A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

高等学校計算数学学報 = Numerical mathematics

Elementary Differential Equations (Fourth Edition). By Earl D. Rainville and Phillip E. Bedient. Pp. xiv, 466. 1969 (The Macmillan Company, New York.)

Statisticians are increasingly posed with thought-provoking and even paradoxical questions, challenging our qualifications for entering the statistical paradises created by Big Data. By developing measures for data quality, this article suggests a framework to address such a question: “Which one should I trust more: a 1% survey with 60% response rate or a self-reported administrative dataset covering 80% of the population?” A 5-element Euler-formula-like identity shows that for any dataset of size $n$, probabilistic or not, the difference between the sample average $\overline{X}_{n}$ and the population average $\overline{X}_{N}$ is the product of three terms: (1) a data quality measure, $\rho_{{R,X}}$, the correlation between $X_{j}$ and the response/recording indicator $R_{j}$; (2) a data quantity measure, $\sqrt{(N-n)/n}$, where $N$ is the population size; and (3) a problem difficulty measure, $\sigma_{X}$, the standard deviation of $X$. This decomposition provides multiple insights: (I) Probabilistic sampling ensures high data quality by controlling $\rho_{{R,X}}$ at the level of $N^{-1/2}$; (II) When we lose this control, the impact of $N$ is no longer canceled by $\rho_{{R,X}}$, leading to a Law of Large Populations (LLP), that is, our estimation error, relative to the benchmarking rate $1/\sqrt{n}$, increases with $\sqrt{N}$; and (III) the “bigness” of such Big Data (for population inferences) should be measured by the relative size $f=n/N$, not the absolute size $n$; (IV) When combining data sources for population inferences, those relatively tiny but higher quality ones should be given far more weights than suggested by their sizes. Estimates obtained from the Cooperative Congressional Election Study (CCES) of the 2016 US presidential election suggest a $\rho_{{R,X}}\approx-0.005$ for self-reporting to vote for Donald Trump. Because of LLP, this seemingly minuscule data defect correlation implies that the simple sample proportion of the self-reported voting preference for Trump from $1\%$ of the US eligible voters, that is, $n\approx2\mbox{,}300\mbox{,}000$, has the same mean squared error as the corresponding sample proportion from a genuine simple random sample of size $n\approx400$, a $99.98\%$ reduction of sample size (and hence our confidence). The CCES data demonstrate LLP vividly: on average, the larger the state’s voter populations, the further away the actual Trump vote shares from the usual $95\%$ confidence intervals based on the sample proportions. This should remind us that, without taking data quality into account, population inferences with Big Data are subject to a Big Data Paradox: the more the data, the surer we fool ourselves.

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Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election

Differential and Integral Calculus. By F. D. Murnaghan. Pp. x, 502. 1947. (Remsen Press, New York)

Treating a boundary value problem in analytical fluid dynamics, translation of 2D steady Navier-Stokes equations to ordinary differential form leads to a second order equation of Riccati type. In the case of a compressible fluid with constant kinematic viscosity along streamlines, it is possible to find an exact solution of the differential problem by rational combination of Airy’s functions and their derivatives.

Exact solution of a differential problem in analytical fluid dynamics: use of Airy's functions

Temporal fringe pattern analysis is invaluable in transient phenomena studies but necessitates long processing times. Here we describe a parallel computing strategy based on the single-program multiple-data model and hyperthreading processor technology to reduce the execution time. In a two-node cluster workstation configuration we found that execution periods were reduced by 1.6 times when four virtual processors were used. To allow even lower execution times with an increasing number of processors, the time allocated for data transfer, data read, and waiting should be minimized. Parallel computing is found here to present a feasible approach to reduce execution times in temporal fringe pattern analysis.

Temporal fringe pattern analysis with parallel computing

In this paper I present a first attempt for a possible description of fluids dynamics by mean of a cellular automata technique. With the use of simple and elementary rules, based on random behaviour either, the model permits to obtain the evolution in time for a two-dimensional grid, where one molecule of the material fluid can ideally place itself on a single geometric square. By mean of computational simulations, some realistic effects, here showed by use of digital pictures, have been obtained. In a subsequent step of this work I think to use a parallel program for a high performances computational simulation, for increasing the degree of realism of the digital rendering by mean of a three-dimensional grid too. For the execution of the simulations, numerical methods of resolution for differential equations have not been used.

A first approach for a possible cellular automaton model of fluids dynamics

In this work we present a matrix generalization of the Euler identity about exponential representation of a complex number. The concept of matrix exponential is used in a fundamental way. We define a notion of matrix imaginary unit which generalizes the usual complex imaginary unit. The Euler-like identity so obtained is compatible with the classical one. Also, we derive some exponential representation for matrix real and imaginary unit, and for the first Pauli matrix.

A matrix generalization of Euler identity e^(ix) = cosx + i sinx

In this work I present a generalization of Amdahl's law on the limits of a parallel implementation with many processors In particular I establish some mathematical relations involving the number of processors and the dimension of the treated problem, and with these conditions I define, on the ground of the reachable speedup, some classes of parallelism for the implementations I also derive a condition for obtaining superlinear speedup The used mathematical technics are those of differential calculus I describe some examples from classical problems offered by the specialized literature on the subject

Gianluca Argentini

Papers

Exact solution of a differential problem in analytical fluid dynamics: use of Airy's functions

Temporal fringe pattern analysis with parallel computing

A first approach for a possible cellular automaton model of fluids dynamics

A matrix generalization of Euler identity e^(ix) = cosx + i sinx

A generalization of Amdahl's law and relative conditions of parallelism