Showing papers on "Distribution (differential geometry) published in 2007"
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02 Jul 2007TL;DR: Through manifold analysis of face pictures, a novel age estimation framework is developed to find a sufficient embedding space and model the low-dimensional manifold data with a multiple linear regression function.
Abstract: Extensive recent studies on human faces reveal significant potential applications of automatic age estimation via face image analysis. Due to the temporal features of age progression, aging face images display sequential pattern of low-dimensional distribution. Through manifold analysis of face pictures, we developed a novel age estimation framework. The manifold learning methods are applied to find a sufficient embedding space and model the low-dimensional manifold data with a multiple linear regression function. Experimental results on a large size age database demonstrate the effectiveness of the framework. To our best knowledge, this is the first work involving the manifold ways of age estimation.
183 citations
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TL;DR: In this article, the authors consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure and show that it fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measures.
Abstract: We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also establish a more general version of this result for general foliations with compact leaves.
37 citations
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TL;DR: In this paper, the authors considered a special odd-dimensional submanifold of the cotangent bundle associated with any rank 2 distribution, which is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D.
Abstract: In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in $\mathbb R^5$. We solve the analogous problem for germs of generic rank 2 distributions in ${\mathbb R}^n$ for n>5. We use a completely different approach based on the symplectification of the problem. The main idea is to consider a special odd-dimensional submanifold $W_D$ of the cotangent bundle associated with any rank 2 distribution D. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. Using the classical theory of curves in projective spaces, we construct the canonical frame of the distribution D on a certain (2n-1)-dimensional fiber bundle over $W_D$ with the structure group of all Mobius transformations, preserving 0.
27 citations
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TL;DR: In this article, the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, and an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution ℜ°.
Abstract: We consider nonholonomic systems with linear, time-independent constraints subject to positional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator distribution ℜ°, the fibers of which are the annihilators of the set of all values taken by the reaction forces on the fibers of the constraint distribution. We show that this distribution, which can be effectively computed in specific cases, plays a central role in the study of first integrals linear in the velocities of this class of nonholonomic systems. In particular we prove that, if the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, then an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution ℜ°. Since the fibers of ℜ° contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved momenta than what was known so far. Some examples are given.
23 citations
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TL;DR: In this paper, exact solutions of the five-dimensional vacuum cosmological field equations based on Lyra geometry are obtained and it is shown that neither dust distribution nor perfect fluid distributions survive for the model.
Abstract: In this paper exact solutions of the five-dimensional vacuum cosmological field equations based on Lyra geometry are obtained. Further it is shown that neither dust distribution nor perfect fluid distributions survive for the model. Some properties of the vacuum model are also discussed.
18 citations
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TL;DR: A logic-based systematic method of designing manifold systems to achieve flowrate uniformity among the channels that interconnect a distribution manifold and a collection manifold has been developed, implemented, and illustrated by case studies.
Abstract: A logic-based systematic method of designing manifold systems to achieve flowrate uniformity among the channels that interconnect a distribution manifold and a collection manifold has been developed, implemented, and illustrated by case studies. The method is based on tailoring the flow resistance of the individual channels to achieve equal pressure drops for all the channels. The tailoring of the flow resistance is accomplished by the use of gate-valve-like obstructions. The adjustment of the valve-like obstructions is determined here by means of numerical simulations. Although the method is iterative, it may converge in one cycle of the iterations. Progress toward the goal of per channel uniformity can be accelerated by tuning a multiplicative constant. The only departure of the method from being fully automatic is the selection of the aforementioned multiplicative constant. The method is described in detail in a step-by-step manner. These steps are illustrated both generically and specifically for four case studies. As an example, in one of the case studies, an original flow imbalance of over 100% in an untailored manifold system was reduced to a flow imbalance of less than 10% in one cycle of the method.
18 citations
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TL;DR: In this article, the first integrals are Kirillov's operators for a representation of the Virasoro algebra and the second integrals form a Lie-Poisson bracket and the first subspace is a bracket-generating distribution of complex dimension two.
15 citations
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TL;DR: In this paper, the authors characterize the C^1 smooth families of maps where the topological dynamics does not change (the smooth deformations) as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space.
Abstract: In the space of C^k piecewise expanding unimodal maps, k>=1, we characterize the C^1 smooth families of maps where the topological dynamics does not change (the "smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of C^{k-1+Lip} deformations tangent to every given C^k horizontal direction, for k>=2.
15 citations
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TL;DR: In this paper, the authors regard a contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution as a bi-Legendrian manifold and study its canonical bi-legendrian structure.
