Cantorian spacetime relations in terms of the Hilbert space of quantum mechanics. Proceeding in this way, we gain a better understanding of the physico-mathematical structure of quantum spacetime which is at the heart of the paradoxical and non-intuitive outcome of the famous quantum two-slit gedanken experiment.     

Computer use in mathematics education – present situation,intentions, and problems in the Federal Republic of Germany†

Computers, and computer‐related thinking structures, are only gradually influencing mathematics education. On the one hand, there is a discrepancy between involved teachers who already have changed their own classroom teaching to a great extent, and a majority of mathematics teachers who have not yet taken notice of the computer for teaching purposes. On the other hand, knowledge of the computer and of algorithms is frequently merely added to the mathematical subject matter. As opposed to that, the authors argue that it is necessary to genuinely integrate such subject matter, and to include general topics such as social impact and changed attitudes toward application. With regard to implementation, they develop concrete ideas which are aligned in a differentiated manner to the specific situation and the opportunities offered in the Federal Republic of Germany. The rationale for that is that only such reference to a specific situation will provide an opportunity for readers abroad to usefully apply approaches and ideas to the situation given in their own cultural environment.

An abbreviated version of this paper for cursory reading or other purposes has been marked by bold lines on the margin.