358 research outputs found
A topologically induced 2-in/2-out operation on loop cohomology
We apply the Transfer Algorithm introduced in arXiv:1106.5090 to transfer an
A_\infty-algebra structure that cannot be computed using the classical Basic
Perturbation Lemma. We construct a space X whose (base pointed) loop cohomology
H = H^*(\Omega X; Z_2) comes equipped with a nontrivial operation \omega : H x
H --> H x H.Comment: 9 pages. The title has changed in version 9, and some indexing issues
in the transfer algorithm have been correcte
The Deformation Complex For DG Hopf Algebras
Let H be a differential graded Hopf algebra over a field k. This paper gives
an explicit construction of a triple cochain complex that defines the
Hochschild-Cartier cohomology of H. A certain truncation of this complex is the
appropriate setting for deforming H as an H(q)-structure. The direct limit of
all such truncations is the appropriate setting for deforming H as a strongly
homotopy associative structure. Sign complications are systematically
controlled. The connection between rational perturbation theory and the
deformation theory of certain free commutative differential graded algebras is
clarified.Comment: 21 pages, 4 figure
A Diagonal on the Associahedra
Let C_*(K) denote the cellular chains on the Stasheff associahedra. We
construct an explicit combinatorial diagonal \Delta : C_*(K) --> C_*(K) \otimes
C_*(K); consequently, we obtain an explicit diagonal on the A_\infty-operad. We
apply the diagonal \Delta to define the tensor product of A_\infty-(co)algebras
in maximal generality
Diagonals on the Permutahedra, Multiplihedra and Associahedra
We construct an explicit diagonal \Delta_P on the permutahedra P. Related
diagonals on the multiplihedra J and the associahedra K are induced by Tonks'
projection P --> K and its factorization through J. We introduce the notion of
a permutahedral set Z and lift \Delta_P to a diagonal on Z. We show that the
double cobar construction \Omega^2(C_*(X)) is a permutahedral set; consequently
\Delta_P lifts to a diagonal on \Omega^2(C_*(X)). Finally, we apply the
diagonal on K to define the tensor product of A_\infty-(co)algebras in maximal
generality.Comment: 45 pages, 13 figures. This (final) version is significantly more
detailed than the previou
An -coalgebra Structure on a Closed Compact Surface
Let be an -gon with There is a formal combinatorial
-coalgebra structure on cellular chains with non-vanishing
higher order structure when . If is a closed compact surface of
genus and is a polygonal decomposition, the quotient map
projects the formal -coalgebra structure on
to a quotient structure on , which persists to homology
, whose operations are determined by
the quotient map , and whose higher order structure is non-trivial if and
only if is orientable or unorientable with . But whether or not
the -coalgebra structure on homology observed here is topologically
invariant is an open question.Comment: 13 pages, 6 figure
Obstructions to Deformations of DG Modules
Let k be a field and n > 0. There exists a DG k-module (V,d) and various
approximations d + t d_1 + t^2 d_2 + ... + t^n d_n to a differential on V[[t]],
one of which is a non-trivial deformation, another is obstructed, and another
is unobstructed at order n. The analogous problem in the category of k-algebras
in characteristic zero remains a long-standing open question.Comment: 9 page
Minimal Paths on Some Simple Surfaces with Singularities
Given two points on a soup can or conical cup with lid, we find and classify
all paths of minimal length connecting them. When the number of minimal paths
is finite, there are at most four on a can and three on a cup. At worst,
minimal paths are piece-wise smooth with three components, each of which is a
classical geodesic. Minimal paths are geodesics in the sense of Banchoff.Comment: 12 pages, 7 figures (v6:typo fixes in Lemma 1) (v5:revised the Ascoli
argument in the first paragraph) (v4:Editorial revisions pursuant to
referee's report) (v3:Revision of Title from "Minimal Paths on Unicone and
Bicylinder Boundaries"; Other editorial revisions
Periodic Orbits of Billiards on an Equilateral Triangle
We give a complete solution of the following problem: Find, classify and
count the (classes of) periodic orbits on an equilateral triangle. We prove
that Fagnano's period 3 orbit is the only periodic orbit with odd period. A
periodic orbit is either prime or some d-fold iterate thereof. We count prime
and iterate periodic orbits of period via a bijection with a certain
partition of , then count only prime orbits using the prime factorization of
.Comment: Version 7: Minor edits made, some figures redrawn, Table 1 moved from
appendix to body of text
A-infinity Bialgebras of Type (m,n)
An A-infinity bialgebra of type (m,n) is a Hopf algebra H equipped with a
"compatible" operation \omega : H^{\otimes m} \to H^{\otimes n} of positive
degree. We determine the structure relations for A-infinity bialgebras of type
(m,n) and construct a purely algebraic example for each m \geq 2 and m+n \geq
4.Comment: 10 pages, 4 figure
Generalizations of the Brachistochrone Problem
Consider a frictionless surface S in a gravitational field that need not be
uniform. Given two points A and B on S, what curve is traced out by a particle
that starts at A and reaches B in the shortest time? This paper considers this
problem on simple surfaces such as surfaces of revolution and solves the
problem two ways: First, we use conservation of mechanical energy and the
Euler-Lagange equation; second, we use geometrical optics and the eikonal
equation. We conclude with a discussion of the relativistic effects at
relativistic velocities.Comment: 12 pages, 6 figure
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