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Register a new userNormal form for renormalization groups
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The results of the renormalization group are commonly advertised as the existence of power law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis. We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables. We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions. We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.
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In this paper we will present the results of Artin--Markov on braid groups by using the Groebner--Shirshov basis. As a consequence we can reobtain the normal form of Artin--Markov--Ivanovsky as an easy corollary.
In a recent paper by L. A. Bokut, V. V. Chaynikov and K. P. Shum in 2007, Braid group $B_n$ is represented by Artin-Buraus relations. For such a representation, it is told that all other compositions can be checked in the same way. In this note, we support this claim and check all compositions.
The superfluid/normal-fluid interface of liquid 4He is investigated in gravity on earth where a small heat current Q flows vertically upward or downward. We present a local space- and time-dependent renormalization-group (RG) calculation based on model F which describes the dynamic critical effects for temperatures T near the superfluid transition T_lambda. The model-F equations are rewritten in a dimensionless renormalized form and solved numerically as partial differential equations. Perturbative corrections are included for the spatially inhomogeneous system within a self-consistent one-loop approximation. The RG flow parameter is determined locally as a function of space and time by a constraint equation which is solved by a Newton iteration. As a result we obtain the temperature profile of the interface. Furthermore we calculate the average order parameter <psi>, the correlation length xi, the specific heat C_Q and the thermal resistivity rho_T where we observe a rounding of the critical singularity by the gravity and the heat current. We compare the thermal resistivity with an experiment and find good qualitative agreement. Moreover we discuss our previous approach for larger heat currents and the self-organized critical state and show that our theory agrees with recent experiments in this latter regime.
The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson-Lie groups $K$ on a symplectic manifold equipped with an $AN$-valued moment map, where $AN$ is the dual Poisson-Lie group of $K$. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of $K$ with $mathfrak{k}^*$-valued moment map to a Hamiltonian action with an $AN$-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a ``delinearization commutes with symplectic quotients theorem which is also of independent interest, and then putting this together with the local normal form theorem for classical Hamiltonian actions wtih $mathfrak{k}^*$-valued moment maps. A key ingredient for our main result is the delinearization $mathcal{D}(omega_{can})$ of the canonical symplectic structure on $T^*K$, so we additionally take some steps toward explicit computations of $mathcal{D}(omega_{can})$. In particular, in the case $K=SU(2)$, we obtain explicit formulas for the matrix coefficients of $mathcal{D}(omega_{can})$ with respect to a natural choice of coordinates on $T^*SU(2)$.
We present mathematical details of derivation of the critical exponents for the free energy and magnetization in the vicinity of the Gaussian fixed point of renormalization. We treat the problem in general terms and do not refer to particular models of interaction energy. We discuss the case of arbitrary dispersion of the fixed point.
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