The document discusses Wiener–Hopf factorization of functionally commutative matrix functions. It is shown that for a matrix function A(t) defined on the contour Γ, which consists of two parts D+ and D- intersecting only at infinity, the factorization A(t) = A+(t)A-(t) is possible, where A+(t) is defined on D+ and A-(t) is defined on D-. Methods for finding such a factorization are considered, in particular, using the properties of the Fourier transform. It is noted that this factorization allows solving various boundary value problems, including those arising in physics.
Strategy and work programs on reform and modernization of The General Department of Customs and Excise of Combodia (2014-2018).
The economic growth and achievements of Cambodia over recent years through the successful implementation of the Rectangular Strategy Phase I and II are testimony to the
sustainable development in all areas. Meanwhile, the General Department of Customs and Excise of Cambodia implemented its Strategy and Work Program on Reform and Modernization (SWRM) 2014-2018 with fruitful results. While much has been accomplished, the work of reform and modernization is a continuing process as the General Department of Customs and Excise prepares to face additional challenges and opportunities in the years ahead.
Introduce by www.vietxnk.com
Strategy and work programs on reform and modernization of The General Department of Customs and Excise of Combodia (2014-2018).
The economic growth and achievements of Cambodia over recent years through the successful implementation of the Rectangular Strategy Phase I and II are testimony to the
sustainable development in all areas. Meanwhile, the General Department of Customs and Excise of Cambodia implemented its Strategy and Work Program on Reform and Modernization (SWRM) 2014-2018 with fruitful results. While much has been accomplished, the work of reform and modernization is a continuing process as the General Department of Customs and Excise prepares to face additional challenges and opportunities in the years ahead.
Introduce by www.vietxnk.com
Formation of the Tree of Life - 2010 EditionAstroQab
This book reveals the secret of how the kabbalistic Tree of Life diagram is systematically built in a series of steps from Ain Sof to Malkut. It shows how each step involves combining the twenty-two Hebrew letters in certain conformations to produce the ten sefirot and the twenty-two pathways. The end-result includes a new path configuration that solves many mysteries of the Sefer Yetzirah and the Zohar. During the course of the book information is drawn from the ancient texts: Sefer Berashit, Sefer Yetzirah, and the Zohar (i.e. Sefer Dtzenioutha, Ha Idra Rabba Qadisha, and Ha Idra Zuta Qadisha).
This book explores (using polyhex mathematics) the special relationship that exists between the kabbalistic Tree of Life diagram and the geometry of the simple hexagon. It reveals how the ten sefirot and the twenty-two pathways of the Tree of Life are each individually and uniquely linked to the hexagonal form. The Hebrew alphabet is also shown to be esoterically associated with the geometry of the hexagon. This book consists of 76 pages and 66 diagrams.
[New Chapter - The GLGL Wheel.] The Sefer Yetzirah ('Book of Formation') is the ancient source text from which the contemporary design of the kabbalistic Tree of Life diagram has evolved. The original 'Tree of Yetzirah' design however, is different in certain important respects to our modern day versions of the Tree diagram. Sefer Yetzirah Magic includes an exposition of the original design of the Tree and develops a system of magic (and 'initiation') that's based on the Sefer Yetzirah's original metaphysical blueprint.
The way we measure time and space reflects the limited perception of the rational mind. The rational mind is born out of the physical world and provides a natural (albeit limited) interface between human consciousness and the physical realm. This book seeks to stimulate a more esoteric understanding of time and space by examining the basic precepts upon which our standard measurement of the universe is based. The aim is to promote a more intuitive perception of the universe by highlighting the esoteric 1/7 and 4/Pi principles of existence.
In this doctoral thesis, a classical, lumped-element model is used to study the cochlea and to simulate click-evoked and spontaneous OAEs. The original parameter values describing the microscopic structures of the cochlea are re-tuned to match several key features of the cochlear response in humans. The frequency domain model is also recast in a formulation known as state space; this permits the calculation of linear instabilities given random perturbations in the cochlea which are predicted to produce spontaneous OAEs. The averaged stability results of an ensemble of randomly perturbed models have been published in [(2008) ‘Statistics of instabilities in a state space model of the human cochlea,’ J. Acoust. Soc. Am. 124(2), 1068-1079]. These findings support one of the prevailing theories of SOAE generation.
Formation of the Tree of Life - 2010 EditionAstroQab
This book reveals the secret of how the kabbalistic Tree of Life diagram is systematically built in a series of steps from Ain Sof to Malkut. It shows how each step involves combining the twenty-two Hebrew letters in certain conformations to produce the ten sefirot and the twenty-two pathways. The end-result includes a new path configuration that solves many mysteries of the Sefer Yetzirah and the Zohar. During the course of the book information is drawn from the ancient texts: Sefer Berashit, Sefer Yetzirah, and the Zohar (i.e. Sefer Dtzenioutha, Ha Idra Rabba Qadisha, and Ha Idra Zuta Qadisha).
This book explores (using polyhex mathematics) the special relationship that exists between the kabbalistic Tree of Life diagram and the geometry of the simple hexagon. It reveals how the ten sefirot and the twenty-two pathways of the Tree of Life are each individually and uniquely linked to the hexagonal form. The Hebrew alphabet is also shown to be esoterically associated with the geometry of the hexagon. This book consists of 76 pages and 66 diagrams.
[New Chapter - The GLGL Wheel.] The Sefer Yetzirah ('Book of Formation') is the ancient source text from which the contemporary design of the kabbalistic Tree of Life diagram has evolved. The original 'Tree of Yetzirah' design however, is different in certain important respects to our modern day versions of the Tree diagram. Sefer Yetzirah Magic includes an exposition of the original design of the Tree and develops a system of magic (and 'initiation') that's based on the Sefer Yetzirah's original metaphysical blueprint.
The way we measure time and space reflects the limited perception of the rational mind. The rational mind is born out of the physical world and provides a natural (albeit limited) interface between human consciousness and the physical realm. This book seeks to stimulate a more esoteric understanding of time and space by examining the basic precepts upon which our standard measurement of the universe is based. The aim is to promote a more intuitive perception of the universe by highlighting the esoteric 1/7 and 4/Pi principles of existence.
In this doctoral thesis, a classical, lumped-element model is used to study the cochlea and to simulate click-evoked and spontaneous OAEs. The original parameter values describing the microscopic structures of the cochlea are re-tuned to match several key features of the cochlear response in humans. The frequency domain model is also recast in a formulation known as state space; this permits the calculation of linear instabilities given random perturbations in the cochlea which are predicted to produce spontaneous OAEs. The averaged stability results of an ensemble of randomly perturbed models have been published in [(2008) ‘Statistics of instabilities in a state space model of the human cochlea,’ J. Acoust. Soc. Am. 124(2), 1068-1079]. These findings support one of the prevailing theories of SOAE generation.
Sign up for the Certified Professional Innovator program at University of Michigan April 28 & 29th, run at the Innovatrium by Professor Jeff DeGraff. Past participants were from Apple, GE, Honeywell, Lenovo, John Hopkins, Toyota and Siemens.... Come get certified with this world class community of Innovators.
El tema de redes es muy amplio y difícil de abarcar en tan poco espacio y es necesario enfocarnos en todo lo bueno que ha hecho por nosotros. En primera ya no existen fronteras por la fácil comunicación. Ayuda a muchas empresas ya que al subir su producto en Internet pueden abarcar zonas inimaginables hace unos años. En conclusión las redes han logrado facilitar nuestras vidas y no sabemos lo que nos va a traer el día de mañana. Debemos disfrutar de todas sus ventajas y beneficios sin olvidarnos de lo más importante, nosotros mismos.
In order to support information regarding arthritis in examinees in the study, x-rays of the wrists
and hands, and knees will be conducted on all examinees sixty years of age and above. The x-rays will be
taken in the following positions and sequence
In order to support information regarding arthritis in examinees in the study, x-rays of the wrists
and hands, and knees will be conducted on all examinees sixty years of age and above. The x-rays will be
taken in the following positions and sequence.
_________________________________________________________________
Windows Live™: Keep your life in sync.
http://windowslive.com/explore?ocid=TXT_TAGLM_BR_life_in_synch_052009
Сытник В. С. Основы расчета и анализа точности геодезических измерений в стро...Иван Иванов
В книге изложены вопросы теории и практики расчета, бценки
и анализа точности геодезических измерений, выполняемых при
возведении промышленных, жилых и общественных зданий й\цн-
женериых сооружений. На основе существующих в теории вероят^~—-
ностей
математической статистики и ошибок измерений рассмат
риваются методы расчета необходимой и достаточной точности гео
дезических измерений
применительно к определенным стадиям
строительно-монтажных работ и конструктивным решениям зданий
и сооружений. Значительное внимание уделено анализу точности
результатов геодезических измерений
Заковряшин А. И. Конструирование РЭА с учетом особенностей эксплуатацииИван Иванов
Показана роль конструкторского проектирования в обеспечении эффективности технического обслуживания РЭА по фактическому состоянию. В книге
взаимосвязанно решаются вопросы обеспечения ремонто- и контролепригодности
при конструировании РЭА. Ремонтопригодность рассматривается лак решающи”
фактор обеспечения эффективности применения аппаратуры. Область значений
конструктивных показателей РЭА определяется как результат решения задачи
оптимизации заданного качества функционирования.
Improving profitability for small businessBen Wann
In this comprehensive presentation, we will explore strategies and practical tips for enhancing profitability in small businesses. Tailored to meet the unique challenges faced by small enterprises, this session covers various aspects that directly impact the bottom line. Attendees will learn how to optimize operational efficiency, manage expenses, and increase revenue through innovative marketing and customer engagement techniques.
The world of search engine optimization (SEO) is buzzing with discussions after Google confirmed that around 2,500 leaked internal documents related to its Search feature are indeed authentic. The revelation has sparked significant concerns within the SEO community. The leaked documents were initially reported by SEO experts Rand Fishkin and Mike King, igniting widespread analysis and discourse. For More Info:- https://news.arihantwebtech.com/search-disrupted-googles-leaked-documents-rock-the-seo-world/
3.0 Project 2_ Developing My Brand Identity Kit.pptxtanyjahb
A personal brand exploration presentation summarizes an individual's unique qualities and goals, covering strengths, values, passions, and target audience. It helps individuals understand what makes them stand out, their desired image, and how they aim to achieve it.
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
Memorandum Of Association Constitution of Company.pptseri bangash
www.seribangash.com
A Memorandum of Association (MOA) is a legal document that outlines the fundamental principles and objectives upon which a company operates. It serves as the company's charter or constitution and defines the scope of its activities. Here's a detailed note on the MOA:
Contents of Memorandum of Association:
Name Clause: This clause states the name of the company, which should end with words like "Limited" or "Ltd." for a public limited company and "Private Limited" or "Pvt. Ltd." for a private limited company.
https://seribangash.com/article-of-association-is-legal-doc-of-company/
Registered Office Clause: It specifies the location where the company's registered office is situated. This office is where all official communications and notices are sent.
Objective Clause: This clause delineates the main objectives for which the company is formed. It's important to define these objectives clearly, as the company cannot undertake activities beyond those mentioned in this clause.
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Liability Clause: It outlines the extent of liability of the company's members. In the case of companies limited by shares, the liability of members is limited to the amount unpaid on their shares. For companies limited by guarantee, members' liability is limited to the amount they undertake to contribute if the company is wound up.
https://seribangash.com/promotors-is-person-conceived-formation-company/
Capital Clause: This clause specifies the authorized capital of the company, i.e., the maximum amount of share capital the company is authorized to issue. It also mentions the division of this capital into shares and their respective nominal value.
Association Clause: It simply states that the subscribers wish to form a company and agree to become members of it, in accordance with the terms of the MOA.
Importance of Memorandum of Association:
Legal Requirement: The MOA is a legal requirement for the formation of a company. It must be filed with the Registrar of Companies during the incorporation process.
Constitutional Document: It serves as the company's constitutional document, defining its scope, powers, and limitations.
Protection of Members: It protects the interests of the company's members by clearly defining the objectives and limiting their liability.
External Communication: It provides clarity to external parties, such as investors, creditors, and regulatory authorities, regarding the company's objectives and powers.
https://seribangash.com/difference-public-and-private-company-law/
Binding Authority: The company and its members are bound by the provisions of the MOA. Any action taken beyond its scope may be considered ultra vires (beyond the powers) of the company and therefore void.
Amendment of MOA:
While the MOA lays down the company's fundamental principles, it is not entirely immutable. It can be amended, but only under specific circumstances and in compliance with legal procedures. Amendments typically require shareholder
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[Note: This is a partial preview. To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations]
Sustainability has become an increasingly critical topic as the world recognizes the need to protect our planet and its resources for future generations. Sustainability means meeting our current needs without compromising the ability of future generations to meet theirs. It involves long-term planning and consideration of the consequences of our actions. The goal is to create strategies that ensure the long-term viability of People, Planet, and Profit.
