Finite Number Of Derivatives Research Articles

Exorcising ghosts in quantum gravity

We remark that Ostrogradsky ghosts in higher-derivative gravity, with a finite number of derivatives, are fictitious as they result from an unjustified truncation performed in a complete theory containing infinitely many curvature invariants. The apparent ghosts can then be projected out of the quadratic gravity spectrum by redefining the boundary conditions of the theory in terms of an integration contour that does not enclose the ghost poles. This procedure does not alter the renormalizability of the theory. One can thus use quadratic gravity as a quantum field theory of gravity that is both renormalizable and unitary.

Limiting shifted homotopy in higher-spin theory and spin-locality

Higher-spin vertices containing up to quintic interactions at the LagTangian level are explicitly calculated in the one-form sector of the non-linear unfolded higher-spin equations using a \U0001d6fd →-∞-shifted contracting homotopy introduced in the paper. The problem is solved in a background independent way and for any value of the complex parameter \U0001d702 in the higher-spin equations. All obtained vertices are shown to be spin-local containing a finite number of derivatives in the spinor space for any given set of spins. The vertices proportional to \U0001d7022 and {overline{eta}}^2 are in addition ultra-local, i.e., zero-forms that enter into the vertex in question are free from the dependence on at least one of the spinor variables y or overline{y} . Also the \U0001d7022 and {overline{eta}}^2 vertices are shown to vanish on any purely gravitational background hence not contributing to the higher-spin current interactions on AdS4. This implies in particular that the gravitational constant in front of the stress tensor is positive being proportional to eta overline{eta} . It is shown that the \U0001d6fd-shifted homotopy technique developed in this paper can be reinterpreted as the conventional one but in the \U0001d6fd-dependent deformed star product.

Bounds on amplitudes in effective theories with massive spinning particles

We consider a procedure for directly constructing general tree-level four-particle scattering amplitudes of massive spinning particles that are consistent with the usual requirements of Lorentz invariance, unitarity, crossing symmetry, and locality. There are infinitely many such amplitudes, but we can isolate interesting theories by bounding the high-energy growth of the tree amplitudes within the effective field theory. This allows us to set model-independent lower bounds on the growth of tree-level amplitudes in any effective field theory with a given particle content and any interaction terms with an arbitrary but finite number of derivatives. In certain common cases this corresponds to finding the highest possible strong coupling scale. When applied to spin 2, we show that the only amplitudes that saturate this bound are generated by the known ghost-free theories of a massive spin-2 particle, namely dRGT massive gravity and the pseudolinear theory. We also make a conjecture for the allowed growth of tree amplitudes in a theory with a single massive particle of any integer spin.

Expansions of non-symmetric toroidal magnetohydrodynamic equilibria

Expansions of non-symmetric toroidal ideal magnetohydrodynamic equilibria with nested flux surfaces are carried out for two cases. The first expansion is in a topological torus in three dimensions, in which physical quantities are periodic of period 2π in y and z. Data is given on the flux surface x = 0. Despite the possibility of magnetic resonances the power series expansion can be carried to all orders in a parameter which measures the flux between x = 0 and the surface in question. Resonances are resolved by appropriate addition resonant fields, as by Weitzner, [Phys. Plasmas 21, 022515 (2014)]. The second expansion is about a circular magnetic axis in a true torus. It is also assumed that the cross section of a flux surface at constant toroidal angle is approximately circular. The expansion is in an analogous flux coordinate, and despite potential resonance singularities, may be carried to all orders. Non-analytic behavior occurs near the magnetic axis. Physical quantities have a finite number of derivatives there. The results, even though no convergence proofs are given, support the possibility of smooth, well-behaved non-symmetric toroidal equilibria.

The scalar Nevanlinna–Pick interpolation problem with boundary conditions

We show that if the Nevanlinna–Pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc, then the problem remains solvable with the addition of any number of boundary interpolation conditions, provided the boundary interpolation values have modulus less than unity. We give new, inductive proofs of the Nevanlinna–Pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation (the right half plane or unit disc), with function values and any finite number of derivatives specified. Our solutions are analytic on the closure of the domain of interpolation. Our proofs only require a minimum of matrix theory and operator theory. We also give new, straightforward algorithms for obtaining minimal H ∞ norm solutions. Finally, some numerical examples are given.

