The Experts below are selected from a list of 234 Experts worldwide ranked by ideXlab platform
Jaume Carot - One of the best experts on this subject based on the ideXlab platform.
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flat Deformation Theorem and symmetries in spacetime
Classical and Quantum Gravity, 2009Co-Authors: Josep Llosa, Jaume CarotAbstract:The flat Deformation Theorem states that given a semi-Riemannian analytic metric g on a manifold, locally there always exists a two-form F, a scalar function c, and an arbitrarily prescribed scalar constraint depending on the point x of the manifold and on F and c, say Ψ(c, F, x) = 0, such that the deformed metric η = cg − F2 is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric g may be written in the extended Kerr–Schild form, namely ηab := agab − 2bk(alb) where η is flat and ka, la are two null covectors such that kala = −1; next we show how the symmetries of g are connected to those of η, more precisely; we show that if the original metric g admits a conformal Killing vector (including Killing vectors and homotheties), then the Deformation may be carried out in a way such that the flat deformed metric η 'inherits' that symmetry.
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Flat Deformation Theorem and symmetries in spacetime
Classical and Quantum Gravity, 2009Co-Authors: Josep Llosa, Jaume CarotAbstract:The \emph{flat Deformation Theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the point $x$ of the manifold and on $F$ and $c$, say $\Psi (c, F, x)=0$, such that the \emph{deformed metric} $\eta = cg -\epsilon F^2$ is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric $g$ may be written in the \emph{extended Kerr-Schild form}, namely $\eta_{ab} := a g_{ab} - 2 b k_{(a} l_{b)}$ where $\eta$ is flat and $k_a, l_a$ are two null covectors such that $k_a l^a= -1$; next we show how the symmetries of $g$ are connected to those of $\eta$, more precisely; we show that if the original metric $g$ admits a Conformal Killing vector (including Killing vectors and homotheties), then the Deformation may be carried out in a way such that the flat deformed metric $\eta$ `inherits' that symmetry.
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Flat Deformation Theorem and symmetries in spacetime
AIP Conference Proceedings, 2009Co-Authors: Jaume Carot, Josep LlosaAbstract:The flat Deformation Theorem states that given a semi‐Riemannian analytic metric g on a manifold, locally it can always be written in terms of a semi‐Riemannian flat metric η, a two‐form F and a scalar function c fulfilling a prescribed scalar constraint, say Ψ(c,F,x) = 0. Here we show that if the original metric g admits a Killing vector, then the Deformation may be carried out in a way such that the flat deformed metric ‘inherits’ that symmetry.
Josep Llosa - One of the best experts on this subject based on the ideXlab platform.
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flat Deformation Theorem and symmetries in spacetime
Classical and Quantum Gravity, 2009Co-Authors: Josep Llosa, Jaume CarotAbstract:The flat Deformation Theorem states that given a semi-Riemannian analytic metric g on a manifold, locally there always exists a two-form F, a scalar function c, and an arbitrarily prescribed scalar constraint depending on the point x of the manifold and on F and c, say Ψ(c, F, x) = 0, such that the deformed metric η = cg − F2 is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric g may be written in the extended Kerr–Schild form, namely ηab := agab − 2bk(alb) where η is flat and ka, la are two null covectors such that kala = −1; next we show how the symmetries of g are connected to those of η, more precisely; we show that if the original metric g admits a conformal Killing vector (including Killing vectors and homotheties), then the Deformation may be carried out in a way such that the flat deformed metric η 'inherits' that symmetry.
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Flat Deformation Theorem and symmetries in spacetime
Classical and Quantum Gravity, 2009Co-Authors: Josep Llosa, Jaume CarotAbstract:The \emph{flat Deformation Theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the point $x$ of the manifold and on $F$ and $c$, say $\Psi (c, F, x)=0$, such that the \emph{deformed metric} $\eta = cg -\epsilon F^2$ is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric $g$ may be written in the \emph{extended Kerr-Schild form}, namely $\eta_{ab} := a g_{ab} - 2 b k_{(a} l_{b)}$ where $\eta$ is flat and $k_a, l_a$ are two null covectors such that $k_a l^a= -1$; next we show how the symmetries of $g$ are connected to those of $\eta$, more precisely; we show that if the original metric $g$ admits a Conformal Killing vector (including Killing vectors and homotheties), then the Deformation may be carried out in a way such that the flat deformed metric $\eta$ `inherits' that symmetry.
