Applying the averaging theory of first, second and third order to one class generalized polynomial Liénard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit.
The space of shapes of quadrilaterals can be identified with $\mathbb{CP}^{2}$. We deal with the subset of $\mathbb{CP}^{2}$ corresponding to convex quadrilaterals and the subset which corresponds to simple (that is, without self-intersections) quadrilaterals. We provide a complete description of the topological closures in $\mathbb{CP}^{2}$ of both spaces. Although the interior of each space is homeomorphic to a disjoint union $\mathbb{R}^{4}\sqcup \mathbb{R}^{4}$, their closures are topologically different. In particular, the boundary of the space corresponding to convex quadrilaterals is homeomorphic to a pair of three-dimensional spheres glued along a Möbius strip while the boundary of the space corresponding to simple quadrilaterals is more complicated.
Abstract. Given real numbers 4π > θ 0 ≥ θ 1 ≥ θ 2 ≥ θ 3 > 0 so that 3 j=0 θ j = 4π, we provide a detailed description of the space of flat metrics on the 2-sphere with 4 conical points of cone angles θ 0 , θ 1 , θ 2 , θ 3 , endowed with a geometric structure arising from the area function.
We compute the mass spectrum of the fermionic sector of the Dirac-Kähler extension of the SM (DK-SM) by showing that there exists a Bogoliubov transformation that transforms the DK-SM into a flavor U(4) extension of the SM (SM-4) with a particular choice of masses and mixing textures. Mass relations of the model allow determination of masses of the 4th generation. Tree level prediction for the mass of the 4 th charged lepton is 370 GeV. The model selects the normal hierarchy for neutrino masses and reproduces naturally the near tri-bimaximal and quark mixing textures. The electron neutrino and the 4th neutrino masses are related via a seesaw like mechanism.
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