Abstract: Poisson Voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices Although this particular space tessellation is intensively studied by mathematicians, in two- and three-dimensional (3D) spaces there is no exact result known for the size distribution of Voronoi cells Motivated by the simple form of the distribution function in the 1D case, a simple and compact analytical formula is proposed for approximating the Voronoi cell's size-distribution function in the practically important 2D and 3D cases as well Denoting the dimensionality of the space by d ( d = 1 , 2 , 3 ) the f ( y ) = Const * y ( 3 d - 1 ) / 2 exp ( - ( 3 d + 1 ) y / 2 ) compact form is suggested for the normalized cell-size distribution function By using large-scale computer simulations the viability of the proposed distribution function is studied and critically discussed

Abstract: A systematic theory is introduced for finding the derivatives of complex-valued matrix functions with respect to a complex-valued matrix variable and the complex conjugate of this variable. In the framework introduced, the differential of the complex-valued matrix function is used to identify the derivatives of this function. Matrix differentiation results are derived and summarized in tables which can be exploited in a wide range of signal processing related situations

Abstract: 1. Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Functional equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3. The Bloch-Wigner function D(z) and its generalizations . . . . . . . . . . . . 10 4. Volumes of hyperbolic 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5. . . . and values of Dedekind zeta functions . . . . . . . . . . . . . . . . . . . . . . . . . 16

Abstract: Using the the concept of cracking we explore the influence of density fluctuations and local anisotropy have on the stability of local and non-local anisotropic matter configurations in general relativity. This concept, conceived to describe the behaviour of a fluid distribution just after its departure from equilibrium, provides an alternative approach to consider the stability of selfgravitating compact objects. We show that potentially unstable regions within a configuration can be identify as a function of the difference of propagations of sound along tangential and radial directions. In fact, it is found that these regions could occur when, at particular point within the distribution, the tangential speed of sound is greater than radial one.

Abstract: We derive an explicit formula for the fine-scale Green’s function arising in variational multiscale analysis. The formula is expressed in terms of the classical Green’s function and a projector which defines the decomposition of the solution into coarse and fine scales. The theory is presented in an abstract operator format and subsequently specialized for the advection-diffusion equation. It is shown that different projectors lead to fine-scale Green’s functions with very different properties. For example, in the advection-dominated case, the projector induced by the $H^1_0$-seminorm produces a fine-scale Green’s function which is highly attenuated and localized. These are very desirable properties in a multiscale method and ones that are not shared by the $L^2$-projector. By design, the coarse-scale solution attains optimality in the norm associated with the projector. This property, combined with a localized fine-scale Green’s function, indicates the possibility of effective methods with local character for dominantly hyperbolic problems. The constructs lead to a new class of stabilized methods, and the relationship between $H^1_0$-optimality and the streamline-upwind Petrov-Galerkin (SUPG) method is described.

Abstract: Flat plate turbulent boundary layers under zero pressure gradient at high Reynolds numbers are studied to reveal appropriate scale relations and asymptotic behaviour. Careful examination of the skin-friction coefficient results confirms the necessity for direct and independent measurement of wall shear stress. We find that many of the previously proposed empirical relations accurately describe the local Cf behaviour when modified and underpinned by the same experimental data. The variation of the integral parameter, H, shows consistent agreement between the experimental data and the relation from classical theory. In accordance with the classical theory, the ratio of D and d asymptotes to a constant. Then, the usefulness of the ratio of appropriately defined mean and turbulent time-scales to define and diagnose equilibrium flow is established. Next, the description of mean velocity profiles is revisited, and the validity of the logarithmic law is re-established using both the mean velocity profile and its diagnostic function. The wake parameter, P, is shown to reach an asymptotic value at the highest available experimental Reynolds numbers if correct values of logarithmic-law constants and an appropriate skin-friction estimate are used. The paper closes with a discussion of the Reynolds number trends of the outer velocity defect which are important to establish a consistent similarity theory and appropriate scaling.