SPACETIME NONCOMMUTATIVITY WITH A BIFERMIONIC PARAMETER

We consider a Moyal plane and propose to make the noncommutativity parameter Θμν bifermionic, i.e. composed of two fermionic (Grassmann odd) parameters. The Moyal product then contains a finite number of derivatives, which avoid the difficulties of the standard approach. As an example, we construct a two-dimensional noncommutative field theory model based on the Moyal product with a bifermionic parameter and show that it has a locally conserved energy–momentum tensor. The model has no problem with the canonical quantization and appears to be renormalizable.

Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces II: numerical stability

In this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence.

Caustic Formation in Tachyon Effective Field Theories

Certain configurations of D-branes, for example wrong dimensional branes or the brane-antibrane system, are unstable to decay. This instability is described by the appearance of a tachyonic mode in the spectrum of open strings ending on the brane(s). The decay of these unstable systems is described by the rolling of the tachyon field from the unstable maximum to the minimum of its potential. We analytically study the dynamics of the inhomogeneous tachyon field as it rolls towards the true vacuum of the theory in the context of several different tachyon effective actions. We find that the vacuum dynamics of these theories is remarkably similar and in particular we show that in all cases the tachyon field forms caustics where second and higher derivatives of the field blow up. The formation of caustics signals a pathology in the evolution since each of the effective actions considered is not reliable in the vicinity of a caustic. We speculate that the formation of caustics is an artifact of truncating the tachyon action, which should contain all orders of derivatives acting on the field, to a finite number of derivatives. Finally, we consider inhomogeneous solutions in p-adic string theory, a toy model of the bosonic tachyon which contains derivatives of all orders acting on the field. For a large class of initial conditions we conclusively show that the evolution is well behaved in this case. It is unclear if these caustics are a genuine prediction of string theory or not.

Integrating the Kuramoto-Sivashinsky equation in polar coordinates: application of the distributed approximating functional approach.

An algorithm is presented to integrate nonlinear partial differential equations, which is particularly useful when accurate estimation of spatial derivatives is required. It is based on an analytic approximation method, referred to as distributed approximating functionals (DAF's), which can be used to estimate a function and a finite number of derivatives with a specified accuracy. As an application, the Kuramoto-Sivashinsky (KS) equation is integrated in polar coordinates. Its integration requires accurate estimation of spatial derivatives, particularly close to the origin. Several stationary and nonstationary solutions of the KS equation are presented, and compared with analogous states observed in the combustion front of a circular burner. A two-ring, nonuniform counter-rotating state has been obtained in a KS model simulation of such a burner.

Spectral multipliers on exponential growth solvable Lie groups

Let M be a measure space and let L be a positive definite operator on L2(M). By the spectral theorem, for any bounded Borel measurable function F : [0, ∞) 7→ C the operator F (L)f = ∫ ∞ 0 F (λ)dE(λ)f is bounded on L2(M). We are interested in sufficient conditions on F for F (L) to be bounded on Lp(M), p 6= 2. We direct the reader to [1], [3], [4], [8], [9], [10], [12] and [13] for more background on various multiplier theorems. In this paper we assume F is compactly supported and have some smoothness (finite number of derivatives) and we consider only the case p = 1. Our measure space G is semidirect product of stratified nilpotent Lie group N and the real line. The operator L is (minus) sublaplacian on G. Our group has exponential volume growth. The earlier theory suggested that one needs holomorphic F for F (L) to be bounded on L1, however the recent results [5], [6], [7] showed that estimates on only a finite number of derivatives of F imply boundedness of F (L) on L1 on some solvable G of exponential growth. In this case we say that G (more precisely L) has Ckfunctional calculus. On the other hand, Christ and Muller give an example of a solvable Lie group on which F must be holomorphic. The problem is to find the condition on G (and possibly L) which decides whether G has a Ck-functional calculus or not. Here, our condition is in terms of roots of adjoint representation of the Lie algebra of G. Our groups are of “rank one”,

Orthogonal matrix polynomials and applications

Orthogonal matrix polynomials, on the real line or on the unit circle, have properties which are natural generalizations of properties of scalar orthogonal polynomials, appropriately modified for the matrix calculus. We show that orthogonal matrix polynomials, both on the real line and on the unit circle, appear at various places and we describe some of them. The spectral theory of doubly infinite Jacobi matrices can be described using orthogonal 2×2 matrix polynomials on the real line. Scalar orthogonal polynomials with a Sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line. Orthogonal matrix polynomials on the unit circle are related to unitary block Hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing. Finally we show how orthogonal matrix polynomials can be used for Gaussian quadrature of matrix-valued functions. Parallel algorithms for this purpose have been implemented (using PVM) and some examples are worked out.

