16 research outputs found
Methodological derivation of the eikonal equation
Usually, when working with the eikonal equation, reference is made to its derivation in the monograph by Born and Wolf. The derivation of this equation was done rather carelessly. Understanding this derivation requires a certain number of implicit assumptions. For a better understanding of the eikonal approximation and for methodological purposes, the authors decided to repeat the derivation of the eikonal equation, explicating all possible assumptions. Methodically, the following algorithm for deriving the eikonal equation is proposed. The wave equation is derived from Maxwellβs equation. In this case, all conditions are explicitly introduced under which it is possible to do this. Further, from the wave equation, the transition to the Helmholtz equation is carried out. From the Helmholtz equation, with the application of certain assumptions, a transition is made to the eikonal equation. After analyzing all the assumptions and steps, the transition from the Maxwellβs equations to the eikonal equation is actually implemented. When deriving the eikonal equation, several formalisms are used. The standard formalism of vector analysis is used as the first formalism. Maxwellβs equations and the eikonal equation are written as three-dimensional vectors. After that, both the Maxwellβs equations and the eikonal equation use the covariant 4-dimensional formalism. The result of the work is a methodically consistent description of the eikonal equation
Chronology of the development of Active Queue Management algorithms of RED family. Part 1: from 1993 up to 2005
This work is the first part of a large bibliographic review of active queue management algorithms of the Random Early Detection (RED) family, presented in the scientific press from 1993 to 2023. The first part will provide data on algorithms published from 1993 to 2005
Π’Π΅Π½Π·ΠΎΡ ΠΏΡΠΎΠ½ΠΈΡΠ°Π΅ΠΌΠΎΡΡΠ΅ΠΉ Π² Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·ΠΎΠ²Π°Π½Π½ΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ ΠΠ°ΠΊΡΠ²Π΅Π»Π»Π°
It is generally accepted that the main obstacle to the application of
Riemannian geometrization of Maxwellβs equations is an insufficient number of parameters defining a geometrized medium. In the classical description of the equations of electrodynamics in the medium, a constitutive tensor with 20 components is used. With Riemannian geometrization, the constitutive tensor is constructed from a Riemannian metric tensor having 10 components. It is assumed that this discrepancy prevents the application of Riemannian geometrization of Maxwellβs equations. It is necessary to study the scope of applicability of the Riemannian geometrization of Maxwellβs equations. To determine whether the lack of components is really critical
for the application of Riemannian geometrization. To determine the applicability of Riemannian geometrization, the most common variants of electromagnetic media are considered. The structure of the dielectric and magnetic permittivity is written out for them, the number of significant components for these tensors is determined.
Practically all the considered types of electromagnetic media require less than ten parameters to describe the constitutive tensor. In the Riemannian geometrization of Maxwellβs equations, the requirement of a single impedance of the medium is critical. It is possible to circumvent this limitation by moving from the complete Maxwellβs equations to the approximation of geometric optics. The Riemannian geometrization of Maxwellβs equations is applicable to a wide variety of media types, but only for approximating geometric optics.Π‘ΡΠΈΡΠ°Π΅ΡΡΡ, ΡΡΠΎ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌ ΠΏΡΠ΅ΠΏΡΡΡΡΠ²ΠΈΠ΅ΠΌ ΠΊ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΡ ΡΠΈΠΌΠ°Π½ΠΎΠ²ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·Π°ΡΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΠ°ΠΊΡΠ²Π΅Π»Π»Π° ΡΠ²Π»ΡΠ΅ΡΡΡ Π½Π΅Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΠΎΠ΅ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²ΠΎ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ², Π·Π°Π΄Π°ΡΡΠΈΡ
Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·ΠΎΠ²Π°Π½Π½ΡΡ ΡΡΠ΅Π΄Ρ. ΠΡΠΈ ΠΊΠ»Π°ΡΡΠΈΡΠ΅ΡΠΊΠΎΠΌ ΠΎΠΏΠΈΡΠ°Π½ΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΡΠ»Π΅ΠΊΡΡΠΎΠ΄ΠΈΠ½Π°ΠΌΠΈΠΊΠΈ Π² ΡΡΠ΅Π΄Π΅ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΡΡΡ ΡΠ΅Π½Π·ΠΎΡ ΠΏΡΠΎΠ½ΠΈΡΠ°Π΅ΠΌΠΎΡΡΠ΅ΠΉ, ΠΈΠΌΠ΅ΡΡΠΈΠΉ 20 ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½Ρ. ΠΡΠΈ ΡΠΈΠΌΠ°Π½ΠΎΠ²ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·Π°ΡΠΈΠΈ ΡΠ΅Π½Π·ΠΎΡ ΠΏΡΠΎΠ½ΠΈΡΠ°Π΅ΠΌΠΎΡΡΠ΅ΠΉ ΡΡΡΠΎΠΈΡΡΡ ΠΈΠ· ΡΠΈΠΌΠ°Π½ΠΎΠ²ΠΎΠ³ΠΎ ΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΡΠ΅Π½Π·ΠΎΡΠ°, ΠΈΠΌΠ΅ΡΡΠ΅Π³ΠΎ ΡΠΎΠ»ΡΠΊΠΎ 10 ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½Ρ.
