Estimating the Hurst Parameter

Let be a fractional Brownian motion with parameter 0 < H < 1. We are interested in the estimation of this parameter. To achieve this goal, we consider certain functionals of the second order increments of b _H (·), using variation technics. Based on an almost-sure convergence theorem for general functionals, we single out particular functionals that allows to construct certain regression models for the parameter H. We show that this regression based estimator for H is asymptotically unbiased, consistent and that it satisfies a Central Limit Theorem.      

Multivariate central limit theorems for averages of fractional Volterra processes and applications to parameter estimation

The purpose of this paper is to establish the multivariate normal convergence for the average of certain Volterra processes constructed from a fractional Brownian motion with Hurst parameter \(H > \frac{1}{2}\). Some applications to parameter estimation are then discussed.     

Periods and (χ, <Emphasis Type="Italic">b</Emphasis>)-factors of cuspidal automorphic forms of symplectic groups

In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations σ of symplectic groups Sp_2n(A), which detects the right-most pole of the L-function L(s, σ × χ) for some character χ of F×A × of order at most 2, and hence the occurrence of a simple global Arthur parameter (χ, b) in the global Arthur parameter ψ attached to σ. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.     

Admissible values of one parameter for maximal Sperner families of subsets of the type (<Emphasis Type="Italic">k,k</Emphasis> + 1)

In this paper we generalize one assertion (obtained by us earlier) on admissible values of a certain parameter for partial maximal Sperner families (m. S. f.) of subsets of a finite set of the type (k, k + 1). We also prove that the minimal value of the parameter under consideration for all m. S. f. of the type (k, k + 1), except for two families, is less than \(\left( {_k^{n - 1} } \right)\) ? 1.     

Analysis of Atkinson’s variable transformation for numerical integration over smooth surfaces in ℝ3

Summary Recently, a variable transformation for integrals over smooth surfaces in ³ was introduced in a paper by Atkinson. This interesting transformation, which includes a grading parameter that can be fixed by the user, makes it possible to compute these integrals numerically via the product trapezoidal rule in an efficient manner. Some analysis of the approximations thus produced was provided by Atkinson, who also stated some conjectures concerning the unusually fast convergence of his quadrature formulas observed for certain values of the grading parameter. In a recent report by Atkinson and Sommariva, this analysis is continued for the case in which the integral is over the surface of a sphere and the integrand is smooth over this surface, and optimal results are given for special values of the grading parameter. In the present work, we give a complete analysis of Atkinsons method over arbitrary smooth surfaces that are homeomorphic to the surface of the unit sphere. We obtain optimal results that explain the actual rates of convergence, and we achieve this for all values of the grading parameter.     

Asymptotic properties of Bayes estimators for Gaussian Itô-processes with noisy observations

The estimation of a real parameter θ in a linear stochastic differential equation of the simple type is investigated, based on noisy, time continuous observations of X_t. Sufficient conditions on the continuous functions β and σ are given such that the (conditionally normal) Bayes estimators of θ satisfy certain error bounds and are strongly consistent.     

Integrated likelihood computation methods

Suppose a model has parameter \(\theta =(\psi , \lambda )\), where \(\psi \) is the parameter of interest and \(\lambda \) is a nuisance parameter. The integrated likelihood method eliminates \(\lambda \) from the likelihood function \(L(\psi , \lambda )\) by integrating with respect to a weight function \(\pi (\lambda | \psi )\). The resulting integrated likelihood function \(\bar{L}(\psi )\) can be used for inference for \(\psi \). However, the analytical form for the integrated likelihood is not always available. This paper discusses 12 different approaches to computing the integrated likelihood. Some methods were originally developed for other computation purposes and they are modified to fit in the integrated likelihood framework. Methods considered include direct numerical integration methods such as Monte Carlo integration method, importance sampling, Laplace method; marginal likelihood computation methods; and methods for computing the marginal posterior density. Simulation studies and real data example are presented to evaluate and compare these methods empirically.     

On the Connection of Uncertainty Principles for Functions on the Circle and on the Real Line

An uncertainty principle for 2-periodic functions and the classical Heisenberg uncertainty principle are shown to be linked by a limit process. Dependent on a parameter, a function on the real line generates periodic functions either by periodization or sampling. It is proven that under certain smoothness conditions, the periodic uncertainty products of the generated functions converge to the real-line uncertainty product of the original function if the parameter tends to infinity. These results are used to find asymptotically optimal sequences for the periodic uncertainty principle, based either on Theta functions or trigonometric polynomials obtained by sampling B-splines.     

Delay Equations with Non-negativity Constraints Driven by a Hölder Continuous Function of Order \beta \in \left(\frac13,\frac12\right)

In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a Hölder continuous function of order \(\beta \in (\frac13,\frac12)\) . We also obtain a bound for the supremum norm of this solution. As an application, we get these results for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H \(\in (\frac13,\frac12)\) .     

