Where results make sense

About us   |   Why use us?   |   Reviews   |   PR   |   Contact us

Explicit formulae (L-function) - Wikipedia, the free encyclopedia In mathematics, the explicit formulae for L-functions are a class of summation formulae, expressing sums taken over the complex number zeroes of a given L-function, typically in terms of quantities studied by number theory by use of the theory of special functions. Therefore the existence of such 'trace formulae' for T means that the explicit formulae essentially encode the nature of T, from the point of view of spectral theory, at least as far as its eigenvalues (spectrum) is concerned. Development of the explicit formulae for a wide class of L-functions took place in papers of André Weil, who first extended the idea to local zeta-functions, and formulated a version of a generalized Riemann hypothesis in this setting, as a positivity statement for a generalized function on a topological group. en.wikipedia.org /wiki/Explicit_formulae_(L-function)   (322 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.maths.ex.ac.uk /~mwatkins/zeta/traceformulaeandNT.htm   (1122 words)

Integral - Wikipedia, the free encyclopedia Many real-world applications of calculus rely on calculating integrals approximately because of the complexity of formulas and since an exact answer is unnecessary. One difficulty in computing definite integrals is that it is not always possible to find "explicit formulae" for antiderivatives. Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. en.wikipedia.org /wiki/Integral   (1728 words)

[No title]   (Site not responding. Last check: 2007-10-31) The purpose of this note is to discuss explicit formulae for the asymptotic coefficients $Q_j$ in \eqref{a.16}. However, it is often the case that the proof does not readily yield explicit formulae for the coefficients $Q_j$. Note that in the one-dimensional case, formulae for $Q_j$ were given in \cite{ShenkShubin}, although not as explicit as \eqref{a.17}: the coefficients $Q_j$ are computed as integrals of a sequence of functions defined by some recurrence relation. www.ma.utexas.edu /mp_arc/html/papers/02-135   (1583 words)

2.3 Explicit determinism The set of atomic logical formulae made on this predicate is a subset of the set of atomic logical formulae made on equality predicate. In contrast to an universally quantified formula, an existentially quantified formula is not deterministic. An atomic formula based on the := predicate can be referenced in THEN and ELSE branches of a conditional operator if the condition of the formula based on the conditional operator is known, i.e. www.cs.iastate.edu /~leavens/FoCBS/cougoulic-node5.html   (583 words)

Hilbert Functions of Ladder Determinantal Varieties   (Site not responding. Last check: 2007-10-31) We consider algebraic varieties defined by the vanishing of all minors of a fixed size of a rectangular matrix with indeterminate entries such that the indeterminates in these minors are restricted to lie in a ladder shaped region of the rectangular array. Explicit formulae for the Hilbert function of such varieties are known in (i) the rectangular case [Abhyankar, 1984], and (ii) the case of 2 ×2 minors in one-sided ladders [Kulkarni, 1985]. We describe here an explicit, albeit complicated, formula for the Hilbert function and the Hilbert series in the case of arbitrary sized minors in two-sided ladders. www.math.iitb.ac.in /~srg/preprints/ladabs.html   (213 words)

Explicit formulae for J-spectral factors for well-posed linear systems (ResearchIndex)   (Site not responding. Last check: 2007-10-31) Abstract: The standard way to obtain explicit formulas for spectral factorization problems for rational transfer functions is to use a minimal realization and then obtain formulae in terms of the generators A, B, C and D. For well-posed linear systems with unbounded generators these formulae will not always be well-defined. Instead, we suggest another approach for the class of well-posed linear systems for which zero is in the resolvent set of A. Such a system is related to a reciprocal system... 6 Explicit formulas for Hankel norm approximations of infinite.. citeseer.ist.psu.edu /573173.html   (404 words)

Encyclopedia :: encyclopedia : Riemann hypothesis   (Site not responding. Last check: 2007-10-31) Later work by Hardy and Littlewood in 1921 and by Selberg in 1942 gave estimates for the average density of zeros on the critical line. Recent work has focused on the explicit calculation of the locations of large numbers of zeros (in the hope of finding a counterexample) and placing upper bounds on the proportion of zeros that can lie away from the critical line (in the hope of reducing this to zero). The zeroes of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes. www.hallencyclopedia.com /Riemann_hypothesis   (1834 words)

Explicit Formulae for Taylor's Functional Calculus (ResearchIndex) In this paper integral formulae, based on Taylor's functional calculus for several operators, are found. Special cases of these formulae include those of Vasilescu and Janas, and an integral formula for commuting operators with real spectra. Let a = (a 1 ; : : : ; an) be a commuting tuple of bounded linear operators on a complex Banach space X, let\Omega be an open subset containing the Taylor's joint spectrum of a, Sp(a; X), and let f be a holomorphic function defined on... citeseer.ist.psu.edu /222566.html   (255 words)