Abstract: This article is concerned with inference for a certain class of inhomogeneous Neyman-Scott point processes depending on spatial covariates. Regression parameter estimates obtained from a simple estimating function are shown to be asymptotically normal when the "mother" intensity for the Neyman-Scott process tends to infinity. Clustering parameter estimates are obtained using minimum contrast estimation based on the K-function. The approach is motivated and illustrated by applications to point pattern data from a tropical rain forest plot.

Abstract: In this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity. The analysis is performed for a general fractional order diffusion operator. The nonlinear term studied is a product of the unknown function and a convolution operator of order 0. Convergence of the approximation and a priori error estimates are given. Numerical computations are included, which confirm the theoretical predictions.

Abstract: Let f be a function transcendental and meromorphic in the plane, and define g(z) by g(z) = ?f(z) = f(z + 1) - f(z). A number of results are proved concerning the existence of zeros of g(z) or g(z)/f(z), in terms of the growth and the poles of f. The results may be viewed as discrete analogues of existing theorems on the zeros of f' and f'/f.

Abstract: We study the Likelihood function of data given fNL for the so-called local type of non-Gaussianity. In this case the curvature perturbation is a non-linear function, local in real space, of a Gaussian random field. We compute the Cramer–Rao bound for fNL and show that for small values of fNL the three-point function estimator saturates the bound and is equivalent to calculating the full Likelihood of the data. However, for sufficiently large fNL, the naive three-point function estimator has a much larger variance than previously thought. In the limit in which the departure from Gaussianity is detected with high confidence, error bars on fNL only decrease as 1/lnNpix rather than Npix−1/2 as the size of the data set increases. We identify the physical origin of this behaviour and explain why it only affects the local type of non-Gaussianity, where the contribution of the first multipoles is always relevant. We find a simple improvement to the three-point function estimator that makes the square root of its variance decrease as Npix−1/2 even for large fNL, asymptotically approaching the Cramer–Rao bound. We show that using the modified estimator is practically equivalent to computing the full Likelihood of fNL given the data. Thus other statistics of the data, such as the four-point function and Minkowski functionals, contain no additional information on fNL. In particular, we explicitly show that the recent claims about the relevance of the four-point function are not correct. By direct inspection of the Likelihood, we show that the data do not contain enough information for any statistic to be able to constrain higher order terms in the relation between the Gaussian field and the curvature perturbation, unless these are orders of magnitude larger than the size suggested by the current limits on fNL. As our main focus is the scaling with Npix of the various quantities, calculations are done in flat sky approximation and without the radiation transfer function.

Abstract: In many different fields of applied statistics the object of interest is depending on some continuous parameter, i.e. continuous time. Typical examples in biostatistics are growth curves or temperature measurements. Although for technical reasons, we are able to measure temperature just in discrete intervals — it is clear that temperature is a continuous process. Temperature during one year is a function with argument “time”. By collecting one-year-temperature functions for several years or for different weather stations we obtain bunch (sample) of functions — functional data set. The questions arising by the statistical analysis of functional data are basically identical to the standard statistical analysis of univariate or multivariate objects. From the theoretical point, design of a stochastic model for functional data and statistical analysis of the functional data set can be taken often one-to-one from the conventional multivariate analysis. In fact the first method how to deal with the functional data is to discretize them and perform a standard multivariate analysis on the resulting random vectors. The aim of this chapter is to introduce the functional data analysis (FDA), discuss the practical usage and implementation of the FDA methods.

Abstract: Complex emergent systems of many interacting components, including complex biological systems, have the potential to perform quantifiable functions. Accordingly, we define “functional information,” I(Ex), as a measure of system complexity. For a given system and function, x (e.g., a folded RNA sequence that binds to GTP), and degree of function, Ex (e.g., the RNA–GTP binding energy), I(Ex) = −log2[F(Ex)], where F(Ex) is the fraction of all possible configurations of the system that possess a degree of function ≥ Ex. Functional information, which we illustrate with letter sequences, artificial life, and biopolymers, thus represents the probability that an arbitrary configuration of a system will achieve a specific function to a specified degree. In each case we observe evidence for several distinct solutions with different maximum degrees of function, features that lead to steps in plots of information versus degree of function.