Leading companies such as Nike, Toyota, and Siemens are prioritizing sustainable innovation in their business models, setting an example for others to follow. In this Sustainability training presentation, you will learn key concepts, principles, and practices of sustainability applicable across industries. This training aims to create awareness and educate employees, senior executives, consultants, and other key stakeholders, including investors, policymakers, and supply chain partners, on the importance and implementation of sustainability.
LEARNING OBJECTIVES
1. Develop a comprehensive understanding of the fundamental principles and concepts that form the foundation of sustainability within corporate environments.
2. Explore the sustainability implementation model, focusing on effective measures and reporting strategies to track and communicate sustainability efforts.
3. Identify and define best practices and critical success factors essential for achieving sustainability goals within organizations.
CONTENTS
1. Introduction and Key Concepts of Sustainability
2. Principles and Practices of Sustainability
3. Measures and Reporting in Sustainability
4. Sustainability Implementation & Best Practices
To download the complete presentation, visit: https://www.oeconsulting.com.sg/training-presentations
Enterprise Excellence is Inclusive Excellence.pdfKaiNexus
Enterprise excellence and inclusive excellence are closely linked, and real-world challenges have shown that both are essential to the success of any organization. To achieve enterprise excellence, organizations must focus on improving their operations and processes while creating an inclusive environment that engages everyone. In this interactive session, the facilitator will highlight commonly established business practices and how they limit our ability to engage everyone every day. More importantly, though, participants will likely gain increased awareness of what we can do differently to maximize enterprise excellence through deliberate inclusion.
What is Enterprise Excellence?
Enterprise Excellence is a holistic approach that's aimed at achieving world-class performance across all aspects of the organization.
What might I learn?
A way to engage all in creating Inclusive Excellence. Lessons from the US military and their parallels to the story of Harry Potter. How belt systems and CI teams can destroy inclusive practices. How leadership language invites people to the party. There are three things leaders can do to engage everyone every day: maximizing psychological safety to create environments where folks learn, contribute, and challenge the status quo.
Who might benefit? Anyone and everyone leading folks from the shop floor to top floor.
Dr. William Harvey is a seasoned Operations Leader with extensive experience in chemical processing, manufacturing, and operations management. At Michelman, he currently oversees multiple sites, leading teams in strategic planning and coaching/practicing continuous improvement. William is set to start his eighth year of teaching at the University of Cincinnati where he teaches marketing, finance, and management. William holds various certifications in change management, quality, leadership, operational excellence, team building, and DiSC, among others.
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India Orthopedic Devices Market: Unlocking Growth Secrets, Trends and Develop...Kumar Satyam
According to TechSci Research report, “India Orthopedic Devices Market -Industry Size, Share, Trends, Competition Forecast & Opportunities, 2030”, the India Orthopedic Devices Market stood at USD 1,280.54 Million in 2024 and is anticipated to grow with a CAGR of 7.84% in the forecast period, 2026-2030F. The India Orthopedic Devices Market is being driven by several factors. The most prominent ones include an increase in the elderly population, who are more prone to orthopedic conditions such as osteoporosis and arthritis. Moreover, the rise in sports injuries and road accidents are also contributing to the demand for orthopedic devices. Advances in technology and the introduction of innovative implants and prosthetics have further propelled the market growth. Additionally, government initiatives aimed at improving healthcare infrastructure and the increasing prevalence of lifestyle diseases have led to an upward trend in orthopedic surgeries, thereby fueling the market demand for these devices.
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Attending a job Interview for B1 and B2 Englsih learnersErika906060
It is a sample of an interview for a business english class for pre-intermediate and intermediate english students with emphasis on the speking ability.
4. CONTENTS
Mathematics
ADUKOV V.M. About Wiener–Hopf factorization of functionally commutative matrix functions.. 6
KOLYASNIKOV S.A. Units of integral group rings of finite groups with a direct multiplier of
order 3.................................................................................................................................................. 13
LESCHEVICH V.V., PENYAZKOV O.G., ROSTAING J.-C., FEDOROV A.V., SHULGIN A.V.
Experimental and mathematical simulation of auto-ignition of iron micro particles .......................... 21
MARTYUSHEV E.V. Algorithmic solution of the five-point pose problem based on the cayley
representation of rotation matrices....................................................................................................... 31
MEDVEDEV S.V. About extension of homeomorphisms over zero-dimensional homogeneous
spaces................................................................................................................................................... 39
OMELCHENKO E.A., PLEKHANOVA M.V., DAVYDOV P.N. Numerical solution of delayed
linearized quasistationary phase-field system of equations................................................................. 45
RUBINA L.I., UL’YANOV O.N. Towards the differences in behaviour of solutions of linear and
non-linear heat-conduction equations.................................................................................................. 52
SOLDATOVA N.G. About optimum on risks and regrets situations in game of two persons ........... 60
TABARINTSEVA E.V. About solving of an ill-posed problem for a nonlinear differential equa-
tion by means of the projection regularization method........................................................................ 65
TANANA V.P., ERYGINA A.A. About the evaluation of inaccuracy of approximate solution of
the inverse problem of solid state physics ........................................................................................... 72
UKHOBOTOV V.I., TROITSKY A.A. One problem of pulse pursuit............................................... 79
USHAKOV A.L. Updating iterative factorization for the numerical solution of two elliptic equa-
tions of the second order in rectangular area ....................................................................................... 88
Physics
VORONTSOV A.G. Structure amendments in heated metal clusters during its process.................... 94
GUREVICH S.Yu., GOLUBEV E.V., PETROV Yu.V. Lamb waves in ferromagnetic metals: laser
exitation and electromagnetic registration........................................................................................... 99
RIDNYI Ya.M., MIRZOEV A.A., MIRZAEV D.A. Ab-initio simulation of influence of short-
range ordering carbon impurities on the energy of their dissolution in the FCC-iron......................... 108
USHAKOV V.L., PYZIN G.P., BESKACHKO V.P. A method for monitoring small surface
movements of a sessile drop during evaporation................................................................................. 117
Short communications
BEREZIN I.Ya., PETRENKO Yu.O. Problems of vibration safety of operator of industrial tractor . 123
BOLSHAKOV M.V., GUSEVA A.V., KUNDIKOVA N.D., POPKOV I.I. Fiber and interferential
method of obtaining non-homogeneous polarized beam..................................................................... 128
BOLSHAKOV M.V., GUSEVA A.V., KUNDIKOVA N.D., POPKOV I.I. Transformation of the
spin moment into orbital moment in laser beam.................................................................................. 133
BOLSHAKOV M.V., KOMAROVA M.A., KUNDIKOVA N.D. Experimental determination of
multimode optical fiber radiation mode composition.......................................................................... 138
BRYZGALOV A.N., VOLKOV P.V. Influence of heat-treating on microhardness of silica glass
KU-1 .................................................................................................................................................... 143
GERASIMOV A.M., KUNDIKOVA N.D., MIKLYAEV Yu.V., PIKHULYA D.G.,
TERPUGOV M.V. Efficiency of second harmonic generation in one-dimentional photonic crystal
from isotropic material......................................................................................................................... 147
IVANOV S.A. Stability of two-layer recursive neural networks ........................................................ 151
KARITSKAYA S.G., KARITSKIY Ya.D. Fluorescent impurity centers for monitoring the physical
state of polymers and a computer simulation of processes of structuring of the polymer matrices .... 155
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
5. 5
KATKOV M.L. Existence of the fixed point in the case of evenly contractive monotonic operator.. 160
LEYVI A.Ya. Analysis of mechanisms of surface pattern formation during the high-power plasma
stream processing................................................................................................................................. 162
MATVEEVA L.V. Behavior of Gronwall polynomials outside the boundary of summability region 166
KHAYRISLAMOV K.Z. Poiseuille flow of a fluid with variable viscosity....................................... 170
KHAYRISLAMOV M.Z., HERREINSTEIN A.W. Explicit scheme for the solution of third
boundary value problem for quasi-linear heat equation....................................................................... 174
EVNIN A.Yu. Polynomial as a sum of periodic functions.................................................................. 178
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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ij
k g G
I i j
h g g I g g I I
i jn G
χ χ χ χ δ−
= ∈
=
= = = =
≠
. (4)
( . [7,
.3, § 4, 2]) , G . -
(4) , 1
sI−
= .
1
A −
, (2)
, 1−
:
( )1 1
, 1 , , 1
1 1
( ) ( ) ( ).
s s
k
ik kl lj k m ml i k j l
ij
l k l k m
A A h a c g g
n n
χ χ− −
= =
= =
( . [8, §109, (8)]), -
:
1
( ) ( ) ( )
s
k m l
k ml i k i m i l
ik
h h
h c g g g
n
χ χ χ
=
= .
, – .
,
( )1
, 1 1 1
1 1
( ) ( ) ( ) ( ) ( ) ( ).
s s s
m l m
m j l i m i l m i m l i l j lij
i il m m l
h h h
A a g g g a g h g g
n n n n
χ χ χ χ χ χ−
= = =
= =
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9. . . –
- -
2013, 5, 2 9
, (4),
:
( )1
1
( ).
s
ij
m m i m
ij
i m
A a h g
n
δ
χ−
=
=
– , -
G . , -
( )1
ij i
ij
A δ λ−
= 1−
= Λ . .
, [ ]A G∈ ⊗ , G – , 1
1 1 1s sh h n n= = = = = = | |s G= .
, A , 1, , sλ λ –
.
2. (1). – -
Γ , – . -
, ( )W
( )H Γ Γ ( . [1]).
( )a g ( )ga t t∈ ∈Γ . - ( )A t
,
1
( ) ( ) ( ), 1, ,j g j
j g G
t a t g j s
n
λ χ
∈
= = Γ .
ind ( )j j tρ λΓ= – Γ ( )j tλ , -
2π , t Γ .
( ) ( ) ( )j
j j jt t t t
ρ
λ λ λ− +
= – – ( )j tλ .
1 1( ) diag [ ( ) , ( )] diag [ ( ) , ( )]s st t t t tλ λ λ λ− − + +
Λ = , ⋅ ,
– – - ( )tΛ 1
.
2. ( ) ( [ ])A t Z G∈ ⊗ – Γ -
(2). – ( ) ( ) ( ) ( )A t A t d t A t− += :
1 1 1 2 2 1 1
1 1 2 2 2 2 2
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
s s
s s
s s s s s
t g t g t g
t g t g t g
A t
t g t g t g
λ χ λ χ λ χ
λ χ λ χ λ χ
λ χ λ χ λ χ
− − −
− − −
−
− − −
= ,
1( ) diag [ ] ind ( ),s
j jd t t t tρρ
ρ λΓ= , , , =
1 1 1 1 2 1 1 2 1 1
1 2 2 1 2 2 2 2 2 2
1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
s s
s s
s s s s s s s s
h t g h t g h t g
h t g h t g h t g
A t
h t g h t g h t g
λ χ λ χ λ χ
λ χ λ χ λ χ
λ χ λ χ λ χ
+ + +
+ + +
+
+ + +
= .
3.
1. 4G V= – . -
4S :
{ }4 ,(12)(3 4),(13)(2 4),(14)(2 3)V e= ,
2 2C C× . ( )A t
2- , 2 2× - ,
2 2×
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10. . « . . »10
3 41 2
4 32 1
3 4 1 2
4 3 2 1
( ) ( )( ) ( )
( ) ( )( ) ( )
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
a t a ta t a t
a t a ta t a t
A t
a t a t a t a t
a t a t a t a t
= .
4V ( . [7, .3, § 5])
e (12)(3 4) (13)(2 4) (14)(2 3)
1χ 1 1 1 1
2χ 1 1− 1 1−
3χ 1 1 1− 1−
3χ 1 1− 1− 1
, ( )A t
:
1 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = + + + , 2 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = − + − ,
3 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = + − − , 4 1 2 3 4( ) ( ) ( ) ( ) ( )t a t a t a t a tλ = − − + .
( )A t .
2. 3G S= – 3. -
3 :
{ } { } { }1 2 3, (12),(13), (2 3) , (12 3),(13 2 .K e K K= = =
jC 3( [ ])Z S :
1 2 1C 2C 3C
1C 1C 2C 3C
2C 2C 1 33 3C C+ 22C
3C 3C 22C 1 32C +
(1) - ( )A t :
1 2 3
2 1 3 2
3 2 1 3
( ) 3 ( ) 2 ( )
( ) ( ) ( ) 2 ( ) 2 ( ) .
( ) 3 ( ) ( ) ( )
a t a t a t
A t a t a t a t a t
a t a t a t a t
= +
+
3S ( . [7, .3, § 5])
e (12) (12 3)
1χ 1 1 1
2χ 1 1− 1
3χ 2 0 1−
, 1 1 2 3( ) ( ) 3 ( ) 2 ( )t a t a t a tλ = + + , 2 1 2 3( ) ( ) 3 ( ) 2 ( )t a t a t a tλ = − + , 3 1 3( ) ( ) ( )t a t a tλ = − ,
1
1 3 2 1 1 2
1 1
1 3 2 , 1 1 0
6 6
2 0 2 1 1 1
−
= − = −
− −
.