Abstract: We regard a contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(\kappa,\mu)$-spaces in terms of a canonical connection which can be naturally defined on them.
13 citations
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TL;DR: In this article, the authors analyzed the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold and showed the existence of normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively.
Abstract: This paper analyses the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold. It shows the existence of normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$. The paper concludes with some holonomy constructions for free $n$-distributions for $n>3$.
12 citations
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TL;DR: In this article, the authors consider metric cones with reducible holonomy over pseudo-Riemannian manifolds and show that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra.
Abstract: By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-K\"ahlerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.
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TL;DR: In this article, the authors studied Ricci flat 4-metric of any signature under the assumption that they allow a Lie algebra of Killing fields with 2-dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable.
Abstract: We study Ricci flat 4-metrics of any signature under the assumption that they allow a Lie algebra of Killing fields with 2-dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable. It turns out that locally there is a unique (up to a sign) metric which satisfies the conditions. This metric is of signature (++−−) and, moreover, homogeneous possessing a 6-dimensional symmetry algebra.
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TL;DR: In this paper, the authors analyzed the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold and showed that certain holonomy reductions of the associated normal Tractor connections imply preferred connections with special properties, along with Riemannian structures on the manifold.
Abstract: This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, however, lie in the free 3-distributions. In these cases, there are normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$.
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12 Apr 2007
TL;DR: A novel face superresolution method using Locality Preserving Projections (LPP), which gives an advantage over other manifold learning methods in that it has well-defined linear projections which allow us to formulate well- defined mappings between highdimensional data and low-dimensional data.
Abstract: In this paper, we propose a novel method for performing robust super-resolution of face images by solving the practical
problems of the traditional manifold analysis. Face super-resolution is to recover a high-resolution face image from a
given low-resolution face image by modeling the face image space in view of multiple resolutions. In particular, face
super-resolution is useful to enhance face images captured from surveillance footage. Face super-resolution should be
preceded by analyzing the characteristics of the face image distribution. In literature, it has been shown that face images
lie on a nonlinear manifold by various manifold learning algorithms, so if the manifold structure is taken into
consideration for modeling the face image space, the results of face super-resolution can be improved. However, there
are some practical problems which prevent the manifold analysis from being applied to super-resolution. Almost all of
the manifold learning methods cannot generate mapping functions for new test images which are absent from a training
set. Also, there exists another significant problem when applying the manifold analysis to super-resolution; superresolution
seeks to recover a high-dimensional image from a low-dimensional one while manifold learning methods
perform the exact opposite for dimensionality reduction.
To break those limitations of applying the manifold analysis to super-resolution, we propose a novel face superresolution
method using Locality Preserving Projections (LPP). LPP gives an advantage over other manifold learning
methods in that it has well-defined linear projections which allow us to formulate well-defined mappings between highdimensional
data and low-dimensional data. Moreover, we show that LPP coefficients of an unknown high-resolution
image can be inferred from a given low-resolution image using a MAP estimator.
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TL;DR: In this article, a canonical bundle with a canonical frame is constructed, and it is shown that two pairs are equivalent if and only if the corresponding frames are diffeomorphic, which is a special case of the problem of contact equivalence of systems of differential equations.
Abstract: We consider a problem of equivalence of generic pairs $(X,V)$ on a manifold $M$, where $V$ is a distribution of rank $m$ and $X$ is a distribution of rank one. We construct a canonical bundle with a canonical frame. We prove that two pairs are equivalent if and only if the corresponding frames are diffeomorphic.
As a particular case, with $V$ integrable, we provide a new solution to the problem of contact equivalence of systems of $m$ ordinary differential equations: $x^{(k+1)}=F(t,x,x',...,x^{(k)})$, where $k>2$ or $k=2$ and $m>1$.
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TL;DR: In this paper, the existence and uniqueness of a linear connection on (Q,g) subject to some conditions on its torsion and on the covariant derivative of g is proved.
Abstract: Let (Q,g) be the configuration space of a nonholonomic mechanical system, where g is a Riemannian metric on Q. Suppose the horizontal distribution D on Q admits a vertical distribution D¯, that is D¯ is an integrable complementary (not necessarily orthogonal) distribution to D in TQ. We prove the existence and uniqueness of a linear connection on (Q,g) subject to some conditions on its torsion and on the covariant derivative of g. Then we show that the solutions of the Lagrange-d’Alembert equations are the geodesics of ∇ and vice versa. All the local components of the torsion and curvature tensor fields of ∇ with respect to an adapted frame field are determined. Finally, two examples are given to illustrate the theory we develop in the paper.