Derivatives as an IR regulator for massless fields

The free propagator for the scalar λφ 4-theory is calculated exactly up to the second derivative of a background field. Using this propagator I compute the one-loop effective action, which then contains all powers of the field but with at most two derivatives acting on each field. The standard derivative expansion, which only has a finite number of derivatives in each term, breaks down for small fields when the mass is zero, while the expression obtained here has a well-defined expansion in φ. In this way the resummation of derivatives cures the naive IR divergence. The extension to finite temperature is also discussed.

On the integrability of intermediate distributions for Anosov diffeomorphisms

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

On eigenvalue placement in linear time-invariant systems with input derivatives

A method using the simplest matrix generalized inverse, the {1} -inverse, is developed to find a constant state feedback matrix K that yields a set of prescribed eigenvalues for time-invariant linear multivariable systems that contain any finite number of derivatives of the input vector u. The method requires neither controllability assumptions nor the knowledge of eigenvalues and eigenvectors of the plant matrix A. The matrix K exists if a consistency condition for an algebraic linear system of equations is satisfied.

Inner functions in the unit ball of [formula omitted

Let B be the open unit ball of C n , n > 1. Let I (for “inner”) be the set of all u ϵ H ∘( B) that have ¦u¦ = 1 a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ϵ I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ϵ X, y ϵ Y, and let ¦X¦ be the norm closure, in L ∞ = L ∞( S), of X. Some results: set I is dense in the unit ball of H ∞( B) in the compact-open topology. On S, Q ̄ Q is weak ∗-dense in L ∞, ¦ Q ̄ Q¦ does not contain H ∞, C(S) ⊂¦ Q ̄ H ∞¦ ≠ ¦ H ̄ ∞H ∞¦ ≠ L ∞ . (When n = 1, ¦Q¦ = H ∞ and ¦ Q ̄ Q¦ = L ∞ .) Every unimodular ƒ ϵ L ∞ is a pointwise limit a.e. of products u v ̄ , u ϵ I, ν ϵ I . The zeros of every ƒ ≢ 0 in the ball algebra (but not of every H ∞ -function) can be matched by those of some u ϵ I, as can any finite number of derivatives at 0 if ∥ƒ∥ ∞ < 1 . However, ƒ u cannot be bounded in B if u ϵ I is non-constant.

Raised cosine function for image restoration

In this paper, a new set of raised cosine functions is proposed. These functions have all the useful properties of the spline functions with the additional advantage of continuous infinite derivatives as against only a finite number of derivatives in case of the spline functions. Because of this property they exhibit a smoother behaviour. The property of smoothness, coupled with convolution, makes the raised cosine functions readily applicable to the image restoragtion problem, where the degradation is through a shift invariant blurring function. The results confirm the superior behaviour of these functions in comparison to spline functions.

On zeros of time-invariant multivariable linear systems containing input derivatives

A method is developed to find input vectors which generate zero output vectors in linear control systems whose state equation contains any finite number of derivatives of the input vector. A general matrix eigenvalue problem is derived that may be reduced to an ordinary one.

Etude des Equations des Fluides Chargés Relativistes Inductifs et Conducteurs

In the first part a system of equations for an inductive charged relativistic fluid with finite conductivity is written in a space time with given metric, taking into account thermodynamic phenomena. Speeds of propagation of various types of waves are determined under a restrictive hypothesis concerning the heat currentq: thatq depends only on the thermodynamical quantities and the gradient of one function of these quantities.In the second part it is shown, by a detailed study of the characteristic polynomial and of its irreducible factors, that, whenq is negligible, the proposed system is non-strictly hyperbolic in the sense ofJ. Leray andY. Ohya and existence and uniqueness theorems of a certain Gevrey class are verified; the relativistic causality principle is satisfied under some physically reasonable assumptions on the thermodynamical quantities. The system becomes strictly hyperbolic (existence and uniqueness theorems obtain in classes of functions with a finite number of derivatives) when the fluid is both non inductive and of zero electrical conductivity.In the third part we show briefly, by the methods of the second part, that the equations of relativistic fluids, with an infinite electrical conductivity is also non-strictly hyperbolic. The linearized equations (in the neighborhood of constant values) are strictly hyperbolic.