2
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2013, 5, 2 11
1 2 3 1 1 1
1 2 2 2 2
1 2 3 3 3
( ) ( ) 2 ( ) ( ) 3 ( ) 2 ( )
( ) ( ) ( ) 0 , ( ) ( ) 3 ( ) 2 ( )
( ) ( ) ( ) 2 ( ) 0 2 ( )
t t t t t t
A t t t A t t t t
t t t t t
λ λ λ λ λ λ
λ λ λ λ λ
λ λ λ λ λ
− − − + + +
− − + + +
− +
− − − + +
= − = −
− −
,
1 1 2 3ind ( ( ) 3 ( ) ( ))a t a t a tρ Γ= + + , 2 1 2 3ind ( ( ) 3 ( ) ( ))a t a t a tρ Γ= − + , 3 1 3ind ( ( ) ( ))a t a tρ Γ= − .
3. { }8 1, i, j, kG Q= = ± ± ± ± – ,
2 2 2
i j k ijk 1= = = = − . 5 :
{ } { } { } { } { }1 2 3 4 51 , 1 , i , j , k .K K K K K= = − = ± = ± = ±
jC 8( [ ])Z Q
1 2 1C 2C 3C 4C 5C
1C 1C 2C 3C 4C 5C
2C 2C 1C 3C 4C 5C
3C 3C 3C 1 22 2C + 52C 42C
4C 4C 4C 52C 1 22 2C + 32C
5C 5C 5C 42C 32C 1 22 2C +
1 2 3 4 5
2 1 3 4 5
3 3 1 2 5 4
4 4 5 1 2 3
5 5 4 3 1 2
( ) ( ) 2 ( ) 2 ( ) 2 ( )
( ) ( ) 2 ( ) 2 ( ) 2 ( )
( ) ( ) ( ) ( ) 2 ( ) 2 ( )( )
( ) ( ) 2 ( ) ( ) ( ) 2 ( )
( ) ( ) 2 ( ) 2 ( ) ( ) ( )
a t a t a t a t a t
a t a t a t a t a t
a t a t a t a t a t a tA t
a t a t a t a t a t a t
a t a t a t a t a t a t
+=
+
+
.
8Q :
1 1− i± j± k±
1χ 1 1 1 1 1
2χ 1 1 1 –1 –1
3χ 1 1 –1 1 –1
4χ 1 1 –1 –1 1
5χ 2 –2 0 0 0
( )A t . -
.
1
( ) ( ) ( ), 1, ,5j g j
j g G
t a t g j
n
λ χ
∈
= = , :
1 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + + + + ,
2 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + + − − ,
3 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) 2 ( )t a t a t a t a t a tλ = + − + − ,
4 1 2 3 4 5( ) ( ) ( ) 2 ( ) 2 ( ) ( )t a t a t a t a t a tλ = + − − + , 5 1 2( ) ( ) ( )t a t a tλ = − .
( )A t .
1. , . . / . . ,
. . . – M.: , 1971. – 352 .
2. Adukov, V.M. On Wiener–Hopf factorization of meromorphic matrix functions / V.M. Adukov //
Integral Equations and Operator Theory. – 1991.– V. 14. – P. 767–774.
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3. , . . – - / . .
// . – 1992. – . 4. – . 1. – . 54–74.
4. , . . - / . . // . -
. . – 1999. – . 118, 3. – . 324–336.
5. , . . – - / . .
// . – 2009. – . 200, 8. – . 3–24.
6. , . . n / . . // -
. . – 1952. – . 7. – . 4(50). – . 3–54.
7. , . . . III / . . . – M.: , 2000. –
272 .
8. , . . / . . . – .: « », 2004.– 624 .
ABOUT WIENER–HOPF FACTORIZATION OF
FUNCTIONALLY COMMUTATIVE MATRIX FUNCTIONS
V.M. Adukov
1
An algorithm of an explicit solution of the Wiener–Hopf factorization problem is proposed for func-
tionally commutative matrix functions of a special kind. Elementary facts of the representation theory of
finite groups are used. Symmetry of the matrix function that is factored out allows to diagonalize it by a
constant linear transformation. Thus, the problem is reduced to the scalar case.
Keywords: Wiener–Hopf factorization, special indexes, finite groups.
References
1. Gokhberg I.Ts., Fel'dman I.A. Uravneniya v svertkakh i proektsionnye metody ikh resheniya
(Convolution equations and projection methods for their solution). Moscow: Nauka, 1971. 352 p. (in
Russ.). [Gohberg I.C., Fel dman I.A. Convolution equations and projection methods for their solution.
American Mathematical Society, Providence, R.I., 1974. Translations of Mathematical Monographs,
Vol. 41. MR 0355675 (50 #8149) (in Eng.).]
2. Adukov V.M. On Wiener–Hopf factorization of meromorphic matrix functions. Integral Equa-
tions and Operator Theory. 1991. Vol. 14. pp. 767–774. DOI: 10.1007/BF01198935.
3. Adukov V.M. Faktorizatsiya Vinera–Khopfa meromorfnykh matrits-funktsiy (Wiener-Hopf fac-
torization of meromorphic matrix function). Algebra i analiz. 1992. Vol. 4. Issue 1. pp. 54–74. (in
Russ.). [Adukov V.M. Wiener–Hopf factorization of meromorphic matrix functions. St. Petersburg
Mathematical Journal. 1993. Vol. 4. Issue 1. pp. 51–69. (in Eng.).]
4. Adukov V.M. O faktorizatsii analiticheskikh matrits-funktsiy (About factorization of analytical
matrix functions) Teoreticheskaya i matematicheskaya fizika. 1999. Vol. 118, no. 3. pp. 324–336. [Adu-
kov V.M. Factorization of analytic matrix-valued functions. Theoretical and Mathematical Physics.
1999. Vol. 118, no. 3. pp. 255–263. DOI: 10.1007/BF02557319].
5. Adukov V.M. Faktorizatsiya Vinera–Khopfa kusochno meromorfnykh matrits-funktsiy (Piece-
wise Wiener–Hopf factorization of meromorphic matrix functions). Matematicheskiy sbornik. 2009.
Vol. 200, no. 8. pp. 3–24. (in Russ.).
6. Gakhov F.D. Kraevaya zadacha Rimana dlya sistemy n par funktsiy (Riemann boundary value
problem for system of n pairs of functions). Uspekhi matematicheskikh nauk. 1952. Vol.7. Issue 4(50).
pp. 3–54. (in Russ.).
7. Kostrikin A.I. Vvedenie v algebru. Chast III. (Introduction into algebra. Part III). Moscow: Fiz-
matlit, 2000. 272 p. (in Russ.).
8. Van der Varden B.L. Algebra. Saint Petersburg: Lan', 2004. 624 p. (in Russ.).
18 2013 .
1
Adukov Victor Mikhailovich is Dr. Sc. (Physics and Mathematics), Professor, Department of Mathematical Analysis, South Ural State Uni-
versity.
E-mail: victor.m.adukov@gmail.com
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13. 2013, 5, 2 13
512.552.7
3
. . 1
A×Z3, A 3. -
(9,3), (9,3,3) (15,3).
: , , -
.
1.
G A b= × – , b – 3, A
a 3. ( )V G -
G .
:
0 : ( ) ( ),V G V Aϕ → ( ),x x x A∀ ∈ 1b ;
1 : ( ) ( ) ( / ),V G V A V G abϕ → ≅ ( ),x x x A∀ ∈ 2
b a ;
2
2 : ( ) ( ) ( / ),V G V A V G a bϕ → ≅ ( ),x x x A∀ ∈ b a ;
3 : ( ) ( ), ( ))V G V G a g g a g Gϕ → ∀ ∈ .
A -
a , H . -
: , ( ).G a G h a h h H→ ∀ ∈
1. ( )( ) ( )V G b V A= × K ,
31 2 11 12 2 2 2 2
3 3 3
1 (1 ) (1 ) (1 )K ab a b a b ab a a G
υυ υ
υ
−− −
= = + + + + + + + + + ∩ , 1 1( )ϕ υ υ= ,
2 2( ) (1 ( )) ( )I a V Aϕ υ υ= ∈ + ∩ , 3 3( ) (1 ( )) ( )I b V G aϕ υ υ= ∈ + ∩ . ( )I a – A ,
1a − , 2
1a − ; ( )I b – G a , -
1b − , 2
1b − ( . [1]).
. 1 [2] ( )( ) ( )V G b V A= × K ,
( )1 ( ) ( ) ( )K A A A b V G= + ∩ , ( ) ( )A A A b – , -
G ( 1)( 1)x b− − , 2
( 1)( 1)x b− − {1}x A∈ . -
, K .
Kυ ∈
2 2
2 2
1h ha h haha ha
h H h H h H h H h H h H
h ha ha h ha ha bυ α α α β β β
∈ ∈ ∈ ∈ ∈ ∈
= + + ⋅ + + + ⋅ +
2
2 2
h ha ha
h H h H h H
h ha ha bγ γ γ
∈ ∈ ∈
+ + + ⋅ .
1 2, ( )V Aυ υ ∈ 3 ( )V G aυ ∈
2
2
1 h ha ha
h H h H h H
h ha haυ λ λ λ
∈ ∈ ∈
= + + ,
1
– , , - -
.
E-mail: koliasnikovsa@susu.ac.ru
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2
2
2 h ha ha
h H h H h H
h ha haυ μ μ μ
∈ ∈ ∈
= + + ,
2
2
3 h hb hb
h H h H h H
h a hb a hb aυ ν ν ν
∈ ∈ ∈
= + + .
, 0 ( ) 1ϕ υ = ( . . Kυ ∈ ) 1 1( )ϕ υ υ= , 2 2( )ϕ υ υ= , 3 3( )ϕ υ υ= ,
.
2
2
2
1 0 0 1 0 0 1 0 0
0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1
1 0 0 0 1 0 0 0 1
0 1 0 0 0 1 1 0 0
0 0 1 1 0 0 0 1 0
1 0 0 0 0 1 0 1 0
0 1 0 1 0 0 0 0 1
0 0 1 0 1 0 1 0 0
1 1 1 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0
0 0 0 0 0 0 1 1 1
h
ha
ha
h
ha
ha
h
ha
ha
α
α
α
β
β
β
γ
γ
γ
2
2
2
0
0
h
h
ha
ha
h
ha
ha
h
hb
hb
ε
λ
λ
λ
μ
μ
μ
ν
ν
ν
= ,
1, 1;
0, 1.
h
h
h
ε
=
=
≠
, , 1υ ,
2υ , 3υ
2h ha hha
λ λ λ ε+ + = ; 2h ha hha
λ λ λ ε+ + = ; 2h ha hha
λ λ λ ε+ + = .
, 1 2, (1 ( )) ( )I a V Aυ υ ∈ + ∩ , 3 (1 ( )) ( )I b V G aυ ∈ + ∩ .
, ,
( ) 3h h h hα λ μ ν= + + ; ( ) 3ha ha ha h hα λ μ ν ε= + + − ; 2 2 2( ) 3h hha ha ha
α λ μ ν ε= + + − ;
2( ) 3h ha hbha
β λ μ ν= + + ; 2( ) 3h h hb hha
β λ μ ν ε= + + − ; 2 ( ) 3ha h hb hha
β λ μ ν ε= + + − ;
2 2( ) 3h ha ha hb
γ λ μ ν= + + ; 2 2( ) 3ha h hha hb
γ λ μ ν ε= + + − ; 2 2( ) 3h ha hha hb
γ λ μ ν ε= + + − .
, υ ,
,
31 2 11 12 2 2 2 2
3 3 3
1 (1 ) (1 ) (1 )ab a b a b ab a a
υυ υ
υ
−− −
= + + + + + + + + + . (1)
: ( ) ( ) ( )K V A V A V G aϕ → × × ,
( )1 2 3( ) ( ), ( ), ( )ϕ υ ϕ υ ϕ υ ϕ υ= .
.
,
( ) ( )1 2 3 1 1 2 2 3 3( ) ( ), ( ), ( ) ( ) ( ), ( ) ( ), ( ) ( )w w w w w w wϕ υ ϕ υ ϕ υ ϕ υ ϕ υ ϕ ϕ υ ϕ ϕ υ ϕ= = .
,
2 2 2
(1 ) 3(1 )a a a a+ + = + + ,
2 2 2 2 2
(1 ) 3(1 )ab a b ab a b+ + = + + ,
2 2 2 2 2
(1 ) 3(1 )a b ab a b ab+ + = + + ,
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2013, 5, 2 15
2 2 2 2
(1 )(1 ) (1 )(1 )a a b b b b a a+ + + + = + + + + =
2 2 2 2 2 2 2 2
(1 )(1 ) (1 )(1 )a b ab ab a b ab a b a b ab= + + + + = + + + + =
2 2 2 2 2 2
(1 )(1 ) (1 )(1 )a a ab a b ab a b a a= + + + + = + + + + =
2 2 2 2 2 2
(1 )(1 ) (1 )(1 ).a b ab a a a a a b ab= + + + + = + + + +
, 1 21, 1 ( )w w I a− − ∈ , 2 1 ( )w I b− ∈ ,
2 2 2
1 2 3( 1)(1 ) ( 1)(1 ) ( 1)(1 ) 0w a a w a a w b b− + + = − + + = − + + = .
