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We linearize and solve the Van der Pol equation (with additional nonlinear terms) by the application of a generalized form of Cole-Hopf transformation. We classify also Lienard equations with low order polynomial coefficients which can be linearized by this transformation.
In the sub-Riemannian manifolds, on the one hand, following Baudoin-Garofalo \cite{BaudoinGarofalo}, the upper bound for heat kernels associated to a class of locally subelliptic operators are given under the generalized curvature-dimension inequality with a negative curvature parameter; on the other hand, the argument combining Grigor'yan's integrated maximum principle with Wang's dimension-free Harnack inequality is also shown to derive the upper bound for the heat kernel under the generalized curvature inequality.
Building on the development of [MOR13], bifurcation of unstable modes that emerge from continuous spectra in a class of infinite-dimensional noncanonical Hamiltonian systems is investigated. Of main interest is a bifurcation termed the continuum Hamiltonian Hopf (CHH) bifurcation, which is an infinite-dimensional analog of the usual Hamiltonian Hopf (HH) bifurcation. Necessary notions pertaining to spectra, structural stability, signature of the continuous spectra, and normal forms are described. The theory developed is applicable to a wide class of 2+1 noncanonical Hamiltonian matter models, but the specific example of the Vlasov-Poisson system linearized about homogeneous (spatially independent) equilibria is treated in detail. For this example, structural (in)stability is established in an appropriate functional analytic setting, and two kinds of bifurcations are considered, one at infinite and one at finite wavenumber. After defining and describing the notion of dynamical accessibility, Kreĭn-like theorems regarding signature and bifurcation are proven. In addition, a canonical Hamiltonian example, composed of a negative energy oscillator coupled to a heat bath, is treated and our development is compared to pervious work in this context. A careful counting of eigenvalues, with and without symmetry, is undertaken, leading to the definition of a degenerate continuum steady state (CSS) bifurcation. It is described how the CHH and CSS bifurcations can be viewed as linear normal forms associated with the nonlinear single-wave model described in [BAL13], which is a companion to the present work and that of [MOR13].
This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is shown that the Riemann hypothesis holds for the Dirichlet beta function $L_{-4}(s)$ if and only if it holds for $ζ(s)$- a particular case of the Generalized Riemann Hypothesis.
This chapter present a fats boundary integral equation method for numerical computing of uniform potential flow past multiple aerofoils. The presented fast multipole-based iterative solution procedure requires only $O(nm\ln n)$ operations where $m$ is the number of aerofoils and $n$ is the number of nodes in the discretization of each aerofoil's boundary. We demonstrate the performance of our methods on several numerical examples.
We study Hamiltonian reductions of the free geodesic motion on a non-compact simple Lie group using as reduction group the direct product of a maximal compact subgroup and the fixed point subgroup of an arbitrary involution commuting with the Cartan involution. In general, we describe the reduced system that arises upon restriction to a dense open submanifold and interpret it as a spin Sutherland system. This dense open part yields the full reduced system in important special examples without spin degrees of freedom, which include the BC(n) Sutherland system built on 3 arbitrary couplings for m<n positively charged and (n-m) negatively charged particles moving on the half-line.
The aim of this review is to outline a full route from the fundamental principles of algebraic quantum field theory on curved spacetime in its present-day form to explicit phenomenological applications which allow for comparison with experimental data. We give a brief account on the quantization of the free scalar field and its Wick powers in terms of an algebra of functionals on configuration space. Afterwards we demonstrate that there exist states on this algebra in which the energy momentum tensor is qualitatively and quantitatively of the perfect fluid form assumed in the standard model of cosmology up to small corrections. We indicate the potential relevance of one of these corrections for the actively debated phenomenon of Dark Radiation.
In this paper, we consider the Cauchy problem of the incompressible liquid crystal equations in $n$ dimensions. We prove the local well-posedness of mild solutions to the liquid crystal equations with $L^\infty$ initial data, in particular, the initial energy may be infinite. We prove that the solutions are smooth with respect to the space variables away from the initial time. Based on this regularity estimate, we employ the blow up argument and Liouville type theorems to establish vorticity direction type blow up criterions for the type I mild solutions established in the present paper.
Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered.
In this article, the new exact travelling wave solutions of the time-and space-fractional KdV-Burgers equation has been found. For this the fractional complex transformation have been implemented to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations, in the sense of the Jumarie's modified Riemann-Liouville derivative. Afterwards, the improved (G'/G)-expansion method can be implemented to celebrate the soliton solutions of KdV-Burger's equation of fractional order.
In many cases a massive nonlinear scalar field can lead to accelerated expansion in cosmological models. This paper contains mathematical results on this subject for flat Robertson-Walker space-time. Global existence to the coupled Einstein-massive scalar field system which rules the dynamics of a kind of pure matter in the presence of a massive scalar field and cosmological constant is proved, under the assumption that the scalar field ? is a non-decreasing function, in Robertson-Walker1 space-time; asymptotic behaviour are investigated in the case of a cosmological constant bounded from below by a strictly negative constant depending only on the massive scalar field.
