scholarly journals Legendre-spectral Dyson equation solver with super-exponential convergence

2020 ◽  
Vol 152 (13) ◽  
pp. 134107
Author(s):  
Xinyang Dong ◽  
Dominika Zgid ◽  
Emanuel Gull ◽  
Hugo U. R. Strand
Keyword(s):  
Exponential Convergence ◽  
Dyson Equation

scholarly journals A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Timon S. Gutleb
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Jacobi Polynomial ◽  
Numerical Experiments ◽  
Exponential Convergence ◽  
Analytic Solutions ◽  
Convolution Type ◽  
Volterra Equations ◽  
Polynomial Bases ◽  
Sparsity Structure

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

scholarly journals Electroweak SU(2)L × U(1)Y model with strong spontaneously fermion-mass-generating gauge dynamics

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Petr Beneš ◽  
Jiří Hošek ◽  
Adam Smetana
Keyword(s):  
Chiral Symmetry Breaking ◽  
Axial Vector ◽  
Higgs Sector ◽  
Dyson Equation ◽  
Mass Splitting ◽  
Fermion Mass ◽  
Goldstone Bosons ◽  
The Standard Model ◽  
Exploratory Stage ◽  
The Masses

Abstract Higgs sector of the Standard model (SM) is replaced by quantum flavor dynamics (QFD), the gauged flavor SU(3)f symmetry with scale Λ. Anomaly freedom requires addition of three νR. The approximate QFD Schwinger-Dyson equation for the Euclidean infrared fermion self-energies Σf(p2) has the spontaneous-chiral-symmetry-breaking solutions ideal for seesaw: (1) Σf(p2) = $$ {M}_{fR}^2/p $$ M fR 2 / p where three Majorana masses MfR of νfR are of order Λ. (2) Σf(p2) = $$ {m}_f^2/p $$ m f 2 / p where three Dirac masses mf = m(0)1 + m(3)λ3 + m(8)λ8 of SM fermions are exponentially suppressed w.r.t. Λ, and degenerate for all SM fermions in f. (1) MfR break SU(3)f symmetry completely; m(3), m(8) superimpose the tiny breaking to U(1) × U(1). All flavor gluons thus acquire self-consistently the masses ∼ Λ. (2) All mf break the electroweak SU(2)L × U(1)Y to U(1)em. Symmetry partners of the composite Nambu-Goldstone bosons are the genuine Higgs particles: (1) three νR-composed Higgses χi with masses ∼ Λ. (2) Two new SM-fermion-composed Higgses h3, h8 with masses ∼ m(3), m(8), respectively. (3) The SM-like SM-fermion-composed Higgs h with mass ∼ m(0), the effective Fermi scale. Σf(p2)-dependent vertices in the electroweak Ward-Takahashi identities imply: the axial-vector ones give rise to the W and Z masses at Fermi scale. The polar-vector ones give rise to the fermion mass splitting in f. At the present exploratory stage the splitting comes out unrealistic.

scholarly journals Singularly Perturbed Oscillators with Exponential Nonlinearities

2021 ◽  
Author(s):  
S. Jelbart ◽  
K. U. Kristiansen ◽  
P. Szmolyan ◽  
M. Wechselberger
Keyword(s):  
Limit Cycles ◽  
Space Dimension ◽  
Blow Up ◽  
Exponential Convergence ◽  
Model System ◽  
Piecewise Smooth ◽  
Singularly Perturbed ◽  
Model Systems ◽  
Smooth System ◽  
Unique Limit

AbstractSingular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

scholarly journals Distributed Solving Sylvester equations with an explicit exponential Convergence

2020 ◽  
Vol 53 (2) ◽  
pp. 3260-3265
Author(s):  
Songsong Cheng ◽  
Xianlin Zeng ◽  
Yiguang Hong
Keyword(s):  
Exponential Convergence

scholarly journals INFRARED DIVERGENCE IN QED3 AT FINITE TEMPERATURE

1999 ◽  
Vol 14 (18) ◽  
pp. 2921-2947 ◽  
Author(s):  
DOMINIC LEE ◽  
GEORGIOS METIKAS
Keyword(s):  
Finite Temperature ◽  
Statistical Physics ◽  
Order Term ◽  
Dyson Equation ◽  
Infrared Divergence ◽  
Fermion Mass ◽  
Photon Propagator ◽  
Transverse Part ◽  
Transverse Photon ◽  
Range Of Values

We consider various ways of treating the infrared divergence which appears in the dynamically generated fermion mass, when the transverse part of the photon propagator in N flavour QED 3 at finite temperature is included in the Matsubara formalism. This divergence is likely to be an artifact of taking into account only the leading order term in the [Formula: see text] expansion when we calculate the photon propagator and is handled here phenomenologically by means of an infrared cutoff. Inserting both the longitudinal and the transverse part of the photon propagator in the Schwinger–Dyson equation, we find the dependence of the dynamically generated fermion mass on the temperature and the cutoff parameters. It turns out that consistency with certain statistical physics arguments imposes conditions on the cutoff parameters. For parameters in the allowed range of values we find that the ratio r=2* Mass (T=0)/critical temperature is approximately 6, consistent with previous calculations which neglected the transverse photon contribution.

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