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# Topic: Dirichlet

###### In the News (Fri 24 May 13)

 Johann Peter Gustav Lejeune Dirichlet - Wikipedia, the free encyclopedia His family hailed from the town of Richelet in Belgium, from which his surname "Lejeune Dirichlet" ("le jeune de Richelet" = "the young chap from Richelet") was derived, and that was where his grandfather lived. Dirichlet was born in Düren, where his father was the postmaster. After his death, Dirichlet's lectures and other results in number theory were collected, edited and published by his friend and fellow mathematician Richard Dedekind under the title Vorlesungen über Zahlentheorie (Lectures on Number Theory). en.wikipedia.org /wiki/Johann_Peter_Gustav_Lejeune_Dirichlet   (305 words)

 Dirichlet series - Wikipedia, the free encyclopedia In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex line, such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes. en.wikipedia.org /wiki/Dirichlet_series   (288 words)

 Dirichlet character - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-07) Dirichlet L-series are straightforward generalizations of the Riemann zeta function and appear prominently in the generalized Riemann hypothesis. A Dirichlet L-series can be expressed as a linear combination of the Hurwitz zeta function, and thus the study of L-series can be unified through a study of the Hurwitz zeta. Dirichlet characters and their L-series were introduced by Dirichlet, in 1831, in order to prove Dirichlet's theorem about the infinitude of primes in arithmetic progressions. www.hartselle.us /project/wikipedia/index.php/Dirichlet_character   (388 words)

 Dirichlet   (Site not responding. Last check: 2007-11-07) Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845. Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties. Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Dirichlet.html   (2043 words)

 Johann Peter Gustav Lejeune Dirichlet - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-11-07) His family hailed from the town of in Belgium, from which his surname "Lejeune Dirichlet" ("le jeune de Richelet" = "the young chap from Richelet") was derived, and that was where his grandfather lived. He married, who came from a distinguished Jewish family, being a granddaughter of the philosopher Moses Mendelssohn, and a sister of the composer Felix Mendelssohn. Dirichlet, Johann Peter Gustav Lejeune, Vorlesungen uber Zahlentheorie. www.hartselle.us /project/wikipedia/index.php/Peter_Gustav_Dirichlet   (332 words)

 Dirichlet tessellation of bark beetle spatial attack points This tessellation was proposed in 1850 by Dirichlet (Upton and Fingleton 1985) and a formal mathematical definition is given by Green and Sibson (1978). Dirichlet tessellations can be thought of as representing the areas of territorial animals, allelochemic-producing plants, or the packing of cells in a tissue. Dirichlet (Voronoi) cells have been delineated by algorithms that use Delaunay triangles and circumscribing circles on a HITAC M-180 computer (Tanemura and Hasegawa 1980) or elimination of intersecting circles according to a set of rules (Honda 1978). vinsonlab.tamu.edu /former/john/papers/jae92a.htm   (6012 words)

 Correct calculation of Dirichlet polygon areas   (Site not responding. Last check: 2007-11-07) The Dirichlet cell, first proposed in 1850, has been useful in many scientific disciplines and thus is known under a variety of names including Voronoi, 1909, Thiessen, 1911, Wigner-Seitz, 1933, cell model, 1953, and S-mosaic, 1977 (Rogers 1964; Mead 1971; Rhynsburger 1973; Upton & Fingleton 1985; David 1988). More recently, a Dirichlet tessellation algorithm was developed to define colonization territories of bark beetles (Coleoptera: Scolytidae) under the bark of host trees (Byers 1992). However, calculations of the area of a Dirichlet cell are undervalued by about 10 to 30 percent. www.wcrl.ars.usda.gov /cec/z65-abs.htm   (974 words)

