Logarithmic order and dual logarithmic order

2001 ◽  
pp. 279-290 ◽  
Author(s):  
Takayuki Furuta
Keyword(s):  
Logarithmic Order

scholarly journals Two-parton scattering amplitudes in the Regge limit to high loop orders

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Simon Caron-Huot ◽  
Einan Gardi ◽  
Joscha Reichel ◽  
Leonardo Vernazza
Keyword(s):  
Scattering Amplitudes ◽  
Scattering Amplitude ◽  
Dimensional Regularization ◽  
High Energy ◽  
Logarithmic Order ◽  
Finite Radius ◽  
High Loop ◽  
Parton Scattering ◽  
Infrared Divergences ◽  
Regge Limit

Abstract We study two-to-two parton scattering amplitudes in the high-energy limit of perturbative QCD by iteratively solving the BFKL equation. This allows us to predict the imaginary part of the amplitude to leading-logarithmic order for arbitrary t-channel colour exchange. The corrections we compute correspond to ladder diagrams with any number of rungs formed between two Reggeized gluons. Our approach exploits a separation of the two-Reggeon wavefunction, performed directly in momentum space, between a soft region and a generic (hard) region. The former component of the wavefunction leads to infrared divergences in the amplitude and is therefore computed in dimensional regularization; the latter is computed directly in two transverse dimensions and is expressed in terms of single-valued harmonic polylogarithms of uniform weight. By combining the two we determine exactly both infrared-divergent and finite contributions to the two-to-two scattering amplitude order-by-order in perturbation theory. We study the result numerically to 13 loops and find that finite corrections to the amplitude have a finite radius of convergence which depends on the colour representation of the t-channel exchange.

scholarly journals Electric dipole moment constraints on CP-violating heavy-quark Yukawas at next-to-leading order

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Joachim Brod ◽  
Emmanuel Stamou
Keyword(s):  
Electric Dipole ◽  
Anomalous Dimension ◽  
Phenomenological Study ◽  
Dipole Moments ◽  
Charm Quark ◽  
Leading Order ◽  
Logarithmic Order ◽  
Electric Dipole Moments ◽  
Yukawa Couplings ◽  
Experimental Bound

Abstract Electric dipole moments are sensitive probes of new phases in the Higgs Yukawa couplings. We calculate the complete two-loop QCD anomalous dimension matrix for the mixing of CP-odd scalar and tensor operators and apply our results for a phenomenological study of CP violation in the bottom and charm Yukawa couplings. We find large shifts of the induced Wilson coefficients at next-to-leading-logarithmic order. Using the experimental bound on the electric dipole moments of the neutron and mercury, we update the constraints on CP-violating phases in the bottom and charm quark Yukawas.

scholarly journals Component Evolution in Random Intersection Graphs

10.37236/935 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael Behrisch
Keyword(s):  
Real World ◽  
Linear Order ◽  
Graph Model ◽  
Vertex Degree ◽  
Intersection Graph ◽  
Giant Component ◽  
Logarithmic Order ◽  
Intersection Graphs ◽  
Similarity Network ◽  
Random Intersection Graph

We study the evolution of the order of the largest component in the random intersection graph model which reflects some clustering properties of real–world networks. We show that for appropriate choice of the parameters random intersection graphs differ from $G_{n,p}$ in that neither the so-called giant component, appearing when the expected vertex degree gets larger than one, has linear order nor is the second largest of logarithmic order. We also describe a test of our result on a protein similarity network.

scholarly journals On Logarithmic Order and Logarithmic Lower Order of Integral Functions

2019 ◽  
Vol 14 (21) ◽  
pp. 7996-7999
Author(s):  
Aseel H. Abed Sadaa ◽  
Mustafa A. Sabri
Keyword(s):  
Lower Order ◽  
Logarithmic Order

scholarly journals The Growth on the Maximum Modulus of Double Dirichlet Series

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Yong-Qin Cui ◽  
Hong-Yan Xu ◽  
Na Li
Keyword(s):  
Dirichlet Series ◽  
Partial Derivative ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Logarithmic Order ◽  
Double Dirichlet Series

The main purpose of this paper is to investigate the growth of several entire functions represented by double Dirichlet series of finite logarithmic order, h-order. Besides, we also study some properties on the maximum modulus of double Dirichlet series and its partial derivative. Our results are extension and improvement of previous results given by Huo and Liang.

scholarly journals On the Sobolev space of functions with derivative of logarithmic order

2019 ◽  
Vol 9 (1) ◽  
pp. 836-849 ◽  
Author(s):  
Elia Brué ◽  
Quoc-Hung Nguyen
Keyword(s):  
Sobolev Space ◽  
Continuity Equation ◽  
Renormalized Solutions ◽  
Logarithmic Order

Abstract Two notions of “having a derivative of logarithmic order” have been studied. They come from the study of regularity of flows and renormalized solutions for the transport and continuity equation associated to weakly differentiable drifts.

