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Showing papers on "Constant (mathematics) published in 2009"


Journal ArticleDOI
TL;DR: This fractional derivative provides a fractional calculus parallel with the classical one, which applies to non-differentiable functions, and the present short article summarizes the main basic formulae so obtained.

420 citations


Journal ArticleDOI
TL;DR: In this paper, a perturbative approach around the Einstein-Hilbert action was used to find static and spherically symmetric black hole solutions in f(R) theories of gravity.
Abstract: In the context of f(R) theories of gravity, we address the problem of finding static and spherically symmetric black hole solutions. Several aspects of constant curvature solutions with and without electric charge are discussed. We also study the general case (without imposing constant curvature). Following a perturbative approach around the Einstein-Hilbert action, it is found that only solutions of the Schwarzschild-(anti) de Sitter type are present up to second order in perturbations. Explicit expressions for the effective cosmological constant are obtained in terms of the f(R) function. Finally, we have considered the thermodynamics of black holes in anti-de Sitter space-time and found that this kind of solution can only exist provided the theory satisfies R(0)+f(R(0))<0. Interestingly, this expression is related to the condition which guarantees the positivity of the effective Newton's constant in this type of theories. In addition, it also ensures that the thermodynamical properties in f(R) gravities are qualitatively similar to those of standard general relativity.

229 citations


Proceedings ArticleDOI
Rahul Garg1, Rohit Khandekar1
14 Jun 2009
TL;DR: The Matlab implementation of GraDeS (Gradient Descent with Sparsification) outperforms previously proposed algorithms like Subspace Pursuit, StOMP, OMP, and Lasso by an order of magnitude and uncovered cases where L1-regularized regression (Lasso) fails but GraDeS finds the correct solution.
Abstract: We present an algorithm for finding an s-sparse vector x that minimizes the square-error ∥y -- Φx∥2 where Φ satisfies the restricted isometry property (RIP), with isometric constant δ2s 1 and Hs sets all but s largest magnitude coordinates to zero. GraDeS converges to the correct solution in constant number of iterations. The condition δ2s

207 citations


Posted Content
TL;DR: The Høst-Madsen-Nosratinia conjecture in the negative is settled, and it is shown that at least 1.2 degrees-of-freedom are achievable for all values of complex channel coefficients except for a subset of measure zero.
Abstract: It has been conjectured by Host-Madsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degree-of-freedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree of freedom, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 degree-of-freedom is achievable. In this paper, we settle the Host-Madsen-Nosratinia conjecture in the negative. We show that at least 1.2 degrees-of-freedom are achievable for all values of complex channel coefficients except for a subset of measure zero. For the class of linear beamforming and interference alignment schemes considered in this paper, it is also shown that 1.2 is the maximum number of degrees of freedom achievable on the complex Gaussian 3 user interference channel with constant channel coefficients, for almost all values of channel coefficients. To establish the achievability of 1.2 degrees of freedom we introduce the novel idea of asymmetric complex signaling - i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian point-to-point, multiple-access and broadcast channels where circularly symmetric complex Gaussian inputs are optimal, for interference channels optimal inputs are in general asymmetric. With asymmetric complex signaling, we also show that the 2 user complex Gaussian X channel with constant channel coefficients achieves the outer bound of 4/3 degrees-of-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed.

184 citations


Proceedings ArticleDOI
04 Jan 2009
TL;DR: The main result is a (3+e)-competitive algorithm for this problem, that holds for essentially any power function, and a model of allowable speeds that generalizes all known models in the literature.
Abstract: All of the theoretical speed scaling research to date has assumed that the power function, which expresses the power consumption P as a function of the processor speed s, is of the form P = sα, where α > 1 is some constant. Motivated in part by technological advances, we initiate a study of speed scaling with arbitrary power functions. We consider the problem of minimizing the total flow plus energy. Our main result is a (3+e)-competitive algorithm for this problem, that holds for essentially any power function. We also give a (2+e)-competitive algorithm for the objective of fractional weighted flow plus energy. Even for power functions of the form sα, it was not previously known how to obtain competitiveness independent of α for these problems. We also introduce a model of allowable speeds that generalizes all known models in the literature.

180 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigate the holographic dark energy scenario with a varying gravitational constant, in flat and non-flat background geometry, and extract the exact differential equations determining the evolution of the dark energy density-parameter, which include G-variation correction terms.

