Error estimation of floating-point summation and dot product

We improve the well-known Wilkinson-type estimates for the error of standard floating-point recursive summation and dot product by up to a factor 2. The bounds are valid when computed in rounding to nearest, no higher order terms are necessary, and they are best possible. For summation there is no restriction on the number of summands. The proofs are short by using a new tool for the estimation of errors in floating-point computations which cures drawbacks of the “unit in the last place (ulp)”. The presented estimates are nice and simple, and closer to what one may expect.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-estimation for the location parameters in stochastic volatility models

L ₁-estimation of a location parameter is studied for the “product type” stochastic volatility models. The asymptotic distribution of the L ₁-estimator is established under general conditions on the behavior of the distribution function of the errors near zero.     

Free limits of forcing and more on Aronszajn trees

We prove that the Souslin Hypothesis does not imply “every Aron. (=Aronszajn) tree is special”. For this end we introduce variants of the notion “special Aron. tree”. We also introduce a limit of forcings bigger than the inverse limit, and prove it preserves properness and related notions not less than inverse limit, and the proof is easier in some respects. The result was announced in [9]. The author thanks Uri Avraham for detecting many errors.     

Asymptotically optimal choice of the smoothing parameter in a nonparametric kernel estimator for a probability density

This paper presents a method of estimation of an “optimal” smoothing parameter (window width) in kernel estimators for a probability density. The obtained estimator is calculated directly from observations. By “optimal” smoothing parameters we mean those parameters which minimize the mean integral square error (MISE) or the integral square error (ISE) of approximation of an unknown density by the kernel estimator. It is shown that the asymptotic “optimality” properties of the proposed estimator correspond (with respect to the order) to those of the well-known cross-validation procedure [1, 2]. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 67–80, Perm, 1990.     

Comparison of time risks based on a multinomial logistic response model in longitudinal studies

Summary The multinomial logistic response model has been used in the analysis of data from longitudinal studies of RERF's mortality cohort population. The model was restricted to linear and quadratic doseresponses for practical as well as biological reasons. The advantages and disadvantages of the multinomial logistic model are pointed out. Numerical comparison is made of the maximum likelihood (ML) estimates of parameters obtained by binomial and multinomial logistic procedures. The dose-response difference between two independent “same age” groups is evaluated from the ML estimates of parameters under a linear logistic response model. A significant dose-response difference between two independent “same age” groups in the years 1950–1959 and 1960–1969 is noted only for the 15–24 age group for all cancers other than leukemia. Radiation Effects Research Foundation     

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”- and “DG”-norm formed by the “L ²(H ¹)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.     

Weyl Laws for Partially Open Quantum Maps

We study a toy model for “partially open” wave-mechanical system, like for instance a dielectric micro-cavity, in the semiclassical limit where ray dynamics is applicable. Our model is a quantized map on the 2-dimensional torus, with an additional damping at each time step, resulting in a subunitary propagator, or “damped quantum map”. We obtain analogues of Weyl’s laws for such maps in the semiclassical limit, and draw some more precise estimates when the classical dynamics is chaotic. Submitted: October 16, 2008. Accepted: April 3, 2009.     

Hull-Volume with Applications to Convergence Analysis

We introduce and study decompositions of finite sets as well as coverings of their convex hulls, and use these objects to develop various estimates of and formulas for the “hull-volume” of the sets (i.e., the volume of their convex hull). We apply our results to the convergence analysis of the “iterate-sets” associated with each iteration of a reduce-or-retreat optimization method (including pattern-search methods like Nelder–Mead as well as model-based methods).     

An analytical study of the sensitivity of a discrete stochastic model for sales dynamics

We consider a discrete model for sales dynamics in the case of a stochastic model of the market. The model includes “fast” and “slow” components of the market situation described by a stochastic process of “white noise” type and the correlated stochastic process. By using an integral representation of the main characteristics of the Kalman filter, we obtain expressions for stochastic parameters of additional errors of the estimate that arise in the case where the characteristics of noises are inexact. We make an asymptotical analysis of these expressions and give recommendations for the price-forming strategy in the case of uncertainty of the market situation. Bibliography: 2 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 110–116.     

Influence measures and robust estimators of dependence in multivariate extremes

We develop a simple influence measure to assess whether Bayesian estimators in multivariate extreme value problems are sensitive to outliers. The proposed measure is easy to compute by importance sampling and successfully captures two effects on the functional: the “data effect” and the “parameter uncertainty effect”. We also propose a new Bayesian estimator which is easy to implement and is robust. The methods are tested and illustrated using simulated data and then applied to stock market data.     