[No title] Hence, the formula for a morphism Di is given in terms of the morphisms AW, SHI, the diagonal and the automorphism t. Therefore, if we expre* *ss the factors of the summands of the formula for Di in a canonical way (see Lemma 2.1* *), those summands of the simplified formula for Di having a factor with a degenera* *cy operator in its expression must be eliminated. It is not hard to notice that we only have to consider the summands of the explicit formula for AW (t SHI)n (see Theorem 3.1), having the same number of face operators in both factors. www.math.purdue.edu /research/atopology/Real/cupi.txt   (4275 words)

Hyperelliptic Curves allowing fast Arithmetic, Koblitz curves The preprint "Efficient Arithmetic on Genus 2 Hyperelliptic Curves over Finite Fields via Explicit Formulae" is concerned with Genus 2 curves over arbitrary finite fields and the most efficient arithmetic on them obtained via explicit formulae. For hardware the choice of characteristic two is especially interesting and using these formulae it is faster than odd characteristic. Furthermore we are concerned with scalar multiplications and look for optimal matches of systems of coordinates to compute them -- this depends on the system of the input and also on the number of precomputations to be stored. www.ruhr-uni-bochum.de /itsc/tanja/preprints.html   (604 words)

4.4 Equations of motion and gravitational waveform Among the results of these approaches are formulae for the equations of motion and gravitational waveform of binary systems of compact objects, carried out to high orders in a PN expansion. Here we shall only state the key formulae that will be needed for this article. Explicit formulae for terms through various orders have been calculated by various authors: non-radiative terms through 2PN order [40, 38, 72, 39, 27], radiation reaction terms at 2.5PN and 3.5PN order [78, 79, 17], and non-radiative 3PN terms [80, 81, 46, 26, 25]. relativity.livingreviews.org /Articles/lrr-2001-4/node18.html   (569 words)

Explicit formulae for time-space Brownian chaos, Giovanni Peccati In this way, we complete the results of previous papers, and provide a time-space counterpart to the classic Stroock formulae for Wiener chaos. Our main tool is an extension of the Clark--Ocone formula in the context of initially enlarged filtrations. A formal connection between our results and the orthogonal decomposition of the space of square-integrable functionals of a standard Brownian bridge -- as proved by Gosselin and Wurzbacher -- is also established. projecteuclid.org /getRecord?id=euclid.bj/1068129009   (261 words)

Listening to College Writers I tried to be as explicit as I knew how about my own expectations and standards for student writing—expectations and standards with sound pedagogical rationale or not. Through what became swings back and forth from very explicit treatment of form to open choice about form, I came to understand better the urgent importance and extraordinary possibility of inviting students to be present in their own writing. Also, through acknowledging that different teachers have different formulae and by giving students practice in various forms, teachers can frame form not as a universal, mysterious set of restrictions handed down from on high by the god of school essays, but rather as a set of tools available for getting something across to an audience. www.writingproject.org /cs/nwpp/lpt/nwpr/343   (4860 words)

4.4 Equations of motion and gravitational waveform Here we shall only state the key formulae that will be needed for this review. Explicit and unambiguous formulae for non-spinning bodies through 3.5PN order have been calculated by various authors (see [34 Formulae for fluxes of angular and linear momentum can also be derived. relativity.livingreviews.org /Articles/lrr-2006-3/articlesu15.html   (560 words)

Explicit formulae for the Bures metric   (Site not responding. Last check: 2007-10-31) The aim of this paper is to derive explicit formulae for the computation of the Riemannian Bures metric g on the manifold Several formulae are known for computing the Bures metric in low dimensions. The formulae presented in this paper allow for computing in finite dimensions without any diagonalization procedures. stacks.iop.org /0305-4470/32/2663   (231 words)

Quantized Feedback Systems Perturbed By White Noise - Brockett, Liberzon (ResearchIndex)   (Site not responding. Last check: 2007-10-31) Abstract: This paper treats a class of nonlinear feedback systems perturbed by white noise, the nonlinearity being given by a piecewise constant function of a certain type. We obtain explicit formulae for steady-state probability densities associated with such systems. This result is used to address a stochastic optimal control problem that can be interpreted as minimization of the cost of implementing a feedback control law. citeseer.ifi.unizh.ch /43175.html   (445 words)

