
	Where results make sense

Topic: Dirichlet function

Related Topics

List of functions

Dirichlet kernel

Dirichlet character

Dirichlet series

Dirichlet boundary condition

Dirichlet

Dirichlet tessellation

Pigeonhole principle

Johann Peter Gustav Lejeune Dirichlet

	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)

	ipedia.com: Nowhere continuous Article (Site not responding. Last check: 2007-10-20)
		More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or the continuity definition by the definition of continuity in a topological space.
		One example of such a function is the indicator function of the rational numbers.
		Functions of this type were originally investigated by Dirichlet.
	www.ipedia.com /nowhere_continuous.html (403 words)

	Nowhere continuous -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		One example of such a function is the (additional info and facts about indicator function) indicator function of the (An integer or a fraction) rational numbers.
		To show the function is not continuous at y, we need find an ε such that no matter how small we choose δ, there will be points z within δ of y such that f(z) is not within ε of f(y) = 1.
		Functions of this type were originally investigated by (additional info and facts about Dirichlet) Dirichlet.
	www.absoluteastronomy.com /encyclopedia/n/no/nowhere_continuous.htm (326 words)

	Dirichlet eta function - Wikipedia, the free encyclopedia
		Hardy gave a simple proof of the functional equation for the eta function, which is
		From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.
		Peter Borwein used approximations involving Chebyshev polynomials to produce a method for efficient evaluation of the eta function.
	en.wikipedia.org /wiki/Dirichlet_eta_function (216 words)

	PlanetMath: Dirichlet's function
		This function has the property that it is continuous at every irrational number and discontinuous at every rational one.
		This is often given as the (amazing!) example of a sequence of everywhere-continuous functions whose limit function is nowhere continuous.
		This is version 4 of Dirichlet's function, born on 2002-12-01, modified 2005-03-25.
	planetmath.org /encyclopedia/DirichletsFunction.html (256 words)

	Dirichlet (Site not responding. Last check: 2007-10-20)
		His family hailed from the town of Richelet in Belgium, from which his surname "Lejeune Dirichlet" ("lejeune de Richelet" = "the young chap from Richelet") was derived, and that was where his grandfather lived.
		Dirichlet was born in Düren, where hisfather was the postmaster.
		After his death, Dirichlet's lectures and other results in numbertheory were collected, edited and published by his friend and fellow mathematician Richard Dedekind under the title Vorlesungen über Zahlentheorie (Lectures on Number Theory).
	www.therfcc.org /dirichlet-34860.html (261 words)

	Dirichlet (Site not responding. Last check: 2007-10-20)
		Dirichlet was the first mathematician to define rigorously the concept of a function.
		While Dirichlet's definition was still a dynamic one based on assignments (as opposed to the static definition as set of pairs which is now accepted as state of the art in what concerns rigor) it brought functions into the modern era.
		To illustrate what weird animal could be a function, he introduced what is now called the Dirichlet function, a function that is 1 at the rationals, 0 at the irrationals.
	www.math.fau.edu /schonbek/Modern_Analysis/calcmath18.html (243 words)

	QMG project: The finite element package
		The conductivity function should take one argument; this argument is a matrix with a number of columns equal to the dimension of the brep, and a variable of rows.
		The argument of this function is understood to be a matrix, where each row of the matrix is a vector of coordinates of points that lie on the boundary face.
		The vector b is depends on the Dirichlet and on the mesh, f_0 is the contribution to the right-hand side of the assembled linear system Ku=f arising from the Dirichlet data.
	www.cs.cornell.edu /home/vavasis/qmg1.0/fe.html (2064 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		That is, it is easy to construct a function which cannot be integrated by the method of Riemann, yet whose integral can be taken easily and unambiguously.
		Dirichlet's function assigns 1 to each rational number and 0 to each irrational number.
		Dirichlet's function cannot be integrated by Riemann's method, unless of course a and b are the same number.
	www.nadn.navy.mil /Users/math/hmk/sm161notes/dirichlet.htm (885 words)

	[No title]
		Whereas the study of the quantum mechanical saddle oscillator by means of periodic orbit theory is an interesting investigation, the similarity of Lorentzian smoothing in the density of states, respectively in the density of zeros for $\zeta(z)$, seems to be coincidental.
		Basic properties of Riemann's zeta-function and Dirichlet's $L$-functions are surveyed; it may be a bit misleading for a non-specialist that the zero-free region for the zeta-function (in Theorem 8.6) is not given in the sharpest known form.
		The method consists in evaluating Dirichlet series, and several of their derivatives, at a set of regularly spaced values; this is done by using the fast Fourier transform to reduce the problem to the evaluation of rational functions.
	www.math.niu.edu /~rusin/known-math/99/zeta (3655 words)

	1.2. Relations and Functions (Site not responding. Last check: 2007-10-20)
		Functions can be classified into three groups: those for which every element in the image has one preimage, those for which the range is the same as the image, and those which have both of these properties.
		A function f from A to B is called onto if for all b in B there is an a in A such that f(a) = b.
		A function f from A to B is called a bijection if it is one to one and onto, i.e.
	web01.shu.edu /projects/reals/logic/relation.html (615 words)

	L-functions and elliptic curves
		For many such series of interest, the corresponding function can be extended to a meromorphic function on the whole plane by a process known as "analytic continuation".
		Dirichlet functions are of great importance in number theory because of the following formal property.
		This function is known as the Hasse-Weil L-function.
	www.mbay.net /~cgd/flt/flt06.htm (2077 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 Beta Function -- from Harry J. Smith (Site not responding. Last check: 2007-10-20)
		Dirichlet beta function of x > 0 is defined by the infinite series 1 − 1/3^x + 1/5^x − 1/7^x +...
		B(k) is a Bernoulli number and J is a large number of the form 4n + 1.
		This is from "An Atlas Of Functions" by Spanier, J. and Oldham, K. 1987, equation 3:3:7.
	www.geocities.com /hjsmithh/Numbers/Beta.html (206 words)