173 citations


Journal ArticleDOI
TL;DR: In this article, a piecewise constant level set (PCLS) method is implemented to solve a structural shape and topology optimization problem, where the boundary is described by discontinuities of PCLS functions.
Abstract: In this paper, a piecewise constant level set (PCLS) method is implemented to solve a structural shape and topology optimization problem. In the classical level set method, the geometrical boundary of the structure under optimization is represented by the zero level set of a continuous level set function, e.g. the signed distance function. Instead, in the PCLS approach the boundary is described by discontinuities of PCLS functions. The PCLS method is related to the phase-field methods, and the topology optimization problem is defined as a minimization problem with piecewise constant constraints, without the need of solving the Hamilton-Jacobi equation. The result is not moving the boundaries during the iterative procedure. Thus, it offers some advantages in treating geometries, eliminating the reinitialization and naturally nucleating holes when needed. In the paper, the PCLS method is implemented with the additive operator splitting numerical scheme, and several numerical and procedural issues of the implementation are discussed. Examples of 2D structural topology optimization problem of minimum compliance design are presented, illustrating the effectiveness of the proposed method.

172 citations


Journal ArticleDOI
Jacopo Stoppa1
TL;DR: In this article, it was shown that a polarised manifold with a constant scalar curvature and discrete automorphisms is K-stable, which refines the K-semistability proved by S.K. Donaldson.

166 citations


Journal ArticleDOI
TL;DR: The fixed point structure of the renormalization flow in higher derivative gravity is investigated in terms of the background covariant effective action using an operator cutoff that keeps track of powerlike divergences.
Abstract: The fixed point structure of the renormalization flow in higher derivative gravity is investigated in terms of the background covariant effective action using an operator cutoff that keeps track of powerlike divergences. Spectral positivity of the gauge fixed Hessian can be satisfied upon expansion in the asymptotically free higher derivative coupling. At one-loop order in this coupling strictly positive fixed points are found for the dimensionless Newton constant g(N) and the cosmological constant lambda, which are determined solely by the coefficients of the powerlike divergences. The renormalization flow is asymptotically safe with respect to this fixed point and settles on a lambda(g(N)) trajectory after O(10) units of the renormalization mass scale to accuracy 10(-7).

155 citations


Journal ArticleDOI
TL;DR: This work introduces a new random variable shape parameter strategy and gives numerical results showing that the new random strategy often outperforms both existing variable shape and constant shape strategies.
Abstract: Several variable shape parameter methods have been successfully used in radial basis function approximation methods. In many cases variable shape parameter strategies produced more accurate results than if a constant shape parameter had been used. We introduce a new random variable shape parameter strategy and give numerical results showing that the new random strategy often outperforms both existing variable shape and constant shape strategies.

147 citations


Journal ArticleDOI
TL;DR: It is shown that Steiner structures are optimal constant dimension codes achieving the Wang-Xing-Safavi-Naini bound and it is pointed out that a family of known Steiner structure is actually afamily of optimal constantdimension codes achieving both the Johnson type bounds I and II.
Abstract: Very recently, an operator channel was defined by Koetter and Kschischang when they studied random network coding. They also introduced constant dimension codes and demonstrated that these codes can be employed to correct errors and/or erasures over the operator channel. Constant dimension codes are equivalent to the so-called linear authentication codes introduced by Wang, Xing and Safavi-Naini when constructing distributed authentication systems in 2003. In this paper, we study constant dimension codes. It is shown that Steiner structures are optimal constant dimension codes achieving the Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Then, we derive two Johnson type upper bounds, say I and II, on constant dimension codes. The Johnson type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known Steiner structures is actually a family of optimal constant dimension codes achieving both the Johnson type bounds I and II.

Journal ArticleDOI
TL;DR: In this paper, a five-dimensional symmetry algebra consisting of Lie point symmetries is computed for the nonlinear Schrodinger equation, which, together with a reflection invariance, generates two five-parameter solution groups.

Journal ArticleDOI
TL;DR: In this article, a SIS epidemic model incorporating media coverage is presented, and the dynamics of this disease model under constant and pulse vaccination are analyzed, and stability analysis of the model with constant vaccination shows that the disease free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and if the endemic equilibrium exists.

Book ChapterDOI
21 Aug 2009
TL;DR: It is shown that adaptively sampled O (k ) centers give a constant factor bi-criteria approximation for the k -means problem, with a constant probability, and can be found using LP-based techniques.
Abstract: We show that adaptively sampled O (k ) centers give a constant factor bi-criteria approximation for the k -means problem, with a constant probability. Moreover, these O (k ) centers contain a subset of k centers which give a constant factor approximation, and can be found using LP-based techniques of Jain and Vazirani [JV01] and Charikar et al. [CGTS02]. Both these algorithms run in effectively O (nkd ) time and extend the O (logk )-approximation achieved by the k -means++ algorithm of Arthur and Vassilvitskii [AV07].

