scholarly journals Mean-Field Expansion, Regularization Issue, and Multi-Quark Functions in Nambu–Jona-Lasinio Model

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 668 ◽  
Author(s):  
Aydan A. Garibli ◽  
Rauf G. Jafarov ◽  
Vladimir E. Rochev
Keyword(s):  
Mean Field ◽  
Legendre Transform ◽  
Third Order ◽  
Transform Method ◽  
Landau Pole ◽  
The Third ◽  
Njl Model ◽  
Symmetry Constraints ◽  
The Mean ◽  
Analytical Regularization

In this paper, the results of the investigation of multi-quark equations in the Nambu–Jona-Lasinio (NJL) model in the mean-field expansion are presented. The multi-quark functions have been considered up to the third order of expansion. One of the purposes of our computations is the study of corrections of higher orders to parameters of the model. The important problem of the application of the NJL model is regularization. We compare the NJL model with 4-dimensional cutoff regularization and the dimensionally analytical regularization. We also discuss so-called “predictive regularization” in the NJL model, and a modification of this regularization, which is free of the Landau pole, is proposed. To calculate the high-order corrections, we use the Legendre transform method in the framework of bilocal-source formalism, which allows one effectively to take into consideration the symmetry constraints. A generalization of the mean-field expansion for other types of multi-quark sources is also discussed.

Analogy between predictions of Kolmogorov and Yaglom

1997 ◽  
Vol 332 ◽  
pp. 395-409 ◽  
Author(s):  
R. A. Antonia ◽  
M. Ould-Rouis ◽  
F. Anselmet ◽  
Y. Zhu
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Longitudinal Velocity ◽  
Turbulent Wake ◽  
Mean Value ◽  
Order Moment ◽  
Third Order ◽  
The Third ◽  
Velocity Increments ◽  
The Mean

The relation, first written by Kolmogorov, between the third-order moment of the longitudinal velocity increment δu1 and the second-order moment of δu1 is presented in a slightly more general form relating the mean value of the product δu1(δui)2, where (δui)2 is the sum of the square of the three velocity increments, to the secondorder moment of δui. In this form, the relation is similar to that derived by Yaglom for the mean value of the product δu1(δuθ)2 where (δuθ)2 is the square of the temperature increment. Both equations reduce to a ‘four-thirds’ relation for inertialrange separations and differ only through the appearance of the molecular Prandtl number for very small separations. These results are confirmed by experiments in a turbulent wake, albeit at relatively small values of the turbulence Reynolds number.

scholarly journals Wave-Generated Current: A Second-Order Formulation

2020 ◽  
Vol 8 (6) ◽  
pp. 418
Author(s):  
Anne Katrine Bratland
Keyword(s):  
Wave Theory ◽  
Second Order ◽  
Stokes Drift ◽  
Wave Number ◽  
Stokes Wave ◽  
Wave Loads ◽  
Third Order ◽  
The Third ◽  
Computational Fluid Dynamics Cfd ◽  
The Mean

In Stokes’ wave theory, wave numbers are corrected in the third order solution. A change in wave number is also associated with a change in current velocity. Here, it will be argued that the current is the reason for the wave number correction, and that wave-generated current at the mean free surface in infinite depth equals half the Stokes drift. To demonstrate the validity of this second-order formulation, comparisons to computational fluid dynamics (CFD) results are shown; to indicate its effect on wave loads on structures, model tests and analyses are compared.

scholarly journals Faddeev Approach to the Nucleon in the NJL model

1997 ◽  
Vol 50 (1) ◽  
pp. 123
Author(s):  
N. Ishii ◽  
H. Asami ◽  
W. Bentz ◽  
K. Yazaki
Keyword(s):  
Magnetic Moments ◽  
Effective Interaction ◽  
Mass Difference ◽  
Mean Field ◽  
Field Approximation ◽  
Coupling Constants ◽  
Mean Field Approximation ◽  
State Matrix ◽  
Njl Model ◽  
The Mean

The nucleon and the delta are described as solutions of the relativistic Faddeev equation in the NJL model. We discuss the dependence of the baryon masses on the particular form of the four-Fermi interaction Lagrangians. Using the quark–diquark wave function, we calculate some bound state matrix elements such as the axial coupling constants, magnetic moments of the nucleon, the pion-nucleon sigma term and the proton-neutron mass difference. We also try to compare two pictures of describing the baryons in the NJL model, i.e. the mean-field approximation and the relativistic Faddeev approach. As a first step, we discuss how to improve the mean-field approximation by introducing an effective interaction. We also discuss the perturbative estimate of the deformation of the `vacuum" in the Faddeev approach.

