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	Selberg trace formula - Wikipedia, the free encyclopedia
		In the 1960s the general thrust of the Selberg trace formula, as a piece of analysis, was taken up by the Israel Gelfand school, by Harish-Chandra and Langlands in Princeton, and by Tomio Kubota in Japan.
		The existence of trace formulae both for the differential operator and Hecke operator cases was a hint of the power (for essentially arithmetic cases) of the adele group approach.
		Contemporary successors of the theory are the Arthur-Selberg trace formula applying to the case of general semisimple G, and the many studies of the trace formula in the Langlands philosophy (dealing with technical issues such as endoscopy).
	en.wikipedia.org /wiki/Selberg_trace_formula (525 words)

	Knowledge King - Atle Selberg (Site not responding. Last check: 2007-11-07)
		Selberg was born in Langesund, Norway and great work of Srinivasa Aaiyangar Ramanujan influenced on him very soon while he was still at school.
		It establishes a duality between the length spectrum of a Riemann surface and the eigenvalues of the Laplacian which is analogous to the duality between the prime numbers and the zeros of the zeta function.
		Selberg and Erdös gave elementary proofs of the prime number theorem, although it was prior believed that such proofs with only real variables can't be found.
	www.knowledgeking.net /encyclopedia/a/at/atle_selberg.html (291 words)

	Atle Selberg - Wikipedia, the free encyclopedia
		Atle Selberg (born June 17, 1917) is a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory.
		Then in 1948 Selberg, working with Paul Erdős, gave an elementary proof of the prime number theorem (although there was a dispute between them about to whom this result should primarily be attributed).
		Selberg moved to the United States and settled at the Institute for Advanced Study in the 1950s where he remains today.
	www.wikipedia.org /wiki/Atle_Selberg (302 words)

	Selberg (Site not responding. Last check: 2007-11-07)
		Selberg used his trace formula to prove that the "Selberg zeta function" of a Riemann surface satisfies an analogue of the Riemann hypothesis.
		Selberg was one of the four editors of Axel Thue's Selected mathematical papers published in Oslo in 1977.
		In 1989 Selberg published Reflections around the Ramanujan centenary which is the text of a talk which he gave at the conclusion of the Ramanujan Centenary Conference in January 1988 at the Tata Institute in Bombay.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Selberg.html (761 words)

	Selberg trace formula and zeta functions
		Selberg discovered that the so called "Poisson summation formula" of classical Fourier analysis had a noncommutative generalization that could be applied to obtain an array of important identities in number theory and the theory of automorphic functions.
		This formula, which was motivated by Riemann's zeta function, relates in an exact way the spectrum of the quantal motion on compact surfaces of negative curvature to the classical motion.
		Its divisor is determined by the eigenvalues and scattering poles of the Laplacian and the Euler characteristic of the surface.
	www.maths.ex.ac.uk /~mwatkins/zeta/physics4.htm (4568 words)

	how Selberg's trace formula and the Riemann-Weil explicit formula are related
		"Selberg noticed this similarity in 1950-51 and was quickly led to a deeper study of trace formulas.
		Among other things, Selberg found that there is a zeta function which corresponds to [his trace formula] in the same way that [the Riemann zeta function] corresponds to [Weil's explicit formula].
		Selberg generalised this, producing a noncommutative analogue of the PSF, which can be interpreted as a trace formula on higher-genus compact Riemannian surfaces.
	www.math.ex.ac.uk /~mwatkins/zeta/STF-WEF.htm (338 words)

	the Riemann-Weil explicit formula
		This is a generalisation of the Riemann-von Mangoldt explicit formula (which relates the zeros of the Riemann zeta function to the distribution of prime numbers).
		Explicit formulae generally are characterised as having the form of "the sum of a certain function extended to the prime powers [being] essentially equal to a sum of its Mellin transform extended to the zeros of the (relevant) zeta function."
		A striking explicit formula was given by Riemann for the prime-counting function in terms of the zeros, and the generalizations obtained later all go by the name of 'Explicit Formulae'.
	www.maths.ex.ac.uk /~mwatkins/zeta/weilexplicitformula.htm (1161 words)

	trace formulae and number theory
		One side of a formula is given by a trace of a quantum object, typically derived from a quantum Hamilitonian, and the other side is described in terms of closed orbits of the corresponding classical Hamiltonian.
		In algebraic situations, such as the original Selberg trace formula, the identities are exact, while in general they hold only in semi-classical or high-energy limits.
		In this paper we present an intermediate trace formula in which the original trace is expressed in terms of traces of quantum monodromy operators directly related to the classical dynamics.
	www.secam.ex.ac.uk /~mwatkins/zeta/traceformulaeandNT.htm (1122 words)

	Atlas: Selberg's trace formula as an explicit formula by Lejla Smajlovic (Site not responding. Last check: 2007-11-07)
		We interprete Selberg's trace formula for a strictly hyperbolic Fuchsian group as an explicit formula for a fundamental class of functions in the symmetric case.
		The explicit formula in this case is proved for a larger class of test functions, not necessarily smooth at zero.
		Thus we obtain that the trace formula holds for a larger class of test functions.
	atlas-conferences.com /cgi-bin/abstract/calm-40 (112 words)

