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## Function F (function + f)
## Selected Abstracts## Dynamical scaling in fractal structures in the aggregation of tetraethoxysilane-derived sonogels JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 5-1 2010Dimas R. VolletDynamical scaling properties in fractal structures were investigated from small-angle X-ray scattering (SAXS) data of the kinetics of aggregation in silica-based gelling systems. For lack of a maximum in the SAXS intensity curves, a characteristic correlation distance , was evaluated by fitting a particle scattering factor model valid for polydisperse coils of linear chains and f -functional branched polycondensates in solution, so the intensity at q = ,,1, I(,,1, t), was considered to probe dynamical scaling properties. The following properties have been found: (i) the SAXS intensities corresponding to different times t, I(q, t), are given by a time-independent function F(q,) = I(q, t),,D/Q, where the scattering invariant Q has been found to be time-independent; (ii) , exhibited a power-law behavior with time as ,,t,, the exponent , being close to 1 but diminishing with temperature; (iii) I(,,1, t) exhibited a time dependence given by I(,,1, t) ,t,, with the exponent , found to be around 2 but diminishing with temperature, following the same behavior as the exponent ,. In all cases, ,/, was quite close to the fractal dimension D at the end of the studied process. This set of findings is in notable agreement with the dynamical scaling properties. [source] ## A new universal approximation result for fuzzy systems, which reflects CNF DNF duality INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 12 2002Irina PerfilievaThere are two main fuzzy system methodologies for translating expert rules into a logical formula: In Mamdani's methodology, we get a DNF formula (disjunction of conjunctions), and in a methodology which uses logical implications, we get, in effect, a CNF formula (conjunction of disjunctions). For both methodologies, universal approximation results have been proven which produce, for each approximated function f(x), two different approximating relations RDNF(x, y) and RCNF(x, y). Since, in fuzzy logic, there is a known relation FCNF(x) , FDNF(x) between CNF and DNF forms of a propositional formula F, it is reasonable to expect that we would be able to prove the existence of approximations for which a similar relation RCNF(x, y) , RDNF(x, y) holds. Such existence is proved in our paper. © 2002 Wiley Periodicals, Inc. [source] ## Exploratory orientation data analysis with , sections JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 5 2004K. Gerald Van Den BoogaartSince the domain of crystallographic orientations is three-dimensional and spherical, insightful visualization of them or visualization of related probability density functions requires (i) exploitation of the effect of a given orientation on the crystallographic axes, (ii) consideration of spherical means of the orientation probability density function, in particular with respect to one-dimensional totally geodesic submanifolds, and (iii) application of projections from the two-dimensional unit sphere onto the unit disk . The familiar crystallographic `pole figures' are actually mean values of the spherical Radon transform. The mathematical Radon transform associates a real-valued function f defined on a sphere with its mean values along one-dimensional circles with centre , the origin of the coordinate system, and spanned by two unit vectors. The family of views suggested here defines , sections in terms of simultaneous orientational relationships of two different crystal axes with two different specimen directions, such that their superposition yields a user-specified pole probability density function. Thus, the spherical averaging and the spherical projection onto the unit disk determine the distortion of the display. Commonly, spherical projections preserving either volume or angle are favoured. This rich family displays f completely, i.e. if f is given or can be determined unambiguously, then it is uniquely represented by several subsets of these views. A computer code enables the user to specify and control interactively the display of linked views, which is comprehensible as the user is in control of the display. [source] ## Gâteaux derivatives and their applications to approximation in Lorentz spaces ,p,w MATHEMATISCHE NACHRICHTEN, Issue 9 2009Maciej CiesielskiAbstract We establish the formulas of the left- and right-hand Gâteaux derivatives in the Lorentz spaces ,p,w = {f: ,0,(f **)pw < ,}, where 1 , p < ,, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, ,), 0 < , , ,. We also find a general form of any supporting functional for each function from ,p,w, and the necessary and sufficient conditions for which a spherical element of ,p,w is a smooth point of the unit ball in ,p,w. We show that strict convexity of the Lorentz spaces ,p,w is equivalent to 1 < p < , and to the condition ,0,w = ,. Finally we apply the obtained characterizations to studies the best approximation elements for each function f , ,p,w from any convex set K , ,p,w (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle MATHEMATISCHE NACHRICHTEN, Issue 1-2 2007Lev AizenbergAbstract Let D be a unit disk andM be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2,. Our first result characterizes the restriction of the holomorphic functions f , ,(D), which are in the Hardy class ,1 near the arcM and are denoted by ,, ,1M(,,), to the open arcM. