Hopf Formula Research Articles

Higher Hopf formulae for homology via Galois Theory

We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A , thus we describe the Barr–Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.

Dirichlet Problems for some Hamilton–Jacobi Equations with Inequality Constraints

We use viability techniques for solving Dirichlet problems with inequality constraints (obstacles) for a class of Hamilton–Jacobi equations. The hypograph of the “solution” is defined as the “capture basin” under an auxiliary control system of a target associated with the initial and boundary conditions, viable in an environment associated with the inequality constraint. From the tangential condition characterizing capture basins, we prove that this solution is the unique “upper semicontinuous” solution to the Hamilton–Jacobi–Bellman partial differential equation in the Barron-Jensen/Frankowska sense. We show how this framework allows us to translate properties of capture basins into corresponding properties of the solutions to this problem. For instance, this approach provides a representation formula of the solution which boils down to the Lax–Hopf formula in the absence of constraints.

Galois Groups, Abstract Commutators, and Hopf Formula

A general description of the Galois group of a “pointed” normal extension in categorical Galois theory is examined under the presence of a suitable commutator operation. In particular, using the Hopf formula for the second homology group of a group, the connection between Galois theory and group homology is clarified.

Complex algebraic plane curves via the Poincaré–Hopf formula, I: Parametric lines

We present a new and very eficient approach to study topology of algebraic curves in C2. It relies on using the Poincare?Hopf formula, applied to a suitable Hamiltonian vector field, to estimate the number of double points and the Milnor numbers of singular points of the curve and on considering finite-dimensional spaces of curves with given asymptotics at infinity. We apply this method to classification of parametric lines with one self-intersection.

On tangential cohomology of Riemannian foliations

We prove Künneth formula, Poincaré duality, Hopf formula and index theorem for tangentially oriented compact Riemannian foliations.

N-Fold Čech Derived Functors and Generalised Hopf Type Formulas

In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-fold Cech derived functors of the lower central series functors Z_k. The paper ends with an application to algebraic K-theory.

Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case

The aim of this paper is twofold. We construct an extension to a non-integrable case of Hopf's formula, often used to produce viscosity solutions of Hamilton-Jacobi equations for $p$-convex integrable Hamiltonians. Furthermore, for a general class of $p$-convex Hamiltonians, we present a proof of the equivalence of the minimax solution with the viscosity solution.

Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations

We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes.

More about the (co)homology of groups and associative algebras

It is proved that the homology and cohomology theories of groups and associative algebras are non-abelian derived functors of the cokernel and kernel groups of higher dimensions of their defining standard chain and cochain complexes respectively. The same results are also obtained for the relative (co)homology of groups, the mod q cohomology of groups and the cohomology of groups with operators. This allowed us to give an alternative approach to higher Hopf formulas for integral homology of groups. An axiomatic characterization of the relative cohomology of groups is given and higher relative (n+1)-th cohomology of groups is described in terms of n-fold extensions.

The regularity of generalized solutions of Hamilton–Jacobi equations

We study the differentiability of generalized solutions for Hamilton–Jacobi equation. Some results on the relationship between Hopf's formulas and solutions constructed via characteristics are established. We also consider some cases where the generalized solutions are continuously differentiable on the strip ( 0 , t 0 ) × R of the domain Ω .

Homology with Trivial Coefficients of Leibniz n-Algebras

We introduce homology K-vector spaces with trivial coefficients for Leibniz n-algebras and we obtain exact sequences relating them. As a consequence we get a Hopf formula and several results on central extensions of Leibniz n-algebras.

Two Consequences of a Path Transform

In a previous paper, the author proved a recent Wiener–Hopf formula, established by Alili and Doney, by means of a path transform. We derive here two other results using this path transform.

Generalized Hopf formulas for the nonautonomous Hamilton-Jacobi equation

Generalized Hopf formulas are provided for minimax (viscosity) solutions of Hamilton–Jacobi equations of the form V t + H(t, D x V) = 0 and V t + H(t, V, D x V) = 0 with the boundary condition V(T, x) = ϕ(x), where ϕ is a convex function. The bounds within which these formulas apply are elucidated.

More about homological properties of precrossed modules

Homology groups moduloq of a precrossed P-module in any dimensions are defined in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second homology group modulo q of a precrossed P-module which shows that

Hopf-Type Estimates and Formulas For Nonconvex Nonconcave Hamilton--Jacobi Equations

Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the Hamilton--Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function. The estimates become equalities whenever a "minmax" equals a "maxmin" and thus a representation formula for the solution is obtained, generalizing the classical Hopf formulas as well as some formulas of Kruzkov [Functional Anal. Appl., 2 (1969), pp. 128--136].