( )
( )
31 2
31 2
3 31 1 2 2
11 12 2 2 2 2
3 3 3
11 12 2 2 2 2
3 3 3
11 12 2 2 2 2
3 3 3
1 (1 ) (1 ) (1 )
1 (1 ) (1 ) (1 )
1 (1 ) (1 ) (1 ).
ww w
ww w
w ab a b a b ab a a
ab a b a b ab a a
ab a b a b ab a a
υυ υ
υυ υ
υ
−− −
−− −
−− −
= + + + + + + + + + ×
× + + + + + + + + + =
= + + + + + + + + +
, 3 3 3 3w wυ υ≠ , , 2 2
3 3 3 3(1 ) (1 )w a a w a aυ υ+ + = + + .
, , ϕ -
. (1), . -
.
.
2. (9,3)
9 3
| 1 | 1G a a b b= = × = . ( )V a [3].
[2].
1 2( ) ,V a a F F u u= × = × ,
8 2 7
1 1 ( ) ( )u a a a a= − + + + , 1 8 3 6 4 5
1 1 ( ) ( ) ( ),u a a a a a a−
= − − + + + + +
2 7 4 5
2 1 ( ) ( ),u a a a a= − + + + 1 2 7 3 6 8
2 1 ( ) ( ) ( )u a a a a a a−
= − − + + + + + .
1
1 2( )V G a b u u K= × × × × ,
Kυ ∈
3 6 2 6 3 21 21 1
1 (1 ) (1 )
3 3
a b a b a b a b
υ υ
υ
− −
= + + + + + + ,
3
1 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 3
( )V G a – ( . [3]) 3( ) 1ϕ υ = .
8
0
i
ii
aλ=
,
8
0
i
ii
aμ=
,
( 2 3 4
0 0 1 1 2 2 3 3 4 4
5 6 7 8
5 5 6 6 7 7 8 8 6 3 7 4
2 3 4 5 6
8 5 0 6 1 7 2 8 3 0
4 1
1 3 (1 ) 1 ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( 1) ( ) ( ) ( 1)
( )
a a a a
a a a a b ab
a b a b a b a b a b
a
υ λ μ λ μ λ μ λ μ λ μ
λ μ λ μ λ μ λ μ λ μ λ μ
λ μ λ μ λ μ λ μ λ μ
λ μ
= + + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ +
+ + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ + + ⋅ +
+ + ⋅ + + − ⋅ + + ⋅ + + ⋅ + + − ⋅ +
+ + ⋅
)
7 8 2 2 2 2
5 2 3 6 4 7 5 8
3 2 4 2 5 2 6 2
6 0 7 1 8 2 0 3
7 2 8 2
1 4 2 5
( ) ( ) ( ) ( )
( 1) ( ) ( ) ( 1)
( ) ( ) .
b a b b ab a b
a b a b a b a b
a b a b
λ μ λ μ λ μ λ μ
λ μ λ μ λ μ λ μ
λ μ λ μ
+ + ⋅ + + ⋅ + + ⋅ + + ⋅ +
+ + − ⋅ + + ⋅ + + ⋅ + + − ⋅ +
+ + ⋅ + + ⋅
(2)
, 3
1 2, (1 ( ))u u I a∈ + ,
2 6 6
1 1 ( 1) ( 1)u a a a a= − − + − , 6 3 2 3
2 1 ( 1) ( 1) ( 1)u a a a a a a= − − + − + − .
: K F Fϕ → × ,
( )1 2( ) ( ), ( )ϕ υ ϕ υ ϕ υ= .
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1 2,u u
1
1 8
a a+ 2 7
a a+ 3 6
a a+ 4 5
a a+
1u 1 –1 1 0 0
2
1u 5 –4 3 –2 1
3
1u 19 –18 15 –9 3
2u 1 0 –1 0 1
2
2u 5 1 –4 –2 3
3
2u 19 3 –18 –9 15
1 2u u –1 0 1 1 –1
2
1 2u u –5 –1 5 3 –4
2
1 2u u 1 1 –2 0 –1
2 2
1 2u u 7 1 –6 –3 5
. 1 (2) ,
2 3 3 3
1 2 1 2( ,1),( ,1),(1, ),(1, ) ( )u u u u Kϕ∈ ,
1 2 1 20 , , , 2i i j j≤ ≤ ( )1 2 1 2
1 2 1 2, ( )i i j j
u u u u Kϕ∈ ,
1 2 1 21, 2, 2, 1i i j j= = = = 1 2 1 22, 1, 1, 2i i j j= = = = .
, .
2. 2 2 3 3 3
1 2 1 2 2 1 2( ) ( , ) ( ,1) (1, ) (1, )K u u u u u u uϕ = × × × . 3 3 3( )F F Kϕ× ≅ × × .
, ( )V G ,
, ( . [1, . 45]),
.
3. ( )W G – , | ( ): ( ) | 1V G W G = ,
1 2 3 4 5 6( )V G a b u u u u u u= × × × × × × × ,
8 2 7
1 1 ( ) ( )u a a a a= − + + + , 2 7 4 5
2 1 ( ) ( ),u a a a a= − + + +
8 2 2 2 7
3 1 ( ) ( )u ab a b a b a b= − + + + , 2 2 7 4 5 2
4 1 ( ) ( )u a b a b a b a b= − + + + ,
2 8 2 7 2
5 1 ( ) ( )u ab a b a b a b= − + + + , 2 7 2 4 2 5
6 1 ( ) ( )u a b a b a b a b= − + + + .
. ( )W G ( )V G , -
( )V C , C G . ( )V C
.
3 3
( )V a a= , ( )V b b= , 3 3
( )V a b a b= , 6 6
( )V a b a b= ,
1 2( )V a a u u= × × , 3 4( )V ab ab u u= × × , 2 2
5 6( )V ab ab u u= × × .
2 ,
1 1 2 2
1 2 5 6 1 2 1 2( ) ( , )u u u u u u u uϕ − −
= , 1 1 3
2 3 5 6 2( ) ( ,1)u u u u uϕ − −
= ,
1 1 3
1 4 5 6 1( ) (1, )u u u u uϕ − −
= , 1 1 3
2 3 4 5 2( ) (1, )u u u u uϕ − −
= .
.
3. (9,3,3)
9 3 3
| 1 | 1 | 1G a a b b c c= = × = × = . A A a b= × .
( )V A 3.
( )V A a b F= × × , 1 6...F u u= × × .
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1 1 6( ) ...V A a b u u K= × × × × × × ,
Kυ ∈
3 6 2 6 3 21 21 1
1 (1 ) (1 )
3 3
a b a b a b a b
υ υ
υ
− −
= + + + + + + ,
3
1 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 3
( )V G a – ( . [3]) 3( ) 1ϕ υ = .
, ( )3
1 ( ) , 1,...,6u I a∈ + = .
: K F Fϕ → × , ( )1 2( ) ( ), ( )ϕ υ ϕ υ ϕ υ= .
2, 3 3
( )i
ab a= , 0,1,2i = , 3 3
( ,1),(1, ) ( )u u Kϕ∈ .
( ) ( ) ( )3 3 5 6 5 61 2 1 2 4 4
5 51 2 1 2 3 4 3 4 6 6, , , , , ( ), , 0,1,2i j i i j ji i j j i j
u u u u u u u u u u u u K i jϕ∈ =
,
1 3 5 2 4 6 1 3 5 2 4 61, 2, 2, 1i i i i i i j j j j j j= = = = = = = = = = = =
1 3 5 2 4 6 1 3 5 2 4 62, 1, 1, 2i i i i i i j j j j j j= = = = = = = = = = = = .
GAP ( . [4]) ,
( )6 3 5 62 4 1 2 4
52 4 6 1 2 3 4 6, ( ), , 0,1,2.i j j ji i j j j
u u u u u u u u u K i jϕ∉ =
, .
4. 1 12...K w w= × × ,
2 2
1 1 2 1 2( ) ( , )w u u u uϕ = , 2 2
2 3 4 3 4( ) ( , )w u u u uϕ = , 2 2
3 5 6 5 6( ) ( , )w u u u uϕ = ,
3
4 2( ) ( ,1)w uϕ = , 3
5 4( ) ( ,1)w uϕ = , 3
6 6( ) ( ,1)w uϕ = ,
3
7 1( ) (1, )w uϕ = , 3
8 2( ) (1, )w uϕ = , 3
9 3( ) (1, )w uϕ = ,
3
10 4( ) (1, )w uϕ = , 3
11 5( ) (1, )w uϕ = , 3
12 6( ) (1, )w uϕ = .
3 3 3 3 3 3 3 3 3( )F F Kϕ× ≅ × × × × × × × × .
, , .
5. ( )W G – , | ( ): ( ) | 1V G W G = .
4. (15,3)
15 3
| 1 | 1G a a b b= = × = . ( )V a
[2]. .
0( )V a a u F= × × , 1 2 3F u u u= × × ,
3 12
0 1 ( ),u a a= − + + 1 6 9
0 1 ( )u a a−
= − + + ,
1 14 2 13 3 12 5 10 6 9 7 8
1 3 3( ) 2( ) ( ) 2( ) 3( ) 3( )u a a a a a a a a a a a a−
= − − + − + − + + + + + + + ,
2 13 3 12 4 11 5 10 6 9 7 8
2 3 2( ) ( ) 2( ) ( ) 2( ) ( )u a a a a a a a a a a a a= − + + + + + − + − + + + ,
1 14 2 13 3 12 4 11 5 10 6 9
2 3 3( ) 3( ) 3( ) 2( ) 2( ) ( )u a a a a a a a a a a a a−
= − + + − + + + − + + + − + ,
14 3 12 4 11 5 10 6 9 7 8
3 3 ( ) 2( ) 2( ) ( ) ( ) 2( )u a a a a a a a a a a a a= + + − + − + − + + + + + ,
1 2 13 3 12 4 11 5 10 6 9 7 8
3 3 3( ) ( ) 3( ) 2( ) 3( ) 2( )u a a a a a a a a a a a a−
= + + − + − + + + + + − + .
1 0 1 2 3( )V G a b u u u u K= × × × × × × , Kυ ∈
5 10 2 10 5 2 5 1031 2 11 1
1 (1 ) (1 ) (1 )
3 3 3
a b a b a b a b a a
υυ υ
υ
−− −
= + + + + + + + + + ,
5
1 2, (1 ( )) ( )I a V aυ υ ∈ + ∩ , 5
3 (1 ( )) ( )I b V aυ ∈ + ∩ .
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ψ ( )V a 5
( )V G a ,
5
: a ab aψ → . 5 5
0( )V G a ab a u F′ ′= × × , 1 2 3F u u u′ ′ ′ ′= × × ,
( ), 0,1,2,3.i iu u iψ′ = =
, 5
1 2 3, , (1 ( ))u u u I a∈ + , 1 2 3, , (1 ( ))u u u I b′ ′ ′ ∈ +
4 5 4 3 2 10
1 1 ( 1)( 1) (2 2 2 1)( 1)u a a a a a a a a= + + − − − − + − + − ,
4 3 2 5 3 2 10
2 1 (2 2 1)( 1) (2 2 1)( 1)u a a a a a a a a a= − − − + + − − − − + − ,
4 3 2 5 4 2 10
3 1 ( 2 2 1)( 1) ( 2 2 1)( 1)u a a a a a a a a a= + + + + − − − − − + − ,
5 4 5 3 5 5 5
1 ( 2 2 )( 1)u a a a a a a a a b′ = + + − − −−
4 5 2 5 5 5 2
(2 2 )( 1)a a a a a a a b− − − + − ,
5 4 5 3 5 2 5 5
2 (2 2 )( 1)u a a a a a a a a b′ = + − + − − +
3 5 2 5 5 5 2
( 2 2 )( 1)a a a a a a a b+ − + − − ,
5 4 5 2 5 5 5
3 (2 2 )( 1)u a a a a a a a a b′ = + − − + − +
4 5 3 5 5 5 2
( 2 2 )( 1)a a a a a a a b+ + − − − .
( )5
0 1 ( )u I a∉ + , ( )0 1 ( )u I b′ ∉ + .
: K F F Fϕ ′→ × × ,
( )1 2 3( ) ( ), ( ), ( )ϕ υ ϕ υ ϕ υ ϕ υ= .
, 9 3× , ,
( ) ( ) ( )3 3 3
,1,1 , 1, ,1 , 1,1, ( ), 1,2,3i i iu u u K iϕ′ ∈ = .
GAP ( . [4]) ,
( )3 3 31 2 1 2 1 2
1 2 3 1 2 3 1 2 3, , ( ), , , 0,1,2( 1,2,3)i j ki i j j k k
u u u u u u u u u K i j kϕ′ ′ ′ ∈ = = .
, . 2.