Abstract: We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.

Abstract: Inverse problems can be described as functional equations where the value of the function is known or easily estimable but the argument is unknown. Many problems in econometrics can be stated in the form of inverse problems where the argument itself is a function. For example, consider a nonlinear regression where the functional form is the object of interest. One can readily estimate the conditional expectation of the dependent variable given a vector of instruments. From this estimate, one would like to recover the unknown functional form. This chapter provides an introduction to the estimation of the solution to inverse problems. It focuses mainly on integral equations of the first kind. Solving these equations is particularly challenging as the solution does not necessarily exist, may not be unique, and is not continuous. As a result, a regularized (or smoothed) solution needs to be implemented. We review different regularization methods and study the properties of the estimator. Integral equations of the first kind appear, for example, in the generalized method of moments when the number of moment conditions is infinite, and in the nonparametric estimation of instrumental variable regressions. In the last section of this chapter, we investigate integral equations of the second kind, whose solutions may not be unique but are continuous. Such equations arise when additive models and measurement error models are estimated nonparametrically.

Abstract: We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accuracy the underlying function in between those edges. If f is a smooth function, say analytic, then classical Fourier projections recover f with exponential accuracy. However, if f contains one or more discontinuities, its global Fourier projections produce spurious Gibbs oscillations which spread throughout the smooth regions, enforcing local loss of resolution and global loss of accuracy. Our aim in the computation of the Gibbs phenomenon, is to detect edges and to reconstruct piecewise smooth f’s, while regaining the high accuracy encoded in the spectral data. To detect edges, we utilize a general family of edge detectors based on concentration kernels. Each kernel forms an approximate derivative of the delta function, which detects edges by separation of scales. We show how such kernels can be adapted to detect edges with one- and two-dimensional discrete data, with noisy data, and with incomplete spectral information. The main feature is concentration kernels which enable us to convert global spectral moments into local information in physical space. To reconstruct f with high accuracy we discuss novel families of mollifiers and filters. The main feature here is making these mollifiers and filters adapted to the local region of smoothness while increasing their accuracy together with the dimension of the data. These mollifiers and filters form approximate delta functions which are properly parameterized to recover f with (root-) exponential accuracy.

Abstract: A new analytic potential energy function form which incorporates the two leading inverse-power terms in the long-range potential is introduced and applied to a recently reported data set for the ground state of Ca2. The new function yields an accurate representation of data which span 99.97% of the well depth and involves only a fraction as many (between 1/3 and 2/3 fewer) parameters as were needed to define published potential functions for these systems based on the same data. Fits using this form also allow a robust determination of the dissociation energy and C 6 dispersion coefficient, and the resulting function is easier to use than other potential functions which have been determined for this system.

Abstract: Stabilization problem of positive discrete-time systems with bounds on the inputs or the states is solved in this paper. First, the synthesis of state-feedback controllers that ensure the nonnegativity and the stability of the unconstrained closed-loop systems is studied for both the nominal and uncertain systems. These results are then extended to the case of bounded controls and also for systems with bounds on the states. All the proposed conditions are solvable in terms of Linear Programming. A cost function is then proposed to synthesis controllers with good performance. An example illustrate the feasibility of the proposed approach.

Abstract: We present Feynman type diagrams for calculating the n-point function of the primordial curvature perturbation in terms of scalar field perturbations during inflation. The diagrams can be used to evaluate the corresponding terms in the n-point function at tree level or any required loop level. Rules are presented for drawing the diagrams and writing down the corresponding terms in real space and Fourier space. We show that vertices can be renormalized to automatically account for diagrams with dressed vertices. We apply these rules to calculate the primordial power spectrum up to two loops, the bispectrum including loop corrections, and the trispectrum.