Journal ArticleDOI
TL;DR: In this paper, the role of several two-component integrable systems in the classical problem of shallow water waves is described, which can be related to three different integrably generalization of the Camassa-Holm equation, the Zakharov-Ito system and the Kaup-Boussinesq system.

Journal ArticleDOI
TL;DR: In this paper, the authors consider the cosmology of modified gravity models in which Newton's constant is distorted by a function of the inverse d'Alembertian acting on the Ricci scalar and derive a technique for choosing the distortion function so as to fit an arbitrary expansion history.
Abstract: We consider the cosmology of modified gravity models in which Newton's constant is distorted by a function of the inverse d'Alembertian acting on the Ricci scalar. We derive a technique for choosing the distortion function so as to fit an arbitrary expansion history. This technique is applied numerically to the case of ΛCDM cosmology, and the result agrees well with a simple hyperbolic tangent.

Journal ArticleDOI
TL;DR: In this article, it was shown that the B-field modified Nahm equation can also be interpreted as a boundary condition of the F1-strings ending on the D3-brane.
Abstract: We show that the Nahm equation which describes a fuzzy D3-brane in the presence of a B-field can be derived as a boundary condition of the F1-strings ending on the D3-brane, and that the modifications of the original Nahm equation by a B-field can be understood in terms of the noncommutative geometry of the D3-brane. Naturally this is consistent with the alternative derivation by quantising the open strings in the B-field background. We then consider a configuration of multiple M2-branes ending on an M5-brane with a constant 3-form C-field. By analogy with the case of strings ending on a D3-brane with a constant B-field, one can expect that this system can be described in terms of the boundary of the M2-branes moving within a certain kind of quantum geometry on the M5-brane worldvolume. By repeating our analysis, we show that the analogue of the B-field modified Nahm equation, the C-field modified Basu-Harvey equation can also be understood as a boundary condition of the M2-branes. We then compare this to the M5-brane BIon description and show that the two descriptions match provided we postulate a new type of quantum geometry on the M5-brane worldvolume. Unlike the D-brane case, this is naturally expressed in terms of a relation between a 3-bracket of the M5-brane worldvolume coordinates and the C-field.

Journal ArticleDOI
Xiannan Li1
TL;DR: The problem of finding upper bounds for Dirichlet L-functions at the edge of the critical strip has a long and interesting history as discussed by the authors, and the main focus of this paper is on the case where the coefficients of these L -functions are known to be small.
Abstract: The problem of finding upper bounds for L-functions at the edge of the critical strip has a long and interesting history. Here, the situation for classical L-functions such as Dirichlet L-functions is relatively well understood. The reason for this is because the size of the coefficients of these L-functions is known to be small. Although L-functions are generally expected to have coefficients which are bounded by a constant at the primes, this has only been proven for a small class of familiar examples. Our main focus here is on the problem of finding upper bounds for L-functions for which we have comparatively bad bounds for the size of the coefficients.

Journal ArticleDOI
TL;DR: In this article, the authors consider the case of modified gravity models in which Newton's constant is distorted by a function of the inverse d'Alembertian acting on the Ricci scalar.
Abstract: We consider the cosmology of modified gravity models in which Newton's constant is distorted by a function of the inverse d'Alembertian acting on the Ricci scalar. We derive a technique for choosing the distortion function so as to fit an arbitrary expansion history. This technique is applied numerically to the case of LambdaCDM cosmology, and the result agrees well with a simple hyperbolic tangent.

Journal ArticleDOI
TL;DR: An exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent is proved.
Abstract: We prove an exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent (a circuit is called multilinear, if the polynomial computed by each of its gates is multilinear). We also prove a super-polynomial separation between the size of product-depth d and product-depth d + 1 multilinear circuits (where d is constant). That is, there exists a polynomial f such that

Journal ArticleDOI
Anders Johansson1
TL;DR: It is shown that keeping a constant lower limit on the net-time headway is the key mechanism behind the dynamics of pedestrian streams and there is no need to come up with an arbitrary fit function (with arbitrary fit parameters) as has traditionally been done.
Abstract: We show that keeping a constant lower limit on the net-time headway is the key mechanism behind the dynamics of pedestrian streams. There is a large variety in flow and speed as functions of density for empirical data of pedestrian streams obtained from studies in different countries. The net-time headway, however, stays approximately constant over all these different data sets. By using this fact, we demonstrate how the underlying dynamics of pedestrian crowds, naturally follows from local interactions. This means that there is no need to come up with an arbitrary fit function (with arbitrary fit parameters) as has traditionally been done. Further, by using not only the average density values but the variance as well, we show how the recently reported stop-and-go waves [Helbing et al., Phys. Rev. E 75, 046109 (2007)] emerge when local density variations take values exceeding a certain maximum global (average) density, which makes pedestrians stop.

