Riemann-Finsler Geometry with Applications to Information Geometry

Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce Riemann-Finsler geometry, by which we establish Information Geometry on a much broader base, so that the potential applications of Information Geometry will be beyond statistics.     

Spaces with Dual Order-Isomorphic to ℓ1

We establish that an ordered Banach space is order-isomorphic to c₀ if and only if it is a -Dedekind complete vector lattice and its norm dual is order-isomorphic to ₁.     

Erratum to “Affine surfaces in R4 with planar geodesics with respect to the affine metric”

In this work a mistake in the paper is corrected. There is also a new proof of the main theorem which classifies the non-degenerate affine surfaces in R ⁴ having planar geodesics with respect to the affine metric.     

General approach to filtering with fractional brownian noises — application to linear systems

At first a general approach is proposed to filtering in systems where the observation noise is a fractional Brownian motion. It is shown that the problem can be handled in terms of some appropriate semimartingale and analogs of the classical innovation process and fundamental filtering theorem are obtained. Then the problem of optimal filtering is completely solved for Gaussian linear systems with fractional Brownian noises. Closed form simple equations are derived both for the mean of the optimal filter and the variance of the filtering error. Finally the results are explicited in various specific cases     

Incremental-like bundle methods with application to energy planning

An important field of application of non-smooth optimization refers to decomposition of large-scale or complex problems by Lagrangian duality. In this setting, the dual problem consists in maximizing a concave non-smooth function that is defined as the sum of sub-functions. The evaluation of each sub-function requires solving a specific optimization sub-problem, with specific computational complexity. Typically, some sub-functions are hard to evaluate, while others are practically straightforward. When applying a bundle method to maximize this type of dual functions, the computational burden of solving sub-problems is preponderant in the whole iterative process. We propose to take full advantage of such separable structure by making a dual bundle iteration after having evaluated only a subset of the dual sub-functions, instead of all of them. This type of incremental approach has already been applied for subgradient algorithms. In this work we use instead a specialized variant of bundle methods and show that such an approach is related to bundle methods with inexact linearizations. We analyze the convergence properties of two incremental-like bundle methods. We apply the incremental approach to a generation planning problem over an horizon of one to three years. This is a large scale stochastic program, unsolvable by a direct frontal approach. For a real-life application on the French power mix, we obtain encouraging numerical results, achieving a significant improvement in speed without losing accuracy.     

Cartan–Eilenberg complexes with respect to cotorsion pairs

Let ${(\mathcal{A}, \mathcal{B})}$ be a cotorsion pair in R-Mod. We introduce and study the notion of Cartan–Eilenberg- ${\mathcal{A}}$ complexes. Specifically, we establish some relationships between Cartan–Eilenberg- ${\mathcal{A}}$ complexes and DG- ${\mathcal{A}}$ complexes. Using these relationships, some applications are given.     

Convergence rates to nonlinear diffusion waves for weak entropy solutions to p-system with damping

This paper studies the asymptotic behavior of weak entropy solutions to the Cauchy problem of the so-called p-system with damping. The convergence rates to nonlinear diffusion waves for weak entropy solutions are obtained in L∞-norm or L2-norm. These convergence rates are the same to the decay rates of smooth solution obtained by Nishihara. They are proved by using the vanishing viscosity method and the elementary L2-energy method.     

On differentiability with respect to the initial data of the solution to an SDE with a Lévy noise and discontinuous coefficients

We construct a stochastic flow generated by an stochastic differential equation with its drift being a function of bounded variation and its noise being a stable process with exponent from (1,2). It is proved that the flow is non-coalescing and Sobolev differentiable with respect to the initial data. The representation for the derivative is given.     

Uniqueness of Solutions to Conservation Laws with Linearly Degenerate Eigenvalue

Consider a hyperbolic system of conservation laws (1) where is a smooth nonlinear mapping. ▽f has two real and distinct eigenvalues λ_1 and λ_2(λ_1<λ_2), one of which is genuinely nonlinear, i. e. (2) the other one of which is linearly degenerate in sense of Lax, i, e,     

Primal—Dual Constraint Aggregation with Application to Stochastic Programming

The special constraint structure and large dimension are characteristic for multistage stochastic optimization. This results from modeling future uncertainty via branching process or scenario tree. Most efficient algorithms for solving this type of problems use certain decomposition schemes, and often only a part of the whole set of scenarios is taken into account in order to make the problem tractable.We propose a primal–dual method based on constraint aggregation, which constructs a sequence of iterates converging to a solution of the initial problem. At each iteration, however, only a reduced sub-problem with smaller number of aggregate constraints has to be solved. Number of aggregates and their composition are determined by the user, and the procedure for calculating aggregates can be parallelized. The method provides a posteriori estimates of the quality of the current solution approximation in terms of the objective function value and the residual.Results of numerical tests for a portfolio allocation problem with quadratic utility function are presented.     