Abstract: We study the geometry of the phase space of spatially flat Friedmann-Lemaitre-Robertson-Walker models in f(R) gravity, for a general form of the function f(R). The equilibrium points (de Sitter spaces) and their stability are discussed, and a comparison is made with the phase space of the equivalent scalar-tensor theory. New effective Lagrangians and Hamiltonians are also presented.

Abstract: Digital images are computed using an approach for correcting lens aberration. According to an example embodiment of the present invention, a digital imaging arrangement implements microlenses to direct light to photosensors that detect the light and generate data corresponding to the detected light. The generated data is used to compute an output image, where each output image pixel value corresponds to a selective weighting and summation of a subset of the detected photosensor values. The weighting is a function of characteristics of the imaging arrangement. In some applications, the weighting reduces the contribution of data from photosensors that contribute higher amounts of optical aberration to the corresponding output image pixel.

Abstract: We show that the second order field equations characterizing extremal solutions for spherically symmetric, stationary black holes are in fact implied by a system of first order equations given in terms of a prepotential W. This confirms and generalizes the results in [14]. Moreover we prove that the squared prepotential function shares the same properties of a c-function and that it interpolates between M^2_{ADM} and M^2_{BR}, the parameter of the near-horizon Bertotti-Robinson geometry. When the black holes are solutions of extended supergravities we are able to find an explicit expression for the prepotentials, valid at any radial distance from the horizon, which reproduces all the attractors of the four dimensional N>2 theories. Far from the horizon, however, for N-even our ansatz poses a constraint on one of the U-duality invariants for the non-BPS solutions with Z eq 0. We discuss a possible extension of our considerations to the non extremal case.

Abstract: Data from future high-precision Cosmic Microwave Background (CMB) measurements will be sensitive to the primordial Helium abundance $Y_p$. At the same time, this parameter can be predicted from Big Bang Nucleosynthesis (BBN) as a function of the baryon and radiation densities, as well as a neutrino chemical potential. We suggest to use this information to impose a self-consistent BBN prior on $Y_p$ and determine its impact on parameter inference from simulated Planck data. We find that this approach can significantly improve bounds on cosmological parameters compared to an analysis which treats $Y_p$ as a free parameter, if the neutrino chemical potential is taken to vanish. We demonstrate that fixing the Helium fraction to an arbitrary value can seriously bias parameter estimates. Under the assumption of degenerate BBN (i.e., letting the neutrino chemical potential $\xi$ vary), the BBN prior's constraining power is somewhat weakened, but nevertheless allows us to constrain $\xi$ with an accuracy that rivals bounds inferred from present data on light element abundances.

Abstract: First, a function projective synchronization of two identical systems is defined. Second, based on the active control method and symbolic computation Maple, the scheme of function projective synchronization is developed to synchronize two identical chaotic systems (two identical classic Lorenz systems) up to a scaling function matrix with different initial values. Numerical simulations are used to verify the effectiveness of the scheme.

Abstract: We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'number on the forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the degree-discrepancy lemma in the recent work of Sherstov (2007). Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ o SYMM o ANYO(1) cannot simulate the circuit class AC0 in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC0 in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gales, can simulate the class ACC0 in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain (2005) for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented bv polynomials of small degree over Zm, when m,q ges 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of Bourgain et al. (2005). It is known that such estimates imply that circuits of type MAJ o MODm o ANDisin log n cannot compute the MODq function in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC0 in polynomial size when the bottom fan-in is increased to poly-logarithmic size.

Abstract: This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterised by a finite set F of non-negative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that the partition function, i.e. the sum of the weights of all configurations, can be computed in polynomial time if either (1) every function in F is of ``product type'', or (2) every function in F is ``pure affine''. For every other fixed set F, computing the partition function is FP^{#P}-complete.

Abstract: We study the geometry of the phase space of spatially flat Friedmann?Lemaitre?Robertson?Walker models in f(R) gravity, for a general form of the function f(R). The equilibrium points (de Sitter spaces) and their stability are discussed, and a comparison is made with the phase space of the equivalent scalar-tensor theory. New effective Lagrangians and Hamiltonians are also presented.