3,632 research outputs found
P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches
While the P vs NP problem is mainly approached form the point of view of
discrete mathematics, this paper proposes reformulations into the field of
abstract algebra, geometry, fourier analysis and of continuous global
optimization - which advanced tools might bring new perspectives and approaches
for this question. The first one is equivalence of satisfaction of 3-SAT
problem with the question of reaching zero of a nonnegative degree 4
multivariate polynomial (sum of squares), what could be tested from the
perspective of algebra by using discriminant. It could be also approached as a
continuous global optimization problem inside , for example in
physical realizations like adiabatic quantum computers. However, the number of
local minima usually grows exponentially. Reducing to degree 2 polynomial plus
constraints of being in , we get geometric formulations as the
question if plane or sphere intersects with . There will be also
presented some non-standard perspectives for the Subset-Sum, like through
convergence of a series, or zeroing of fourier-type integral for some natural . The last discussed
approach is using anti-commuting Grassmann numbers , making nonzero only if has a Hamilton cycle. Hence,
the PNP assumption implies exponential growth of matrix representation of
Grassmann numbers. There will be also discussed a looking promising
algebraic/geometric approach to the graph isomorphism problem -- tested to
successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure
Calculating the energy spectra of magnetic molecules: application of real- and spin-space symmetries
The determination of the energy spectra of small spin systems as for instance
given by magnetic molecules is a demanding numerical problem. In this work we
review numerical approaches to diagonalize the Heisenberg Hamiltonian that
employ symmetries; in particular we focus on the spin-rotational symmetry SU(2)
in combination with point-group symmetries. With these methods one is able to
block-diagonalize the Hamiltonian and thus to treat spin systems of
unprecedented size. In addition it provides a spectroscopic labeling by
irreducible representations that is helpful when interpreting transitions
induced by Electron Paramagnetic Resonance (EPR), Nuclear Magnetic Resonance
(NMR) or Inelastic Neutron Scattering (INS). It is our aim to provide the
reader with detailed knowledge on how to set up such a diagonalization scheme.Comment: 29 pages, many figure
Computations in formal symplectic geometry and characteristic classes of moduli spaces
We make explicit computations in the formal symplectic geometry of Kontsevich
and determine the Euler characteristics of the three cases, namely commutative,
Lie and associative ones, up to certain weights.From these, we obtain some
non-triviality results in each case. In particular, we determine the integral
Euler characteristics of the outer automorphism groups Out F_n of free groups
for all n <= 10 and prove the existence of plenty of rational cohomology
classes of odd degrees. We also clarify the relationship of the commutative
graph homology with finite type invariants of homology 3-spheres as well as the
leaf cohomology classes for transversely symplectic foliations. Furthermore we
prove the existence of several new non-trivalent graph homology classes of odd
degrees. Based on these computations, we propose a few conjectures and problems
on the graph homology and the characteristic classes of the moduli spaces of
graphs as well as curves.Comment: 33 pages, final version, to appear in Quantum Topolog
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