
	Where results make sense

Topic: Automorphic function

Related Topics

functional analysis

functional programming

trigonometric function

functional group

exponential function

probability density function

continuous functions

graph of a function

Riemann zeta function

function domain

holomorphic function

In the News (Mon 28 May 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	Introduction to social network methods: Chapter 14: Automorphic equivalence
		Automorphic equivalence is not as demanding a definition of similarity as structural equivalence, but is more demanding than regular equivalence.
		Automorphic equivalence begins to change the focus of our attention, moving us away from concern with individual's network positions, and toward a more abstracted view of the network.
		Automorphic equivalence asks if the whole network can be re-arranged, putting different actors at different nodes, but leaving the relational structure or skeleton of the network intact.
	faculty.ucr.edu /~hanneman/nettext/C14_Automorphic_Equivalence.html (2581 words)

	Springer Online Reference Works
		The study of modular functions began in the 19th century in connection with the study of elliptic functions and preceded the appearance of the general theory of automorphic functions.
		Modular functions have also been applied in the study of conformal mapping; boundary properties of analytic functions and cluster sets (cf.
		The modular group (1) is then replaced by the modular group of automorphisms of the unit disc.
	eom.springer.de /m/m064430.htm (682 words)

	Springer Online Reference Works
		The Riemann surface of an elliptic function that satisfies a third- or fourth-degree equation is a torus; the genus of such a function is one.
		In the first case the algebraic function is a rational, in the second case it is an elliptic, while in the third case it is a general function.
		Each algebraic function field in one variable is the field of fractions of a Dedekind ring, so that many results and concepts of the theory of divisibility in algebraic number fields can be applied to function fields [12].
	eom.springer.de /A/a011490.htm (1619 words)

	Open Questions: Glossary of Mathematics
		The set A is called the "domain" of the function, while the "range" of the function consists of all b ∈ B that occur as the second element of a pair.
		A function is usually denoted symbolically with a name such as "f" and written in the form f(a) = b if (a,b) is a pair of corresponding elements.
		A function is "injective" or "1-to-1" if there is no b ∈ B that occurs more than once as the second element of a pair.
	www.openquestions.com /oq-magl.htm (335 words)

	Robert Langlands' work - automorphics forms
		Some of their consequences were explained in a graduate course given at Princeton in the spring of 1967, and then things were put in a somewhat wider context in a series of lectures at Yale later that Spring.
		These two observations underline that functoriality arose in an attempt to find a nonabelian class-field theory under the influence of the view, which arose in the early sixties, that much of the theory of automorphic forms could and should be treated in the context of group representations.
		The formula for Whittaker functions for unramified representations suggested in the letter was proved by Casselman and Shalika.
	www.sunsite.ubc.ca /DigitalMathArchive/Langlands/automorphic.html (829 words)

	Elliptic Curves and Modular Functions
		Modular and elliptic functions are both special cases of the concept of an automorphic function, which is a meromorphic function of 1 or more complex variables defined on a particular complex manifold and invariant under a particular group of analytic transformations (symmetries) of the manifold.
		the modular functions, are essentially the meromorphic functions on D considered as a Riemann surface by its isomorphism with H/ We have stressed this idea of symmetry because of the way it relates the analytic and geometric properties of an object like a Riemann surface to the algebraic properties of a group.
		Thus the elliptic functions are essentially the automorphic functions on the extended complex plane corresponding to the group of translations by two non-collinear values.
	www.mbay.net /~cgd/flt/flt05.htm (2994 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		The Laplace-Beltrami operator $\Del$ acts in the space of functions on Y. When $Y$ is compact it has discrete spectrum; we denote by $\mu_1 \leq \mu_2 \leq...$ its eigenvalues on $Y$ and by $\phi_i$ the corresponding eigenfunctions.
		The study of automorphic functions and the corresponding eigenvalues is important in many areas of representation theory, number theory and geometry.
		Fix one automorphic function, $\phi$, and consider the function $\phi^2$ on $Y$.
	www.math.technion.ac.il /~techm/20000323160020000323ber (361 words)

	PlanetMath: Poincaré, Jules Henri
		Poincaré's first area of interest in mathematics was the Fuchsian function that he named after the mathematician Lazarus Fuch because Fuch was known for being a good teacher and done alot of research in differential equations and in the theory of functions.
		The functions did not keep the name fuchsian and are today called automorphic.
		He also did work in analytic functions, algebraic geometry, and Diophantine problems where he made important contributions not unlike most of the areas he studied in.
	planetmath.org /encyclopedia/JulesHenriPoincare.html (2261 words)

	Visualizing Automorphic Functions (Site not responding. Last check: 2007-10-12)
		We use Farris' methods for visualizing complex-valued functions in the plane.
		Because that group is infinite, we can't possibly use every element, but the convergence properties for these functions are nice enough that the error isn't too bad when we only include some hundreds of terms.
		We have multiplied the function by a constant (25) to scale it into a range of values that looks pretty.
	math.scu.edu /~ffarris/auto/auto.html (249 words)

