In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.

374 Results for the subject "Logarithms":

AbstractThe issue of computing a real logarithm of a real matrix is addressed. After a brief review of some known methods, more attention is paid to three: (1) Padé approximation techniques, (2) Newton's method, and (3) a series expansion method. Newton's method has not been previously treated in the literature; we address commutativity issues, and simplify the algorithmic formulation. We also address general structure-preserving issues for two applications in which we are interested: finding the real Hamiltonian logarithm of a symplectic matrix, and finding the skew-symmetric logarithm of an orthogonal matrix. The diagonal Padé approximants and the proposed series expansion technique are proven to be structure-preserving. Some algorithmic issues are discussed.

Luca Dieci Publication date: 1996/09/01AbstractThe logarithms ∼αsnln2n(Q2/μ2) and ∼αsnln2n−1(Q2/μ2) in the asymptotics of the quark form factor in the Sudakov region Q2 = |q2|>|p2| = |p′2| = μ2 are summed. We get the simple result S(Q2, μ23) = exp − αs(μ22φCF 1n2 Q2μ2 + 3αs(μ2)CF4φ 1n Q2μ2 The same expression S(Q2, M2) describes the jet cross section in the e+e−→qq process where the jet is defined as a set of particles whose total invariant mass does not exceed M2. The QCD predictions are compared with the experimental data available and the jets observed are shown to be much broader than QCD predicts.

A.V. Smilga Publication date: 1979/12/17AbstractDuring inflation explicit perturbative computations of quantum field theories which contain massless, nonconformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. Starobinskiĭ's technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. We use explicit perturbative computations in two simple scalar *Read more...*

Publisher SummaryThis chapter discusses exponentials and logarithms. It presents two of the most important classes of functions of mathematics: the exponential and logarithmic functions. There are two different ways to define the exponential and logarithmic functions. In a sense to be made precise shortly, exponential and logarithmic functions are inverses of one another. If a be a positive number, then an exponential function is a function of the form f(x) = ax, where x can be any real number. In discussing the logarithm to the base a, if x = ay then the logarithm to the base a of x is y, written as, y = loga x.

STANLEY I. GROSSMAN Publication date: 1984/01/01During inflation explicit perturbative computations of quantum field theories which contain massless, non-conformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. Starobinskiı̆'s technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. We use explicit perturbative computations in two simple scalar model *Read more...*

AbstractTo send the message to the recipient securely, authenticated encryption schemes were proposed. In 2008, Wu et al. [T.S. Wu, C.L. Hsu, K.Y. Tsai, H.Y. Lin, T.C. Wu, Convertible multi-authenticated encryption scheme, Information Sciences 178 (1) 256–263.] first proposed a convertible multi-authenticated encryption scheme based on discrete logarithms. However, the author finds that the computational complexity of this scheme is rather high and the message redundancy is used. To improve the computational efficiency and remove the message redundancy, the author proposes a new convertible multi-authenticated encryption scheme based on the intractability of one-way hash functions and discrete logarithms. As for efficiency, the computation cost of the proposed scheme is smaller than Wu et al.’s scheme.

Jia-Lun Tsai Publication date: 2009/03/27AbstractThis chapter tells why it is important to study soil–plant–water relations. Water is the most important substance necessary for food production. People depend upon plants for food, so the challenge of feeding a growing population is discussed. The human population growth curve, a logarithmic one, is presented. Rules of logarithms are then given. A calculation is done to show that the human population is limited by the productivity of the land. The calculation shows that it requires two square yards (16,700 cm2) to feed one person. The sigmoid plant growth curve is presented followed by a mathematical analysis of Blackman's compound interest law for plant growth. Finally, data from corn and soybean are analyzed to show that their growth rate is exponential. A biography of Napier, the inventor of logarithms, is given in the appendix.

M.B. Kirkham Publication date: 2014/01/01AbstractThe standard MS renormalization prescription is inadequate for dealing with multi-scale problems. To illustrate this we consider the computation of the effective potential in the Higgs-Yukawa model. It is argued that it is natural to employ a two-scale renormalization group. We give a modified version of a two-scale scheme introduced by Einhorn and Jones. In such schemes the beta functions necessarily contain potentially large logarithms of the RG scale ratios. For credible perturbation theory one must implement a large logarithms resummation on the beta functions themselves. We show how the integrability condition for the two RG equations allows one to perform this resummation.