	7.4. Lebesgue Integral
		Just as step functions were used to define the Riemann integral of a bounded function f over an interval [a, b], simple functions are used to define the Lebesgue integral of f over a set of finite measure.
		That is somewhat comparable to one of the equivalent definitions of continuous functions: a function f is continuous if the inverse image of every open interval is open.
		Measurable functions that are bounded are equivalent to Lebesgue integrable functions.
	web01.shu.edu /projects/reals/integ/lebes.html (1687 words)

	week217
		Dirichlet used these L-functions to prove that there are infinitely many primes equal to k mod n as long as k is relatively prime to n.
		Also: such zeta functions are quotients of polynomials, they satisfy a functional equation, and a lot of information about their zeroes and poles can be recovered from the topology of the corresponding complex algebraic varieties.
		These p functions on Z form a basis of the vector space of periodic functions on Z with period p.
	math.ucr.edu /home/baez/week217.html (4450 words)

	Dirichlet Eta Function -- from Harry J. Smith (Site not responding. Last check: 2007-10-20)
		Dirichlet Eta Function -- from Harry J. Smith
		The Dirichlet eta function eta(x) = 1 − 2^−x + 3^−x −...
		Dirichlet eta function of x > 0, is defined by the series 1 − 1/2^x + 1/3^x − 1/4^x +...
	www.geocities.com /hjsmithh/Numbers/Eta.html (171 words)

	[No title]
		Two functions Ns and Ss are created to help in the study because of the complexity involved.
		To accomplish this task, we construct two functions Ns and Ss, which represent the normalized, and “smoothed” sums respectively of ap ‘s defined as p+1 minus the number of solutions modulo p, where p is a prime.
		In designing and running the program which generated primes and computed the values for the functions Ns and Ss, the PARI package, which is capable of doing formal computations at high speed, was imported into a C program as a library.
	www.math.princeton.edu /mathlab/projects/ellcurves/op/curvecubic.doc (2824 words)

	Duality (Site not responding. Last check: 2007-10-20)
		Earlier I mentioned the Dirichlet conditions for Fourier series, and I noted that these guarantee a Fourier transform is possible.
		Well, it would be nice to extend that to functions which go to any constant value, instead of to zero.
		A function which approaches a constant can now be expressed as the sum of a constant and a function that approaches zero.
	www.sunlightd.com /Fourier/Duality.htm (412 words)

	abstract (Site not responding. Last check: 2007-10-20)
		N THE PARITY OF THE PARTITION FUNCTION AND A CONJECTURE OF The subject of the partition function congruences originated in the work of Ramanajun.
		The log function among the additive,the power function among the multiplicative functions can be characterized by their regular behaviour.New results and problems will be treated.
		This transform is of interest in view of applications to the fourth moment of the zeta-function, and it is closely related, on one hand, to the spectral theory of automorphic functions, and one the other hand, to the additive divisor problem.
	www.imsc.ernet.in /~antzeta/abstract/abstract.html (1165 words)

	Encyclopedia: Chinese remainder theorem
		In abstract algebra, it is possible to combine several rings into one large product ring.
		In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.
		The inverse isomorphism can be constructed as follows.
	www.nationmaster.com /encyclopedia/Chinese-remainder-theorem (977 words)

	Math 539, Fall 2003
		The main goal will be to prove Dirichlet's theorem on primes in arithmetic progressions and the prime number theorem.
		We introduce Dirichlet series in a formal way earlier than is done in the text since this makes some of the manipulations of the earlier chapters more transparent.
		But the serious use of analytic functions does not begin until we begin to study the Riemann zeta function and Dirichlet L-functions.
	www.math.ubc.ca /~boyd/wwwold/math539 (413 words)

	L-Functions (Site not responding. Last check: 2007-10-20)
		is the data for the Riemann zeta function.
		is the data for the Dirichlet L-function with the character which is the Legendre symbol modulo 5
		is the data for the Dirichlet L-function with the character
	www.math.okstate.edu /~loriw/start/start.html (196 words)

	Dirichlet Function (Site not responding. Last check: 2007-10-20)
		The Dirichlet Function is not continuous at any point on the real line.
		Show that g can not be continuous at a rational number by considering sequences of irrational numbers.
		Then show that g can not be continuous at an irrational number by considering sequences of rational numbers.
	pirate.shu.edu /~wachsmut/ira/cont/fp_diric.html (48 words)

	The Dirichlet/Riemann zeta Function (Site not responding. Last check: 2007-10-20)
		If f is omited, the constant function 1 is assumed.
		There are some functions, such as f(n) = n factorial, that never converge, no matter the value of s.
		Yet there are nonzero functions that converge everywhere as well.
	www.mathreference.com /lc-z,drz.html (173 words)

	Mathematics of Computation
		By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if
		P. Bateman and E. Grosswald, On Epstein's zeta function.
		M. Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field.
	www.ams.org /mcom/2004-73-245/S0025-5718-03-01537-0/home.html (144 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		~machine precision (for 32-bit word) integer :: checkstatus !file open/close status report character(40) :: savefilename !output file containing sim output !NOTE: Provide following user defined functions as Fortran function calls: !
		Function f(x) :initial condition - VANISHES when x is "close" to left/right boundaries !
		Function lb(t) :left (Dirichlet) boundary function u(x=xleft,t) = lb(t) !
	unr.edu /homepage/mcubed/TwoSided_Test_Case.f90 (333 words)

Try your search on: Qwika (all wikis)