This study analyses the relationship between Generalized Functions and influence functions in Winkler base. Muller-Breslau's principle is difficult for getting the exact value in influence lines at which we are concerned. But the meaning of this principle can be expressed in terms of the Generalized Functions in governing differential equations, EIy''+ky = q, and due to the very convenient characteristics of the Generalized Functions in q , we can get the particular solutions of the D.E. easily. The conclusion are followings 1) The influence functions for the deflection, deflection angle, moment and shear at ξ when unit force is applied at x , are the deflections which is the solutions in EIy''+ky = q, when the q s are expressed in terms of the δ?, δ?₁, EIδ?₂, EIδ?₃ respectively. 2) The influence functions for the deflection, deflection angle, moment and shear at ξ when unit moment is applied at x are those deflections which are the solution in EIy''+ky = q when the q s are expressed in terms of the -δ?₁, δ?₂,- EIδ?₃,- kδ? respectively. 3) The influence functions for the deflection, deflection angle, moment and shear at ξ when unit deflection angle discontinuity is applied at x are those deflections which are the solutions in D.E. when the q s are expressed in terms of the EIδ?₂, EIδ?₃,- kEIδ?,- kEI δ?₁ respectively. 4) For the case of unit deflection discontinuity is applied at x , we get the influence functions for the deflection, deflection angle, moment and shear at ξ from the solutions of D.E. where q is expressed in terms of -EIδ?₃, -kδ?, kEIδ?₁,- kEIδ?₂ respectively. Here we can see the rotation of δ?, δ?₁, δ?₂, δ?₃ depending on the cases of loads in q.

Generalized Functions와 영향함수와의 관계

The Monte Carlo method.- Sampling from known distributions.- Pseudorandom number generators.- Variance reduction techniques.- Quasi-Monte Carlo constructions.- Using quasi-Monte Carlo constructions.- Using quasi-Monte Carlo in practice.- Financial applications.- Beyond numerical integration.- Review of algebra.- Error and variance analysis for Halton sequences.- References.- Index.

Monte Carlo and Quasi-Monte Carlo Sampling

A conditionally Gaussian channel is a vector channel in which the channel output, given the channel input, has a Gaussian distribution with (well-behaved) input-dependent mean and covariance. We study the capacity-achieving probability measure for conditionally Gaussian channels subject to bounded-input constraints and average cost constraints. Many practical communication systems, including additive Gaussian noise channels, certain optical channels, fading channels, and interference channels fall within this framework. Subject to bounded-input constraint (and average cost constraints), we show that the channel capacity is achievable and we derive a necessary and sufficient condition for a probability measure to be capacity achieving. Under certain conditions, the capacity-achieving measure is proved to be discrete.

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Capacity-achieving probability measure for conditionally Gaussian channels with bounded inputs

We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N≫k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of $\hat{\mathbf{A}}$, and estimate their coefficients, in polynomial(k,log N) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002; Proceedings of SPIE Wavelets XI, 2005). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008).

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Combinatorial Sublinear-Time Fourier Algorithms

Managing, processing and understanding big healthcare data is challenging, costly and demanding. Without a robust fundamental theory for representation, analysis and inference, a roadmap for uniform handling and analyzing of such complex data remains elusive. In this article, we outline various big data challenges, opportunities, modeling methods and software techniques for blending complex healthcare data, advanced analytic tools, and distributed scientific computing. Using imaging, genetic and healthcare data we provide examples of processing heterogeneous datasets using distributed cloud services, automated and semi-automated classification techniques, and open-science protocols. Despite substantial advances, new innovative technologies need to be developed that enhance, scale and optimize the management and processing of large, complex and heterogeneous data. Stakeholder investments in data acquisition, research and development, computational infrastructure and education will be critical to realize the huge potential of big data, to reap the expected information benefits and to build lasting knowledge assets. Multi-faceted proprietary, open-source, and community developments will be essential to enable broad, reliable, sustainable and efficient data-driven discovery and analytics. Big data will affect every sector of the economy and their hallmark will be ‘team science’.

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Methodological challenges and analytic opportunities for modeling and interpreting Big Healthcare Data

Introduction.- Preparations.- Laplace and Z Transforms.- Fourier Series.- L^2 Theory.- Separation of Variables.- Fourier Transforms.- Distributions.- Multi-Dimensional Fourier Analysis.- Appendix A: The ubiquitous convolution.- Appendix B: The Discrete Fourier Transform.- Appendix C: Formulae.- Appendix D: Answers to exercises.- Appendix E: Literature.

Anders Vretblad

Papers

Fourier Analysis and Its Applications