	Selberg (Site not responding. Last check: 2007-11-07)
		In 1950 Selberg was awarded a Fields Medal at the International Congress of Mathematicians at Harvard.
		Selberg is also well known for his elementary proof of the prime number theorem, with a generalisation to prime numbers in an arbitrary arithmetic progression.
		Probably Selberg's best and most important work is his trace formula for SL), which was done several years after the work for which he was awarded the Fields Medal.
	www.bg-rams.ac.at /intranet/Physik/history/Selberg.html (792 words)

	Encyclopedia: Poisson summation formula
		In mathematics, the Poisson summation formula is a relation holding between a sum of a function F over all integers, and a corresponding sum for the Fourier transform G.
		In fact in more recent work on counting lattice points in regions it is routinely used − summing the indicator function of a region D over lattice points is exactly the question, so that the LHS of the summation formula is what is sought and the RHS something that can be attacked by mathematical analysis.
		In non-commutative harmonic analysis, the idea is taken even further in the Selberg trace formula, but takes on a much deeper character.
	www.nationmaster.com /encyclopedia/Poisson-summation-formula (461 words)

	Atle Selberg books ; 0387183892 Misspelled: atle selberg adle selperg selbreg tle ale ate atl atleselberg elberg slberg ...
		Atle Selberg (born June 17, 1917) is a Norwegian mathematician, one of the greatest analyticnumber theorists of all time.Selberg was born in Langesund, Norway.
		However Selberg in the early 1950s proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian.
		This so-called Selberg trace formula bore a striking resemblance to the explicit formulae, which gave credibility to the speculation of Hilbert and Pólya.Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property.
	www.americanenglishliterature.com /62300_atle-selberg_0387183892atleselbergcollectedpapersbookcover.html (612 words)

	Selberg trace formula - Encyclopedia Glossary Meaning Explanation Selberg trace formula (Site not responding. Last check: 2007-11-07)
		Selberg trace formula - Encyclopedia Glossary Meaning Explanation Selberg trace formula.
		Here you will find more informations about Selberg trace formula.
		The orginal Selberg trace formula article can be editet
	www.encyclopedia-glossary.com /en/Selberg-trace-formula.html (582 words)

	Selberg (Site not responding. Last check: 2007-11-07)
		Ramanujan and reading his work, Selberg began to make his own mathematical explorations.
		Selberg used his trace formula to prove that the "Selberg zeta function" of a
		Selberg was one of the four editors of Axel
	www.educ.fc.ul.pt /icm/icm2003/icm14/Selberg.htm (718 words)

	Atle Selberg (Site not responding. Last check: 2007-11-07)
		Atle Selberg, born June 17, 1917, a Norwegian mathematician is one of the greatest analyticnumber theorists of all time.
		All is still licensed under the GNU FDL.
		His first duty was to go to Constantine Jopp, and speak his regret like a.
	www.termsdefined.net /at/atle-selberg.html (571 words)

	Lectures on the Arthur-Selberg Trace Formula - (American Mathematical Society Bookstore) (Site not responding. Last check: 2007-11-07)
		-- Mathematical Reviews The Arthur-Selberg trace formula is an equality between two kinds of traces: the geometric terms given by the conjugacy classes of a group and the spectral terms given by the induced representations.
		The formulas are difficult in general and even the case of $GL$ (2) is nontrivial.
		Explains why the truncation formula reduces to a simple formula involving only the elliptic terms on the geometric sides with the representations appearing cuspidally on the spectral side (applies to Tamagawa numbers).
	mirror.math.nankai.edu.cn /mirror/www.ams.org/ULECT-9.html (626 words)

	DC MetaData for: Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function (Site not responding. Last check: 2007-11-07)
		DC MetaData for: Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function
		Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function
		formulas of number theory as a trace formula on the noncommutative space of
	www.esi.ac.at /Preprint-shadows/esi620.html (120 words)

	Oxford University Press (Site not responding. Last check: 2007-11-07)
		The Arthur-Selberg trace formula is an equality between two kinds of traces; the geometric terms given by the conjugacy classes of a group and the spectral terms given by the induced representations.
		The formulas are difficult in general and even the case of GL(2) is nontrivial.
		The problem is that when the truncated terms converge, they are also shown to be polynomial in the truncation variable and expressed as "weighted" orbital and "weighted" characters.
	www5.oup.com /isbn/0-8218-0571-1 (322 words)

	Atle Selberg - Wikipedia
		Atle Selberg, born 1917, is one of the greatest analytic number theorists of all time.
		However Hardy proved that an infinite number of zeros do exist on this line.
		Selberg and Erdos gave elementary proofs of the prime number theorem.
	nostalgia.wikipedia.org /wiki/Atle_Selberg (239 words)