Journal ArticleDOI
TL;DR: In this article, a two-parameter family of exact solutions for cylindrically symmetric vacuum solutions in Weyl coordinates in the context of the metric f(R) theories of gravity was introduced, which corresponds to a constant Ricci scalar.
Abstract: In the previous work we introduced a new static cylindrically symmetric vacuum solution in Weyl coordinates in the context of the metric f(R) theories of gravity.1 Now we obtain a two-parameter family of exact solutions which contains a cosmological constant and a new parameter as β. This solution corresponds to a constant Ricci scalar. We proved that in f(R) gravity the constant curvature solution in cylindrically symmetric cases is only one member of the most generalized Tian family in GR. We show that our constant curvature exact solution is applicable to the exterior of a string. The sensibility of stability under initial conditions is discussed.

Journal ArticleDOI
TL;DR: This work investigates whether the constant c can be made even smaller when one allows constant factor approximation, and describes a kind of approximation schemes-trade-offs between approximation factor and the time complexity.

Journal ArticleDOI
TL;DR: All the solutions of a rational difference equation from Putnam's mathematical competition are described, which are eventually equal to its positive equilibrium, and a new, elegant and short proof of the fact that the equation has a positive solution which is not eventuallyequal to one.

Journal ArticleDOI
TL;DR: A method to construct closed curves, including knotted curves, of constant curvature and continuous torsion using pieces of Salkowski curves is outlined.

Journal ArticleDOI
TL;DR: In this paper, the authors suggest that the solution to the cosmological vacuum energy puzzle does not require any new field beyond the standard model, but rather can be explained as a result of the interaction of the infrared sector of the effective theory of gravity with standard model fields.
Abstract: We suggest that the solution to the cosmological vacuum energy puzzle does not require any new field beyond the standard model, but rather can be explained as a result of the interaction of the infrared sector of the effective theory of gravity with standard model fields. The cosmological constant in this framework can be presented in terms of QCD parameters and the Hubble constant H as follows, vac H⋅mqq/mη' (4.3⋅10−3eV)4, which is amazingly close to the observed value today. In this work we explain how this proposal can be tested by analyzing CMB data. In particular, knowing the value of the observed cosmological constant fixes univocally the smallest size of the spatially flat, constant time 3d hypersurface which, for instance in the case of an effective 1-torus, is predicted to be around 74 Gpc. We also comment on another important prediction of this framework which is a violation of cosmological isotropy. Such anisotropy is indeed apparently observed by WMAP, and will be confirmed (or ruled out) by future PLANCK data.

Journal ArticleDOI
TL;DR: In this article, complete Riemannian manifolds satisfying the equation $Ric+ abla^2 f-\frac{1}{m}df\otimes df=0$ by studying the associated PDE (Delta_f f + m\mu e^{2f/m} = 0) were studied and it was shown that there are no nonconstant solutions.
Abstract: We study complete Riemannian manifolds satisfying the equation $Ric+ abla^2 f-\frac{1}{m}df\otimes df=0$ by studying the associated PDE $\Delta_f f + m\mu e^{2f/m}=0$ for $\mu\leq 0$. By developing a gradient estimate for $f$, we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ricci flat warped products with fibers which have nonpositive Einstein constant. We also show that for nontrivial steady gradient Ricci solitons, the quantity $R+| abla f|^2$ is a positive constant.

Journal ArticleDOI
TL;DR: In this article, the authors prove the existence of Kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant curvature Kahler metric.
Abstract: In this paper we prove the existence of Kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature Kahler metric. In the case where the manifold has nontrivial holomorphic vector fields with zeros, we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature Kahler metrics.

Journal ArticleDOI
TL;DR: A comprehensive asymptotic theory for the estimation of a change-point in the mean function of functional observations is developed and it is shown how the limit distribution of a suitably defined change- point estimator depends on the size and location of the change.

Journal ArticleDOI
TL;DR: In this article, a 2-parameter family of exact solutions for cylindrically symmetric vacuum solutions in Weyl coordinates in the context of the metric f(R) theories of gravity was introduced.
Abstract: In the previous work we introduced a new static cylindrically symmetric vacuum solutions in Weyl coordinates in the context of the metric f(R) theories of gravity\cite{1}. Now we obtain a 2-parameter family of exact solutions which contains cosmological constant and a new parameter as $\beta$. This solution corresponds to a constant Ricci scalar. We proved that in $f(R)$ gravity, the constant curvature solution in cylindrically symmetric cases is only one member of the most generalized Tian family in GR. We show that our constant curvature exact solution is applicable to the exterior of a string. Sensibility of stability under initial conditions is discussed.