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Flat Deformation Theorem and symmetries in spacetime
AIP Conference Proceedings, 2009Co-Authors: Jaume Carot, Josep LlosaAbstract:The flat Deformation Theorem states that given a semi‐Riemannian analytic metric g on a manifold, locally it can always be written in terms of a semi‐Riemannian flat metric η, a two‐form F and a scalar function c fulfilling a prescribed scalar constraint, say Ψ(c,F,x) = 0. Here we show that if the original metric g admits a Killing vector, then the Deformation may be carried out in a way such that the flat deformed metric ‘inherits’ that symmetry.
N. S. Papageorgiou - One of the best experts on this subject based on the ideXlab platform.
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Existence and multiplicity of solutions for the noncoercive Neumann -Laplacian.
2010Co-Authors: N. S. Papageorgiou, Eugénio M. RochaAbstract:We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator with a nonsmooth potential (hemivariational inequality). Using variational techniques based on the smooth critical point theory and the second Deformation Theorem, we prove an existence Theorem and a multiplicity Theorem, under hypothesis that in general do not imply the coercivity of the Euler functional.
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Existence of five nonzero solutions with exact sign for A p-laplacian equation
Discrete & Continuous Dynamical Systems - A, 2009Co-Authors: Michael E. Filippakis, Alexandru Kristály, N. S. PapageorgiouAbstract:We consider nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential depending on a parameter $\lambda\ >\ 0$. The main result guarantees the existence of two positive, two negative and a nodal (sign-changing) solution for the studied problem whenever $\lambda\ >\ 0$ belongs to a small interval (0, λ*) and $p$ ≥ 2. We do not impose any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by using variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass Theorem, and a Brezis-Nirenberg type result for C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second Deformation Theorem.
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ON MULTIPLE SOLUTIONS FOR STRONGLY RESONANT PROBLEMS WITH THE p-LAPLACIAN
Dynamic Systems and Applications, 2007Co-Authors: Evgenia H. Papageorgiou, N. S. PapageorgiouAbstract:We study a nonlinear elliptic problem driven by the p-Laplacian and with a non- smooth potential function (hemivariational inequality). On the nonsmooth potential we impose conditions of strong resonance. Following a variational approach based on the nonsmooth criti- cal point theory and the second Deformation Theorem, we establish the existence of at least two nontrivial smooth solutions. AMS (MOS) Subject Classication. 35J20, 35J60. 1. PRELIMINARIES Let Z R N be a bounded domain with a C 2 -boundary @Z. We consider the
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Multiple solutions for strongly resonant periodic systems
Nonlinear Analysis: Theory Methods & Applications, 2007Co-Authors: N. S. Papageorgiou, Vasile StaicuAbstract:Abstract We considered a semilinear, second order periodic system. We assumed that the differential operator x → − x ″ − A x has zero as an eigenvalue and has no negative eigenvalues. Also we imposed a strong resonance condition (with respect to the zero eigenvalue) on the potential function F ( t , x ) . Using the second Deformation Theorem, we established the existence of at least two nontrivial solutions. To do this we needed to conduct a detailed analysis of the Cerami compactness condition, which is actually of independent interest.
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A multiplicity Theorem for problems with the p-Laplacian
Journal of Functional Analysis, 2007Co-Authors: Evgenia H. Papageorgiou, N. S. PapageorgiouAbstract:We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter λ∈R and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ2=the second eigenvalue of (−Δp,W01,p(Z)), then the problem has at least three nontrivial solutions. Our approach combines the method of upper–lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p=2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990].
Sharif Ibrahim - One of the best experts on this subject based on the ideXlab platform.
