Relating a gluon mass scale to an infrared fixed point in pure gauge QCD

The condition for the global minimum of the vacuum energy for a non-Abelian gauge theory with a dynamically generated gauge boson mass scale which implies the existence of a nontrivial IR fixed point of the theory was shown. Thus, this vacuum energy depends on the dynamical masses through the nonperturbative propagators of the theory. The results show that the freezing of the QCD coupling constant observed in the calculations can be a natural consequence of the onset of a gluon mass scale, giving strong support to their claim.

Chiral-Scale Perturbation Theory About an Infrared Fixed Point

We review the failure of lowest order chiral SU(3)L ×SU(3)R perturbation theory χPT3 to account for amplitudes involving the f0(500) resonance and O(mK) extrapolations in momenta. We summarize our proposal to replace χPT3 with a new effective theory χPTσ based on a low-energy expansion about an infrared fixed point in 3-flavour QCD. At the fixed point, the quark condensate ⟨q̅q⟩vac ≠ 0 induces nine Nambu-Goldstone bosons: π,K,η and a QCD dilaton σ which we identify with the f0(500) resonance. We discuss the construction of the χPTσ Lagrangian and its implications for meson phenomenology at low-energies. Our main results include a simple explanation for the ΔI = 1/2 rule in K-decays and an estimate for the Drell-Yan ratio in the infrared limit.

Δ I = 1 / 2 rule for kaon decays derived from QCD infrared fixed point

This article gives details of our proposal to replace ordinary chiral SU(3)L×SU(3)R perturbation theory χPT3 by three-flavor chiral-scale perturbation theory χPTσ. In χPTσ, amplitudes are expanded at low energies and small u,d,s quark masses about an infrared fixed point αIR of three-flavor QCD. At αIR, the quark condensate ⟨q¯q⟩vac≠0 induces nine Nambu-Goldstone bosons: π,K,η, and a 0++ QCD dilaton σ. Physically, σ appears as the f0(500) resonance, a pole at a complex mass with real part ≲ mK. The ΔI=1/2 rule for nonleptonic K decays is then a consequence of χPTσ, with a KSσ coupling fixed by data for γγ→ππ and KS→γγ. We estimate RIR≈5 for the nonperturbative Drell-Yan ratio R=σ(e+e−→hadrons)/σ(e+e−→μ+μ−) at αIR and show that, in the many-color limit, σ/f0 becomes a narrow qq¯ state with planar-gluon corrections. Rules for the order of terms in χPTσ loop expansions are derived in Appendix A and extended in Appendix B to include inverse-power Li-Pagels singularities due to external operators. This relates to an observation that, for γγ channels, partial conservation of the dilatation current is not equivalent to σ-pole dominance.

Asymptotic freedom, dimensional transmutation, and an infrared conformal fixed point for the δ-function potential in one-dimensional relativistic quantum mechanics

We consider the Schrödinger equation for a relativistic point particle in an external one-dimensional δ-function potential. Using dimensional regularization, we investigate both bound and scattering states, and we obtain results that are consistent with the abstract mathematical theory of self-adjoint extensions of the pseudodifferential operator H=p2+m2−−−−−−−√. Interestingly, this relatively simple system is asymptotically free. In the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point. Thus it can be used to illustrate nontrivial concepts of quantum field theory in the simpler framework of relativistic quantum mechanics.

Walking near a conformal fixed point

We investigate the 2-d O(3) model with a q-term as a toy model for slowly walking 4-d non-Abelian gauge theories. Using the very efficient meron-cluster algorithm, an accurate investigation of the scale dependence of the renormalized coupling is carried out for different values of the vacuum angle q. Approaching q = p, the infrared dynamics of the 2-d O(3) model is determined by a non-trivial conformal fixed point. We provide evidence for a slowly walking behavior near the fixed point and we perform a finite-size scaling analysis of the mass gap.

Fixed-point DSP implementation of nonlinear H(infinity) controller for large gap electromagnetic suspension system

Electromagnetic suspension systems are inherently nonlinear and often face hardware limitation when digitally controlled. The main contributions of this paper are: the design of a nonlinear H(infinity) controller. including dynamic weighting functions, applied to a large gap electromagnetic suspension system and the presentation of a procedure to implement this controller on a fixed-point DSP, through a methodology able to translate a floating-point algorithm into a fixed-point algorithm by using l(infinity) norm minimization due to conversion error. Experimental results are also presented, in which the performance of the nonlinear controller is evaluated specifically in the initial suspension phase. (C) 2009 Elsevier Ltd. All rights reserved.

Fixed point theorems of upper semicontinuous multivalued mappings with applications in hyperconvex metric spaces

In the present paper, we establish two fixed point theorems for upper semicontinuous multivalued mappings in hyperconvex metric spaces and apply these to study coincidence point problems and minimax problems. (C) 2002 Elsevier Science (USA). All rights reserved.

