Joint cumulative distribution functions for Dempster–Shafer belief structures using copulas

We first introduce the Dempster–Shafer belief structure and highlight its role in the representation of information about a random variable for which our knowledge of the probabilities is interval-valued. We investigate the formation of the cumulative distribution function (CDF) for these types of variables. It is noted that this is also interval-valued and is expressible in terms of plausibility and belief measures. The class of aggregation operators known as copulas are introduced and a number of their properties are provided. We discuss Sklar’s theorem, which provides for the use of copulas in the formulation of joint CDFs from the marginal CDFs of classic random variables. We then look to extend these ideas to the case of joining the marginal CDFs associated with Dempster–Shafer belief structures. Finally we look at the formulation CDFs obtained from functions of multiple D–S belief structures.     

Smooth distribution function estimation for lifetime distributions using Szasz–Mirakyan operators

Annals of the Institute of Statistical Mathematics - In this paper, we introduce a new smooth estimator for continuous distribution functions on the positive real half-line using...     

Individual loss reserving using paid–incurred data

This paper develops a stochastic model for individual claims reserving using observed data on claim payments as well as incurred losses. We extend the approach of Pigeon et al. (2013), designed for payments only, towards the inclusion of incurred losses. We call the new technique the individual Paid and Incurred Chain (iPIC) reserving method. Analytic expressions are derived for the expected ultimate losses, given observed development patterns. The usefulness of this new model is illustrated with a portfolio of general liability insurance policies. For the case study developed in this paper, detailed comparisons with existing approaches reveal that iPIC method performs well and produces more accurate predictions.     

Brill–Noether theory for cyclic covers

We recall that the Brill–Noether Theorem gives necessary and sufficient conditions for the existence of a ${\mathfrak{g}}_d^r$. Here we consider a general n-fold, étale, cyclic cover $p:\stackrel{?}{C}\to C$ of a curve C of genus g and investigate for which numbers $r,d$ a ${\mathfrak{g}}_d^r$ exists on $\stackrel{?}{C}$. For $r=1$ this means computing the gonality of $\stackrel{?}{C}$. Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of limit linear series for tree-like curves we show that the Plücker formula yields a necessary condition for the existence of a ${\mathfrak{g}}_d^r$ which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24], for all $n,r,d$.     

A polynomial algorithm for a two parameter extension of Wythoff NIM based on the Perron–Frobenius theory

For any positive integer parameters a and b, Gurvich recently introduced a generalization mex _b of the standard minimum excludant mex = mex₁, along with a game NIM(a, b) that extends further Fraenkel’s NIM = NIM(a, 1), which in its turn is a generalization of the classical Wythoff NIM = NIM(1, 1). It was shown that P-positions (the kernel) of NIM(a, b) are given by the following recursion: $$x_n = {\rm mex}_b(\{x_i, y_i \;|\; 0 \leq i < n\}), \;\; y_n = x_n + an; \;\; n \geq 0,$$ and conjectured that for all a, b the limits ?(a, b) = x _n (a, b)/n exist and are irrational algebraic numbers. In this paper we prove that showing that ${\ell(a,b) = \frac{a}{r-1}}$ , where r > 1 is the Perron root of the polynomial $$P(z) = z^{b+1} - z - 1 - \sum_{i=1}^{a-1} z^{\lceil ib/a \rceil},$$ whenever a and b are coprime; furthermore, it is known that ?(ka, kb) = k?(a, b). In particular, ${\ell(a, 1) = \alpha_a = \frac{1}{2} (2 - a + \sqrt{a^2 + 4})}$ . In 1982, Fraenkel introduced the game NIM(a) = NIM(a, 1), obtained the above recursion and solved it explicitly getting ${x_n = \lfloor \alpha_a n \rfloor, \; y_n = x_n + an = \lfloor (\alpha_a + a) n \rfloor}$ . Here we provide a polynomial time algorithm based on the Perron–Frobenius theory solving game NIM(a, b), although we have no explicit formula for its kernel.     