2
1i 2i 3i 1j 2j 3j 1k 2k 3k
0 0 0 1 0 1 2 0 2
0 0 0 2 0 2 1 0 1
0 0 1 0 0 2 1 0 2
0 0 1 1 0 0 0 0 1
0 0 1 2 0 1 2 0 0
0 0 2 0 0 1 2 0 1
0 0 2 1 0 2 1 0 0
0 0 2 2 0 0 0 0 2
1i 2i 3i 1j 2j 3j 1k 2k 3k 1i 2i 3i 1j 2j 3j 1k 2k 3k
0 1 0 0 2 0 2 1 2 0 2 0 0 1 0 1 2 1
0 1 0 1 2 1 1 1 1 0 2 0 1 1 1 0 2 0
0 1 0 2 2 2 0 1 0 0 2 0 2 1 2 2 2 2
0 1 1 0 2 2 0 1 1 0 2 1 0 1 2 2 2 0
0 1 1 1 2 0 2 1 0 0 2 1 1 1 0 1 2 2
0 1 1 2 2 1 1 1 2 0 2 1 2 1 1 0 2 1
0 1 2 0 2 1 1 1 0 0 2 2 0 1 1 0 2 2
0 1 2 1 2 2 0 1 2 0 2 2 1 1 2 2 2 1
0 1 2 2 2 0 2 1 1 0 2 2 2 1 0 1 2 0
1 0 0 0 0 1 1 0 0 1 1 0 0 2 1 0 1 2
1 0 0 1 0 2 0 0 2 1 1 0 1 2 2 2 1 1
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1 0 0 2 0 0 2 0 1 1 1 0 2 2 0 1 1 0
1 0 1 0 0 0 2 0 2 1 1 1 0 2 0 1 1 1
1 0 1 1 0 1 1 0 1 1 1 1 1 2 1 0 1 0
1 0 1 2 0 2 0 0 0 1 1 1 2 2 2 2 1 2
1 0 2 0 0 2 0 0 1 1 1 2 0 2 2 2 1 0
1 0 2 1 0 0 2 0 0 1 1 2 1 2 0 1 1 2
1 0 2 2 0 1 1 0 2 1 1 2 2 2 1 0 1 1
1 2 0 0 1 1 2 2 1 2 0 0 0 0 2 2 0 0
1 2 0 1 1 2 1 2 0 2 0 0 1 0 0 1 0 2
1 2 0 2 1 0 0 2 2 2 0 0 2 0 1 0 0 1
1 2 1 0 1 0 0 2 0 2 0 1 0 0 1 0 0 2
1 2 1 1 1 1 2 2 2 2 0 1 1 0 2 2 0 1
1 2 1 2 1 2 1 2 1 2 0 1 2 0 0 1 0 0
1 2 2 0 1 2 1 2 2 2 0 2 0 0 0 1 0 1
1 2 2 1 1 0 0 2 1 2 0 2 1 0 1 0 0 0
1 2 2 2 1 1 2 2 0 2 0 2 2 0 2 2 0 2
2 1 0 0 2 2 1 1 2 2 2 0 0 1 2 0 2 1
2 1 0 1 2 0 0 1 1 2 2 0 1 1 0 2 2 0
2 1 0 2 2 1 2 1 0 2 2 0 2 1 1 1 2 2
2 1 1 0 2 1 2 1 1 2 2 1 0 1 1 1 2 0
2 1 1 1 2 2 1 1 0 2 2 1 1 1 2 0 2 2
2 1 1 2 2 0 0 1 2 2 2 1 2 1 0 2 2 1
2 1 2 0 2 0 0 1 0 2 2 2 0 1 0 2 2 2
2 1 2 1 2 1 2 1 2 2 2 2 1 1 1 1 2 1
2 1 2 2 2 2 1 1 1 2 2 2 2 1 2 0 2 0
, .
6. 1 9...K w w= × × ,
1 1 3 1( ) ( , , )w u u uϕ ′= , ( )2 2 2
2 2 2 1 2 3( ) , ,w u u u u uϕ ′ ′ ′= , ( )2 2
3 3 3 1 3( ) , ,w u u u uϕ ′ ′= ,
( )2 2
4 1 3 1 3( ) 1, ,w u u u uϕ ′ ′= , 3
5 2( ) (1, ,1)w uϕ = , 3
6 3( ) (1, ,1)w uϕ = ,
3
7 1( ) (1,1, )w uϕ ′= , ( )3
3 8 2( ) 1,1,w uϕ ′= , ( )3
9 3( ) 1,1,w uϕ ′= ,
3 3 3 3 3( )F F Kϕ× ≅ × × × × .
, , -
.
7. ( )W G – , | ( ): ( ) | 1V G W G = ,
0 1 12( ) ...V G a b u u u= × × × × × ,
3 12
0 1 ( )u a a= − + + , 14 2 13 3 12 4 11 5 10 6 9
( ) 3 2( ) 2( ) 2( ) ( ) ( ) ( )u x x x x x x x x x x x x x= − + + + − + + + − + + + ,
1 ( )u u a= , 2
2 ( )u u a= , 2
3 ( )u u a= , 4 ( )u u ab= , 2 2
5 ( )u u a b= , 4
6 ( )u u a b= , 2
7 ( )u a b ,
4 2
8 ( ),u u a b= 8
9 ( )u u a b= , 3
10 ( )u u a b= , 6 2
11 ( )u u a b= , 12
12 ( )u u a b= .
. ( )V C , ( )W G , C -
G .
5 5
( )V a a= , ( )V b b= , 5 5
( )V a b a b= , 10 10
( )V a b a b= , 3 3
0( )V a a u= × ,
0 1 2 3( )V a a u u u u= × × × × , 0 4 5 6( )V ab ab u u u u= × × × × ,
2 2
0 7 8 9( )V a b a b u u u u= × × × × , 3 3
0 10 11 12( )V a b a b u u u u= × × × × .
6 ,
1 1 1
10 11 12 1 3 1( ) ( , , )u u u u u uϕ − − −
′= , ( )1 1 1 2 2 2
1 2 3 4 6 7 2 2 1 2 3( ) , ,u u u u u u u u u u uϕ − − −
′ ′ ′= ,
( )1 1 1 1 1 1 2 2
1 7 8 9 10 12 3 3 1 3( ) , ,u u u u u u u u u uϕ − − − − − −
′ ′= , ( )1 1 1 1 1 1 2 2
1 3 7 9 10 12 1 3 1 3( ) 1, ,u u u u u u u u u uϕ − − − − − −
′ ′= ,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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3
2 4 6 9 12 2( ) (1, ,1)u u u u u uϕ = , ( )1 1 1 1 1 1 1 3
3 4 6 7 8 9 10 12 3( ) 1, ,1u u u u u u u u uϕ − − − − − − −
= ,
( )1 1 1 1 1 3
1 3 4 8 10 11 12 1( ) 1,1,u u u u u u u uϕ − − − − −
′= ( )3
1 3 5 9 10 2( ) 1,1,u u u u u uϕ ′= ,
( )1 1 1 1 1 3
1 3 6 7 8 9 11 3( ) 1,1,u u u u u u u uϕ − − − − −
′= .
.
1. , . . / . . . –
: . . ., 1987. – 210 . ( . 24.09.87, 2712– 87).
2. , . .
/ . . . – , . . . .,
2000. – 45 . ( . 30.03.00, 860– 00).
3. , . . 7 9 / . . , . . // -
, . – 1999. – 11(450). – . 81–84.
4. Martin Schönert et al. GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathe-
matik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, sixth edition, 1997.
UNITS OF INTEGRAL GROUP RINGS OF FINITE GROUPS
WITH A DIRECT MULTIPLIER OF ORDER 3
S.A. Kolyasnikov
1
The description of units of integral group rings of finite groups of type A×Z3 was obtained, where A
contains a central subgroup of order 3. For example, the unit groups of integral group rings of Abelian
groups of the types (9,3), (9,3,3) and (15,3) were found.
Keywords: Abelian group, group ring, unit group of group ring.
References
1. Bovdi A.A. Mul'tiplikativnaya gruppa tselochislennogo gruppovogo kol'tsa (Multiplicative group
of integral group ring). Uzhgorod: Uzhgorodskiy gosudarstvennyy universitet, 1987. 210 p. (in Russ.).
2. Kolyasnikov S.A. O gruppe edinits tselochislennogo gruppovogo kol'tsa konechnykh grupp
razlozhimykh v pryamoe proizvedenie (About group of units of integral group ring of finite groups fac-
torable into direct composition). Novosibirsk, Red. Sibirskogo Matematicheskogo Zhurnala, 2000. 45 p.
(in Russ.).
3. Alev R.Zh., Panina G.A. Russian Mathematics (Izvestiya VUZ. Matematika). 1999. Vol. 43,
no. 11. pp. 80–83.
4. Martin Schönert et al. GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathe-
matik. Rheinisch Westfälische Technische Hochschule, Aachen, Germany, sixth edition, 1997.
6 2013 .
1
Kolyasnikov Sergey Andreevich is Senior Lecturer, General Mathematics Department, South Ural State University.
E-mail: koliasnikovsa@susu.ac.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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, -
( (6))
20–
40 . ,
,
-
-
.
-
-
.
-
(1). . 5.
. 5 -
20–40 . , -
, -
. -
.
, -
. ,
.
. -
. , -
, , .
, , ,
.
-
-
,
2–25 600–1100 K -
.
1. Allen, C.M. Energetic-nanoparticle enhanced combustion of liquid fuels in a rapid compression
machine / C.M. Allen, T. Lee // Proc. 47th AIAA Aerospace Sciences Meeting Including The New Ho-
rizons Forum and Aerospace Exposition. – 2009. – AIAA 2009-227.
2. Trunov, M.A. Ignition of aluminum powders under different experimental conditions /
M.A. Trunov, M. Schoenitz, E.L. Dreizin // Propellants, Explosives, Pyrotechnics. – 2005. – Vol. 30. –
P. 36–43.
3. Utilization of iron additives for advanced control of NOx emissions from stationary combustion
sources / V.V. Lissianski, P.M. Maly, V.M. Zamansky, W.C. Gardiner // Ind. Eng. Chem. Res. – 2001. –
Vol. 20, 15. – P. 3287–3293.
4. / . . , . . ,
. . . // . – 1996. – T. 32, 3. – C. 24–34.
5. Cashdollar, K.L. Explosion temperatures and pressures of metals and other elemental dust clouds /
K.L. Cashdollar, I.A. Zlochower // Journal of Loss Prevention in the Process Industries. – 2007. – Vol. 20.
– P. 337–348.
. 5.
. (1, 3) –
, (2, 4) – -
(5). 1, 2 – 30 , 3, 4 – 2 , 5 –
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
29. . ., . .,
.- ., . ., . .
2013, 5, 2 29
6. Dynamics of dispersion and ignition of dust layers by a shock wave / V.M. Boiko, A.N. Papyrin,
M. Wolinski , P. Wolanski // Progress in Astronautics and Aeronautics. – 1984. – Vol. 94. – P. 293–301.
7. Ignition of dust suspensions behind shock waves / A.A. Borisov, B.E. Gel’fand, E.I. Timofeev et
al. // Progress in Astronautics and Aeronautics. – 1984. – Vol. 94. – P. 332–339.
8. Photoemission measurements of soot particles temperature at pyrolysis of ethylene /
Y.A. Baranyshyn, L.I. Belaziorava, K.N. Kasparov, O.G. Penyazkov // Nonequilibrium Phenomena:
Plasma, Combustion, Atmosphere. – Moscow: Torus Press Ltd, 2009. – P. 87–93.
9. Optical properties of gases at ultra high pressures / Yu.N. Ryabinin, N.N. Sobolev,
A.M. Markevich, I.I. Tamm // Journal of Experimental and Theoretical Physics. – 1952. – Vol. 23, 5.
– P. 564–575.
10. , . . / . .
// . – 2004. – T. 40, 1. – . 75–77.
11. , . . /
. . , . . // . – 2011. – . 47, 6. – . 98–100.
12. /
. . , . . , . . . // - . – 2012. –
. 85, 1. – . 139–144.
13. , . .
/ . . , . . // . – 2001. – . 37, 6.
– . 46–55.
14. /
. . , . . – .: , 1979. – 312 .
15. , . . / . . , . . // -
. – 2003. – . 39, 5. – . 65–68.
EXPERIMENTAL AND MATHEMATICAL SIMULATION OF AUTO-IGNITION
OF IRON MICRO PARTICLES
V.V. Leschevich
1
, O.G. Penyazkov
2
, J.-C. Rostaing
3
, A.V. Fedorov
4
, A.V. Shulgin
5
The experimental and numerical studies of autoignition and burning of iron microparticles in oxy-
gen were introduced. Parameters of high-temperature gas environment were created by rapid compres-
sion machine. Powders with a particle size 1–125 microns were studied at the oxygen pressure of 0,5–28
MPa and the temperature of 500–1100 K. Ignition delay time was measured by registration of radiation
from the side wall of the combustion chamber and by measuring the pressure on the end wall. Critical
auto-ignition conditions are determined according to size of the particles, oxygen temperature and pres-
sure. For description of ignition mechanisms a punctual semiempirical simulator was introduced. The
simulator gave the opportunity to describe the experimental data on dependence of ignition delay time of
iron particles on the ambient temperature, taking into account correlations of critical ignition tempera-
tures with pressure.
