The scale factor of matter theory or the position of the matter brane in the auxiliary (ﬁfth) dimension can be found as solution to the ﬁve-dimensional equations of motion (on the ”gravitational side”), or by the renormalization group equations (on the ”matter” or gauge theory side).
String Theory or Field Theory?
Interpretation of the scale factor as an auxiliary co-ordinate of space-time allows one to consider the problems of conﬁnement in the theory of elementary particles and the problems of gravity and cosmology on equal footing.
String Theory or Field Theory?
Schramm, trying to reproduce by a continuum stochastic process both the conformal invariance and Markov properties of the scaling limit of loop-erased random walks, invented in 1999 the so-called “Stochastic L¨owner Evolution” (SLE) , a process parametrized by an auxiliary one-dimensional Brownian motion of speed κ.
Conformal Fractal Geometry and Boundary Quantum Gravity
From the other side the effective conductivity of the auxiliary plane on scales l ≫ lL must be determined by the universal Keller – Dykhne formula (22).
On universality of conductivity of planar random self-dual systems
It will be often convenient to work with the following auxiliary ﬁeld, which is geometrically slightly easier to tackle while for small scales it does not distinguish between w and its periodic counterpart W .
Random Conformal Weldings
The analysis of the processes on their relevant time scales will lead us to study a number of auxiliary processes on geographic spaces different from ΩN .
Renormalisation of hierarchically interacting Cannings processes
It is convenient to introduce in their procedure an auxiliary variable µ for each odd bond still surviving at scale Ω.
Percolation Transition in the random antiferromagnetic spin-1 chain
Associated with the generalized renormalization of induced divergences to construct ﬁnite pdfs, etc, are typically auxiliary scales like the renormalization scale µ of MS renormalization.
Equality of Two Definitions for Transverse Momentum Dependent Parton Distribution Functions
In applications, the auxiliary parameters are to be evolved to values that give good or optimal accuracy for the perturbative calculations that are relevant for a given process at a given hard scale.
Equality of Two Definitions for Transverse Momentum Dependent Parton Distribution Functions
For systems that are inhomogeneous under a suitable scaling symmetry we use the following trick: We introduce one (or more) auxiliary parameter(s) with appropriate scaling.
Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations
For systems that lack scaling invariance and have no parameters, introducing one (or more) auxiliary parameter(s) with appropriate scaling provides a solution.
Integrability Tests for Nonlinear Evolution Equations
Lattices that do not admit a dilation symmetry can be made scaling invariant by extending the set of dependent variables using auxiliary parameters with scaling.
Symbolic Computation of Conserved Densities, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations
With two time-scales, one can potentially improve the rate of convergence of θk (cf. to a single-time-scale algorithm) by sacriﬁcing the rate of convergence of the auxiliary parameters.
Convergence rate of linear two-time-scale stochastic approximation
Therefore the number of auxiliary ﬁelds scales similarly to the discrete scheme.
Continuous Time Quantum Monte Carlo Method for Fermions: Beyond Auxiliary Field Framework
Therefore the number of auxiliary ﬁelds scales similarly as the discrete scheme, so that the method remains local.
Continuous Time Quantum Monte Carlo method for fermions
***