ΠΡΠ΅Π΄ΠΏΠΎΠ»Π°Π³Π°Π΅ΡΡΡ, ΡΡΠΎ Π΄Π°Π½Π½ΠΎΠ΅ Π½Π΅ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΠΈΠ΅ ΠΌΠ΅ΡΠ°Π΅Ρ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΡ ΡΠΈΠΌΠ°Π½ΠΎΠ²ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·Π°ΡΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΠ°ΠΊΡΠ²Π΅Π»Π»Π°. Π ΡΡΠ°ΡΡΠ΅ ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ΠΎ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΠΈΡΡ, Π΄Π΅ΠΉΡΡΠ²ΠΈΡΠ΅Π»ΡΠ½ΠΎ Π»ΠΈ Π½Π΅Π΄ΠΎΡΡΠ°ΡΠΎΠΊ ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½Ρ ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΌ Π΄Π»Ρ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΡ ΡΠΈΠΌΠ°Π½ΠΎΠ²ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·Π°ΡΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΠ°ΠΊΡΠ²Π΅Π»Π»Π°. ΠΠ»Ρ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ ΠΎΠ±Π»Π°ΡΡΠΈ ΠΏΡΠΈΠΌΠ΅Π½ΠΈΠΌΠΎΡΡΠΈ ΡΠΈΠΌΠ°Π½ΠΎΠ²ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·Π°ΡΠΈΠΈ ΡΠ°ΡΡΠΌΠΎΡΡΠ΅Π½Ρ Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ ΡΠ°ΡΠΏΡΠΎΡΡΡΠ°Π½ΡΠ½Π½ΡΠ΅ Π²Π°ΡΠΈΠ°Π½ΡΡ ΡΠ»Π΅ΠΊΡΡΠΎΠΌΠ°Π³Π½ΠΈΡΠ½ΡΡ
ΡΡΠ΅Π΄. ΠΠ»Ρ Π½ΠΈΡ
Π²ΡΠΏΠΈΡΠ°Π½Π° ΡΡΡΡΠΊΡΡΡΠ° Π΄ΠΈΡΠ»Π΅ΠΊΡΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΈ ΠΌΠ°Π³Π½ΠΈΡΠ½ΠΎΠΉ ΠΏΡΠΎΠ½ΠΈΡΠ°Π΅ΠΌΠΎΡΡΠ΅ΠΉ, ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΎ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²ΠΎ Π·Π½Π°ΡΠ°ΡΠΈΡ
ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½Ρ Π΄Π»Ρ ΡΡΠΈΡ
ΡΠ΅Π½Π·ΠΎΡΠΎΠ². ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈ Π²ΡΠ΅ ΡΠ°ΡΡΠΌΠΎΡΡΠ΅Π½Π½ΡΠ΅ ΡΠΈΠΏΡ ΡΠ»Π΅ΠΊΡΡΠΎΠΌΠ°Π³Π½ΠΈΡΠ½ΡΡ
ΡΡΠ΅Π΄ ΡΡΠ΅Π±ΡΡΡ ΠΌΠ΅Π½Π΅Π΅ Π΄Π΅ΡΡΡΠΈ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² Π΄Π»Ρ ΠΎΠΏΠΈΡΠ°Π½ΠΈΡ ΡΠ΅Π½Π·ΠΎΡΠ° ΠΏΡΠΎΠ½ΠΈΡΠ°Π΅ΠΌΠΎΡΡΠ΅ΠΉ. ΠΡΠΈ ΡΠΈΠΌΠ°Π½ΠΎΠ²ΠΎΠΉ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·Π°ΡΠΈΠΈ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΠ°ΠΊΡΠ²Π΅Π»Π»Π° ΠΊΡΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΌ ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΡΠ΅Π±ΠΎΠ²Π°Π½ΠΈΠ΅ Π΅Π΄ΠΈΠ½ΠΈΡΠ½ΠΎΠ³ΠΎ ΠΈΠΌΠΏΠ΅Π΄Π°Π½ΡΠ° ΡΡΠ΅Π΄Ρ. ΠΠ±ΠΎΠΉΡΠΈ Π΄Π°Π½Π½ΠΎΠ΅ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΠ΅ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ ΠΏΡΡΡΠΌ ΠΏΠ΅ΡΠ΅Ρ