Discrete boundary element methods on general meshes in 3D

Abstract. This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in . Our theory covers equations with very general operators, provided the associated weak form is bounded and elliptic on , for some . In contrast to other studies on this topic, we do not assume our meshes to be quasiuniform, and therefore the analysis admits locally refined meshes. To achieve such generality, standard inverse estimates for the quasiuniform case are replaced by appropriate generalised estimates which hold even in the locally refined case. Since the approximation of singular integrals on or near the diagonal of the Galerkin matrix has been well-analysed previously, this paper deals only with errors in the integration of the nearly singular and smooth Galerkin integrals which comprise the dominant part of the matrix. Our results show how accurate the quadrature rules must be in order that the resulting discrete Galerkin method enjoys the same stability properties and convergence rates as the true Galerkin method. Although this study considers only continuous piecewise linear basis functions on triangles, our approach is not restricted in principle to this case. As an example, the theory is applied here to conventional “triangle-based” quadrature rules which are commonly used in practice. A subsequent paper [14] introduces a new and much more efficient “node-based” approach and analyses it using the results of the present paper. Received December 10, 1997 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Space-time discontinuos Galerkin method for solving nonstationary convection-diffusion-reaction problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”-and “ ”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution, the errors are of order O(h ^p + τ ^q ). The estimates hold true even in the hyperbolic case when ɛ = 0.     

The ideal of separants in the ring of differential polynomials

We obtained the criterion of existence of a quasi-linear polynomial in a differential ideal in the ordinary ring of differential polynomials over a field of characteristic zero. We generalized the “going up” and “going down” theorems onto the case of Ritt algebras. In particular, new finiteness criteria for differential standard bases and estimates that characterize calculation complexity were obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 215–227, 2007.     

Analysis of rounded data from dependent sequences

Observations on continuous populations are often rounded when recorded due to the precision of the recording mechanism. However, classical statistical approaches have ignored the effect caused by the rounding errors. When the observations are independent and identically distributed, the exact maximum likelihood estimation (MLE) can be employed. However, if rounded data are from a dependent structure, the MLE of the parameters is difficult to calculate since the integral involved in the likelihood equation is intractable. This paper presents and examines a new approach to the parameter estimation, named as “short, overlapping series” (SOS), to deal with the α-mixing models in presence of rounding errors. We will establish the asymptotic properties of the SOS estimators when the innovations are normally distributed. Comparisons of this new approach with other existing techniques in the literature are also made by simulation with samples of moderate sizes.     

Estimators for the long-memory parameter in LARCH models, and fractional Brownian motion

This paper investigates several strategies for consistently estimating the so-called Hurst parameter H responsible for the long-memory correlations in a linear class of ARCH time series, known as LARCH(∞) models, as well as in the continuous-time Gaussian stochastic process known as fractional Brownian motion (fBm). A LARCH model’s parameter is estimated using a conditional maximum likelihood method, which is proved to have good stability properties. A local Whittle estimator is also discussed. The article further proposes a specially designed conditional maximum likelihood method for estimating the H which is closer in spirit to one based on discrete observations of fBm. In keeping with the popular financial interpretation of ARCH models, all estimators are based only on observation of the “returns” of the model, not on their “volatilities”.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Stability and bifurcation for periodic perturbations of the equilibrium of an oscillator with infinite or infinitesimal oscillation frequency

We consider small perturbations periodic in time of an oscillator whose restoring force has a leading term with exponent 3 or 1/3. The first case corresponds to oscillations with infinitesimal frequency and the second case to oscillations with infinite frequency. The smallness of the perturbation is determined both by the smallness of the considered neighborhood of the equilibrium point and by a small nonnegative parameter ε. For ε=0, the stability of the equilibrium point is studied. For ε>0, we find conditions for an invariant two-dimensional torus to branch off with “soft” or “rigid” loss of stability with loss index 1/2. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 323–335, March, 1999.     

Mixing “in the sense of ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. II

In the first part of the present paper, we established estimates for the rate of approach of the integrals of a family of “physical” white noises to a family of Wiener processes. We use this result to establish the estimate for the rate of approach of a family of solutions of ordinary differential equations perturbed by some “physical” white noises to a family of solutions of the corresponding It? equations. We consider both the case where the coefficient of random perturbation is separated from zero and the case where it is not separated from zero.     

Weighted norm inequalities,off-diagonal estimates and elliptic operators Part II: Off-diagonal estimates on spaces of homogeneous type

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant L^p − L^q estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators. This work was partially supported by the European Union (IHP Network “Harmonic Analysis and Related Problems” 2002-2006, Contract HPRN-CT-2001-00273-HARP). The second author was also supported by MEC “Programa Ramón y Cajal, 2005” and by MEC Grant MTM2004-00678.     

On the linear problem related to the stability of uniformly rotating self-gravitating liquid

The paper is concerned with “maximal regularity” estimates of Sobolev norms of solutions of a linearized evolution problem for the perturbations of the velocity and pressure of a uniformly rotating liquid. Bibliography: 12 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 139–160