Keywords: iron micro particles, oxygen, rapid compression machine, ignition delay time, mathe-
matical simulation.
1
Leschevich Vladimir Vladimirovich is Junior Researcher, A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences
of Belarus.
2
Penyazkov Oleg Glebovich is Corresponding Member of the National Academy of Sciences of Belarus, director, A.V. Luikov Heat and Mass
Transfer Institute of the National Academy of Sciences of Belarus.
3
Rostaing Jean-Christophe is L’Air Liquide, Centre de Recherche Claude-Delorme, France.
4
Fedorov Alexander Vladimirovich is Doctor of Physical and Mathematical Sciences, Head of Laboratory, S.A. Khristianovich Institute of
Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences.
5
Shulgin Alexey Valentinovich is Candidate of Physical and Mathematical Sciences, Senior Researcher, S.A. Khristianovich Institute of Theo-
retical and Applied Mechanics of the Siberian Branch of the Russian Academy of Sciences.
E-mail: shulgin@itam.nsc.ru
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References
1. Allen C.M., Lee T. Energetic-nanoparticle enhanced combustion of liquid fuels in a rapid com-
pression machine // Proc. 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum
and Aerospace Exposition. 2009. AIAA 2009-227.
2. Trunov M.A., Schoenitz M., Dreizin E.L. Ignition of aluminum powders under different experi-
mental conditions. Propellants, Explosives, Pyrotechnics. 2005. Vol. 30. pp. 36–43.
3. Lissianski V.V., Maly P.M., Zamansky V.M., Gardiner W.C. Utilization of iron additives for ad-
vanced control of NOx emissions from stationary combustion sources. Ind. Eng. Chem. Res. 2001. Vol.
20, no. 15. pp. 3287–3293.
4. Zolotko A.N., Vovchuk Ya.I., Poletaev N.I., Frolko A.V., Al'tman I.S. Fizika goreniya i vzryva. 1996.
Vol. 32, no. 3. pp. 24–34. (in Russ.).
5. Cashdollar K.L., Zlochower I.A. Explosion temperatures and pressures of metals and other elemen-
tal dust clouds. Journal of Loss Prevention in the Process Industries. 2007. Vol. 20. pp. 337–348.
6. Boiko V.M., Papyrin A.N., Wolinski M., Wolanski P. Dynamics of dispersion and ignition of dust
layers by a shock wave. Progress in Astronautics and Aeronautics. 1984. Vol. 94. pp. 293–301.
7. Borisov A.A., Gel’fand B.E., Timofeev E.I., Tsyganov S.A., Khomic S.V. Ignition of dust sus-
pensions behind shock waves. Progress in Astronautics and Aeronautics. 1984. Vol. 94. pp. 332–339.
8. Baranyshyn Y.A., Belaziorava L.I., Kasparov K.N., Penyazkov O.G. Photoemission measure-
ments of soot particles temperature at pyrolysis of ethylene. Nonequilibrium Phenomena: Plasma, Com-
bustion, Atmosphere. Moscow: Torus Press Ltd, 2009. pp. 87–93.
9. Ryabinin Yu.N., Sobolev N.N., Markevich A.M., Tamm I.I. Optical properties of gases at ultra
high pressures. Journal of Experimental and Theoretical Physics. 1952. Vol. 23, no. 5. pp. 564–575.
10. Al'tman I.S. Fizika goreniya i vzryva. 2004. Vol. 40, no. 1. pp. 75–77.
11. Fedorov A.V., Shul'gin A.V. Fizika goreniya i vzryva. 2011. Vol. 47, no. 6. pp. 98–100.
12. Leshchevich V.V., Penyaz'kov O.G., Fedorov A.V., Shul'gin A.V., Rosten Zh.-K. Inzhenerno-
fizicheskiy zhurnal. 2012. Vol. 85, no. 1. pp. 139–144. (in Russ.).
13. Bolobov V.I., Podlevskikh N.A. Fizika goreniya i vzryva. 2001. Vol. 37, no. 6. pp. 46–55. (in Russ.).
14. Kholla Dzh. (ed.), Uatta Dzh. (ed.) Sovremennye chislennye metody resheniya obyknovennykh
differentsial'nykh uravneniy (Contemporary numerical methods of solution of ordinary differential eqau-
tion). Moscow: Mir, 1979. 312 p. (in Russ.). [Hall G., Watt J.M. Modern Numerical Methods for Ordi-
nary Differential Equations. Oxford: Clarendon press. 312 p.]
15. Fedorov A.V., Kharlamova Yu.V. Fizika goreniya i vzryva. 2003. Vol. 39, no. 5. pp. 65–68. (in
Russ.).
8 2013
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1, [10]. -
,
, 10- . -
,
5- .
1.1. . , ,a b - , ,A B
. A ijA , T
A – A , det A
– A . a b ×a b T
a b
. a [ ]×a -
, [ ]× = ×a b a b b . I , 0
– , ⋅ – .
2.
2.1 . -
jix , jiy , jiz iQ j - , 1, 2j = ,
1, , 5i = ( . 1).
1 1 2 0j j jx y x= = = 1, 2j = .
.
3 5× :
1 5
1 5
1 5
j j
j j j
j j
x x
y y
z z
=A (1)
2 2 1j j j j j j′′ ′= =A H A H H A , (2)
1jH 2jH – , 1jx , 1jy 2jx . -
:
1
1 1
2 2 2
1 1 1 1 1sign( )
j
j j
j j j j j
x
y
z z x y z
=
+ + +
h ,
2
2 2
2 2 2 2 2sign( )
0
j
j j j j j
x
y y x y
′
′ ′ ′ ′= + +h .
, (2), , -
. ,
20- (12) 10- (13).
2.2. . -
[11–13]. - – ( , , )O π P ,
π – , P – 3-
π , O – ( P ). –
O π , O π . -
, (
) .
- ( , , )j j jO π P 1, 2j = . -
[ ]1 =P I 0 , [ ]2 =P R t , SO(3)∈R –
[ ]T
1 2 3t t t=t – , 1=t .
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
33. . .
,
2013, 5, 2 33
1 iO Q 2 iO Q ( . 1) 3
, -
R t . [12]:
[ ]
1
2 2 2 1
1
0
i
i i i i
i
x
x y z y
z
=E , (3)
1, , 5i = , [ ]×
=E t R .
2.3. .
.
1 ([10]). SO(3)∈R
2 kπ π+ , k ∈ , R :
1
u u
v v
w w
−
× ×
= − +R I I , (4)
, ,u v w∈ .
R (4) [ ]( , , , )u v w ×
=E t t R – .
1.
1 3 2
2 1 3
3 2 1
,
,
,
t vt wt
u
t wt ut
v
t ut vt
w
δ
δ
δ
− − +
′ =
− − +
′ =
− − +
′ =
(5)
1 2 3ut vt wtδ = + + , ( , , , ) ( , , , )u v w u v w′ ′ ′ = −E t E t .
. SO(3)′ = − ∈tR H R ,
T
2= −tH I tt . [ ]×
′ ′= = −E t R E . ′R (4) , ,u v w′ ′ ′ ,
, ,u v w′ ′ ′ :
' '
' '
' '
u u u u
v v v v
w w w w× × × ×
− + = − − +tI I H I I ,
, , (5).
(3) t , -
:
=S t 0 , (6)
i - 5 3× S [ ]
2
T
1 1 1 2
2
i
i i i i
i
x
x y z y
z ×
R .
R (4) 3 3×
S . :
4 3 2 2 3 4 3 2
2 3 2 2
[0] [0] [0] [0] [0] [1] [1]
[1] [1] [2] [2] [2] [3] [3] [4] 0,
if u u v u v uv v u u v
uv v u uv v u v
= + + + + + +
+ + + + + + + + =
(7)
1, ,10i = , [ ]n n w , [0] – . -
, if , 4.
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
34. . « . . »34
1. 3 3× S 3
/iF Δ ,
2 2 2
1 u v wΔ = + + + iF – 6- . , iF
i iF f= Δ , if -
iF .
2.4. . (7) :
=B m 0 , (8)
B – 10 35×
T4 3 3
1u u v u w v w=m –
.
(8) iuf , ivf 1, , 5i = , iwf
1, ,10i = . , :
′
′ =
m
B 0
m
, (9)
′B – 30 50×
T4 3 3 2 2 3 4 5
u w u vw u w v w vw w′ =m
– . , (9) (8).
′B –
( ).
( ):
3 2
u w 3
u w 3
u 3 2
v w 3
v w 3
v uv u v 1
1g 1 [3] [4] [4] [5]
2g 1 [3] [4] [4] [5]
3g 1 [3] [4] [4] [5]
4g 1 [3] [4] [4] [5]
5g 1 [3] [4] [4] [5]
6g 1 [3] [4] [4] [5]
28 .
1 6, ,g g :
1 21
2 32
4 53
5 64
( )
1
g gh uv
g gh u
w w
g gh v
g gh
≡ − = =C 0 , (10)
( )wC :
[4] [5] [5] [6]
[4] [5] [5] [6]
( )
[4] [5] [5] [6]
[4] [5] [5] [6]
w =C . (11)
2. ′B ,
« » – ′B .
24 ′B . -
.
(10) , (8) ,
det ( ) 0w =C . det ( )W w= C . 20- -
w .
2. :
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
35. . .
,
2013, 5, 2 35
10
10 10
0
( )k k
k
k
W p w w+ −
=
= + − , (12)
kp ∈ .
. 1 1 0j jx y= = , 33 0E = . ,
23
2 1 1
13
R vw u
t t t
R uw v
+
= =
−
.
(5), 1
w w−
′ = − . , w
1
( )w−
− ( ) . ,
10 10
1 10 10
10
01
( )( ) ( )k k
i i k
ki
W p w w w w p w w− + −
==
= − + = + −∏ .
1
w w w−
= − 10- :
10
0
k
k
W p w
=
= , (13)
kp 2
0
( 1)
k
k i i k
i
k
w w
i
−
=
= − . :
10 1
2
2
k i
i k
i k
i
p p
i kk
'
=
+
−
=
−
, (14)
i k 10, mod 2 0i k− = . ,
0k = (14)
10
0
2 i
i
p'
=
.
2.5. . -
, (4), – , -
. .
, , [14]
[15] . -
.
0w – W . 2
0 0 0 0/ 2 sign( ) ( / 2) 1w w w w= + +
W , 0| | 1w ≥ . , u - v -
-
0( )wC (11).
, (4) R . R , t -
0 0 0( , , )u v wS (6). 1=t .
T
2= −tH I tt ′ = − tR H R . [12, 6],
: [ ]A =P R t , [ ]B = −P R t , [ ]C ′=P R t [ ]D ′= −P R t . -
(3). , -
.
, 2P , -
, , -
, . , 1Q .
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
36. . « . . »36
1 2
1 2 1 33 3
13 23
,
t t
c c c R t
R R
= − = − = + . (15)
,
• 1 0c > 2 0c > , 2 A=P P ;
• 1 0c < 2 0c < , 2 B=P P ;
• 1 0c′ > 2 0c′ > , 2 C=P P ;
• 2 D=P P .
1c′ 2c′ 1c 2c (15) R ′R .
,
12 11ini T
2 22 21 2( )
1
=
H H 0
P H H P
0
,
1jH 2jH 2.1.
3.
5- [6]. -
C/C++. -
. [6], . .
1
0,5
0,1
352 288×
45°
6
10 35 -
/ , – 24 / Intel Core i5 2,3 Ghz.
2 2ε = −P P , 2P – -
.
. 2. -
6
10 . , - , –
0,5 , , - , , -
, – -
. , -
, ,
.
. 2. . : , , −
× 13
1 56 10
, −
× 10
2 94 10 . : -
, , −
× 2
1 52 10 , −
× 3
7 17 10
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
37. . .
,
2013, 5, 2 37
. 3 ( )
.
, 0 1 . ,
.
4.
5- . -
. -
, 5-
. ,
.
,
/ . , , ( , , )ϕ θ ψ ,
R , ,
2ctg( )w ϕ ψ= − + .
10- W .
1. Kruppa, E. Zur Ermittlung eines Objektes aus zwei Perspektiven mit Innerer Orientierung /
E. Kruppa // Sitz.-Ber. Akad. Wiss., Wien, Math. Naturw. Kl. Abt. IIa. – 1913. – Vol. 122. – P. 1939–
1948.
2. Demazure, M. Sur Deux Problemes de Reconstruction / M. Demazure // INRIA. – 1988. –
RR-0882. – P. 1–29.
3. Faugeras, O. Motion from Point Matches: Multiplicity of Solutions / O. Faugeras, S. Maybank //
International Journal of Computer Vision. – 1990. – Vol. 4. – P. 225–246.
4. Heyden, A. Reconstruction from Calibrated Camera – A New Proof of the Kruppa-Demazure
Theorem / A. Heyden, G. Sparr // Journal of Mathematical Imaging and Vision. – 1999. – Vol. 10. –
P. 1–20.
5. Philip, J. A Non-Iterative Algorithm for Determining all Essential Matrices Corresponding to
Five Point Pairs / J. Philip // Photogrammetric Record. – 1996. – Vol. 15. – P. 589–599.
