984,373 research outputs found
Some Properties of and Open Problems on Hessian Nilpotent Polynomials
In the recent progress [BE1], [M], [Z1] and [Z2], the well-known Jacobian
conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent)
polynomials (the polynomials whose Hessian matrix are nilpotent) and their
(deformed) inversion pairs. In this paper, we prove several results on HN
polynomials, their (deformed) inversion pairs as well as the associated
symmetric polynomial or formal maps. We also propose some open problems for
further study of these objects.Comment: Latex, 33 pages. Some misprints have been corrected; English and also
terminology for some newly defined notion have been improved. To appear in
Annales Polonici Mathematic
Deformations of trianguline B-pairs
The aim of this article is to study deformation theory of trianguline B-pairs
for any p-adic field. For benign B-pairs, a special good class of trianguline
B-pairs, we prove a main theorem concerning tangent spaces of these deformation
spaces. These are generalizations of Bellaiche-Chenevier's and Chenevier's
works in the case of K=Q_p, where they used (phi,Gamma)-modules over Robba ring
instead of using B-pairs. The main theorem, the author hopes, will play crucial
roles in some problems of Zariski density of modular points or of crystalline
points in deformation spaces of global or local p-adic Galois representations.Comment: 30page
Stable marriage and roommates problems with restricted edges: complexity and approximability
In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs.
Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs.
Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs
Singular dual pairs
We generalize the notions of dual pair and polarity introduced by S. Lie and
A. Weinstein in order to accommodate very relevant situations where the
application of these ideas is desirable. The new notion of polarity is designed
to deal with the loss of smoothness caused by the presence of singularities
that are encountered in many problems. We study in detail the relation between
the newly introduced dual pairs, the quantum notion of Howe pair, and the
symplectic leaf correspondence of Poisson manifolds in duality. The dual pairs
arising in the context of symmetric Poisson manifolds are treated with special
attention. We show that in this case and under very reasonable hypotheses we
obtain a particularly well behaved kind of dual pairs that we call von Neumann
pairs. Some of the ideas that we present in this paper shed some light on the
so called optimal momentum maps.Comment: 38 pages, Theorem 7.6 has been upgrade
Schulze and Ranked-Pairs Voting are Fixed-Parameter Tractable to Bribe, Manipulate, and Control
Schulze and ranked-pairs elections have received much attention recently, and
the former has quickly become a quite widely used election system. For many
cases these systems have been proven resistant to bribery, control, or
manipulation, with ranked pairs being particularly praised for being NP-hard
for all three of those. Nonetheless, the present paper shows that with respect
to the number of candidates, Schulze and ranked-pairs elections are
fixed-parameter tractable to bribe, control, and manipulate: we obtain uniform,
polynomial-time algorithms whose degree does not depend on the number of
candidates. We also provide such algorithms for some weighted variants of these
problems
Separation of variables for soliton equations via their binary constrained flows
Binary constrained flows of soliton equations admitting Lax
matrices have 2N degrees of freedom, which is twice as many as degrees of
freedom in the case of mono-constrained flows. For their separation of
variables only N pairs of canonical separated variables can be introduced via
their Lax matrices by using the normal method. A new method to introduce the
other N pairs of canonical separated variables and additional separated
equations is proposed. The Jacobi inversion problems for binary constrained
flows are established. Finally, the factorization of soliton equations by two
commuting binary constrained flows and the separability of binary constrained
flows enable us to construct the Jacobi inversion problems for some soliton
hierarchies.Comment: 39 pages, Amste
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