Homotopy Operator Research Articles

Solutions of Exterior Differential Equations by Quadratures

Homotopy operators are used to obtain unconstrained local solutions of systems of exterior differential equations by quadratures. An analysis is given of when given systems of constraints on the solutions can be realized. The results are applied to several classes of classical problems. A collection of superconductor singular solutions of the free gauge field equations for matrix Lie groups is also obtained, and the general forms of gauge covariant constant fields are given.

Holomorphy minimal homotopy and the 4D, N = 1 supersymmetric Bardeen–Gross–Jackiw anomaly term

By use of a special homotopy operator, we present an explicit, closed-form and simple expression for the left-right Bardeen–Gross–Jackiw anomalies described as the proper superspace integral of a superfunction.

A homotopy operation spectral sequence for the computation of homotopy groups

A homotopy operation spectral sequence for the computation of homotopy groups

Homotopy Structures for Algebras over a Monad

Let T be a monad over a category A. Then a homotopy structure for A, defined by a cocylinder P : A → A, or path-endofunctor, can be lifted to the category AT of Eilenberg–Moore algebras over T, provided that P is consistent with T in a natural sense, i.e. equipped with a natural transformation λ : T P → P T satisfying some obvious axioms. In this way, homotopy can be lifted from well-known, basic situations to various categories of ‘algebras’ for instance, from topological spaces to topological semigroups, or spaces over a fixed space (fibrewise homotopy), or actions of a fixed topological group (equivariant homotopy); from categories to strict monoidal categories; from chain complexes to associative chain algebras. The interest is given by the possibility of lifting the ‘homotopy operations’ (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to AT, just by verifying the consistency between these operations and λ : T P → P T. When this holds, the structure we obtain on our category of algebras is sufficiently powerful to ensure the main general properties of homotopy.

Boundary regularity for the local $$\bar \partial _b - complex$$ on certain weakly pseudoconvex surfaceson certain weakly pseudoconvex surfaces

Let U be a neighborhood of 0in ℂn, 1 ≤s ≤ n − 1,m ∈ ℕ,M = {(z1,..., zn) ∈ U:Imzn = ¦z1¦2 + ...¦zs¦2 + (¦zs+1¦2 +...+ ¦zn−1¦2)m +some error} term be a hypersurface in U. In this paper, the author constructs homotopy operators for the local tangential Cauchy-Riemann operator on the neighborhoods Mδ:= M ∩ {¦zn¦ < δ} of 0in M. C0 -estimates up to the boundary are proved for the operators. The constants in the estimates do not depend on δ.

Microlocalisation de l'homologie de Hochschild

We construct a homotopy operator on the. Hochschild complex of the algebra of smooth functions, which concentrates the chains of the complex, living on the various powers of the base manifold, onto jets of functions all along the principal diagonals of the powers. The construction can be adapted to compute the Hochschild homology of the algebras of continuous functions having partial derivatives of finite order in L p. In particular, our method allows one to lift the compactness hypothesis in A. Connes' theorem concerning the Hochschild homology of the algebra of smooth functions on smooth manifolds.

An equivariant van Kampen spectral sequence

This paper constructs an equivariant homotopy spectral sequence for any finite group G, any finite dimensional representation V, and two suitably connected G-CW complexes X and Y. The spectral sequence converges to the collection of equivariant homotopy groups of the wedge of X and Y, while the E 2 term depends only on the equivariant homotopy groups of X and of Y, along with primary homotopy operations. The edge homomorphism of the spectral sequence is actually an isomorphism in a range, which is the equivariant van Kampen theorem of L.G. Lewis Jr. When G is the trivial group, the spectral sequence reduces to that of C.R. Stover.

Anomalies in Quantum Field Theory

1.: Introduction. 2.: Differential geometry, topology and fibre bundles. 3.: Path integrals, FP method and BRS transformation. 4.: Anomalies in QFT. 5.: Path integral and anomaly. 6.: Physics in terms of differential forms. 7.: Chern-Simons form, homotopy operator and anomaly. 8.: Consistent anomaly. 9.: Stora-Zumino chain of descent equations. 10.: Convenient anomaly. 11.: Index and anomaly. 12.: Gravitation. Bibliography

Loop spaces and homotopy operations

We describe two obstruction theories for a given topological space X to be a loop space, the rst requiring a given H-space structure on X, and the second not. Both are dened in terms of higher homotopy operations.

FUNCTORIAL PROPERTIES OF THE HOPF INVARIANT I

Abstract Homotopy operations Θ: [ΣY, U] → [ΣY, V] which are natural in Y are considered. In particular a technique used in the definition of the Hopf invariant (as treated by Berstein-Hilton) shows that any fibration p: E → B with fiber V, when provided with a homotopy section of Ωp, determines such a homotopy operation [ΣY, E] → [ΣY, V]. More generally, starting from a track class of homotopies α o f ≃ β o g we adapt this fibration technique to construct a homotopy operation [ΣY, M(f,g)] → [ΣY, F α * F β] called a Hopf invariant. The intervening fibration in the definition of this Hopf invariant arises via the fiberwise join construction.