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Data-inspired advances in geometric measure theory: generalized surface and shape metrics
arXiv: Differential Geometry, 2014Co-Authors: Sharif IbrahimAbstract:Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Some of its tools (e.g., flat norm distance and decomposition in generalized surface space) hold interest from a theoretical perspective but computational infeasibility prevented practical use. Others, like nonasymptotic densities as shape signatures, have been developed independently for data analysis (e.g., the integral area invariant). The flat norm measures distance between currents (generalized surfaces) by decomposing them in a way that is robust to noise. The simplicial Deformation Theorem shows currents can be approximated on a simplicial complex, generalizing the classical cubical Deformation Theorem and proving sharper bounds than Sullivan's convex cellular Deformation Theorem. Computationally, the discretized flat norm can be expressed as a linear programming problem and solved in polynomial time. Furthermore, the solution is guaranteed to be integral for integral input if the complex satisfies a simple topological condition (absence of relative torsion). This discretized integrality result yields a similar statement for the continuous case: the flat norm decomposition of an integral 1-current in the plane can be taken to be integral, something previously unknown for 1-currents which are not boundaries of 2-currents. Nonasymptotic densities (integral area invariants) taken along the boundary of a shape are often enough to reconstruct the shape. This result is easy when the densities are known for arbitrarily small radii but that is not generally possible in practice. When only a single radius is used, variations on reconstruction results (modulo translation and rotation) of polygons and (a dense set of) smooth curves are presented.
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Simplicial Flat Norm with Scale
Journal of Computational Geometry, 2013Co-Authors: Sharif Ibrahim, Bala Krishnamoorthy, Kevin R. VixieAbstract:We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that MSFN is NP-complete when homology is defined over integers. We cast MSFN as an instance of integer linear optimization. Following recent results on related problems, the MSFN integer program can be solved in polynomial time by solving its linear programming relaxation, when the simplicial complex satisfies a simple topological condition (absence of relative torsion). Our most significant contribution is the simplicial Deformation Theorem, which states that one may approximate a general current with a simplicial current while bounding the expansion of its mass. We present explicit bounds on the quality of this approximation, which indicate that the simplicial current gets closer to the original current as we make the simplicial complex finer. MSFN opens up the possibilities of using flat norm to denoise or extract scale information of large data sets in arbitrary dimensions. On the other hand, it allows one to employ the large body of algorithmic results on simplicial complexes to address more general problems related to currents.
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Simplicial Flat Norm with Scale
arXiv: Differential Geometry, 2011Co-Authors: Sharif Ibrahim, Bala Krishnamoorthy, Kevin R. VixieAbstract:We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial flat norm is NP-complete when homology is defined over integers. We cast the multiscale simplicial flat norm as an instance of integer linear optimization. Following recent results on related problems, the multiscale simplicial flat norm integer program can be solved in polynomial time by solving its linear programming relaxation, when the simplicial complex satisfies a simple topological condition (absence of relative torsion). Our most significant contribution is the simplicial Deformation Theorem, which states that one may approximate a general current with a simplicial current while bounding the expansion of its mass. We present explicit bounds on the quality of this approximation, which indicate that the simplicial current gets closer to the original current as we make the simplicial complex finer. The multiscale simplicial flat norm opens up the possibilities of using flat norm to denoise or extract scale information of large data sets in arbitrary dimensions. On the other hand, it allows one to employ the large body of algorithmic results on simplicial complexes to address more general problems related to currents.
Vijay Gunasekaran - One of the best experts on this subject based on the ideXlab platform.