Alexis Fontaine's route to the calculus of several variables

In the years around 1732 Alexis Fontaine des Bertins (1704–1771) member of the French Royal Academy of Sciences from 1733 on, worked out a dualoperator differential calculus to solve problems in “families of curves.” Within six years a calculus of several variables had emerged out of this work. These developments, discussed at greater length elsewhere [Greenberg, 1981, Greenberg, 1981, Greenberg, 1981], are summarized, with additional general reflections and the correction of one error. Together the discussions complement the excellent, recently published account of “families of curves” and the origins of partial differentiation in the works of Leibniz, the various Bernoullis, and Euler [Engelsman 1982]. Common concerns motivate the works of all of these mathematicians. At the same time, certain differences in conception in Fontaine's work highlight the creativity of one of the lesser known eighteenth-century mathematicians.     

Linearization of boundary eigenvalue problems

It is shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigenvalue parameter can be linearized by making use of the theory of operator colligations. As examples, first order systems with boundary conditions depending polynomially on and Sturm-Liouville problems with -holomorphic boundary conditions are considered.     

On systems of bilinear forms whose minimal division-free algorithms are all bilinear

In this paper we prove that for a certain class of systems of bilinear forms, all minimal division-free algorithms are essentially bilinear. This class includes systems for computing products in finite algebraic extension fields, and systems for computing the products of Toeplitz and Hankel matrices with vectors. Our results, together with the classification theorems of 10., 12., 169–180) completely describe all minimal division-free algorithms for computing these systems. We also prove, as an immediate consequence of our results, that the multiplicative complexity of the quaternion product over a real field is 8.     

A solution of the abstract Dirichlet problem for Baire-one functions

Let X be a compact convex subset of a locally convex space. We show that any bounded Baire-one function defined on can be extended to an affine Baire-one function on X if and only if X is a Choquet simplex and satisfies a certain topological property.     

RETRACTED ARTICLE: Effect of electromagnetic field on the stability of viscoelastic fluid film flowing down an inclined plane

The stability of thin electrically conducting viscoelastic fluid film flowing down on a non-conducting inclined plane in the presence of electromagnetic field is investigated under induction-free approximation. Surface evolution equation is derived by long-wave expansion method. The stabilizing role of Hartman number M (magnetic field) and the destabilizing role of the viscoelastic property \({\varGamma}\) and the electric parameter E on such fluid film are established through the linear stability analysis of the surface evolution equation. Investigation shows that at small values of Hartman number (M), the influence of electric parameter (E) on the viscoelastic parameter \({(\varGamma)}\) is insignificant, while for large values of M, E introduces more destabilizing effect at low values of \({\varGamma}\) than that at high values of \({\varGamma }\). An interesting result also perceived from our analysis is that the stabilizing effect of Hartman number (M) is decreasing with the increase of the values of \({\varGamma}\) and E, even it gives destabilizing effect after a certain high value of the electric field depending on the high value of \({\varGamma}\). The weakly nonlinear study reveals that the increase of \({\varGamma}\) decreases the explosive and subcritical unstable zones but increases the supercritical stable zone keeping the unconditional zone almost constant.     

On the asymptotic linearization of acoustic waves

The initial value problem of a certain generalization of the nonlinear, dispersive wave equations with dissipation is rigorously studied. The solutions of the equations can be found exactly up to in certain norms. The essential use is made of the fact that this equation is asymptotically linearizable to i.e., the equations can be mapped to an equation which differs from a linearizable equation only in terms which are of An application of the equations to unidirectional small amplitude acoustic waves is discussed. The general methodology used here can also be applied to other asymptotically linearizable equations.

     

Strong test modules and multiplier ideals

We introduce the notion of strong test module and show that a large number of such modules appear in the tight closure theory of complete domains: the test ideal (this has already been known), the parameter test module, and the module of relative test elements. They also appear as certain multiplier ideals, a concept of interest in algebraic geometry. Mathematics Subject Classification (2000):13A35     

Positive solutions for semi-positone three-point boundary value problems

Using a fixed point theorem of generalized cone expansion and compression we present in this paper criteria which guarantee the existence of at least two positive solutions for semi-positone three-point boundary value problems with parameter λ>0$\lambda >0$ belonging to a certain interval.     

Stability of Discontinuous Groups Acting on Homogeneous Spaces

Suppose given a nilpotent connected simply connected Lie group G, a connected Lie subgroup H of G, and a discontinuous group Γ for the homogeneous space M = G/H. In this work we study the topological stability of the parameter space R(Γ,G,H) in the case where G is three-step. We prove a stability theorem for certain particular pairs (Γ,H). We also introduce the notion of strong stability on layers making use of an explicit layering of Hom(Γ,G) and study the case of Heisenberg groups.