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M. As an application of the above characterization, we present an extension theorem for a function f , L1(M) from any symmetric sub-arc L , M of the unit circle, such that , M, to a function f , ,, ,1L(,,). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Detecting and creating oscillations using multifractal methods MATHEMATISCHE NACHRICHTEN, Issue 11 2006Stéphane SeuretAbstract By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f, we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ,j ,kdj,k,j,k , L2(,) the function series ft whose wavelet coefficients are dtj,k = dj,k1, for some fixed real number , > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series ft may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f , L2(,) such that the associated thresholded function series ft effectively possesses oscillating singularities which were not present in the initial function f . This series ft is a typical example of function with homogeneous non-concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On the convergence of Fourier series of computable Lebesgue integrable functions MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 5 2010Philippe MoserAbstract This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp -computable functions (computable Lebesgue integrable functions) with a size notion, by introducing Lp -computable Baire categories. We show that Lp -computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is Lp -computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp -computable functions is not meager. We give an alternative characterization of meager sets via Banach-Mazur games. We study the convergence of Fourier series for Lp -computable functions and show that whereas for every p > 1, the Fourier series of every Lp -computable function f converges to f in the Lp norm, the set of L1 -computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Approximate L(,1,,2,,,,t)-coloring of trees and interval graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2007Alan A. BertossiAbstract Given a vector (,1,,2,,,,t) of nonincreasing positive integers, and an undirected graph G = (V,E), an L(,1,,2,,,,t)-coloring of G is a function f from the vertex set V to a set of nonnegative integers such that ,f(u) , f(v), , ,i, if d(u,v) = i, 1 , i , t, where d(u,v) is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal L(,1,,2,,,,t)-coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(,1,,2,,,,t)-coloring of two relevant classes of graphs,trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t + ,1)) and O(nt2,1) time algorithms are proposed to find ,-approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and , is a constant depending on t and ,1,,,,t. Moreover, an O(n) time algorithm is given for the L(,1,,2)-coloring of unit interval graphs, which provides a 3-approximation. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 204,216 2007 [source] ## The fractional matching numbers of graphs NETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2002Yan LiuAbstract A fractional matching of a graph G is a function f that assigns to each edge a number in [0, 1] such that, for each vertex v, , f(e) , 1, where the sum is taken over all edges incident to v. The fractional matching number of G is the supremum of ,e,E(G)f(e) over all fractional matchings f. In this paper, we provide a new formula for calculating the fractional matching numbers of graphs using the Gallai,Edmonds Structure Theorem. Thus, we characterize graphs for which the fractional matching number equals the matching number and graphs for which the fractional matching number is the maximum possible (one-half the number of vertices). © 2002 Wiley Periodicals, Inc. [source] ## Estimation of regression parameters in missing data problems THE CANADIAN JOURNAL OF STATISTICS, Issue 2 2006Donald L. McleishAbstract Let Y be a response variable, possibly multivariate, with a density function f (y|x, v; ,) conditional on vectors x and v of covariates and a vector , of unknown parameters. The authors consider the problem of estimating , when the values taken by the covariate vector v are available for all observations while some of those taken by the covariate x are missing at random. They compare the profile estimator to several alternatives, both in terms of bias and standard deviation, when the response and covariates are discrete or continuous. Estimation des paramètres de régression en I'absence de certaines données Soit Y une variable réponse uni- ou multi-dimensionnelle et soit f(y|x, v; ,) sa densité étant donné des vecteurs x et v de covariables et un vecteur , de paramètres inconnus. Les auteurs s'intéressent à l'estimation de , lorsque la valeur de v est disponible pour toutes les observations, mais que certaines valeurs de x sont manquantes au hasard. Us comparent l'estimateur profil à diverses autres solutions, tant en terme de biais que d'écart-type, selon que la variable réponse et les covariables sont discrètes ou continues. [source] ## Global continuation for first order systems over the half-line involving parameters MATHEMATISCHE NACHRICHTEN, Issue 8 2009Gilles EvéquozArticle first published online: 21 JUL 200Abstract Let X be one of the functional spaces W1,p ((0, ,), ,N) or C01 ([0, ,), ,N), we study the global continuation in , for solutions (,, u, ,) , , × X × ,k of the following system of ordinary differential equations: where ,N = X1 , X2 is a given decomposition, with associated projection P: ,N , X1. Under appropriate