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08 Mar 2007
TL;DR: In this paper, the authors proposed a manifold structure for a fuel cell stack, which is able to distribute gas between the manifold and inlet of each channel uniformly, and thus cause no difference in voltage between cells, and ensure high quality of the fuel cell.
Abstract: Provided is a manifold structure for a fuel cell stack, which is able to distribute gas between the manifold and inlet of each channel uniformly, and thus cause no difference in voltage between cells, and ensures high quality of a fuel cell. The manifold structure for a fuel cell comprises a distribution port(P) for a uniform distribution of gas between the manifold(M) of a fuel cell stack and the inlet of each channel(C1-C5). The distribution port is formed in such a manner that the distance between the manifold and the inlet of each channel is the same or different. The distribution port is symmetrically formed based on the central axis of the manifold. Otherwise, the distribution port is formed only at one side of both sides of the central axis of the manifold.
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TL;DR: In this paper, it was shown that any 3-Sasakian manifold admits a canonical transversal, projectable quaternionic-Kahler structure and a 3-α-Sakian structure.
Abstract: quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic ge- ometries. In this paper many geometric properties of this class of almost 3-contact metric manifolds are found. In particular, it is proved that the only 3-quasi-Sasakian manifolds of rank 4l+1 are the 3-cosymplectic manifolds and any 3-quasi-Sasakian manifold of maximal rank is necessarily 3-α-Sasakian. Furthermore, the trans- verse geometry of a 3-quasi-Sasakian manifold is studied, proving that any 3-quasi- Sasakian manifold admits a canonical transversal, projectable quaternionic-Kahler structure and a canonical transversal, projectable 3-α-Sasakian structure. The well-known classes of 3-Sasakian and 3-cosymplectic manifolds belong to the wider family of almost 3-contact metric manifolds. Nevertheless, both classes sit also perfectly into the narrower class of 3-quasi-Sasakian manifolds which, as we will see, is a very natural framework for a unified study of the aforementioned geometries. A similar chain of inclusions takes place in the case of a single almost contact metric structure, whereas the class of quasi- Sasakian manifolds encloses both Sasakian and cosymplectic manifolds, but in the setting of 3-structures the interrelations between the triples of tensors produce key additional properties making the choice of the 3-quasi-Sasakian framework still more natural. 3-quasi-Sasakian manifolds were introduced long ago but their first systematic study was carried out by the authors in (5). There, it was proven that in any 3-quasi-Sasakian manifold (M,�α,�α,�α,g) of dimension 4n + 3 the vertical distribution V generated by the three Reeb vector fields is completely integrable determining a canonical totally geodesic and Riemannian foliation. The characteristic vector fields obey the commu- tation relations (� α ,� β ) = c� γ for any even permutation (�,�,) of {1,2,3} and some c ∈ R. Furthermore, it was shown that the ranks of the 1-forms
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TL;DR: In this article, it was shown that any harmonic vector field tangent to the Levi distribution of the foliation by level sets of the defining function φ ( z ) = − K ( z, z ) − 1 / ( n + 1 ) of a strictly pseudoconvex bounded domain Ω ⊂ C n which is smooth up to the boundary must vanish on ∂ Ω.
Abstract: We show that any harmonic (with respect to the Bergman metric) vector field tangent to the Levi distribution of the foliation by level sets of the defining function φ ( z ) = − K ( z , z ) − 1 / ( n + 1 ) of a strictly pseudoconvex bounded domain Ω ⊂ C n which is smooth up to the boundary must vanish on ∂ Ω . If n ≠ 5 and u T is a harmonic vector field with u ∈ C 2 ( Ω ¯ ) then u | ∂ Ω = 0 .
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TL;DR: In this article sufficient and necessary conditions were given for integrabilities of distribution D{ξ} on a nearly Sasakian manifold, and generalized Bejancu's result was given also.
Abstract: Some sufficient and necessary conditions were given for integrabilities of distribution D{ξ} on a nearly Sasakian manifold,and generalized Bejancu's result was given also.
01 Jan 2007
TL;DR: In this paper, it was shown that a symmetric tensor fleld of type (0,m) and a geodesic determine a function which is homogeneous of degree m i 1.