Coupon collector’s problems with statistical applications to rankings

Some new exact distributions on coupon collector’s waiting time problems are given based on a generalized Pólya urn sampling. In particular, usual Pólya urn sampling generates an exchangeable random sequence. In this case, an alternative derivation of the distribution is also obtained from de Finetti’s theorem. In coupon collector’s waiting time problems with $m$ kinds of coupons, the observed order of $m$ kinds of coupons corresponds to a permutation of $m$ letters uniquely. Using the property of coupon collector’s problems, a statistical model on the permutation group of $m$ letters is proposed for analyzing ranked data. In the model, as the parameters mean the proportion of the $m$ kinds of coupons, the observed ranking can be intuitively understood. Some examples of statistical inference are also given.     

Randomized first order algorithms with applications to ℓ 1-minimization

In this paper we propose randomized first-order algorithms for solving bilinear saddle points problems. Our developments are motivated by the need for sublinear time algorithms to solve large-scale parametric bilinear saddle point problems where cheap online assessment of the solution quality is crucial. We present the theoretical efficiency estimates of our algorithms and discuss a number of applications, primarily to the problem of ? ₁ minimization arising in sparsity-oriented signal processing. We demonstrate, both theoretically and by numerical examples, that when seeking for medium-accuracy solutions of large-scale ? ₁ minimization problems, our randomized algorithms outperform significantly (and progressively as the sizes of the problem grow) the state-of-the art deterministic methods.     

Singular solutions with exponential growth to Protter’s problems

Four-dimensional boundary value problems which were formulated by Proter for the nonhomogeneous wave equation are studied. They can be considered as multidimensional versions of the Darboux problems in ?². Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solution may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Some known results suggest that the solution may have at most exponential growth. We construct an infinitely smooth right-hand side function such that the corresponding generalized solution to Protter’s problem has an exponential singularity.     

Bochner–Riesz Means with Respect to a Generalized Cylinder

We study the L ^p boundedness of the generalized Bochner–Riesz means S ^λ which are defined as $$S^{\lambda}f(x) = \mathcal{F}^{-1} \left[\left(1 - \rho \right)_{+}^{\lambda} \widehat{f} \right](x)$$ where ${\rho(\xi) = {\rm max}\{|\xi_{1}|, \ldots, |\xi_{\ell}|\}}$ for ${\xi = (\xi_{1},\ldots, \xi_{\ell}) \in \mathbb{R}^{{d}_{1}} \times \cdots \times \mathbb{R}^{{d}_{\ell}}}$ and ${\mathcal{F}^{-1}}$ is the inverse Fourier transform.     

Fundamental Gaps in Numerical Semigroups with Respect to Their Multiplicity

For a numerical semigroup, we introduce the concept of a fundamental gap with respect tothe multiplicity of the semigroup.The semigroup is fully determined by its multiplicity and these gaps.We study the case when a set of non-negative integers is the set of fundamental gaps with respect to themultiplicity of a numerical semigroup.Numerical semigroups with maximum and mininmm number ofthis kind of gaps are described.     

Yan Theorem in L∞ with Applications to Asset Pricing

We prove an L~∞ version of the Yan theorem and deduce from it a necessary condition for theabsence of free lunches in a model of financial markets,in which asset prices are a continuous R~d valued processand only simple investment strategies are admissible.Our proof is based on a new separation theorem for convexsets of finitely additive measures.     

Finite Mixture Normal Models, with Application to Dose-Response Studies

In this paper, we consider the risk assessment problem under multi-levels and multiple mixture subpopulations. Our result is the generalization of the results of [1-5].1 Finite Mixture Normal ModelsIn dose-response studies, a class of phenomena that frequently occur are that experimental subjects (e.g., mice) may have different responses like 'none, mild, severe' after a toxicant experiment, or 'getting worse, no change, getting better' after a medical treatment, etc. These phenomena have attracted the attention of many researchers in recent years. Finite     

The normal ΔH-matrices with connection to some Jacobi-like methods

The ΔH-matrices appear in the context of certain maximization and minimization problems. The following purely algebraic problem, arising in connection with certain norm-reducing Jacobi-type algorithms is treated: When is a normal ΔH-matrix diagonal? Final results are obtained in the cases n = 3 and n = 4, and in general for the attractive normal ΔH-matrices.