A smoothing algorithm for finite min–max–min problems

We generalize a smoothing algorithm for finite min–max to finite min–max–min problems. We apply a smoothing technique twice, once to eliminate the inner min operator and once to eliminate the max operator. In mini–max problems, where only the max operator is eliminated, the approximation function is decreasing with respect to the smoothing parameter. Such a property is convenient to establish algorithm convergence, but it does not hold when both operators are eliminated. To maintain the desired property, an additional term is added to the approximation. We establish convergence of a steepest descent algorithm and provide a numerical example.     

A majorization–minimization algorithm for split feasibility problems

The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range constraints. Split feasibility generalizes important inverse problems including convex feasibility, linear complementarity, and regression with constraint sets. When a feasible point does not exist, solution methods that proceed by minimizing a proximity function can be used to obtain optimal approximate solutions to the problem. We present an extension of the proximity function approach that generalizes the linear split feasibility problem to allow for non-linear mappings. Our algorithm is based on the principle of majorization–minimization, is amenable to quasi-Newton acceleration, and comes complete with convergence guarantees under mild assumptions. Furthermore, we show that the Euclidean norm appearing in the proximity function of the non-linear split feasibility problem can be replaced by arbitrary Bregman divergences. We explore several examples illustrating the merits of non-linear formulations over the linear case, with a focus on optimization for intensity-modulated radiation therapy.     

Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators

We develop singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators. In particular, we establish existence of a spectral transformation as well as local Borg–Marchenko and Hochstadt–Lieberman type uniqueness results. Finally, we give some applications to the case of radial Dirac operators.     

A nonmonotone derivative-free algorithm for nonlinear complementarity problems based on the new generalized penalized Fischer–Burmeister merit function

In this paper, we propose a new generalized penalized Fischer–Burmeister merit function, and show that the function possesses a system of favorite properties. Moreover, for the merit function, we establish the boundedness of level set under a weaker condition. We also propose a derivative-free algorithm for nonlinear complementarity problems with a nonmonotone line search. More specifically, we show that the proposed algorithm is globally convergent and has a locally linear convergence rate. Numerical comparisons are also made with the merit function used by Chen (J Comput Appl Math 232:455–471, 2009), which confirm the superior behaviour of the new merit function.     

Harder–Narasimhan theory for linear codes (with an appendix on Riemann–Roch theory)

In this text we develop some aspects of Harder–Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder–Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids.We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions toward a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed.Last, we study the two important questions of the preservation of semistability (or more generally the behavior of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights—and its matroid version—which we revisit in an appendix, developing analogues of the Riemann–Roch, Serre duality, Clifford, and gap and gonality sequence theorems. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.     

Mathematical model of liquid–liquid equilibrium for a ternary system using the GMDH-type neural network and genetic algorithm

A GMDH type-neural network was used to predict liquid phase equilibrium data for the (water + ethanol + trans-decalin) ternary system in the temperature range of 300.2–315.2 K. In order to accomplish modeling, the experimental data were divided into train and test sections. The data set was divided into two parts: 70% were used as data for “training” and 30% were used as a test set. The predicted values were compared with those of experimental values in order to evaluate the performance of the GMDH neural network method. The results obtained by using GMDH type neural network are in excellent agreement with the experimental results.     

A modified Polak–Ribière–Polyak conjugate gradient algorithm for unconstrained optimization

A modified Polak–Ribière–Polyak conjugate gradient algorithm which satisfies both the sufficient descent condition and the conjugacy condition is presented. These properties are independent of the line search. The algorithms use the standard Wolfe line search. Under standard assumptions, we show the global convergence of the algorithm. Numerical comparisons with conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that this computational scheme outperforms the known Polak–Ribière–Polyak algorithm, as well as some other unconstrained optimization algorithms.     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

An inexact primal–dual path following algorithm for convex quadratic SDP

We propose primal–dual path-following Mehrotra-type predictor–corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: , where is a self-adjoint positive semidefinite linear operator on , b∈R ^m , and is a linear map from to R ^m . At each interior-point iteration, the search direction is computed from a dense symmetric indefinite linear system (called the augmented equation) of dimension m + n(n + 1)/2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust. Research supported in part by Academic Research Grant R146-000-076-112.