	Glossary of terms for Fermat's Last Theorem
		A meromorphic function of a complex variable that is invariant under a group of transformations of the function's domain.
		A function of the coefficients of a polynomial of one variable.
		A function in a vector space of functions whose image under some linear transformation on the space is a constant multiple of itself.
	cgd.best.vwh.net /home/flt/flt10.htm (2633 words)

	bibliography for automorphic and modular forms, L-functions, representations, and number theory
		[Gelfand 1962] I.M. Gelfand, `Automorphic functions and the theory of representations', in Proceedings, International Congress of Mathematicians, Stockholm, 1962, pp.
		Piatetski-Shapiro, 'On zeta functions of infinite-dimensional representations', Mat.
		[Satake 1966] I. Satake, `Spherical functions and Ramanujan conjecture', in Proc.
	www.math.umn.edu /~garrett/m/b/bib.html (3638 words)

	Schwarz biography
		Plateau published a famous memoir on the topic in 1866 and in the same year Weierstrass established a bridge between the theory of minimal surfaces and the theory of analytic functions.
		In answering the problem of when Gauss's hypergeometric series was an algebraic function Schwarz, as he had done so many times, developed a method which would lead to much more general results.
		This function is an early example of an automorphic function and in this work Schwarz was looking at ideas which led Klein and Poincaré to develop the theory of automorphic functions.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Schwarz.html (1154 words)

	This Week's Finds in Mathematical Physics (Week 217)
		Also: such zeta functions are quotients of polynomials, they satisfy a functional equation, and they can be computed in terms of the topology of the corresponding complex algebraic varieties.
		To be precise: if we form the function pi^{-s/2} Gamma(s/2) zeta(s) then this function is unchanged by the transformation s -> 1 - s This symmetry maps the line Re(s) = 1/2 to itself, and the Riemann Hypothesis says all the zeta zeros in the critical strip actually lie on this magic line.
		The fact that they satisfy a "functional equation" is just another way of saying their Mellin transforms are automorphic forms...
	www.lns.cornell.edu /spr/2005-05/msg0069239.html (2922 words)

	OUP: UK General Catalogue
		Automorphic forms are one of the central topics of analytic number theory.
		In the book, Iwaniec treats the spectral theory of automorphic forms as the study of the space $L^2 (H\Gamma)$, where $H$ is the upper half-plane and $\Gamma$ is a discrete subgroup of volume-preserving transformations of $H$.
		Among the topics discussed are Eisenstein series, estimates for Fourier coefficients of automorphic forms, the theory of Kloosterman sums, the Selberg trace formula, and the theory of small eigenvalues.
	www.oup.com /uk/catalogue/?ci=9780821831601 (368 words)

	Felix Klein - Wikipedia, the free encyclopedia
		Building on the methods of Hermite and Kronecker, he produced similar results to those of Brioschi and went on to completely solve the problem by means of the icosahedral group.
		However Poincaré published an outline of his theory of automorphic functions in 1881, which led to a friendly rivalry between the two men.
		Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke over a period of about 20 years.
	en.wikipedia.org /wiki/Felix_Klein (1655 words)

	Daniel Bump
		Moments of the Riemann Zeta Function and Eisenstein Series II with Jennifer Beineke (in Journal of Number Theory).
		On the dimension of the space of theta functions joint with Alex Pekker.
		Automorphic Summation Formulae and Moments of Zeta, based on work of Beineke and Bump.
	math.stanford.edu /~bump (716 words)

	[No title]
		In recent years, new and important connections have emerged between discrete subgroups of Lie groups, automorphic forms and arithmetic on the one hand, and questions in discrete mathematics, combinatorics, and graph theory on the other.
		In the 1980's, results from the theory of automorphic forms were used to construct explicit families of Ramanujan graphs, that is, graphs for which Laplace eigenvalues satisfy strong inequalities.
		This is already clear from the point of view of automorphic forms, where naive generalizations of the Ramanujan conjecture are know to be false and the correct formulations involve Langlands functoriality and the Arthur conjectures.
	www.ipam.ucla.edu /programs/agg2004 (814 words)

	Read This: Briefly Noted
		At the back, there are dozens of tables giving all sorts of interesting and useful functions (one wonders why, in an age of calculators, one would need tables of common logarithms or trigonometric functions, but there they are).
		This little book should be of interest both to historians seeking to understand the evolution of the theory of automorphic functions and to mathematicians working in the area, and thus it is a valuable addition to the (rather short) list of original source material available in English translation.
		Jacques Hadamard, Non-Euclidean Geometry in the Theory of Automorphic Functions, edited by Jeremy J. Gray and Abe Shenitzer (American Mathematical Society and London Mathematics Society series in the History of Mathematics, volume 17).
	www.maa.org /reviews/brief_nov99.html (900 words)