Applications of the Hopf--Lax Formula for ut+H(u,Du)=0

The $\g$ (sub)level set of the solution to $w_t+H(\g,D_xw)=0$ is the same as the $\g$ (sub)level set of the solution to $u_t+H(u,D_xu)=0$, and the solution u may be built from w. This result is applied to determining upper and lower bounds for a solution of ut+H1 (u,Du)+H2 (u,Du)=0, with H1 convex and H2 concave, as well as ut+H(u,Du)=0, but with initial data $u(0,x)=g_1(x) \vee g_2(x)$ or $g_1(x) \wedge g_2(x)$, with g1 quasi-convex and g2 quasi-concave. A differential game in $L^{\i}$ is constructed giving a new proof of the Hopf formula.

Generalizing Hopf and Lax-Ole�nik formulae via conjugate integral

Using the second Fenchel conjugate transform the conjugate integral sums and the conjugate integral are introduced. An estimate of speed of convergence of the sums to the integral is obtained. In the case of a convex integrant the conjugate integral reduces to the Riemannian one. It is proved that the Fenchel conjugate transform of the conjugate integral with variable upper limit provides a formula for the viscosity solution to a Hamilton-Jacobi equation in which the Hamiltonian depends both on time and the gradient of the unknown function. In the autonomous case the obtained formula coincides with Hopf's one. Two examples are considered in which an application of the conjugate integral allows to find viscosity solutions explicitly. It is shown how the extension of the Lax-Oleinik formula to the nonautonomous case may be obtained using the generalized Hopf formula.

Front Speed in the Burgers Equation with a Random Flux

We study the large-time asymptotic shock-front speed in an inviscid Burgers equation with a spatially random flux function. This equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well-known Buckley–Leverett equation for two-phase flows, and the contaminant transport equation in groundwater flows. The initial condition is a shock located at the origin (the indicator function of the negative real line). We first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a Cole–Hopf formula. Using the invariance principle of the underlying random processes and the Laplace method, we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution. However, the front locations are random, and we show explicitly the probability of observing the head or tail of the fronts. Finally, we pass to the inviscid limit, and establish the same results for the inviscid shock fronts.

On Hopf's formula for lipschitz solutions of the cauchy problem for Hamilton-Jacobi equations

On Hopf's formula for lipschitz solutions of the cauchy problem for Hamilton-Jacobi equations

Isoperimetric inequalities and the homology of groups

An isoperimetric function for a finitely presented group G bounds the number of defining relations needed to prove that a word w =G 1 in terms of the length of w. Suppose G = F/R is a finitely presented group where F is a finitely generated free group and R is the normal closure of a finite set of relators. For each w ∈ R we define the area ∆(w) to be the least number of uses of the given relators needed to prove that w =G 1. The isoperimetric function ΦG(n) is defined to be the maximum of the areas of all the words in R of length ≤ n. One compares the size of such functions using a suitable order 1 on functions. Different presentations give rise to ' equivalent isoperimetric functions where ' is the equivalence relation determined by 1 . Interest in such functions was stimulated by Gromov [13] who characterized word hyperbolic groups as those with linear isoperimetric function, that is, ΦG(n) ' n. Further information was obtained by Cannon et al [8] who showed that automatic groups satisfy ΦG 1 n and that asynchronously automatic groups satisfy ΦG 1 2. They also showed the free nilpotent group of class 2 and rank 2 has ΦG(n) ' n. Gersten [10] used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups. He exploited the fact that, for a torsion-free one relator groupG, the usual 2 dimensional complex having fundamental group isomorphic to G is aspherical. The general technique for giving upper bounds for ΦG(n) is to describe a solution to the word problem for G and analyze its use of defining relations. It seems quite difficult to establish lower bounds for ΦG(n) in this manner. In this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for ΦG. Instead of trying to analyze R directly, we consider some of its quotient groups for which additional machinery is available. For instance, we consider for each w ∈ R, its image w[R,F ] in R/[R,F ] and define a suitable “centralized” isoperimetric function Φ G (n) such that Φ G (n) 1 ΦG(n). Using the Hopf formula it can be shown that R/[R,F ] is a direct sum of the second homology group H2(G,Z) with a free abelian