6. Nistér, D. An Efficient Solution to the Five-Point Relative Pose Problem / D. Nistér // IEEE
Transactions on Pattern Analysis and Machine Intelligence. – 2004. – Vol. 26. – P. 756–777.
7. Li, H. Five-Point Motion Estimation Made Easy / H. Li, R. Hartley // IEEE 18th Int. Conf. of Pat-
tern Recognition. – 2006. – P. 630–633.
8. Kukelova, Z. Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems /
Z. Kukelova, M. Bujnak, T. Pajdla // Proceedings of the British Machine Conference. – 2008. – P. 56.1–
56.10.
9. Stewénius, H. Recent Developments on Direct Relative Orientation / H. Stewénius, C. Engels,
D. Nistér // ISPRS Journal of Photogrammetry and Remote Sensing. – 2006. – Vol. 60. – P. 284–294.
10. Cayley, A. Sur Quelques Propriétés des Déterminants Gauches / A. Cayley // J. Reine Angew.
Math. – 1846. – Vol. 32. – P. 119–123.
. 3. ( ) ( ) -
,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
38. . « . . »38
11. Faugeras, O. Three-Dimensional Computer Vision: A Geometric Viewpoint / O. Faugeras //
MIT Press, 1993. – 663 p.
12. Hartley, R. Multiple View Geometry in Computer Vision. / R. Hartley, A. Zisserman. –
Cambridge University Press, 2004. – 655 p.
13. Maybank, S. Theory of Reconstruction from Image Motion / S. Maybank // Springer-Verlag,
1993. – 261 p.
14. Hook, D.G. Using Sturm Sequences to Bracket Real Roots of Polynomial Equations /
D.G. Hook, P.R. McAree // Graphic Gems I : . . . – Academic Press, 1990. – P. 416–422.
15. Numerical Recipes in C. Second Edition / W. Press, S. Teukolsky, W. Vetterling, B. Flannery. –
Cambridge University Press, 1993. – 1020 p.
ALGORITHMIC SOLUTION OF THE FIVE-POINT POSE PROBLEM
BASED ON THE CAYLEY REPRESENTATION OF ROTATION MATRICES
E.V. Martyushev
1
A new algorithmic solution to the five-point relative pose problem is introduced. Our approach is
not connected with or based on the famous cubic constraint on an essential matrix. Instead, we use the
Cayley representation of rotation matrices in order to obtain a polynomial system of equations from epi-
polar constraints. Solving that system, we directly obtain positional relationships and orientations of the
cameras through the roots for a 10th degree polynomial.
Keywords: five-point pose problem, calibrated camera, epipolar constraints, Cayley representation.
References
1. Kruppa E. Zur Ermittlung eines Objektes aus zwei Perspektiven mit Innerer Orientierung. Sitz.-
Ber. Akad. Wiss., Wien, Math. Naturw. Kl. Abt. IIa. 1913. Vol. 122. pp. 1939–1948.
2. Demazure M. Sur Deux Problemes de Reconstruction. INRIA. 1988. no. RR-0882. pp. 1–29.
3. Faugeras O., Maybank S. Motion from Point Matches: Multiplicity of Solutions. International
Journal of Computer Vision. 1990. Vol. 4. pp. 225–246.
4. Heyden A., Sparr G. Reconstruction from Calibrated Camera – A New Proof of the Kruppa-
Demazure Theorem. Journal of Mathematical Imaging and Vision. 1999. Vol. 10. pp. 1–20.
5. Philip J. A Non-Iterative Algorithm for Determining all Essential Matrices Corresponding to Five
Point Pairs. Photogrammetric Record. 1996. Vol. 15. pp. 589–599.
6. Nistér D. An Efficient Solution to the Five-Point Relative Pose Problem. IEEE Transactions on
Pattern Analysis and Machine Intelligence. 2004. Vol. 26. pp. 756–777.
7. Li H., Hartley R. Five-Point Motion Estimation Made Easy. IEEE 18th Int. Conf. of Pattern Rec-
ognition. 2006. pp. 630–633.
8. Kukelova Z., Bujnak M., Pajdla T. Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative
Pose Problems. Proceedings of the British Machine Conference. 2008. pp. 56.1–56.10.
9. Stewénius H., Engels C., Nistér D. Recent Developments on Direct Relative Orientation. ISPRS
Journal of Photogrammetry and Remote Sensing. 2006. Vol. 60. pp. 284–294.
10. Cayley A. Sur Quelques Propriétés des Déterminants Gauches. J. Reine Angew. Math. 1846.
Vol. 32. pp. 119–123.
11. Faugeras O. Three-Dimensional Computer Vision: A Geometric Viewpoint. MIT Press, 1993.
663 p.
12. Hartley R., Zisserman A. Multiple View Geometry in Computer Vision (Second Edition). Cam-
bridge University Press, 2004. 655 p.
13. Maybank S. Theory of Reconstruction from Image Motion. Springer-Verlag, 1993. 261 p.
14. Hook D.G., McAree P.R. Using Sturm Sequences To Bracket Real Roots of Polynomial Equa-
tions. Graphic Gems I (A. Glassner ed.). Academic Press, 1990. pp. 416–422.
15. Press W., Teukolsky S., Vetterling W., Flannery B. Numerical Recipes in C (Second Edition).
Cambridge University Press, 1993. 1020 p.
18 2013 .
1
Martyushev Evgeniy Vladimirovich is Cand.Sc. (Physics and Mathematics), Mathematical analysis department, South Ural State University.
E-mail: zhmart@mail.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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515.126
. . 1
– ,
. -
g:A →→→→ B -
f : X →→→→ X. , -
, , g
f : X →→→→ X .
: , ,
, .
. -
[1]. ,
.
. X ≈ Y , X Y . w(X) – -
X. ω; {0,1,2, }ω = .
, -
. { : }na n ω∈
a X∈ , ( ) { } { : }nS a a a n ω= ∪ ∈ .
X , a b X -
:f X X→ , ( )f a b= . -
, 1- - .
, . -
, -
, , . , -
.
[2].
1. X – , -
. X
{ : }na n ω∈ { : }nb n ω∈ , limn na a→∞ = , limn nb b→∞ = ( ) ( )S a S b = ∅ .
- U a -
V a :f X X→ , , : V U⊂ , ( )f a b= ,
( )f V V = ∅ , Xf f id= , ( )f x x= x V∉ ( ( ))i ii a V b f V∀ ∈ ⇔ ∈ .
. X ,
:g X X→ , a b . - W
a , W U⊂ ( )g W W = ∅ .
: { : , ( )}a i iJ i a W b g Wω= ∈ ∈ ∉ { : , ( )}b i iJ i a W b g Wω= ∈ ∉ ∈ .
aJ bJ – , .
ai J∈ - iQ ia ,
iQ W⊂ ( ) { }i iQ S a a= . , { : }i aQ i J∈
, { : }na n ω∈ .
1
– - , , -
, - .
E-mail: medv@math.susu.ac.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
40. . « . . »40
, ( )ig a ( )S b .
1 2a a aJ J J= ∪ , 1 { : ( ) ( )}a a iJ i J g a S b= ∈ ∉ 2 { : ( ) ( )}a a iJ i J g a S b= ∈ ∈ .
: 1ai J∈ . - iD
ia , i iD Q⊂ ( ) ( )ig D S b = ∅ . ig g= .
: 2ai J∈ , . . ( )i jg a b= j . –
, ( ) ( )ig Q S b – , i ic Q∈ ,
( ) ( )ig c S b∉ . :i X Xψ → , ia ic . -
- i iD Q⊂ ia , ( )i i i iD D Qψ ⊂ ,
( )i i iD Dψ = ∅ ( ( )) ( )i ig D S bψ = ∅ . :ig X X→ :
1
( ), ,
( ) ( ), ( ),
( ), ( ( )).
i i
i i i i
i i i
g x x D
g x g x x D
g x x X D D
ψ
ψ ψ
ψ
−
∈
= ∈
∈
, ig ( ) ( )i ig D S b = ∅ .
*:g X X→ :
( ), ,
*( )
( ), { : }.
i i a
i a
g x x D i J
g x
g x x X D i J
∈ ∈
=
∈ ∈
, *g – . , * { : }i aW W D i J= ∈ – -
- a . *( )g a b= ( ) ( ) ( ) *( *)S b g W S b g W= .
, aJ = ∅ , *W W= *g g= .
, bi J∈ - iE ib
, *( *)iE g W⊂ , ( ) { }i iE S b b= , 1
( ) ( )i iS a h E−
= ∅ , { : }i bE i J∈
. :ih X X→ *g
, ig g .
1
* { ( ): }i i bV W h E i J−
= ∈ – - a .
:h X X→ , :
1
1
( ), ( ) ,
( )
*( ), { ( ): }.i
i i i b
i b
h x x h E i J
h x
g x x X h E i J
−
−
∈ ∈
=
∈ ∈
:f X X→ :
1
( ), ,
( ) ( ), ( ),
, ( ( )).
h x x V
f x h x x h V
x x X V h V
−
∈
= ∈
∈
, f – . 1 .
1. X – , -
. X
{ : }na n ω∈ { : }nb n ω∈ , limn na a→∞ = , limn nb b→∞ = ( ) ( )S a S b = ∅ .
:f X X→ , Xf f id= , ( )f a b= ( )n nf a b=
n ω∈ .
. *
{ : }nU n ω∈ a , -
. 1 - 0U a
0 :g X X→ , , : *
0 0U U⊂ , 0( )g a b= , 0 0 0( )g U U = ∅ , 0 0 Xg g id= ,
0( )g x x= 0x U∉ 0 0 0( ( ))i ii a U b g U∀ ∈ ⇔ ∈ . -
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
41. . .
2013, 5, 2 41
, 1n ≥ - nU a -
:ng X X→ , , : *
1n n nU U U −⊂ , ( )ng a b= , ( )n n ng U U = ∅ , n n Xg g id= ,
( )ng x x= nx U∉ ( ( ))i n i n ni a U b g U∀ ∈ ⇔ ∈ . , -
{ : }nU n ω∈ a , { ( ): }n ng U n ω∈
b . n ω∈ { : }n i nJ i a Uω= ∈ ∉ .
j ω∈ :j X Xχ → , ( )j j ja bχ = .
- jO ja . ( )j jOχ – - -
jb . , { : }jO j ω∈
{ ( ): }j jO jχ ω∈ , lim { }j jO a→∞ = , lim ( ) { }j j jO bχ→∞ = , 1j k kO U U +⊂
1 1( ) ( ) ( )j j k k k kO g U g Uχ + +⊂ , 1j k ka U U +⊂ k ω∈ .
n ω∈ nJ , { : }n n j nV U O j J= ∪ ∈
( ) { ( ): }n n n j j nW g U O j Jχ= ∪ ∈ - , ( ) nS a V⊂ , ( ) nS b W⊂ , 1n nV V+ ⊂ ,
1n nW W+ ⊂ n nV W = ∅ . , ( )n n nf V W= :nf X X→ , -
:
1
1
( ), ,
( ), ( ),
( ) ( ), ,
( ), ( ) ,
, .
n
j
n n
n n
n j j n
j j n
g x x U
g x x g U
f x x x O j J
x x O j J
x
χ
χ χ
−
−
∈
∈
= ∈ ∈
∈ ∈
, n n Xf f id= .
:f X X→ ( ) lim ( )n nf x f x→∞= . , ( )f a b=
( )f b a= . , . , { : } { }nU n aω∈ =
{ ( ): } { }n ng U n bω∈ = , { , }x X a b∈ n , ( )n n nx U g U∉ .
1( ) ( )n nf x f x+ = ; lim ( )n nf x→∞ . ,
f – . 1 .
, { }a A a∈ ,
. d
A
. n ω∈ -
(n)
n : (0)
A A= ( 1) ( )
( )n n d
A A+
= . ,
n, ( 1)n
A −
≠ ∅ , ( )n
A = ∅ . ( )n
A = ∅ n ω∈ ,
, .
1.
2. X – .
X
. :g A B→ -
:f X X→ , Xf f id= .
. – , –
. f .
, . –
. n .
: 1n = . (1) (1)
A B= = ∅ , – .
, 1{ , , }kA a a= 1{ , , }kB b b= , ( )i ib g a= 1 i k≤ ≤ . ,
i :if X X→ , ( )i i if a b= . -
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
42. . « . . »42
- iU jV ia jb . -
, ( )i i if U V= . :f X X→ :
1
( ), ,
( ) ( ), ,
, { :1 }.
i
i i
i
i i
f x x U i
f x f x x V i
x x X U V i k
−
∈
= ∈
∈ ∪ ≤ ≤
, , n.
1n + . (1) (1)
n. , (1) (1)
( )g A B= .
g
:h X X→ , Xh h id= , (1) (1)
( )h B A= (1) (1)
( )h A B= .
(1)
A A { : }ma m ω∈ . (1)
{ ( ): }mB B g a m ω= ∈ . -
1, , ( ) ( )m kg a h a≠ m k.