Approximate Feedback Linearization: A Homotopy Operator Approach

In this paper, we present an approach for finding feedback linearizable systems that approximate a given single-input nonlinear system on a given compact region of the state space. First, we show that if the system is close to being involutive, then it is also close to being linearizable. Rather than working directly with the characteristic distribution of the system, we work with characteristic one-forms, i.e., with the one-forms annihilating the characteristic distribution. We show that homotopy operators can be used to decompose a given characteristic one-form into an exact and an antiexact part. The exact part is used to define a change of coordinates to a normal form that looks like a linearizable part plus nonlinear perturbation terms. The nonlinear terms in this normal form depend continuously on the antiexact part, and they vanish whenever the antiexact part does. Thus, the antiexact part of a given characteristic one-form is a measure of nonlinearizability of the system. If the nonlinear terms are small, by neglecting them we obtain a linearizable system approximating the original system. One can design control for the original system by designing it for the approximating linearizable system and applying it to the original one. We apply this approach for design of locally stabilizing feedback laws for nonlinear systems that are close to being linearizable.

Homotopy operations and the obstructions to being anH-space

We describe an obstruction theory for a given topological space X to be an H-space, in terms of higher homotopy operations, and show how this theory can be used to calculate such operations in certain cases.

Q-deformed Chern characters for quantum groups SUq(N)

In this paper, we introduce an N×N matrix εab̄ in the quantum groups SUq(N) to transform the conjugate representation into the standard form so that we are able to compute the explicit forms of the important quantities in the bicovariant differential calculus on SUq(N), such as the q-deformed structure constant CKIJ and the q-deformed transposition operator Λ. From the q-gauge covariant condition we study the generalized q-deformed Killing form and construct the mth q-deformed Chern class Pm for the quantum groups SUq(N). In terms of the q-deformed homotopy operator we are able to compute the q-deformed Chern–Simons Q2m−1 by the condition dQ2m−1=Pm. Furthermore, the q-deformed cocycle hierarchy, the q-deformed gauge covariant Lagrangian, and the q-deformed Yang–Mills equation are derived.

Relations Among Homotopy Operations for Simplicial Commutative Algebras

Relations Among Homotopy Operations for Simplicial Commutative Algebras

Q-deformed Chern class, Chern-Simons and cocycle hierarchy

In this paper, from the $q$-gauge covariant condition we define the $q$-deformed Killing form and the second $q$-deformed Chern class for the quantum group $SU_{q}(2)$. Developing Zumino's method we introduce a $q$-deformed homotopy operator to compute the $q$-deformed Chern-Simons and the $q$-deformed cocycle hierarchy. Some recursive relations related to the generalized $q$-deformed Killing forms are derived to prove the cocycle hierarchy formulas directly. At last, we construct the $q$-gauge covariant Lagrangian and derive the $q$-deformed Yang-Mills equation. We find that the components of the singlet and the adjoint representation are separated in the $q$-deformed Chern class, $q$-deformed cocycle hierarchy and the $q$-deformed Lagrangian, although they are mixed in the commutative relations of BRST algebra.

Higher Homotopy Operations and The Realizability of Homotopy Groups

We describe an obstruction theory for the realization of a Π-algebra—that is, a graded group G* with a prescribed action of the primary homotopy operations—as the homotopy groups of some space. The obstructions consist of higher homotopy operations, for which we provide an explicit definition in terms of certain sequences of polyhedra. There is a similar theory for realizing morphisms between Π-algebras, and thus, in particular, for distinguishing different realizations of a fixed Π-algebra. As an application we show that, for all primes p, the Π-algebra π*Sr⊗Z/p cannot be realized

Relations among homotopy operations for simplicial commutative algebras

The homotopy groups of a simplicial commutative algebra over the field with two elements support natural operations. Here we write the relations among these operations in an admissible form.

On the integrability of differential forms related to nonequilibrium entropy and irreversible thermodynamics

The Boltzmann equation has been used in the literature to show that the entropy differential consists of a Pfaffian form in the space of conserved and nonconserved variables (moments) and a term related to the energy dissipation due to the irreversible processes in the system. The said Pfaffian form is called the compensation differential. In this paper, the integrability of the compensation differential is examined by means of the theory of differential forms. The integrability conditions turn out to be generalized forms of the Maxwell relations in equilibrium thermodynamics. It is also shown that the generalized form of the Gibbs–Duhem relation can be seen as an equivalent of the integrability conditions. This conclusion is drawn by using the notion of homotopy operator. The Caratheodory principle is also applied to make the connection with the second law of thermodynamics more intimate than the direct but mathematically more abstract approach using the integrability conditions. The meaning of the integrating factor is clarified by using the notion of contact temperature since the integrating factor, a mathematical function, is endowed with a thermodynamically operational meaning when it is identified with the inverse local absolute temperature through the notion of contact temperature, and the compensation differential is thereby made a thermodynamically meaningful equation governing nonequilibrium processes. Through this study it is shown that in irreversible thermodynamics the compensation differential can play the role parallel to the equilibrium Gibbs relation for the entropy in equilibrium thermodynamics and thus can serve as the foundation on which to formulate a theory of irreversible processes.

Topological strings, flat coordinates and gravitational descendants

We discuss physical spectra and correlation functions of topological minimal models coupled to topological gravity. We first study the BRST formalism of these theories and show that their BRST operator Q = Q s + Q v can be brought to Q s by a certain homotopy operator U, UQU −1 = Q s ( Q s and Q v are the N = 2 and diffeomorphism BRST operators, respectively). The reparametrization (anti)-ghost b mixes with the supercharge operator G under this transformation. The existence of this transformation enables us to use matter fields to represent cohomology classes of the operator Q. We explicitly construct gravitational descendants and show that they generate the higher-order KdV flows. We also evaluate genus-zero correlation functions and rederive basic recursion relations of two-dimensional topological gravity.

Homotopy operations in symplectic and orthogonal groups

Homotopy operations in symplectic and orthogonal groups