 Dirichlet mixtures In Section 6, we'll see that Dirichlet mixtures are superior to all the other regularizers examined, and that there is not much room for improvement to better regularizers. One way to view the posterior counts of Dirichlet mixtures is as a linear combination of pseudocount regularizers, where the weights on the combination vary from one sample to another, but the underlying regularizers are fixed. Thus a 9-component Dirichlet mixture for the amino acids will have 168 degrees of freedom: 9 pseudocount vectors with 20-components each and 9 mixture coefficients (whose sum can be normalized to 1.0). cse.ucsc.edu /research/compbio/html_format_papers/tr-95-11/node14.html   (343 words)

 Dirichlet conditions Dirichlet gave the necessary conditions for existence of a fourier spectra for any signal Dirichlet's condition is to have finite number of "jump discontinuities" for piecewise smooth function. From my understanding, Dirichlet's convergence depends on the property that you can cut the function into small monotonic(either increase or decrease) section, so that the average value at any point [f(x+)+f(x-)]/2 can be calculated. www.edaboard.com /ftopic93912.html   (418 words)

 Dirichlet Forms And Markov Processes: A Generalized Framework Including Both Elliptic And Parabolic Cases - Stannat ...   (Site not responding. Last check: 2007-11-07) We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. Dirichlet forms and Markov processes: a generalized framework including both elliptic and parabolic cases. citeseer.ist.psu.edu /370433.html   (547 words)

 Dirichlet characters   (Site not responding. Last check: 2007-11-07) The first zeros on the critical line of some Dirichlet L-series... Johan Bosman, Lenny Taelman On sums of sums of values of Dirichlet characters NA... Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet charac... www.scienceoxygen.com /math/324.html   (140 words)

 Dirichlet, Peter Gustav Lejeune --  Encyclopædia Britannica   (Site not responding. Last check: 2007-11-07) He taught at the universities of Breslau (1827) and Berlin (1828–55) and in 1855 succeeded Carl Friedrich Gauss at the University of Göttingen. More results on "Dirichlet, Peter Gustav Lejeune" when you join. German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic (unique factorization of every integer into a product of primes) to complex number fields. www.britannica.com /eb/article-9030603?tocId=9030603   (735 words)

 GETWGT - Dirichlet Mixture Estimation   (Site not responding. Last check: 2007-11-07) GETWGT is a set of routines for Dirichlet Mixture estimation. The main requirement in the design of this library was that there be a single routine with a simple interface through which the user interacts. This routine, called getwgt, accepts a set of nucleic acid counts, updates an internal Dirichlet mixture model, and returns its current estimate for the pseudocounts. orion.math.iastate.edu /burkardt/b_src/getwgt/getwgt.html   (297 words)

 Dirichlet Mixtures and other Regularizers   (Site not responding. Last check: 2007-11-07) The optimization problem for Dirichlet mixtures is quite difficult, since there are many local minima of similar quality, and it takes a lot of optimization to get the value of the local minimum determined well enough to distinguish it from the others. The component that 9-component mixtures have for "highly conserved residues" is also split into two components (one mainly for C and W, the other mainly for P and G---these should probably both be split further, to get down to components dominated by a single residue). The data used for training the Dirichlet mixtures described above is provided (in gzipped files of count vectors). www.soe.ucsc.edu /research/compbio/dirichlets   (1699 words)

 BDMLIB - Estimate Dirichlet Mixture Weights   (Site not responding. Last check: 2007-11-07) BDMLIB addresses the problem of determining the weights W that define a Dirichlet mixture. To generate "events", first one of the component Dirichlet PDF's is chosen based on the weights. Then the Dirichlet PDF, whose parameters ALPHA are known, is used to generate a probability vector P. This probability vector is then used with the multinomial PDF to generate an integer vector X. Sound confusing? orion.math.iastate.edu /burkardt/b_src/bdmlib/bdmlib.html   (280 words)

 LEJEUNE-DIRICHLET, PETER GUSTAV (1805 - 1859)   (Site not responding. Last check: 2007-11-07) Dirichlet proved that any arithmetic progression without trivial obstructions contains infinitely many primes. Dedekind edited Dirichlet's lectures on number theory and published these as Vorlesungen uber Zahlentheorie in 1863. It is noted by H. Edwards that: "Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." www.scs.uiuc.edu /~mainzv/exhibitmath/bkgd/dirichlet-bkgd.htm   (150 words)