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Analytical Solution for Sound Radiation Characteristics of Graphene Nanocomposites Plate: Effect of Porosity and Variable Edge Load
International Journal of Structural Stability and Dynamics, 2021Co-Authors: Vijay Gunasekaran, Jeyaraj Pitchaimani, Lenin Babu Mailan Chinnapandi, Ashish KumarAbstract:The effects of graded dispersion of graphene platelets and porosity on vibro-acoustics of nanocomposite plate exposed to variable edge loads are analytically investigated. Voigt and Halpin–Tsai micromechanics model is used to obtain effective properties of the porous graphene nanocomposites. The strain energy technique is implemented to estimate the buckling load ([Formula: see text]). By means of Reddy’s third-order shear Deformation Theorem and Rayleigh Integral, vibration and acoustic responses are obtained. After validating the present analysis with the published results, the nature of edge loads on buckling and vibro-acoustic response is significant. It is noted that an increase in the intensity of non-uniform in-plane loads leads to changes in free vibration modes and resonant amplitude of response. The weight percentage and grading pattern of graphene reinforcement cause the stiffness hardening effect, whereas porosity distribution and coefficients cause the stiffness softening effect on the nanocomposite plate. It is found that the plate with symmetric distribution of graphene platelets with more concentration at the surface and symmetric porosity variation with more porosity at the center radiates less sound power.
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Analytical investigation on free vibration frequencies of polymer nano composite plate: Effect of graphene grading and non-uniform edge loading
Materials Today Communications, 2020Co-Authors: Vijay Gunasekaran, Jeyaraj Pitchaimani, M.c. Lenin BabuAbstract:Abstract An analytical investigation carried out on free vibration characteristics of functionally graded graphene reinforced nanocomposite (FG-GRC) plate under different non-uniform edge loads is presented. Graphene nano-platelets (GPLs) are homogeneously dispersed and graded by varying weight fraction through the thickness. An analytical method based on strain energy approach is adopted to estimate the buckling load. Natural frequencies of the FG-GRC plate are attained using analytical solutions derived based on Reddy's third-order shear Deformation Theorem (TDST). Results revealed that buckling and free vibration behavior of the plate is influenced by the GPLs dispersion pattern and weight fraction under non-uniform edge loads. It is also observed that buckling mode and the fundamental vibration mode of the plate under combined tensile-compression load is entirely different from the other non-uniform edge load cases.
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Vibro-acoustics response of an isotropic plate under non-uniform edge loading: An analytical investigation
Aerospace Science and Technology, 2020Co-Authors: Vijay Gunasekaran, Jeyaraj Pitchaimani, M.c. Lenin BabuAbstract:Abstract Analytical studies carried out on the vibro-acoustic response behavior of an isotropic plate under non-uniform edge loads subjected to steady-state mechanical excitation is presented. An analytical method based on the energy approach is used to calculate the buckling load ( P c r ). Free and forced vibration responses of the plate are obtained using an analytical method based on Reddy's third-order shear Deformation Theorem (TSDT) while sound radiation behavior is analyzed using Rayleigh Integral. Results revealed that P c r is significantly influenced by the nature of non-uniform edge load. Similarly, natural frequencies reduce with an increase in axial compressive load due to a reduction in structural stiffness. Vibration and acoustic resonant amplitudes are affected by the intensity of the compressive load. Sound transmission loss reduces with an increase in compressive load magnitude and the effect is significant in the stiffness dominant region.
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Acoustic radiation and transmission loss of FG-Graphene composite plate under nonuniform edge loading
European Journal of Mechanics - A Solids, 1Co-Authors: Vijay Gunasekaran, Jeyaraj Pitchaimani, Lenin Babu Mailan ChinnapandiAbstract:Abstract The influence of nonuniform edge loads on the acoustic response of a functionally graded graphene reinforced composite plate is investigated analytically. The energy method is implemented to calculate the buckling load ( P c r ). An analytical method based on Reddy's third-order shear Deformation Theorem is used to obtain the vibration response, and acoustic response is obtained using Rayleigh Integral. The nature of edge load variation on buckling and vibro-acoustic response is significant. Free vibration mode shape changes with an increase in edge load and consequently affects the resonant amplitude of responses also especially for the plates with a higher aspect ratio. Volume fraction and dispersion pattern of graphene nano-platelets also influences the resonance amplitudes. Plate with F G − G R C C dispersion pattern has improved buckling and vibro-acoustic response behavior. Similarly, change in sound transmission loss level is significant in the stiffness region compared to the damping and mass dominated region.