C. Ford Publication date: 1997/04/17AbstractThis paper investigates all 8 variants of the He’s digital signature scheme based on factoring and discrete logarithms. Instead of using three modular exponentiation computation, the two most optimal schemes of the generalized He’s signature require only two modular exponentiation for signature verification.

Shun-Fu Pon Publication date: 2005/06/06No abstract

C.W. SCHOFIELD Publication date: 1966/01/01No abstract

G.A. PRATT Publication date: 1966/01/01No abstract

SHEILA PAGE Publication date: 2002/01/01No abstract

Colin McGregor Publication date: 2010/01/01AbstractA shorter proof for an explicit formula for discrete logarithms in finite fields is given.

Zhe-Xian Wan Publication date: 2008/11/06AbstractThe influence of non-specific intermolecular interactions on conformational equilibria of organic molecules is investigated with the help of the London-Debye-Keesom pair coupling potentials. It is shown that in a series of apolar solvents equilibrium constant logarithms are proportional to ζα ≡ Zαs/R6s,d, and in a series of polar solvents equilibrium constants logarithms are proportional to ζμ ≡ Zμ2s/R6s,d, where Z is the average number of neighbours of a solute molecule in the first coordination sphere, αs is the polarizability, μs the dipole moment of solvent molecules, and Rs,d = Rs + Rd is the sum of the radii of spherical volumes per molecule of solvent (s) and dissolved substances (d).

V.V. Prezhdo Publication date: 1994/03/03No abstract

Steven Roman Publication date: 1993/05/01AbstractThe complexation reactions of beryllium with the disodium salts of catecholdisulphonic and chromotropic acids have been studied by potentiometric and conductometric methods.In the acidic range, the formation of mono-derivatives only was indicated in both the systems even in the presence of a large excess of chelating agents. The equilibrium constants “K” of the reaction: Be2+ + H2A2− ⇌ BeA2− + 2H+ and the logarithms of the formation constant (K1) of the complex anion [BeA2−] in catecholdisulphonic and chromotropic acids systems were found to be 9.33 × 10−8 and 13.23, and 2.95 × 10−5 and 16.43, respectively. The logarithms of the formation constants (log Kn) in both the systems for 1:1 and 1:2 complexes have also been determined by the alternative methods to be 13.52 ± 0.12; 12.54 ± 0.14 and 16.89 ± 0.09; 15.91 ± 0.11 respectively.

S.N. Dubey Publication date: 1965/08/01AbstractThis article is an overview of the key mathematical concepts required to understand computer science, physics and physiology as applied to anaesthesia.Topics include classification of numbers, base systems, scientific notation and rounding, logarithms, plain geometry, conic sections, graphical representation and periodic functions. Particular emphasis is given to exponential curves, as these relate to many phenomena in anaesthesia including uptake and elimination of drugs, and the flow of heat, gases, liquids and electrical current.The role of complex functions such as calculus and Fourier analysis is described; however a detailed understanding of these topics is not necessary to appreciate their application.

David Williams Publication date: 2011/09/01AbstractWe obtain the leading divergences at two-loop order for the decays KS→γγ and KS→γl+l− using only one-loop diagrams. We then find the double chiral logarithmic corrections to the decay branching ratio of KS→γγ and to the decay rate for KS→γl+l−. It turns out that these effects are numerically small and therefore make a very small enhancement on the branching ratio and decay rate. We also derive an expression for the corrections of type logμ×LEC. Numerical analysis done for the process KS→γγ shows that these single logarithmic effects can be sizable but come with opposite signs with respect to the double chiral logarithms.

Karim Ghorbani Publication date: 2014/08/01Publisher SummaryThis chapter discusses the evaluation of complex logarithms and related functions with interval arithmetic. It presents a general algorithm that was suggested by Gauss in 1800. However, the algorithm bears the name of Borchardt who rediscovered and published it in 1880. Carlson, 1972, coupled Borchardts algorithm with Richardson extrapolation to obtain a viable technique for programmed computation. The chapter presents a complex version of Carlson's treatment under the assumption that the four operations of complex arithmetic and the principal complex square root are available; the latter is the single-valued function whose range is + = {x + iy : x > 0 or x = 0 with y ≥ 0}.

George J. Miel Publication date: 1980/01/01