 Bretton Woods Workshop   (Site not responding. Last check: 2007-11-07) The aim of this workshop is to explore the modern theory of Dirichlet series in several complex variables (multiple Dirichlet series) and to bring together experts in this and allied fields. During this period we will not have regularly scheduled talks, but conference participants will be encouraged to continue their mathematical interactions and take advantage of the proximity of so many researchers interested in the topic of multiple Dirichlet series. Function field example of a multiple Dirichlet Series, notes to a lecture by Gautam Chinta. sporadic.stanford.edu /bump/mds_workshop.html   (320 words)

 Abstract for Markov chain sampling methods for Dirichlet process mixture models''   (Site not responding. Last check: 2007-11-07) Markov chain methods for sampling from the posterior distribution of a Dirichlet process mixture model are reviewed, and two new classes of methods are presented. One new approach is to make Metropolis-Hastings updates of the indicators specifying which mixture component is associated with each observation, perhaps supplemented with a partial form of Gibbs sampling. These methods are simple to implement and are more efficient than previous ways of handling general Dirichlet process mixture models with non-conjugate priors. www.cs.utoronto.ca /~radford/mixmc.abstract.html   (270 words)

 Amazon.ca: Books: New Directions in Dirichlet Forms   (Site not responding. Last check: 2007-11-07) The theory of Dirichlet forms brings together methods and insights from the calculus of variations, sotchastic analysis, partial differential and difference equations, potential theory, Riemannian geometry and more. This book features contributions by leading experts and provides up-to-date, authoritative accounts on exciting developments in the field and on new research perspectives. the topics are featured, and it is demonstarted that the theory of Dirichlet forms provides the proper framework for exploring these connections. www.amazon.ca /exec/obidos/ASIN/0821810618   (208 words)

 Random Dirichlet functions   (Site not responding. Last check: 2007-11-07) We show that if one inserts random plus and minus signs before the coefficients of the power series of a function in the Dirichlet space, then the resulting series is almost surely a multiplier of the Dirichlet space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip_{1/2} of functions obeying a Lipschitz condition with exponent 1/2. Versions of our results hold for weighted Dirichlet spaces, and much of our work is carried out in this more general setting. www.mth.msu.edu /~shapiro/Pubvit/Downloads/RanDir/RanDir.html   (140 words)

 Amazon.ca: Books: The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics   (Site not responding. Last check: 2007-11-07) Amazon.ca: Books: The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics Top of Page : The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics www.amazon.ca /exec/obidos/ASIN/0821813846   (152 words)

 Solutions of stochastic partial differential equations considered as Dirichlet processes, Denis Laurent   (Site not responding. Last check: 2007-11-07) We consider the parabolic stochastic partial differential equation $$u(t,x)=\Phi(x)+\int_0^t Lu(s,x)+f(s,x,u(s,x),Du(s,x))\,\d s$$ $$+\int_0^t g_i(s,x,u(s,x),Du(s,x))\,\d B^i_s,$$/ \noindent where f and g are supposed to be Lipschitzian and L is a self-adjoint operator associated with a Dirichlet form defined on a finite- or infinite-dimensional space. We prove that it admits a unique solution which is a Dirichlet process and, thanks to Itô's formula for Dirichlet processes, we prove a comparison theorem. [5] Bouleau, N. and Hirsch, F. (1991) Dirichlet Forms and Analysis on Wiener Space, de Gruyter Stud. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.bj/1099579156   (467 words)

 Weighted Extended B-Spline Approximation of Dirichlet Problems Weighted Extended B-Spline Approximation of Dirichlet Problems: SIAM Journal on Numerical Analysis Vol. We describe a new finite element method which uses weighted extended B-splines on a regular grid as basis functions for solving Dirichlet problems on bounded domains in arbitrary dimensions. This web-method does not require any grid generation and can be implemented very efficiently. epubs.siam.org /sam-bin/dbq/article/37320   (116 words)

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