Weak-lensing observables in relativistic N-body simulations

ABSTRACT We present a numerical weak-lensing analysis that is fully relativistic and non-perturbative for the scalar part of the gravitational potential and first order in the vector part, frame dragging. Integrating the photon geodesics backwards from the observer to the emitters, we solve the Sachs optical equations and study in detail the weak-lensing convergence, ellipticity and rotation. For the first time, we apply such an analysis to a high-resolution relativistic N-body simulation, which consistently includes the leading-order corrections due to general relativity on both large and small scales. These are related to the question of gauge choice and to post-Newtonian corrections, respectively. We present the angular power spectra and one-point probability distribution functions for the weak-lensing variables, which we find are broadly in agreement with comparable Newtonian simulations. Our geometric approach, however, is more robust and flexible, and can therefore be applied consistently to non-standard cosmologies and modified theories of gravity.

The Research on the Application of Intelligent Robots in Small Storage and Logistics Based on Improved PID Control Algorithm

Studying on PID control algorithm and through the analysis of the control parameter of scalar part, integration part and derivation part, the author investigates systematically on the motion control system of the intelligent robots in small storage and logistics, and presents an improved PID control algorithm. The improved PID control algorithm not only maintains the merits of the original one, but also simplifies the multifarious adjustment process of the control parameters of scalar part, integration part and derivation part. The improved algorithm turns out to be effective after the simulation and verification by the MATLAB and can be applied and promoted to other practical controls field.

Evaluation of supertraces for a model operator

This chapter evaluates the supertrace of the heat kernel of a hypoelliptic operator acting over p × g, given a semisimple element γ‎ ∈ G. It begins by introducing a hypoelliptic operator, its heat kernel, and a corresponding supertrace Jᵧ(Y₀ᵗ), if Y₀ᵗ ∈ t(γ‎). Then, by a conjugation of the hypoelliptic operator, the chapter obtains a simpler operator where p × p and t have been decoupled. This new operator splits naturally into a scalar part and a matrix part. Hereafter, the chapter evaluates the trace of the heat kernel of the scalar part, and computes the supertrace of the matrix part of the heat kernel. This chapter concludes with an explicit formula for Jᵧ(Y₀ᵗ).

A COVARIANT APPROACH TO THE GENERALIZED MULTI-INFLATON COSMOLOGICAL PERTURBATION

Following the formalism developed in Astrophys. J.375, 443 (1991), differential equations for the gauge invariant scalar part of the metric perturbation in the Friedmann–Robertson–Walker background with multiple inflatons with arbitrary field metric are obtained without any specific choice of gauge. Subsequently, an algorithm for the solution of these equations in the slow-roll approximation is given without any prior choice of the basis system in the field manifold. Vector and tensor perturbations are also briefly reviewed.

RELATIVISTIC QUANTUM FIELD THEORY FOR CONDENSED SYSTEMS-(III): AN EXPLICIT EXPRESSION OF ATOMIC SCHWINGER–DYSON METHOD

A new explicit expressions of self-energies Πμν and Σ are introduced for photons and electrons based on the particle-hole-antiparticle representation (PHA) of Atomic Schwinger–Dyson formalism (ASD). The PHA representation describes exactly the physical processes such as particle-hole excitations (electron-hole) and particle-antiparticle excitations (electron-positron). The self-energy Σ includes both the quantum component and the classical component (classical external field and Coulomb's field), which are divided into the scalar part Σs and 4-dimensional vector parts Σ0, Σj. The electron propagators are composed of the particle part, hole part and antiparticle part in PHA representation. The general representation of photon self-energy Πμν with 16 elements is expressed in terms of only two components (transverse and longitudinal) Πt and Πl. The general form of the photon propagators are written in terms of free propagator D0 and two independent propagators Dl and Dt, which include two independent photon self-energies. The tensor part of the electron self-energy does not appear in ASD formalism which makes perfectly the closed self-consistent system, when we take the bare vertex approximation, Γμ→γμ.

EXACT SOLUTIONS OF THE DIRAC EQUATION FOR MODIFIED COULOMBIC POTENTIALS

Exact solutions are found for the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic potentials, with the scalar part dominating, can be chosen to give exact analytic Dirac wave functions. The method works for the ground state or for the lowest orbital state with l=j-½, for any j.

A SIMPLE DIRAC WAVE FUNCTION FOR A COULOMB POTENTIAL WITH LINEAR CONFINEMENT

A simple analytical solution is found to the Dirac equation for the combination of a Coulomb potential with a linear confining potential. An appropriate linear combination of Lorentz scalar and vector linear potentials, with the scalar part dominating, can be chosen to give a simple Dirac wave function. The binding energy depends only on the Coulomb strength and is not affected by the linear potential. The method works for the ground state, or for the lowest state with l=j-1/2, for any j.

On the spectra of prespectral operators

The spectrum of a prespectral operator was investigated by Dowson in [4]. The question was left open there whether, if a prespectral operator has closed range, the same is true for its scalar part. In this paper we answer this in the affirmative and point out some consequences concerning the essential spectra of prespectral operators. Also, following Taylor and Halberg [11], we present the state diagram of a prespectral operator, which will show, in a sense, the sharpness of the results of the spectral theory of such operators.