A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion...

Does a minimal harmonic function h remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes D ⊂ R of variable width and minimal harmonic functions h corresponding to the boundary point of D “at infinity.” Suppose f(u) is the width of the tube u units away from its endpoint and f is a Lipschitz function. The answer to the que...

A non-elementary Möbius group generated by twoparabolics is determined up to conjugation by one complex parameter and the parameter space has been extensively studied. In this paper, we use the results of [7] to obtain an additional structure for the parameter space, which we term the two-parabolic space. This structure allows us to identify groups that contain additional conjugacy classes of p...

We formulate the Nahm transform in the context of parabolic Higgs bundles on P 1 and extend its scope in completely algebraic terms. This transform requires parabolic Higgs bundles to satisfy an admissibility condition and allows Higgs fields to have poles of arbitrary order and arbitrary behavior. Our methods are constructive in nature and examples are provided. The extended Nahm transform is ...

In this paper, stability and error estimates for time discretizations of linear semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to question: whether upper bound step-size ratios $l^{\infty }(0,T;H)$ -stability BDF2 identical zero-stability. The }(0,T;V)$ also established under m...

Abstract This study presents an improvement of the graphical method for plotting shear and moment diagrams structural members under linearly varying loads (triangular trapezoidal loads). Based on parabolic nature function, when loading varies linearly, relations among load, shear, moment, a mathematical equation is developed to locate zero-shear point, while geometric technique presented calcul...

In this paper we review various numerical homogenization methods for monotone parabolic problems with multiple scales. The spatial discretisation is based on finite element methods and the multiscale strategy relies on the heterogeneous multiscale method. The time discretization is performed by several classes of Runge-Kutta methods (strongly A−stable or explicit stabilized methods). We discuss...

Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory. Abstract We obtain solutions of the nonlinear degenerate parabolic equation ∂ ρ ∂ t = div ρ ∇c ⋆ [ ∇ (F ′ (ρ) + V) ] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires less uniform convexity assumption than that imposed by Alt and ...

Let S (respectively, S 0) be a nite subset of a compact connected Riemann surface X (respectively, X 0) of genus at least two. Let M (respectively, M 0) denote a moduli space of parabolic stable bundles of rank two over X (respectively, X 0) with xed determinant of degree one, having a nontrivial quasi-parabolic structure over each point of S (respectively, S 0), and of parabolic degree less th...

in this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. in the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations.  also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. error estimates were also obtaine...