Paper 4 - Issue 1 It can be either explicit or tacit in nature. Explicit knowledge can be easily articulated and transferred to others. In contrast tacit knowledge, which is personal knowledge, residing in individual’s heads, is very difficult to articulate, codified and communicate (Gupta and McDaniel, 2002). www.ejkm.com /volume-2/v2i1/v2-i1-art4.htm   (4482 words)

formulae - OneLook Dictionary Search Tip: Click on the first link on a line below to go directly to a page where "formulae" is defined. Formulae : Online Plain Text English Dictionary [home, info] Phrases that include formulae: addition formulae, differentiation formulae, double-angle formulae, explicit formulae, formulae as types correspondence, more... www.onelook.com /?w=formulae   (164 words)

Atlas: Tchebotarov's Extremal Problem by Promarz M. Tamrazov Up to now formulae for extremals were known only for some problems under some special conditions: 1) When a quadratic differential has a single in C pole (we mean the well-known Bieberbach problem on coefficients of univalent functions, and it is why this problem required many decades before have been solved by de Branges). These formulae are explicit, and it is a consequence of extremely surprising fact that the derivative of the extremal is a product of simple factors representing separate singularities and zeroes of the derivative. We emphasize that among numerous extremal problems generating quadratic differentials with a number of arbitrarily distributed fixed poles, it is the first case when the extremals are found in explicit form. atlas-conferences.com /cgi-bin/abstract/caqw-70   (535 words)

Mathematics of Computation The first algorithm, inspired by Cantor's reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data. www.ams.org /mcom/2005-74-249/S0025-5718-04-01699-0/home.html   (686 words)

Explicit formulae for a class of Dirichlet series., Don Redmond Explicit formulae for a class of Dirichlet series., Don Redmond Explicit formulae for a class of Dirichlet series. [14] C. Moreno, Explicit formulasin the theory of automorphic forms, Number Theory Day, Lecture Notes in Math., Vol. projecteuclid.org /getRecord?id=euclid.pjm/1102724452   (281 words)

The Fibonomials The material on this page assumes that you already know a little about the Fibonacci Numbers from Fibonacci Numbers, Golden Section and the Golden String (which contains a much fuller treatment of Fibonacci numbers and their mathematical properties) and are familiar with polynomials and algebra at school level age 14-15. Such a formula for a series which defines each number in it in terms of some combination of previous terms is called a recurrence relation or recurrence formula. The formula for the pattern in our new table is similar to that of the Binomial numbers. www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/Fibonomials.html   (3702 words)

A state-space calculus for rational probability density functions and applications to non-Gaussian filtering The existence of moments is investigated using the Markov parameters of the density summand. Moreover, explicit formulae are given for the existing moments in terms of these Markov parameters. One of the main contributions of the paper are explicit state space descriptions for products and convolutions of rational densities. ideas.repec.org /p/dgr/vuarem/1997-21.html   (376 words)

Number theory   (Site not responding. Last check: 2007-10-31) Chebyshev (1850) gave useful bounds for the number of primes between two givenlimits. Riemann (1859) conjectured the limit of the number of primes not exceeding agiven number (the prime number theorem), introduced complex analysis into the theoryof the Riemann zeta function, and derived the explicit formulae of prime number theory from its zeroes. The theory of congruences may be said to start withGauss's Disquisitiones. www.therfcc.org /number-theory-86.html   (1144 words)

On the Bias and MSE of the IV Estimator Under Weak Identification In this paper we provide further results on the properties of the IV estimator in the presence of weak instruments. We begin by formalizing the notion of weak identification within the local-to-zero asymptotic framework of Staiger and Stock (1997), and deriving explicit analytical formulae for the asymptotic bias and mean square error (MSE) of the IV estimator. Because our analytical formulae for bias and MSE are complex functionals of confluent hypergeometric functions, we also derive approximations for these formulae which are based on an expansion that allows the number of instruments to grow to infinity while keeping the population analogue of the first stage F-statistic fixed. ideas.repec.org /p/ecm/wc2000/1622.html   (374 words)

Explicit formulae for the generalized Hermite polynomials in superspace Explicit formulae for the generalized Hermite polynomials in superspace We provide explicit formulae for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero–Moser–Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. In particular, reselling and systematic downloading of files is prohibited. stacks.iop.org /0305-4470/37/1251   (228 words)

Bates College Math Dept Faculty: Eric Towne   (Site not responding. Last check: 2007-10-31) I graduated from Harvard College with a degree in mathematics in 1993; my senior thesis was on explicit formulae in analytic number theory. After that, I taught math at Harvard for five years, although I took some time off to go to Manhattan to do bond derivatives for Goldman, Sachs. I enjoy working with the students here at Bates and encourage any of you to come see me with math (or non-math) questions or concerns. abacus.bates.edu /acad/depts/math/towne.html   (132 words)

Try your search on: Qwika (all wikis)

About us   |   Why use us?   |   Reviews   |   Press   |   Contact us   Copyright © 2005-2007 www.factbites.com Usage implies agreement with terms.