scholarly journals Estimating linear and quadratic forms via indirect observations

Bernoulli ◽  
2020 ◽  
Vol 26 (4) ◽  
pp. 2639-2669
Author(s):  
Anatoli Juditsky ◽  
Arkadi Nemirovski
Keyword(s):  
Quadratic Forms ◽  
Linear And Quadratic Forms

scholarly journals Dynamics of f(R) gravity models and asymmetry of time

2018 ◽  
Vol 27 (02) ◽  
pp. 1850002 ◽  
Author(s):  
Murli Manohar Verma ◽  
Bal Krishna Yadav
Keyword(s):  
Fixed Points ◽  
Quadratic Forms ◽  
Scale Factor ◽  
Gravity Models ◽  
Field Equations ◽  
Classical Level ◽  
The Matrix ◽  
The Stability ◽  
Effects Of Radiation ◽  
Linear And Quadratic Forms

We solve the field equations of modified gravity for [Formula: see text] model in metric formalism. Further, we obtain the fixed points of the dynamical system in phase-space analysis of [Formula: see text] models, both with and without the effects of radiation. The stability of these points is studied against the perturbations in a smooth spatial background by applying the conditions on the eigenvalues of the matrix obtained in the linearized first-order differential equations. Following this, these fixed points are used for analyzing the dynamics of the system during the radiation, matter and acceleration-dominated phases of the universe. Certain linear and quadratic forms of [Formula: see text] are determined from the geometrical and physical considerations and the behavior of the scale factor is found for those forms. Further, we also determine the Hubble parameter [Formula: see text], the Ricci scalar [Formula: see text] and the scale factor [Formula: see text] for these cosmic phases. We show the emergence of an asymmetry of time from the dynamics of the scalar field exclusively owing to the [Formula: see text] gravity in the Einstein frame that may lead to an arrow of time at a classical level.

IV.—On the Independence of Linear and Quadratic Forms in Samples of Normally Distributed Variates

1940 ◽  
Vol 60 (1) ◽  
pp. 40-46 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Quadratic Forms ◽  
Current Mode ◽  
The Mean ◽  
Normal Law ◽  
Image Position ◽  
Linear And Quadratic Forms ◽  
Normally Distributed

It is well known, and fundamentally important in the justification of the current mode of deriving Student's t-distribution, that the mean m of n values xi all obeying the same normal law of probability, and the associated estimate of variance, namelyare statistically independent.

scholarly journals Freeness of linear and quadratic forms in von Neumann algebras

2011 ◽  
Vol 261 (10) ◽  
pp. 2829-2844 ◽  
Author(s):  
G.P. Chistyakov ◽  
F. Götze ◽  
F. Lehner
Keyword(s):  
Quadratic Forms ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Linear And Quadratic Forms
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