A uniform white-noise model for fixed-point roundoff errors in digital systems

Fixed-point roundoff noise in digital implementation of linear systems arises due to overflow, quantization of coefficients and input signals, and arithmetical errors. In uniform white-noise models, the last two types of roundoff errors are regarded as uniformly distributed independent random vectors on cubes of suitable size. For input signal quantization errors, the heuristic model is justified by a quantization theorem, which cannot be directly applied to arithmetical errors due to the complicated input-dependence of errors. The complete uniform white-noise model is shown to be valid in the sense of weak convergence of probabilistic measures as the lattice step tends to zero if the matrices of realization of the system in the state space satisfy certain nonresonance conditions and the finite-dimensional distributions of the input signal are absolutely continuous.

Fixed point transformations in the adaptive control of fractional-order MIMO systems

Though the formal mathematical idea of introducing noninteger order derivatives can be traced from the 17th century in a letter by L’Hospital in which he asked Leibniz what the meaning of D n y if n = 1/2 would be in 1695 [1], it was better outlined only in the 19th century [2, 3, 4]. Due to the lack of clear physical interpretation their first applications in physics appeared only later, in the 20th century, in connection with visco-elastic phenomena [5, 6]. The topic later obtained quite general attention [7, 8, 9], and also found new applications in material science [10], analysis of earth-quake signals [11], control of robots [12], and in the description of diffusion [13], etc.

Application of robust fixed point transformations for technological operation of robots

A robot’s drive has to exert appropriate driving forces that can keep its arm and end effector at the proper position, velocity and acceleration, and simultaneously has to compensate for the effects of the contact forces arising between the tool and the workpiece depending on the needs of the actual technological operation. Balancing the effects of a priori unknown external disturbance forces and the inaccuracies of the available dynamic model of the robot is also important. Technological tasks requiring well prescribed end effector trajectories and contact forces simultaneously are challenging control problems that can be tackled in various manners.

Proper actions, fixed-point algebras and naturality in nonabelian duality

"Vegeu el resum a l'incici del document del fitxer adjunt."

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is constructed for two different models exhibiting self-organized criticality: the Bak-Tang-Wiesenfeld (BTW) sandpile model [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] and the Zhang model [Phys. Rev. Lett. 63, 470 (1989)]. The dynamic renormalization group (DRG) enables one to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

On Commutation and Conjugacy of Rational Languages and the Fixed Point Method

The research on language equations has been active during last decades. Compared to the equations on words the equations on languages are much more difficult to solve. Even very simple equations that are easy to solve for words can be very hard for languages. In this thesis we study two of such equations, namely commutation and conjugacy equations. We study these equations on some limited special cases and compare some of these results to the solutions of corresponding equations on words. For both equations we study the maximal solutions, the centralizer and the conjugator. We present a fixed point method that we can use to search these maximal solutions and analyze the reasons why this method is not successful for all languages. We give also several examples to illustrate the behaviour of this method.

Fixed point results for multivalued contractions on graphs and their applications

Nous présentons dans cette thèse des théorèmes de point fixe pour des contractions multivoques définies sur des espaces métriques, et, sur des espaces de jauges munis d’un graphe. Nous illustrons également les applications de ces résultats à des inclusions intégrales et à la théorie des fractales. Cette thèse est composée de quatre articles qui sont présentés dans quatre chapitres. Dans le chapitre 1, nous établissons des résultats de point fixe pour des fonctions multivoques, appelées G-contractions faibles. Celles-ci envoient des points connexes dans des points connexes et contractent la longueur des chemins. Les ensembles de points fixes sont étudiés. La propriété d’invariance homotopique d’existence d’un point fixe est également établie pour une famille de Gcontractions multivoques faibles. Dans le chapitre 2, nous établissons l’existence de solutions pour des systèmes d’inclusions intégrales de Hammerstein sous des conditions de type de monotonie mixte. L’existence de solutions pour des systèmes d’inclusions différentielles avec conditions initiales ou conditions aux limites périodiques est également obtenue. Nos résultats s’appuient sur nos théorèmes de point fixe pour des G-contractions multivoques faibles établis au chapitre 1. Dans le chapitre 3, nous appliquons ces mêmes résultats de point fixe aux systèmes de fonctions itérées assujettis à un graphe orienté. Plus précisément, nous construisons un espace métrique muni d’un graphe G et une G-contraction appropriés. En utilisant les points fixes de cette G-contraction, nous obtenons plus d’information sur les attracteurs de ces systèmes de fonctions itérées. Dans le chapitre 4, nous considérons des contractions multivoques définies sur un espace de jauges muni d’un graphe. Nous prouvons un résultat de point fixe pour des fonctions multivoques qui envoient des points connexes dans des points connexes et qui satisfont une condition de contraction généralisée. Ensuite, nous étudions des systèmes infinis de fonctions itérées assujettis à un graphe orienté (H-IIFS). Nous donnons des conditions assurant l’existence d’un attracteur unique à un H-IIFS. Enfin, nous appliquons notre résultat de point fixe pour des contractions multivoques définies sur un espace de jauges muni d’un graphe pour obtenir plus d’information sur l’attracteur d’un H-IIFS. Plus précisément, nous construisons un espace de jauges muni d’un graphe G et une G-contraction appropriés tels que ses points fixes sont des sous-attracteurs du H-IIFS.