ΠΎΠ΄Π° ΠΎΡ ΠΏΠΎΠ»Π½ΡΡ
ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΠ°ΠΊΡΠ²Π΅Π»Π»Π° ΠΊ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΡ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΎΠΏΡΠΈΠΊΠΈ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΡΠΈΠΌΠ°Π½ΠΎΠ²Π° Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΠ·Π°ΡΠΈΡ ΡΡΠ°Π²Π½Π΅Π½ΠΈΠΉ ΠΠ°ΠΊΡΠ²Π΅Π»Π»Π° ΠΏΡΠΈΠΌΠ΅Π½ΠΈΠΌΠ° Π΄Π»Ρ Π±ΠΎΠ»ΡΡΠΎΠ³ΠΎ ΡΠ°Π·Π½ΠΎΠΎΠ±ΡΠ°Π·ΠΈΡ ΡΠΈΠΏΠΎΠ² ΡΡΠ΅Π΄Ρ, Π½ΠΎ ΡΠΎΠ»ΡΠΊΠΎ Π΄Π»Ρ ΠΏΡΠΈΠ±Π»ΠΈΠΆΠ΅Π½ΠΈΡ Π³Π΅ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΎΠΏΡΠΈΠΊΠΈ
ΠΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΠΈ ΡΠ·ΡΠΊΠ° Julia Π΄Π»Ρ ΠΎΠ±ΡΠ°Π±ΠΎΡΠΊΠΈ ΡΡΠ°ΡΠΈΡΡΠΈΡΠ΅ΡΠΊΠΈΡ Π΄Π°Π½Π½ΡΡ
The Julia programming language is a specialized language for scientific computing. It is relatively new, so most of the libraries for it are in the active development stage. In this article, the authors consider the possibilities of the language in the field of mathematical statistics. Special emphasis is placed on the technical component, in particular, the process of installing and configuring the software environment is described in detail. Since users of the Julia language are often not professional programmers, technical issues in setting up the software environment can cause difficulties that prevent them from quickly mastering the basic features of the language. The article also describes some features of Julia that distinguish it from other popular languages used for scientific computing. The third part of the article provides an overview of the two main libraries for mathematical statistics. The emphasis is again on the technical side in order to give the reader an idea of the general possibilities of the language in the field of mathematical statistics.Π―Π·ΡΠΊ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΡ Julia ΡΠ²Π»ΡΠ΅ΡΡΡ ΡΠΏΠ΅ΡΠΈΠ°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΠΌ ΡΠ·ΡΠΊΠΎΠΌ Π΄Π»Ρ Π½Π°ΡΡΠ½ΡΡ
Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ. Π―Π·ΡΠΊ ΡΡΠ°Π²Π½ΠΈΡΠ΅Π»ΡΠ½ΠΎ Π½ΠΎΠ²ΡΠΉ, ΠΏΠΎΡΡΠΎΠΌΡ Π±ΠΎΠ»ΡΡΠΈΠ½ΡΡΠ²ΠΎ