Overall properties of a cracked solid

1980 ◽  
Vol 88 (2) ◽  
pp. 371-384 ◽  
Author(s):  
J. A. Hudson
Keyword(s):  
Integral Equation ◽  
Solid Material ◽  
Equation Of Motion ◽  
Second Order ◽  
Number Density ◽  
Scattering Operator ◽  
Third Order ◽  
The Third ◽  
Differential Integral Equation ◽  
The Mean

AbstractThe differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.

scholarly journals A Tighter Bound for Graphical Models

2001 ◽  
Vol 13 (9) ◽  
pp. 2149-2171 ◽  
Author(s):  
M. A. R. Leisink ◽  
H. J. Kappen
Keyword(s):  
Graphical Models ◽  
Mean Field ◽  
Third Order ◽  
First Order ◽  
Direct Extension ◽  
Odd Order ◽  
Order Polynomial ◽  
The Mean ◽  
Mean Field Bound ◽  
Better Than

We present a method to bound the partition function of a Boltzmann machine neural network with any odd-order polynomial. This is a direct extension of the mean-field bound, which is first order. We show that the third-order bound is strictly better than mean field. Additionally, we derive a third-order bound for the likelihood of sigmoid belief networks. Numerical experiments indicate that an error reduction of a factor of two is easily reached in the region where expansion-based approximations are useful.

scholarly journals GAUGE INDEPENDENT PHASE STRUCTURE OF GAUGED NAMBU-JONA-LASINIO AND YUKAWA MODELS

1993 ◽  
Vol 08 (32) ◽  
pp. 3031-3047 ◽  
Author(s):  
KEI-ICHI KONDO
Keyword(s):  
Chiral Symmetry Breaking ◽  
Mean Field ◽  
Dyson Equation ◽  
Chiral Condensate ◽  
Inversion Method ◽  
Gauge Parameter ◽  
Logarithmic Correction ◽  
Chiral Phase ◽  
Njl Model ◽  
The Mean

We investigate the critical behavior of the gauged NJL model (QED plus four-fermion interaction) and the gauged Yukawa model by the use of the inversion method. By calculating the gauge-invariant chiral condensate in the inversion method to the lowest order, we derive the critical line which separates the spontaneous chiral-symmetry breaking phase from the chiral symmetric one. The critical exponent for the chiral order parameter associated with the second order chiral phase transition is shown to take the mean field value together with possible logarithmic correction to the mean-field prediction. All the above results are gauge-parameter independent and are compared with the previous results obtained from the Schwinger-Dyson equation for the fermion propagator.

ASYMPTOTIC PROPERTIES OF A THIRD ORDER NEURAL NETWORK

1991 ◽  
Vol 02 (04) ◽  
pp. 315-322
Author(s):  
Mats Bengtsson
Keyword(s):  
Neural Network ◽  
Storage Capacity ◽  
Hebbian Learning ◽  
Mean Field Theory ◽  
Asymptotic Properties ◽  
Mean Field ◽  
Network Size ◽  
Third Order ◽  
The Third ◽  
Large N

We have investigated the storage capacity in the limit of large N (the network size) for a third order recurrent artificial neural network with Hebbian learning. Numerical results for the relation between the overlap to stored patterns, and the fraction of the number of stored patterns and N2 (the m—α relation), agree well with replica symmetric predictions. A comparative study is made of the m—α relation for a third and a second order network. Large differences exist between these two models, usually to the favour of the third order network. This result stands in some contrast to previous investigations. The phase transition temperature is investigated numerically and compared with mean field theory predictions.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

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