Non-vanishing Constant Research Articles

Second-Order Asymptotically Optimal Outlier Hypothesis Testing

We revisit the outlier hypothesis testing framework of Li <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (TIT 2014) and derive fundamental limits for the optimal test under the generalized Neyman-Pearson criterion. In outlier hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated from anomalous distributions. The nominal and anomalous distributions are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unknown</i> . We study the tradeoff among the probabilities of misclassification error, false alarm and false reject for tests that satisfy weak conditions on the rate of decrease of these error probabilities as a function of sequence length. Specifically, we propose a threshold-based test that ensures exponential decay of misclassification error and false alarm probabilities. We study two constraints on the false reject probability, with one constraint being that it is a non-vanishing constant and the other being that it has an exponential decay rate. For both cases, we characterize bounds on the false reject probability, as a function of the threshold, for each pair of nominal and anomalous distributions and demonstrate the optimality of our test under the generalized Neyman-Pearson criterion. We first consider the case of at most one outlying sequence and then generalize our results to the case of multiple outlying sequences where the number of outlying sequences is unknown and each outlying sequence can follow a different anomalous distribution.

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

This paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) stating that Penrose’s proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity i^-. Modelling gravitational radiation by scalar radiation, we then take a first step towards a dynamical understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, rphi sim t^{-1} as trightarrow -infty , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, rpartial _vphi =0, on past null infinity. We show that if the initial Hawking mass at i^- is nonzero, then, in accordance with the non-smoothness of {mathcal {I}}^+, the asymptotic expansion of partial _v(rphi ) near {mathcal {I}}^+ reads partial _v(rphi )=Cr^{-3}log r+{mathcal {O}}(r^{-3}) for some non-vanishing constant C. In fact, the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting compactly supported scattering data for the linear (or coupled) wave equation on {mathcal {I}}^- and on {mathcal {H}}^-, we find that the asymptotic expansion of partial _v(rphi ) near {mathcal {I}}^+ generically contains logarithmic terms at second order, i.e. at order r^{-4}log r.

Stress of a Spatially Uniform Dislocation Density Field

It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising a linear elastic material under no loads vanishes. We prove that the same result does not hold in general in the geometrically nonlinear case. This problem of mechanics establishes the purely geometrical result that the $\mathop{\mathrm{curl}}\nolimits $ of a sufficiently smooth two-dimensional rotation field cannot be a non-vanishing constant on a domain. It is classically known in continuum mechanics, stated first by the brothers Cosserat (SIAM J. Appl. Math. 25(3):483–491, 1973), that if a second order tensor field on a simply connected domain is at most a curl-free field of rotations, then the field is necessarily constant on the domain. It is shown here that, at least in dimension 2, this classical result is in fact a special case of a more general situation where the curl of the given rotation field is only known to be at most a constant.

Absence of solid angle deficit singularities in beyond-generalized proca theories

In Gleyzes-Langlois-Piazza-Vernizzi (GLPV) scalar-tensor theories, which are outside the domain of second-order Horndeski theories, it is known that there exists a solid angle deficit singularity in the case where the parameter $\alpha_{\rm H}$ characterizing the deviation from Horndeski theories approaches a non-vanishing constant at the center of a spherically symmetric body. Meanwhile, it was recently shown that second-order generalized Proca theories with a massive vector field $A^{\mu}$ can be consistently extended to beyond-generalized Proca theories, which recover shift-symmetric GLPV theories in the scalar limit $A^{\mu} \to \nabla^{\mu} \chi$. In beyond-generalized Proca theories up to quartic-order Lagrangians, we show that solid angle deficit singularities are generally absent due to the existence of a temporal vector component. We also derive the vector-field profiles around a compact object and show that the success of the Vainshtein mechanism operated by vector Galileons is not prevented by new interactions in beyond generalized Proca theories.

Gravitational waves versus cosmic strings

The equation which governs the temporal evolution of a gravitational wave (GW) in curved space-time can be treated as the Schrodinger equation for a particle moving in the presence of an effective potential. When GWs propagate in an expanding Universe with constant effective potential, there is a critical value (kc) of the comoving wave-number which discriminates the metric perturbations into oscillating (k > kc) and non-oscillating (k < kc) modes. The effective potential is reduced to a non-vanishing constant in a cosmological model which is driven by a two-component fluid, consisting of radiation (dominant) and cosmic strings (subdominant). However, the cosmological evolution (gradually) results in the scaling of any long-cosmic-string network and, therefore, after some time (Δτ) the Universe enters in the pure-radiation epoch. The evolution of the non-oscillatory GW modes during Δτ, results in the distortion of the low-frequency part of the stochastic GW power-spectrum, which, therefore, departs from scale invariance (anticipated in the pure-radiation case). As regards the corresponding high-frequency part (which is determined by the evolution of the oscillating modes), we find that the presence of cosmic strings gives rise to the quantum-gravitational creation of gravitons, leading to the amplification of the GW signal by (almost) two orders of magnitude.

