A multi-frequency inverse source problem

This paper is concerned with an inverse source problem that determines the source from measurements of the radiated fields away at multiple frequencies. Rigorous stability estimates are established when the background medium is homogeneous. It is shown that the ill-posedness of the inverse problem decreases as the frequency increases. Under some regularity assumptions on the source function, it is further proven that by increasing the frequency, the logarithmic stability converts to a linear one for the inverse source problem.     

On the Gerber-Shiu function and change of measure

We consider several models for the surplus of an insurance company mainly under some light-tail assumptions. We are interested in the expected discounted penalty at ruin. By a change of measure we remove the discounting, which simplifies the expression. This leads to (defective) renewal equations as they had been found by different methods in the literature. If we use the change of measure such that ruin becomes certain, the renewal equations simplify to ordinary renewal equations. This helps to discuss the asymptotics as the initial capital goes to infinity. For phase-type claim sizes, explicit formulae can be derived.     

Authentication schemes from actions on graphs, groups, or rings

We propose a couple of general ways of constructing authentication schemes from actions of a semigroup on a set, without exploiting any specific algebraic properties of the set acted upon. Then we give several concrete realizations of this general idea, and in particular, we describe several authentication schemes with long-term private keys where forgery (a.k.a. impersonation) is NP-hard. Computationally hard problems that can be employed in these realizations include the Graph Colorability problem, the Diophantine problem, and many others.     

Continuous version of Filippov’s theorem for fractional differential inclusions

Using Bressan-Colombo results, concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values, we prove a continuous version of Filippov’s theorem for a fractional differential inclusion.     

Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods

This paper is devoted to the numerical simulation of two-dimensional stationary Bingham fluid flow by semismooth Newton methods. We analyze the modeling variational inequality of the second kind, considering both Dirichlet and stress-free boundary conditions. A family of Tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified. By using Fenchel’s duality, optimality systems which characterize the original and regularized solutions are obtained. The regularized optimality systems are discretized using a finite element method with (cross-grid P₁)-Q₀ elements for the velocity and pressure, respectively. A semismooth Newton algorithm is proposed in order to solve the discretized optimality systems. Using an additional relaxation, a descent direction is constructed from each semismooth Newton iteration. Local superlinear convergence of the method is also proved. Finally, we perform numerical experiments in order to investigate the behavior and efficiency of the method.     

Addison-type series representation for the Stieltjes constants

The Stieltjes constants γk(a) appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about its only pole at s=1. We generalize a technique of Addison for the Euler constant γ=γ₀(1) to show its application to finding series representations for these constants. Other generalizations of representations of γ are given.     

Monodromy for principal bundles

Given a strongly semistable principal bundle E_G over a curve, in Biswas et al. (2006) [4], a group-scheme for it was constructed, which was named as the monodromy group-scheme. Here we extend the construction of the monodromy group-scheme to principal bundles over higher dimensional varieties.     

Numerical ranges of reducible companion matrices

In this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A⊕B, where A∈M_n-2, B∈M₂ with σ(B)={aω₁,aω₂}, , ω₁≠ω₂, and .     

Model selection by sequentially normalized least squares

Model selection by means of the predictive least squares (PLS) principle has been thoroughly studied in the context of regression model selection and autoregressive (AR) model order estimation. We introduce a new criterion based on sequentially minimized squared deviations, which are smaller than both the usual least squares and the squared prediction errors used in PLS. We also prove that our criterion has a probabilistic interpretation as a model which is asymptotically optimal within the given class of distributions by reaching the lower bound on the logarithmic prediction errors, given by the so called stochastic complexity, and approximated by BIC. This holds when the regressor (design) matrix is non-random or determined by the observed data as in AR models. The advantages of the criterion include the fact that it can be evaluated efficiently and exactly, without asymptotic approximations, and importantly, there are no adjustable hyper-parameters, which makes it applicable to both small and large amounts of data.