2,015 research outputs found
Morse theory, closed geodesics, and the homology of free loop spaces
This is a survey paper on Morse theory and the existence problem for closed
geodesics. The free loop space plays a central role, since closed geodesics are
critical points of the energy functional. As such, they can be analyzed through
variational methods. The topics that we discuss include: Riemannian background,
the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of
subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index
formulas, homological computations using Morse theory, - vs.
-symmetries, Katok's examples and Finsler metrics, relations to
symplectic geometry, and a guide to the literature.
The Appendix written by Umberto Hryniewicz gives an account of the problem of
the existence of infinitely many closed geodesics on the -sphere.Comment: 45 pages, 5 figures. Appendix by Umberto Hryniewic
MARKETING COMMUNICATION IN THE BAKERY INDUSTRY IN ROMANIA IN THE CONTEXT OF CURRENT CRISIS
This year, the crisis still persists in all areas of business models in the market and its potential for growth, customer behaviour will change clearly. In this respect, the importance of efficiency will be materialized in the results and objectives. Cutting marketing budgets is not necessarily a solution for companies in this period. Romanian consumers are changing, and if it changes, and then business/SMEs must adapt and therefore communication must adapt. The purpose of this work is to highlight the importance to be given to a fair and consistent communication in time of crisis. This crisis could be an excellent opportunity for many companies who will be held successfully in the mind of the consumer. Therefore, any company must communicate more and better with its customers in time of crisis than in times of economic growth.marketing communication, market segments, crisis, budget marketing, the bakery industry
Fredholm theory and transversality for the parametrized and for the -invariant symplectic action
We study the parametrized Hamiltonian action functional for
finite-dimensional families of Hamiltonians. We show that the linearized
operator for the -gradient lines is Fredholm and surjective, for a generic
choice of Hamiltonian and almost complex structure. We also establish the
Fredholm property and transversality for generic -invariant families of
Hamiltonians and almost complex structures, parametrized by odd-dimensional
spheres. This is a foundational result used to define -equivariant Floer
homology. As an intermediate result of independent interest, we generalize
Aronszajn's unique continuation theorem to a class of elliptic
integro-differential inequalities of order two.Comment: 63 page
Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces
We define Floer homology for a time-independent, or autonomous Hamiltonian on
a symplectic manifold with contact type boundary, under the assumption that its
1-periodic orbits are transversally nondegenerate. Our construction is based on
Morse-Bott techniques for Floer trajectories. Our main motivation is to
understand the relationship between linearized contact homology of a fillable
contact manifold and symplectic homology of its filling.Comment: Final version, 92 pages, 7 figures, index of notations. Remark 4.9 in
Version 1 is wrong. To correct it we modified the weights for the Sobolev
spaces along the gradient trajectories (Figure 3). Proposition 4.8 in Version
1 is now split into Propositions 4.8 and 4.9. This version contains full
details for the second derivative estimates in Lemma 4.2
Symplectic homology and the Eilenberg-Steenrod axioms
We give a definition of symplectic homology for pairs of filled Liouville
cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod
axioms except for the dimension axiom. The resulting long exact sequence of a
pair generalizes various earlier long exact sequences such as the handle
attaching sequence, the Legendrian duality sequence, and the exact sequence
relating symplectic homology and Rabinowitz Floer homology. New consequences of
this framework include a Mayer-Vietoris exact sequence for symplectic homology,
invariance of Rabinowitz Floer homology under subcritical handle attachment,
and a new product on Rabinowitz Floer homology unifying the pair-of-pants
product on symplectic homology with a secondary coproduct on positive
symplectic homology.
In the appendix, joint with Peter Albers, we discuss obstructions to the
existence of certain Liouville cobordisms.Comment: v3: corrected Lemma 7.11. Various other minor modifications and
reformatting. Final version to be published in Algebraic and Geometric
Topolog
Posets of annular non-crossing partitions of types B and D
We study the set \sncb (p,q) of annular non-crossing permutations of type
B, and we introduce a corresponding set \ncb (p,q) of annular non-crossing
partitions of type B, where and are two positive integers. We prove
that the natural bijection between \sncb (p,q) and \ncb (p,q) is a poset
isomorphism, where the partial order on \sncb (p,q) is induced from the
hyperoctahedral group , while \ncb (p,q) is partially ordered by
reverse refinement. In the case when , we prove that \ncb (p,1) is a
lattice with respect to reverse refinement order.
We point out that an analogous development can be pursued in type D, where
one gets a canonical isomorphism between \sncd (p,q) and \ncd (p,q). For
, the poset \ncd (p,1) coincides with a poset ``''
constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by
the results of that paper.Comment: Revised version (shortened Introduction, corrected typos), 31 pages,
4 figures, to appear in Discrete Mathematic
The space of paths in complex projective space with real boundary conditions
We compute the integral homology of the space of paths in
with endpoints in , and its algebra structure with
respect to the Pontryagin-Chas-Sullivan product with
-coefficients.Comment: 33 pages, 6 figure
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