Gaussian Distribution of Trie Depth for Strongly Tame Sources

2014 ◽  
Vol 24 (1) ◽  
pp. 54-103 ◽  
Author(s):  
EDA CESARATTO ◽  
BRIGITTE VALLÉE
Keyword(s):  
Dirichlet Series ◽  
Polynomial Growth ◽  
Transfer Operator ◽  
Second Step ◽  
Speed Of Convergence ◽  
Logarithmic Order ◽  
Natural Class ◽  
Free Region ◽  
General Source ◽  
Precise Asymptotic

The depth of a trie has been deeply studied when the source which produces the words is a simple source (a memoryless source or a Markov chain). When a source is simple but not an unbiased memoryless source, the expectation and the variance are both of logarithmic order and their dominant terms involve characteristic objects of the source, for instance the entropy. Moreover, there is an asymptotic Gaussian law, even though the speed of convergence towards the Gaussian law has not yet been precisely estimated. The present paper describes a ‘natural’ class of general sources, which does not contain any simple source, where the depth of a random trie, built on a set of words independently drawn from the source, has the same type of probabilistic behaviour as for simple sources: the expectation and the variance are both of logarithmic order and there is an asymptotic Gaussian law. There are precise asymptotic expansions for the expectation and the variance, and the speed of convergence toward the Gaussian law is optimal. The paper first provides analytical conditions on the Dirichlet series of probabilities of a general source under which this Gaussian law can be derived: a pole-free region where the series is of polynomial growth. In a second step, the paper focuses on sources associated with dynamical systems, called dynamical sources, where the Dirichlet series of probabilities is expressed with the transfer operator of the dynamical system. Then, the paper extends results due to Dolgopyat, already generalized by Baladi and Vallée, and shows that the previous analytical conditions are fulfilled for ‘most’ dynamical sources, provided that they ‘strongly differ’ from simple sources. Finally, the present paper describes a class of sources not containing any simple source, where the trie depth has the same type of probabilistic behaviour as for simple sources, even with more precise estimates.

scholarly journals THE ORDER OF DEATH OF ORGANISMS LARGER THAN BACTERIA

1931 ◽  
Vol 14 (3) ◽  
pp. 315-337 ◽  
Author(s):  
Otto Rahn
Keyword(s):  
Quantum Theory ◽  
Standard Method ◽  
Biological Significance ◽  
Ultra Violet ◽  
Logarithmic Order ◽  
Violet Light ◽  
Unstable Molecules ◽  
Multicellular Organisms ◽  
Survivor Curve ◽  
Living Bacteria

In a previous paper it has been shown that the logarithmic order of death of bacteria can be accounted for by the assumption of some very unstable molecules so essential for reproduction that the inactivation of only one such molecule per cell prevents reproduction and makes the cell appear "dead" according to the standard method of counting living bacteria. In the present paper dealing with the order of death of larger organisms, only a motile alga, Chlamydomonas, is shown to have the same order of death. The very scant material on the order of death of yeasts is contradictory. It seems possible that more than one molecule must be destroyed to kill a yeast cell. With the spores of a mold, Botrytis cinerea, the number of "reacting molecules" is decidedly larger than 1. A flagellate, Colpidium, gave a survivor curve suggesting the destruction of two molecules before motility ceases. Erythrocytes exposed to ultra-violet light also follow the formula for two reacting molecules. The analysis of the survivor curves of multicellular organisms is not possible because no distinction between the number of essential molecules and the number of essential cells seems possible. Besides, variability of resistance changes the shape of the survivor curves in such a way that it becomes impossible to differentiate between variability and actual survivor curve. The general results seem to justify the assumption that in each cell, the number of molecules which are really essential for life and reproduction, is quite limited, and that, therefore, equal cells will not react simultaneously, though ultimately the reaction will be the same. This theory of lack of continuity in cell reactions owing to the limited number of reacting molecules is an analogon to the quantum theory where continuity ceases because division of matter reaches a limit. It seems probable that this lack of continuity in cell reactions has a general biological significance reaching beyond the order of death.

scholarly journals Some Remarks on Nonlinear Electrodynamics

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Patricio Gaete
Keyword(s):  
Interaction Energy ◽  
Long Range ◽  
Coulomb Potential ◽  
Strong Field ◽  
Nonlinear Electrodynamics ◽  
Field Limit ◽  
Logarithmic Order ◽  
Heisenberg Type ◽  
Path Dependent ◽  
Strong Field Limit

By using the gauge-invariant, but path-dependent, variables formalism, we study both massive Euler-Heisenberg-like and Euler-Heisenberg-like electrodynamics in the approximation of the strong-field limit. It is shown that massive Euler-Heisenberg-type electrodynamics displays the vacuum birefringence phenomenon. Subsequently, we calculate the lowest-order modifications to the interaction energy for both classes of electrodynamics. As a result, for the case of massive Euler-Heisenbeg-like electrodynamics (Wichmann-Kroll), unexpected features are found. We obtain a new long-range (1/r3-type) correction, apart from a long-range(1/r5-type) correction to the Coulomb potential. Furthermore, Euler-Heisenberg-like electrodynamics in the approximation of the strong-field limit (to the leading logarithmic order) displays a long-range (1/r5-type) correction to the Coulomb potential. Besides, for their noncommutative versions, the interaction energy is ultraviolet finite.

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