Π±ΠΈΠ±Π»ΠΈΠΎΡΠ΅ΠΊ Π΄Π»Ρ Π½Π΅Π³ΠΎ Π½Π°Ρ
ΠΎΠ΄ΠΈΡΡΡ Π² Π°ΠΊΡΠΈΠ²Π½ΠΎΠΉ ΡΡΠ°Π΄ΠΈΠΈ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠΊΠΈ. Π ΡΡΠ°ΡΡΠ΅ Π°Π²ΡΠΎΡΡ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°ΡΡ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΠΈ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΡ ΡΠ·ΡΠΊΠ° Π² ΠΎΠ±Π»Π°ΡΡΠΈ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΡΠ°ΡΠΈΡΡΠΈΠΊΠΈ. ΠΡΠΎΠ±ΡΠΉ Π°ΠΊΡΠ΅Π½Ρ Π΄Π΅Π»Π°Π΅ΡΡΡ Π½Π° ΡΠ΅Ρ
Π½ΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΎΡΡΠ°Π²Π»ΡΡΡΠ΅ΠΉ, Π² ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ ΠΏΠΎΠ΄ΡΠΎΠ±Π½ΠΎ ΠΎΠΏΠΈΡΡΠ²Π°Π΅ΡΡΡ ΠΏΡΠΎΡΠ΅ΡΡ ΡΡΡΠ°Π½ΠΎΠ²ΠΊΠΈ ΠΈ Π½Π°ΡΡΡΠΎΠΉΠΊΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠ½ΠΎΠ³ΠΎ ΠΎΠΊΡΡΠΆΠ΅Π½ΠΈΡ. Π’Π°ΠΊ ΠΊΠ°ΠΊ ΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΠ΅Π»ΠΈ ΡΠ·ΡΠΊΠ° Julia Π·Π°ΡΠ°ΡΡΡΡ Π½Π΅ ΡΠ²Π»ΡΡΡΡΡ ΠΏΡΠΎΡΠ΅ΡΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΠΌΠΈ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΡΠ°ΠΌΠΈ, ΡΠ΅Ρ
Π½ΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΌΠΎΠΌΠ΅Π½ΡΡ Π² Π½Π°ΡΡΡΠΎΠΉΠΊΠ΅ ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠ½ΠΎΠ³ΠΎ ΠΎΠΊΡΡΠΆΠ΅Π½ΠΈΡ ΠΌΠΎΠ³ΡΡ Π²ΡΠ·ΡΠ²Π°ΡΡ Ρ Π½ΠΈΡ
ΡΡΡΠ΄Π½ΠΎΡΡΠΈ, ΠΏΡΠ΅ΠΏΡΡΡΡΠ²ΡΡΡΠΈΠ΅ Π±ΡΡΡΡΠΎΠΌΡ ΠΎΡΠ²ΠΎΠ΅Π½ΠΈΡ Π±Π°Π·ΠΎΠ²ΡΡ
Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΠ΅ΠΉ ΡΠ·ΡΠΊΠ°. Π ΡΡΠ°ΡΡΠ΅ ΠΎΠΏΠΈΡΡΠ²Π°ΡΡΡΡ Π½Π΅ΠΊΠΎΡΠΎΡΡΠ΅ ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎΡΡΠΈ Julia, ΠΊΠΎΡΠΎΡΡΠ΅ ΠΎΡΠ»ΠΈΡΠ°ΡΡ Π΅Π³ΠΎ ΠΎΡ Π΄ΡΡΠ³ΠΈΡ
ΠΏΠΎΠΏΡΠ»ΡΡΠ½ΡΡ
ΡΠ·ΡΠΊΠΎΠ², ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΡ
Π΄Π»Ρ Π½Π°ΡΡΠ½ΡΡ
Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ. Π’Π°ΠΊΠΆΠ΅ Π΄Π°ΡΡΡΡ ΠΎΠ±Π·ΠΎΡ Π΄Π²ΡΡ
ΠΎΡΠ½ΠΎΠ²Π½ΡΡ
Π±ΠΈΠ±Π»ΠΈΠΎΡΠ΅ΠΊ Π΄Π»Ρ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΡΠ°ΡΠΈΡΡΠΈΠΊΠΈ. Π£ΠΏΠΎΡ ΠΎΠΏΡΡΡ-ΡΠ°ΠΊΠΈ Π΄Π΅Π»Π°Π΅ΡΡΡ Π½Π° ΡΠ΅Ρ
Π½ΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΡΠΎΡΠΎΠ½Π΅, ΡΡΠΎΠ±Ρ Π΄Π°ΡΡ ΡΠΈΡΠ°ΡΠ΅Π»Ρ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΠ΅ ΠΎΠ± ΠΎΠ±ΡΠΈΡ
Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΡΡ
ΡΠ·ΡΠΊΠ° Π² ΠΎΠ±Π»Π°ΡΡΠΈ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΡΠ°ΡΠΈΡΡΠΈΠΊΠΈ