The importance of the fusion relation of loops was recognized in the context of spin structures on the loop space by Stolz and Teichner and further developed by Waldorf. On a spin manifold M the equivalence classes of `fusive' spin structures on the loop space LM, incorporating the fusion property, strong regularity and reparameterization-invariance, are shown to be in 1-1 correspondence with equivalence classes of string structures on M. The identification is through the affine space of `string' cohomology classes considered by Redden.
In this thesis we study field theories written on a particular model of noncommutative spacetime, the Groenewold-Moyal (GM) plane. We start with briefly reviewing the novel features of field theories on GM plane e.g. the $\ast$-product, restoration of Poincaré-Hopf symmetry and twisted commutation relations. We then discuss our work on renormalization of field theories on GM plane. We show that any generic noncommutative theory involving pure matter fields with polynomial interactions, is a renormalizable theory if the analogous commutative theory is renormalizable. We further show that all such noncommutative theories will have same fixed points and $β$-functions for the couplings, as that of the analogous commutative theory. The unique feature of these field theories is the twisted statistics obeyed by the particles. Motivated by it, we look at the possibility of twisted statistics by deforming internal symmetries instead of spacetime symmetries. We construct two different twisted theories which can be viewed as internal symmetry analogue of the GM plane and dipole field theories which arise in the low energy limit of certain string configurations. We further study their various properties like the issue of causality and the scattering formalism. Having studied the mathematical properties of noncommutative and twisted internal symmetries we move on to discuss their potential phenomenological signatures. We first discuss the noncommutative thermal correlation functions and show that because of the twisted statistics, all correlation functions except two-point function get modified. Finally we discuss the modifications in Hanbury-Brown Twiss (HBT) correlation functions due to twisted statistics on GM plane and the potential of observing signatures of noncommutativity by doing a HBT correlation experiment with Ultra High Energy Cosmic Rays (UHECRs).
The modified Dirac equations for the massive particles with the replacement of the physical mass $m$ with the help of the relation $m\rightarrow m_1 + γ_5 m_2$ are investigated. It is shown that for a free fermion theory with a $γ_5$ mass term, the finiteness of the mass spectrum at the value $ m_{max}= {m_1}^2/2m_2$ takes place. In this case the region of the unbroken $\cal PT$-symmetry may be expressed by means of the simple restriction of the physical mass $m\leq m_{max}$. Furthermore, we have that the areas of unbroken $\cal PT$-symmetry $m_1\geq m_2\geq 0$, which guarantees the reality values of the physical mass $m$, consists of three different parametric subregions: i) $0\leq m_2 < m_1/\sqrt{2}$, \,\,ii) $m_2=m_1/\sqrt{2}=m_{max},$ \,\,(iii)$m_1/\sqrt{2}< m_2 \leq m_1$. It is vary important, that only the first subregion (i) defined mass values $m_1,m_2,$ which correspond to the description of traditional particles in the modified models, because this area contain the possibility transform the modified model to the ordinary Dirac theory. The second condition (ii) is defined the "maximon" - the particle with maximal mass $m=m_{max}$. In the case (iii) we have to do with the unusual or "exotic" particles for description of which Hamiltonians and equations of motion have no a Hermitian limit. The formulated criterions may be used as a major test in the process of the division of considered models into ordinary and "exotic fermion theories".
We consider some modifications of the two dimensional canonical commutation relations, leading to {\em non commutative bosons} and we show how biorthogonal bases of the Hilbert space of the system can be obtained out of them. Our construction extends those recently introduced by one of us (FB), modifying the canonical anticommutation relations. We also briefly discuss how bicoherent states, producing a resolution of the identity, can be defined.
If the dimension $d$ is even, the resonances of the Schrödinger operator $-Δ+V$ on ${\mathbb R}^d$ with $V$ bounded and compactly supported are points on $Λ$, the logarithmic cover of ${\mathbb C} \setminus \{0\}$. We show that for fixed sign potentials $V$ and for nonzero integers $m$, the resonance counting function for the $m$th sheet of $Λ$ has maximal order of growth.
We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.
We extend the construction of a spectral triple for k-Minkowski space, previously given for the two-dimensional case, to the general n-dimensional case. This takes into account the modular group naturally arising from the symmetries of the geometry, and requires the use of notions that have been recently developed in the frameworks of twisted and modular spectral triples. First we compute the spectral dimension, using an appropriate weight, and show that in general it coincides with the classical one. We also study the classical limit and the analytic continuation of the associated zeta function. Then we compare this notion of dimension with the one coming from homology. To this end, we compute the twisted Hochschild dimension of the universal enveloping algebra underlying k-Minkowski space. The result is that twisting avoids the dimension drop, similarly to other examples coming from quantum groups. In particular, the simplest such twist is given by the inverse of the modular group mentioned above.