Search results for "Conjecture"
showing 10 items of 217 documents
Poincaré's role in the Crémieu-Pender controversy over electric convection
1989
Summary In the course of 1901, V. Cremieu published the results of some experiments carried out to test the magnetic effects of electric convection currents. According to Cremieu, his experiments had proved that convection currents had no magnetic effects and consequently they were not equivalent to conduction currents, that is they were not ‘real’ electric currents. These negative results conflicted with those of well-known experiments carried out by other researchers, in particular with Rowland's experiments, and with Maxwell's, Hertz's and Lorentz's theories, which was more shocking. The publication of Cremieu's experiments raised a controversy which involved directly or indirectly some …
Cosmic censorship conjecture in some matching spherical collapsing metrics
2017
A physically plausible Lema{\^{\i}}tre-Tolman-Bondi collapse in the marginally bound case is considered. By "physically plausible" we mean that the corresponding metric is ${\cal C}^1$ matched at the collapsing star surface and further that its {\em intrinsic} energy is, as due, stationary and finite. It is proved for this Lema{\^{\i}}tre-Tolman-Bondi collapse, for some parameter values, that its intrinsic central singularity is globally naked, thus violating the cosmic censorship conjecture with, for each direction, one photon, or perhaps a pencil of photons, leaving the singularity and reaching the null infinity. Our result is discussed in relation to some other cases in the current liter…
-Poincaré supergravities from Lie algebra expansions
2012
Abstract We use the expansion of superalgebras procedure (summarized in the text) to derive Chern–Simons (CS) actions for the ( p , q ) -Poincare supergravities in three-dimensional spacetimes. After deriving the action for the ( p , 0 ) -Poincare supergravity as a CS theory for the expansion osp ( p | 2 ; R ) ( 2 , 1 ) of osp ( p | 2 ; R ) , we find the general ( p , q ) -Poincare superalgebras and their associated D = 3 supergravity actions as CS gauge theories from an expansion of the simple osp ( p + q | 2 , R ) superalgebras, namely osp ( p + q | 2 , R ) ( 2 , 1 , 2 ) .
Nilpotence of orbits under monodromy and the length of Melnikov functions
2021
Abstract Let F ∈ ℂ [ x , y ] be a polynomial, γ ( z ) ∈ π 1 ( F − 1 ( z ) ) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation d F + e ω = 0 of the integrable foliation given by F . We study different invariants: the orbit depth k , the nilpotence class n , the derivative length d associated with the couple ( F , γ ) . These invariants bind the length l of the first nonzero Melnikov function of the deformation d F + e ω along γ . We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this b…
Proof of a Conjecture on Contextuality in Cyclic Systems with Binary Variables
2015
We present a proof for a conjecture previously formulated by Dzhafarov, Kujala, and Larsson (Foundations of Physics, in press, arXiv:1411.2244). The conjecture specifies a measure for the degree of contextuality and a criterion (necessary and sufficient condition) for contextuality in a broad class of quantum systems. This class includes Leggett-Garg, EPR/Bell, and Klyachko-Can-Binicioglu-Shumovsky type systems as special cases. In a system of this class certain physical properties $q_{1},...,q_{n}$ are measured in pairs $(q_{i},q_{j})$; every property enters in precisely two such pairs; and each measurement outcome is a binary random variable. Denoting the measurement outcomes for a proper…
Poincaré Surface of Sections, Mappings
2001
We consider a system with two degrees of freedom, which we describe in four-dimensional phase space. In this (finite) space we define an (oriented) two-dimensional surface. If we then consider the trajectory in phase space, we are interested primarily in its piercing points through this surface. This piercing can occur repeatedly in the same direction. If the motion of the trajectory is determined by the Hamiltonian equations, then the n + 1-th piercing point depends only on the nth. The Hamiltonian thus induces a mapping n → n + 1 in the “Poincare surface of section” (PSS). The mapping transforms points of the PSS into other (or the same) points of the PSS. In the following we shall limit …
Some remarks on Geometric simple connectivity in dimension Four. Part A
2007
The present paper contains some complements and comments to the longer article Geometric simple connectivity in smooth four dimensional differential Topology, Part A. Its aim is to be a useful companion when reading that article,and also to help in understand how it fits into the first author’s programforthe Poincar´e conjecture.
Characterization of Orlicz–Sobolev space
2007
We give a new characterization of the Orlicz–Sobolev space W1,Ψ(Rn) in terms of a pointwise inequality connected to the Young function Ψ. We also study different Poincaré inequalities in the metric measure space.
The Lie algebra of polynomial vector fields and the Jacobian conjecture
1998
The Jacobian conjecture for polynomial maps ϕ:Kn→Kn is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra\(\mathbb{D}\) of formal vector fields inn variables. To be precise, let\(\mathbb{D}_0 \) be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of\(\mathbb{D}_0 \),Hλ the root spaces corresponding to linear forms λ onH and\(A = \oplus _{\lambda \in {\rm H}^ * } H_\lambda \). Then every polynomial map ϕ:Kn→Kn with invertible Jacobian matrix is an automorphism if and only if every automorphism Φ of\(\mathbb{D}\) with Φ(A)\( \subseteq A\) satisfies Φ(A)=A.
More limit cycles than expected in Liénard equations
2007
The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x"+ f(x)x' + x = 0 where f is a polynomial. We prove that for a well-chosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n ≥ 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classic…