Imprints of cosmic strings on the cosmological gravitational wave background

The equation which governs the temporal evolution of a gravitational wave (GW) in curved space-time can be treated as the Schrodinger equation for a particle moving in the presence of an effective potential. When GWs propagate in an expanding Universe with constant effective potential, there is a critical value (k_c) of the comoving wave-number which discriminates the metric perturbations into oscillating (k > k_c) and non-oscillating (k < k_c) modes. As a consequence, if the non-oscillatory modes are outside the horizon they do not freeze out. The effective potential is reduced to a non-vanishing constant in a cosmological model which is driven by a two-component fluid, consisting of radiation (dominant) and cosmic strings (sub-dominant). It is known that the cosmological evolution gradually results in the scaling of a cosmic-string network and, therefore, after some time (\Dl \ta) the Universe becomes radiation-dominated. The evolution of the non-oscillatory GW modes during \Dl \ta (while they were outside the horizon), results in the distortion of the GW power spectrum from what it is anticipated in a pure radiation-model, at present-time frequencies in the range 10^{-16} Hz < f < 10^5 Hz.

Dynamics of superhorizon photons during inflation with vacuum polarization

We study asymptotic dynamics of photons propagating in the polarized vacuum of a locally de Sitter Universe. The origin of the vacuum polarization is fluctuations of a massless, minimally coupled, scalar, which we model by the one-loop vacuum polarization tensor of scalar electrodynamics. We show that late time dynamics of the electric field on superhorizon scales approaches that of an Airy oscillator. The magnetic field amplitude, on the other hand, asymptotically approaches a nonvanishing constant (plus an exponentially small oscillatory component), which is suppressed with respect to the initial (vacuum) amplitude. This implies that the asymptotic photon dynamics is more intricate than that of a massive photon obeying the local Proca equation.

Dynamics of superhorizon photons during inflation with vacuum polarization

We study asymptotic dynamics of photons propagating in the polarized vacuum of a locally de Sitter Universe. The origin of the vacuum polarization is fluctuations of a massless, minimally coupled, scalar, which we model by the one-loop vacuum polarization tensor of scalar electrodynamics. We show that late time dynamics of the electric field on superhorizon scales approaches that of an Airy oscillator. The magnetic field amplitude, on the other hand, asymptotically approaches a nonvanishing constant (plus an exponentially small oscillatory component), which is suppressed with respect to the initial (vacuum) amplitude. This implies that the asymptotic photon dynamics is more intricate than that of a massive photon obeying the local Proca equation.

Solution of the Goldfish N-Body Problem in the Plane with (Only) Nearest-Neighbor Coupling Constants All Equal to Minus One Half

The (Hamiltonian, rotation- and translation-invariant) “goldfish” N-body problem in the plane is characterized by the Newtonian equations of motion written here in their complex version, entailing the identification of the real “physical” plane with the complex plane. In this paper we exhibit in completely explicit form the solution of the initial-value problem for this N-body model in the special case in which the two-body interaction only acts among “nearest neighbors” (namely, only among particles whose labels differ by one unit: a n,m = 0 unless |n − m| = 1) and the corresponding coupling constants all equal minus one half, a n,n+1 = a n+1,n = −1/2, n = 1, 2, ..., N − 1. This result implies that, if ω is a real nonvanishing constant, say, without loss of generality, ω > 0, then all the solutions of this N-body model are completely periodic indeed isochronous with period T = 2π / ω. An analogous conclusion holds as well for the model in which also the first and last particle interact with the same coupling constant, namely a 1,N = a N,1 = −1/2 (rather than vanishing).

Chern-Simons modification of general relativity

General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate ${v}_{\ensuremath{\mu}}$---an external quantity, which we fix to be a nonvanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo)tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed or enhanced by the extension.

Optimized effective potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potential

The optimized effective potential (OEP) is the exact Kohn-Sham potential for explicitly orbital-dependent energy functionals, e.g., the exact exchange energy. We give a proof for the OEP equation which does not depend on the chain rule for functional derivatives and directly yields the equation in its simplest form: a certain first-order density shift must vanish. This condition explains why the highest-occupied orbital energies of Hartree-Fock and exact exchange OEP are so close. More importantly, we show that the exact OEP can be constructed iteratively from the first-order shifts of the Kohn-Sham orbitals and that these can be calculated easily. The exact exchange potential ${v}_{\mathrm{x}}(\mathbf{r})$ for spherical atoms and three-dimensional sodium clusters is calculated. Its long-range asymptotic behavior is investigated, including the approach of ${v}_{\mathrm{x}}(\mathbf{r})$ to a nonvanishing constant in particular spatial directions. We calculate total and orbital energies and static electric dipole polarizabilities for the sodium clusters employing the exact exchange functional. Exact OEP results are compared to the Krieger-Li-Iafrate and local density approximations.

P−p′ system with B field and projection operator noncommutative solitons

We study the system of the Dp'-brane with Dp-brane (p<p') inside, in the case where B_{ij} field is a nonvanishing constant. In order to understand how the Dp-brane is viewed from the Dp'-brane worldvolume theory, we investigate the process in which the Dp-brane is probed with p'-p' open string. We calculate the scattering amplitudes among p-p' open strings and p'-p' open strings and show that not only the Weyl transform of the projection operator onto the ground state but also those onto higher excited states emerge as multiplicative factors of the amplitudes.