conditions upon the given functions F and ,, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of , × X × ,k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A Semiparametric Estimator of the Crossing Point in the Two-Sample Linear Shift Function: Application to Crossing Lifetime Distributions BIOMETRICAL JOURNAL, Issue 2 2004Chi-tsung WuAbstract Let X and Y be two random variables with continuous distribution functions F and G. Consider two independent observations X1, , , Xm from F and Y1, , , Yn from G. Moreover, suppose there exists a unique x* such that F(x) > G(x) for x < x* and F(x) < G(x) for x > x* or vice versa. A semiparametric model with a linear shift function (Doksum, 1974) that is equivalent to a location-scale model (Hsieh, 1995) will be assumed and an empirical process approach (Hsieh, 1995) is used to estimate the parameters of the shift function. Then, the estimated shift function is set to zero, and the solution is defined to be an estimate of the crossing-point x*. An approximate confidence band of the linear shift function at the crossing-point x* is also presented, which is inverted to yield an approximate confidence interval for the crossing-point. Finally, the lifetime of guinea pigs in days observed in a treatment-control experiment in Bjerkedal (1960) is used to demonstrate our procedure for estimating the crossing-point. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On the solutions of the linear integral equations of Volterra type MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2007smet ÖzdemirAbstract Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1,K*f were obtained as |f(t)|,1, |f(t)|,2 and |f(t)|,4 in (J. Math. Anal. Appl. 1978; 64:381,397; Int. J. Math. Math. Sci. 1982; 5(1):123,131). The boundary of the solution function of an equation in this type was found as |f(t)|,2n in (Integr. Equ. Oper. Theory 2002; 43:466,479), where t,[0, ,) and n is a natural number such that n,2. In (Math. Comp. 2006; 75:1175,1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43:466,479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43:466,479). In the present paper, the sufficient conditions for the boundedness of functions f, f,, f,,, ,, f(n+3), (n,,) defined on the infinite interval [0, ,) are given by our method, where f is the solution of the equation f(t)=1,K*f. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle MATHEMATISCHE NACHRICHTEN, Issue 1-2 2007Lev AizenbergAbstract Let D be a unit disk andM be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l (M) < 2,. Our first result characterizes the restriction of the holomorphic functions f , ,(D), which are in the Hardy class ,1 near the arcM and are denoted by ,, ,1M(,,), to the open arcM. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M. As an application of the above characterization, we present an extension theorem for a function f , L1(M) from any symmetric sub-arc L , M of the unit circle, such that , M, to a function f , ,, ,1L(,,). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A new investigation of the extended Krylov subspace method for matrix function evaluations NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2010L. KnizhnermanAbstract For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## On the noise sensitivity of monotone functions RANDOM STRUCTURES AND ALGORITHMS, Issue 3 2003Elchanan MosselIt is known that for all monotone functions f : {0, 1}n , {0, 1}, if x , {0, 1}n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability , = n,,, then P[f(x) , f(y)] < cn,,+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[fn(x) , fn(y)] , ,, where 0 < , < 1/2, required , , c(,)n,,, where , = 1 , ln 2/ln 3 = 0.36907 ,, and c(,) > 0. We improve this result by achieving for every 0 < , < 1/2, P[fn(x) , fn(y)] , ,, with: , = c(,)n,, for any , < 1/2, using the recursive majority function with arity k = k(,); , = c(,)n,1/2logtn for t = log2 = .3257 ,, using an explicit recursive majority function with increasing arities; and , = c(,)n,1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand. We also study the problem of achieving the best dependence on , in the case that the noise rate , is at least a small constant; the results we obtain are tight to within logarithmic factors. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 333,350, 2003 [source] ## Non-isothermal Decomposition Reaction Kinetics of the Magnesium Oxalate Dihydrate CHINESE JOURNAL OF CHEMISTRY, Issue 3 2006Jian-Jun ZhangAbstract The thermal decomposition of the magnesium oxalate dihydrate in a static air atmosphere was investigated by TG-DTG techniques. The intermediate and residue of each decomposition were identified from their TG curve. The kinetic triplet, the activation energy E, the pre-exponential factor A and the mechanism functions f(,) were obtained from analysis of the TG-DTG curves of thermal decomposition of the first stage and the second stage by the Popescu method and the Flynn-Wall-Ozawa method. [source] ## Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2007Patrick CheriditoFor a d -dimensional diffusion of the form dXt = ,(Xt)dt + ,(Xt)dWt and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, ,, and A solving the second-order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes solve the 2BSDE, where ,, is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in , and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,,, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Yt = v(t, Xt), t , [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc. [source] |