	Trace formula 285B (Site not responding. Last check: 2007-11-07)
		The Trace formula, introduced by Selberg and greatly refined by Arthur, is an important tool for computing the trace of a certain type of linear operator in the regular representation of a group G. It has many applications in number theory.
		For example, it can been used to calculate the traces of Hecke operators, and to verify cases of functoriality in Langlands' program.
		Students learning the trace formula are usually overwhelmed by all the notation and ideas from different fields (number theory, automorphic forms, Lie Algebras, Group theory, Representation theory, etc.) This course will be an attempt to bridge the gap.
	www.math.ucla.edu /~ccli/math/trace (355 words)

	index
		Joint with M. Horton and D. Newland, The Contest between the Kernels in the Selberg Trace Formula for the (q+1)-regular Tree.
		The Roelcke-Selberg spectral resolution of the Laplacian, and the Selberg trace formula.
		The trace formula for a tree and Ihara’s zeta function.
	math.ucsd.edu /~aterras (1059 words)

	Atle Selberg: Definition and Links by Encyclopedian.com - All about Atle Selberg
		Atle Selberg: Definition and Links by Encyclopedian.com - All about Atle Selberg
		Atle Selberg, born June 17, 1917, a Norwegian mathematician is one of the greatest analytic number theorists of all time.
		He established the importance of Viggo Brun's sieve methods[?] in number theory, inventing a method that now bears his name, as well as workng on the large sieve[?].
	www.encyclopedian.com /at/Atle-Selberg.html (310 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Polyakov formula for variation of the determinant of Laplacian within a conformal class of the metric.
		Using an appropriate version of the Selberg Trace Formula, we explain the relationship among the Laplace spectrum, the length spectrum, and the singular points in a hyperbolic orbisurface.
		Analysis of the geodesic exponential map on the group of volume preserving diffeomorphisms; in 2-d case this map is smooth, Fredholm and quasiruled.
	www.math.mcgill.ca /jakobson/analysish/seminar.html (1934 words)

	A. Zaharescu (Site not responding. Last check: 2007-11-07)
		Abstract: Using a reformulation of the Eichler-Selberg trace formula, due to Frechette, Ono, and Papanikolas, we consider the problem of the vanishing (resp.
		For example, we show that for a fixed operator and weight, the set of levels for which the trace vanishes is effectively computable.
		Also, for a fixed operator the set of weights for which the trace vanishes (for any level) is finite.
	www.math.uiuc.edu /%7Ezaharesc/seminar.html (2066 words)

	Amazon.de: Bücher: Groups Acting on Hyperbolic Space (Site not responding. Last check: 2007-11-07)
		Starting off with several models of hyperbolic space and its group of motions the authors discuss the spectral theory of the Laplacian and Selberg's theory for cofinite groups.
		This culminates in explicit versions of the Selberg trace formula and the Selberg zeta-function.
		The interplay with arithmetic is demonstrated by means of the groups PSL(2) over rings and of quadratic integers, their Eisenstein series and their associated Hermitian forms.
	www.amazon.de /exec/obidos/ASIN/3540627456 (383 words)

	Combinatorics of traces of Hecke operators -- Frechette et al. 101 (49): 17016 -- Proceedings of the National Academy ...
		We investigate the combinatorial properties of the traces of
		congruences for traces are obtained in ref. 4.
		Proof: We proceed by using the trace formula from Theorem 2.1.
	www.pnas.org /cgi/content/full/101/49/17016 (1440 words)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-11-07)
		math.GR/0309104: 5 Sep 2003, Trace forms of Galois extensions in the presence of a fourth root of unity, by J. Minac and Z. Reichstein.
		ANT-0188: 15 Jun 1999, The Mordell-Weil rank of the Jacobian of a curve of genus 2 with multiplication by a square root of 2, by Peter R. Bending.
		ANT-0185: 7 Jun 1999, An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z), by Avner Ash and Warren Sinnott.
	front.math.ucdavis.edu /ANT (12251 words)

	International Journal of Mathematics and Mathematical Sciences (Site not responding. Last check: 2007-11-07)
		We survey graph theoretic analogues of the Selberg trace and pretrace formulas along with some applications.
		The spherical and horocycle transforms are considered (along with three basic examples, which may be viewed as a short table of these transforms).
		Finally, the Selberg trace formula is deduced and applied to the Ihara zeta function of
	www.hindawi.com /journals/ijmms/volume-2003/S016117120311126X.html (595 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Time: 5:00 PM (Tea at 4:30 PM) ABSTRACT The Arthur-Selberg trace formula is one of the most powerful tools in modern representation theory and automorphic forms.
		It was used by Arthur to derive an L2 Lefschetz formula for the action of a Hecke correspondence on L2 cohomology.
		Arthur's formula, proven using difficult techniques from harmonic analysis, involves orbital integrals, Weyl character formula, and characters of discrete series.
	www.math.technion.ac.il /%7Etechm/oldmessages/2246 (164 words)

	Bunke and Olbrich (1995) Selberg zeta and theta functions: A differential operator approach
		Bunke and Olbrich (1995) Selberg zeta and theta functions: A differential operator approach
		Selberg zeta and theta functions: A differential operator approach
		To view the the latter's ratings, click on Chapters/Papers/Articles in the STATISTICS box, select a publication from the list that appears, and then click on either Quality or Interest in that publication's STATISTICS box.
	www.getcited.org /?PUB=103275772&showStat=Ratings (87 words)

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