	Letter A
		A function (PL) whose value is given by an algebraic expression ({L) of the functand (PL) of the function.
		Study of geometries derived from algebra, particularly from ring (PL); classically, the algebra is the ring of polynomials and the geometry is the set of zeros of the polynomials ("varieties").
		A function (PL) is an alias of another function if the two are indistinguishable in having the same values at a finite set of points.
	members.fortunecity.com /jonhays/letterA.htm (3975 words)

	Landau Center: Preprints
		Abstract: The main achievement of this paper is the establishment of precise relations among various geometric quantities such as: the uniform radii of simply-connected and of convex hyperbolic discs, the curvature of hyperbolic geodesics in multiply-connected hyperbolic plane domains and some analytic properties of the analytic covering mappings of the unit disc onto such domains.
		Abstract: In this paper two types of differential operators are constructed and studied, both of which produce automorphic forms of any order form any give automorphic function or form, for any Kleinian group.
		The first type of these operators is related to the hyperbolic metric in a given invariant region under the group, while the other one is independent of the choice of the invariant region.
	www.ma.huji.ac.il /~landau/preprint90.html (2455 words)

	[No title]
		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 v space of periodic functions on Z with period p.
		Let K be a number field and assume K is Galois over Q (equivalently, that there is a polynomial f with rational coefficients such that K is the smallest subfield of the complex numbers containing all the roots of f; K is called the "splitting field" of f).
	math.ucr.edu /home/baez/twf_ascii/week217 (4253 words)

	W. Li Abstract (Site not responding. Last check: 2007-10-12)
		For a curve C defined over Z or over the polynomial ring k[x] over a finite field k, the Hasse-Weil L-function attached to C is an infinite product with character sums appearing in each factor.
		Automorphic L-functions for GL(n) over k(x) are generalizations of the Riemann zeta function, and they look just like Hasse-Weil L-functions.
		Particular attention will be given to the case where the character sums are Kloosterman sums, in this case the existence of the corresponding automorphic L-function is called the Kloosterman conjecture.
	www.math.uiuc.edu /Bulletin/Abstracts/September/sep26-97notheory.html (137 words)

	Journal of the American Mathematical Society
		Abstract: We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group
		The existence of such a function is predicted by the Langlands conjecture.
		), is geometric: the automorphic function is obtained via Grothendieck's ``faisceaux-fonctions'' correspondence from a complex of sheaves on an algebraic stack.
	www.ams.org /jams/1998-11-02/S0894-0347-98-00260-4/home.html (640 words)

	Fields Institute - Automorphic Forms
		The theory of automorphic forms is a wide and deep subject touching many areas of mathematics.
		An important problem is to express the Hasse-Weil zeta function of a Shimura variety in terms of automorphic L-functions.
		Here in order to define the local factors not just at primes of good reduction, we need to study the variety at the finite set of primes of bad reduction.
	www.fields.utoronto.ca /programs/scientific/02-03/automorphic_forms (423 words)

	Spectral Theory of the Riemann Zeta-Function - Cambridge University Press
		The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance.
		In this book, based on his own recent research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself.
		In this book, readers will find a detailed account of one of the most fascinating stories in the recent development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521445205 (267 words)

	Professor David Drasin Abstract
		If D is any unbounded domain with Dirichlet boundary, there is always a positive harmonic function which is zero at every finite boundary point (Martin function).
		D is Lipschitz, this generates the cone of Martin functions.
		This work (joint with V. arin and P. Poggi-Corradini) has applications to non-self-adjoint operators on the torus, and to entire functions of classes more general than functions of completely regular growth.
	www.math.uiuc.edu /hilda/htmlcalendars/May01_00/drasin_may04-00.html (120 words)

	automorphic - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "automorphic" is defined.
		Automorphic : Online Plain Text English Dictionary [home, info]
		Phrases that include automorphic: automorphic function, automorphic number, automorphic cuspidal representation, automorphic forms, automorphic granular, more...
	www.onelook.com /?w=automorphic (139 words)

	This Week's Finds in Mathematical Physics (Week 217)
		You can\nfind a precise statement and a version of Riemann\'s second proof here:\n\n2) Daniel Bump, Zeta Function, lecture notes on "the functional\nequation" available at http://math.stanford.edu/~bump/zeta.html\nand http://www.maths.ex.ac.uk/~mwatkins/zeta/fnleqn.htm\n\nThis proof is a beautiful application of Fourier analysis.
		We thought we were interested in\nthe Riemann zeta function for its own sake, or what it could tell us\nabout prime numbers.
		function is almost a modular form, but not quite.
	www.physicsforums.com /showthread.php?t=77435 (4658 words)

	week217
		The theta function transforms in a very simple way when we replace t by -1/t, as one can show using Fourier analysis.
		and extended to functions on Z by χ(n)=0 for n a multiple of p.
		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)

Try your search on: Qwika (all wikis)