Abstract: This paper reformulates a problem of Sharafutdinov (4) regard- ing the kinetic equation on a Riemannian manifold and extends the new variant from the single-time context to the multi-time context. Section 1 is dedicated to the single time case. It begins by pointing out that a symmetric tensor fleld of type (0;m) and a geodesic determine a function which is homogeneous of degree m i 1. Also, it is proved that the vector fleld H associated to the geodesic spray G t is tangent to the spherical bundle SM. The kinetic equation on J 1 (R;M) associated to a symmetric tensor fleld is deduced. Then the paper presents the connection between the geodesic vector fleld (difierential operator) H and the inner difierenti- ation operator d. Section 2 extends the single-time case to the multi-time case. It is proved that p symmetric d-tensor flelds and a minimal subman- ifold determine a function which is homogeneous of degree m i 1. Also,it points out that the a-ne vector flelds Hfi generate a completely integrable distribution tangent to the spherical bundle. The kinetic PDEs system is written explicitly and the vector flelds Hfi are connected with the inner difierential operator d.
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TL;DR: In this paper, it was shown that 3-quasi-Sasakian manifolds have a rank-based classification and a splitting theorem for these manifolds assuming the integrability of one of the almost product structures.
Abstract: In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.
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TL;DR: In this paper, the authors propose a method to single out connections with the help of a special set of 1-forms by the condition that the 1-form become parallel with respect to this connection.
Abstract: It is well-known that a torsion-free linear connection on a light-like manifold $(M,g)$ compatible with the degenerate metric $g$ exists if and only if $Rad(TM)$ is a Killing distribution. In case of existence, there is an infinitude of connections with none distinguished. We propose a method to single out connections with the help of a special set of 1-forms by the condition that the 1-forms become parallel with respect to this connection. Such sets of 1-forms could be regarded as an additional structure imposed upon the light-like manifold. We consider also connections with torsion and with non-metricity on light-like manifolds.
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TL;DR: In this paper, a general idea for describing spatially finite physical objects with a consistent nontrivial translational-rotational dynamical structure and evolution as a whole, making use of the mathematical concepts and structures connected with the Frobenius integrability/nonintegrability theorems and electromagnetic strain quantities, is presented.
Abstract: This paper aims to present a general idea for description of spatially finite physical objects with a consistent nontrivial translational-rotational dynamical structure and evolution as a whole, making use of the mathematical concepts and structures connected with the Frobenius integrability/nonintegrability theorems and electromagnetic strain quantities. The idea is based on consideration of {\it nonintegrable} subdistributions of some appropriate completely integrable distribution (differential system) on a manifold and then to make use of the corresponding curvatures and correspondingly directed strains as measures of interaction, i.e. of energy-momentum exchange among the physical subsystems mathematically represented by the nonintegrable subdistributions. The concept of photon-like object is introduced and description (including lagrangian) of such objects in these terms is given.
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TL;DR: In this paper, it was shown that any Kahler manifold admitting a flat complex conformal connection is a Bochner-Kahler manifold with special scalar distribution and zero geometric constants.
Abstract: We prove that any Kahler manifold admitting a flat complex conformal connection is a Bochner-Kahler manifold with special scalar distribution and zero geometric constants. Applying the local structural theorem for such manifolds we obtain a complete description of the Kahler manifolds under consideration.
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TL;DR: In this article, the Weyl structures on light-like hypersurfaces endowed with a conformal structure of certain type and specific screen distribution were investigated and necessary and sufficient conditions for a Weyl structure defined by the $1-$form of an almost contact structure given by an additional complex structure in case of an ambiant Kaehler manifold to be closed.
Abstract: We study Weyl structures on lightlikes hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambiant Lorentzian space $\mathbb{R}^{n+2}_{1}$ and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves. Finally, we establish necessary and sufficient conditions for a Weyl structure defined by the $1-$form of an almost contact structure given by an additional complex structure in case of an ambiant Kaehler manifold to be closed.
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01 Jan 2007
TL;DR: In this article, the authors use averaged properties on a disc of variable radius to detect the Painleve property of a difference equation, which is used as a tool for detecting the integrability of difference equations.
Abstract: The theme running throughout this thesis is the Painleve equations, in their differential,
discrete and ultra-discrete versions The differential Painleve equations have
the Painleve property If all solutions of a differential equation are meromorphic
functions then it necessarily has the Painleve property Any ODE with the Painleve
property is necessarily a reduction of an integrable PDE
Nevanlinna theory studies the value distribution and characterizes the growth
of meromorphic functions, by using certain averaged properties on a disc of variable
radius We shall be interested in its well-known use as a tool for detecting
integrability in difference equations—a difference equation may be integrable if it
has sufficiently many finite-order solutions in the sense of Nevanlinna theory This
does not provide a sufficient test for integrability; additionally it must satisfy the
well-known singularity confinement test [Continues]
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TL;DR: In this paper, a global existence result for nonlinear Klein-Gordon equations with small data in infinite homogeneous waveguids, R2×M, where M=(M,g) is a Zoll manifold, was proved.