(1)
( )h A A B = ∅ (1)
( )h B B A = ∅ . m ω∈
:m X Xχ → , ma ( )mg a . ( ) ( )A h A B h B
, - mU ( )m m mV Uχ= -
ma ( )mg a , mU , mV , 1
( ) ( )m mh U h U−
= 1
( ) ( )m mh V h V−
= -
, (1) (1)
A B . , (1)
(1)
– , , { : }jmU j ω∈
(1)
*a A∈ ⇔ { : }jmV j ω∈ (1)
( *) ( *)g a h a B= ∈ ⇔ -
{ ( ): }jmh V j ω∈ (1)
* ( *)a h h a A= ∈ . -
f :
1
1 1
1 1
( ), ,
( ), ,
( ) ( ), ( ) ,
( ), ( ) ,
( ), .
m
m m
m
m m
m m
x x U m
x x V m
f x h h x x h V m
h h x x h U m
h x
χ
χ
χ
χ
−
− −
− −
∈
∈
= ∈
∈
. 2 .
:g A B→
: f X X A→ .
2. X
{ : }nU n ω∈ { : }nV n ω∈ - , :
1) 0 1 nU U U⊃ ⊃ ⊃ ⊃ ,
2) 0U V = ∅ , { : }nV V n ω= ∈ ,
3) { : }nV n ω∈ X,
4) n ω∈ :n n ng U V→ .
{ : }nA U n ω= ∈ : f X X A→ ,
0 0( ) ( )f A g A V= ⊂ .
. : f X X A→ :
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
43. . .
2013, 5, 2 43
0 0
1
1 1
1
1
0
( ), ,
( ), ( ) ,
( )
( ), ( ) ,
, .
n n n n
n n n n
g x x U
g g x x g U n
f x
g x x V g U n
x x U V
ω
ω
−
+ +
−
+
∈
∈ ∈
=
∈ ∈
∉
, 0 1 { : }n nU A U U n ω+= ⊕ ∈ 1 1( ) ( )n n n nf V U U V+ += n ω∈ , -
, f X X A . -
1( )n ng U + 1 ( )n n nV g U + - nV ( , X),
0U V - X, ng , -
f . , 0 0( ) ( )f A g A V= ⊂ . 2 .
3. X
a b.
: { }f X X a→ , ( )f a b= .
. X , X -
{ : }nW n ω∈ , - . ,
{ : }nW W n ω= ∈ - X.
, a W∉ 0b W∈ . a *
{ : }nU n ω∈ , -
- *
0W U = ∅ .
X , 0 :g X X→ ,
a b . - 0U a , *
0 0U U⊂
0 0 0 0( )V g U W= ⊂ . , , 1n ≥ :ng X X→
- nU a : *
1n n nU U U−⊂
( )n n nV g U= - nW . , { : } { }nU n aω∈ = . -
3 2.
1. X
. X – .
3. - -
*U { : }iW i ω∈ , -
- . -
*U U⊂ { : }iV W i ω⊂ ∈ , A U⊂ .
. , { : }iA a i ω= ∈ . i
i ib W∈ :ig X X→ , ia ib . -
iU ia , : *iU U⊂ ( )i i ig U W⊂ .
{ : }iU i ω∈ { : }iU i k≤ .
iU { : }i jU U j i< , , { : }iU i k≤
- . { : }iU U i k= ≤
{ ( ): }i iV g U i k= ≤ – . 3 .
4. X – , -
. X -
.
: f X X A→ , ( )f A B= .
. ,
, - , A B .
{ : 1, }niW n i ω≥ ∈ . { : 1, }niW W n i ω= ≥ ∈ .
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
44. . « . . »44
2 :g X X→ , ( )g A B= . – -
, - 0U ,
: 0( )B g U⊂ , 0U W = ∅ , 0( )g U W = ∅ 0 0( )U g U = ∅ .
3 1n ≥ -
nU , nV :n n ng U V→ , , nA U⊂ , 1n nU U+ ⊂ { : }n niV W i ω⊂ ∈ .
, { : }nA U n ω= ∈ ,
. 4 2.
. 2
, 4 .
5. X – -
, . -
Z * , -
*X X Z, * .
. Z, , -
, -
Z. 4 : f X X A→ , ( )f A B= .
X A . 1
* ( )X f X A−
= , -
.
1. van Douwen, E.K. A compact space with a measure that knows which sets are homeomorphic /
E.K. van Douwen // Adv. in Math. – 1984. – Vol. 52. – Issue 1. – P. 1–33.
2. Engelking, R. General topology / R. Engelking. – Berlin: Heldermann Verlag, 1989. – 540 p.
ABOUT EXTENSION OF HOMEOMORPHISMS
OVER ZERO-DIMENSIONAL HOMOGENEOUS SPACES
S.V. Medvedev
1
Let X be a zero-dimensional homogeneous space satisfying the first axiom of countability. We
prove the theorem about an extension of a homeomorphism g: A → B to a homeomorphism f: X → X,
where A and B are countable disjoint compact subsets of the space X. If, additionally, X is a non-
pseudocompact space, then the homeomorphism g is extendable to a homeomorphism f: X → X A.
Keywords: homogeneous space, homeomorphism, first axiom of countability, pseudocompact space.
References
1. van Douwen E.K. A compact space with a measure that knows which sets are homeomorphic.
Adv. in Math. 1984. Vol. 52. Issue 1. pp. 1–33.
2. Engelking R. General topology. Berlin: Heldermann Verlag, 1989. 540 p.
30 2013 .
1
Medvedev Sergey Vasiljevich is Cand.Sc. (Physics and Mathematics), Associate Professor, Mathematical and Functional Analysis Depart-
ment, South Ural State University.
E-mail: medv@math.susu.ac.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
45. 2013, 5, 2 45
519.633
. . 1
, . . 2
, . . 3
,
,
.
.
: ,
, .
1.
-
[1]
[ ]11 12( , ) ( , ) ( , ) ( , ) ( , ), ( , ) [0, ] 0,t t
tv x t v x t w x t v x w x x t Tπ= Δ − Δ + Φ ⋅ + Φ ⋅ ∈ × , (1)
[ ]21 220 ( , ) ( ) ( , ) ( , ) ( , ), ( , ) [0, ] 0,t t
v x t w x t v x w x x t Tβ π= + + Δ + Φ ⋅ + Φ ⋅ ∈ × , (2)
[ ](0, ) ( , ) (0, ) ( , ) 0, 0,v t v t w t w t t Tπ π= = = = ∈ , (3)
[ ]( , ) ( , ), ( , ) ( , ), ( , ) [0, ] ,0v x t x t w x t x t x t rϕ ψ π= = ∈ × − , (4)
( , ) ( , ), ( , ) ( , )t t
v x s v x t s w x s w x t s= + = + [ ],0 , 0.s r r∈ − >
1 1 2 2: ( , ) ( , ), : ( , ) ( , )t t
i i i iv x z x t w x z x tΦ ⋅ → Φ ⋅ → [ ]0,x π∈ , [ ]0,t T∈ -
[ ]( ,0 ; )C r R− R.
[ ]( ) ( ), ,0 ,u t h t t r= ∈ − (5)
-
[ ]( ) ( ) ( ), 0,t
Lu t Mu t u f t t T= + Φ + ∈ , (6)
. . . . [2, 3]. U, F –
, ( ) ( )t
u s u t s= + [ ],0s r∈ − , :L U F→ , [ ]: ( ,0 ; )C r U FΦ − →
, { }ker 0L ≠ , M: domM F→ , U,
[ ]: 0,f T F→ . (6) ,
,
. -
(5), (6), -
.
,
- - ,
(6), . . [4–6]. ,
, -
1
– , - ,
« ».
E-mail: omelchenko_ea@mail.ru
2
– , , - -
.
E-mail: mariner79@mail.ru
3
– , , .
E-mail: davydov@csu.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
46. . « . . »46
, ,
.
[2, 3, 5, 6]
(1)–(4). , -
,
( 0, , 1, 2).ij i jΦ = =
, .
, -
, [5, 6]
. , , -
.
2.
-
[ ]( , ) ( , ) ( , ), ( , ) [0, ] 0,tv x t v x t w x t x t Tπ= Δ − Δ ∈ × , (7)
[ ]0 ( , ) ( ) ( , ), ( , ) [0, ] 0,v x t w x t x t Tβ π= + + Δ ∈ × , (8)
[ ](0, ) ( , ) (0, ) ( , ) 0, 0,v t v t w t w t t Tπ π= = = = ∈ , (9)
( ,0) ( ), [0, ],v x x xϕ π= ∈ (10)
0β < , v, w – . , ( ),0w x -
w . , [7], (7)–(10) .
( ),0v x , ( ),0w x -
(7)–(10) -
.
[ ]0,π / ,h Nπ=
, 0, , .nx nh n N= = [ ]0,T
0,τ > ,mt mτ= 0, , .m M=
v, w ( , )n mx t m
nv , m
nw . -
1
1 1 1 1
2 2
2 2
, 0, , 1,
m m m m m m m m
n n n n n n n nv v v v v w w w
m M
h hτ
+
+ − + −− − + − +
= − = − (11)
1 1
2
2
0 , 0, , ,
m m m
m m n n n
n n
w w w
v w m M
h
β + −− +
= + + = (12)
1, , 1n N= − ,
0
( ), 0, , ,n nv x n Nϕ= =
0 0 0, 0, , .m m m m
N Nv v w w m M= = = = =
(11), (12) ( ),m m m
n n nξ ηΨ = ,
1 1 1
2
1 1
2
( , ) ( , ) ( , ) 2 ( , ) ( , )
( , ) 2 ( , ) ( , )
,
m n m n m n m n m n m
n
n m n m n m
v x t v x t v x t v x t v x t
h
w x t w x t w x t
h
ξ
τ
+ + −
+ −
− − +
= − +
− +
+
1 1
2
( , ) 2 ( , ) ( , )
( , ) ( , ) ,m n m n m n m
n n m n m
w x t w x t w x t
v x t w x t
h
η β + −− +
= + +
( , )n mv x t , ( , )n mw x t v, w (7)–(10) -
. , 1 2p p
hτ + ,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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. .
2013, 5, 2 47
C, τ h, 1 2
2
( )p pm
n
R
C hτΨ ≤ + ,
1, , 1,n N= − 0, , 1m M= − .
1. v, w (7)–(10) , v -
t, v, w x. -
(11), (12) 2
.hτ +
. ( ),v x t , ( ),w x t
2 21 1 1
( ), .
2 12 12
m m
n tt xxxx xxxx n xxxxv h v w h wξ τ η= − + + = −
.
1. 0β < . (11), (12) , 2
.hτ ≤
. m
qρ , m
qσ q- m- ,
( , ) , ( , ) , 0, 1, 2,n niqx iqxm m
n m q n m qv x t e w x t e qρ σ= = = ± ±
(11), (12)
1
2
(2 (cos 1) 2 (cos 1))m m m m
q q q qqh qh
h
τ
ρ ρ ρ σ+
− = − − − , (13)
2
2
0 (cos 1)
m
qm m
q q qh
h
σ
ρ βσ= + + − .
2
2
(1 cos )
m
qm
q
qh
h
ρ
σ
β
=
− −
(14)
(13),
2
1
2 2
2
1 (1 cos ) 1 .
2(1 cos )
m m
q q
h
qh
h qh h
τ
ρ ρ
β
+
= − − −
− −
(15)
, 0β < ,
2
2 2
2
1 (1 cos ) 1 1
2(1 cos )
q
h
r qh
h qh h
τ
β
≡ − − − ≤
− −
q N∈ . , 1qr ≥ − q N∈ . ,
2
2 2
(1 cos ) 1 1.
2(1 cos )
h
qh
h qh h
τ
β
− − ≤
− −
2
hτ ≤ .
(14) (15)
2
1 2 2
1
2 2
2
2
2
1 (1 cos ) 1
2(1 cos )
2 2
(1 cos ) (1 cos )
2 1
1 (1 cos ) 1 .
2
(1 cos )
m
qm
qm
q
m
q
h
qh
h qh h
qh qh
h h
qh
h qh
h
τ
ρ
ρ β
σ
β β
τ
σ
β
+
+
− − −
− −
= = =
− − − −
= − − −
− −
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
48. . « . . »48
qr -
2
hτ ≤ .
1. 1 , 0β ≥ -
.
, -
[8], 1 1.
2. 0β < (11), (12) ,τ h , 2
hτ ≤ . -
(7) – (10) 2
hτ + .
3.
- (1)–(4).
[ ]0,π /h Nπ= , 0, ,nx nh n N= = .
[ ],r T− . , /T r – -
, / /T M r Lτ = = , ,L M N∈ ,
, , ,0, , .mt m m L Mτ= = − ,m m
n nv w v, w
nx kt :
( ){ } ( ){ }, , : , 0, , , 0, , .m m m m
n n n n
k
v w v w k L m k n N k M= − ≤ ≤ = =
,
( ){ } ( ) [ ] [ ]: , , ,0 ,0m m k k
n n n n
k
I v w g h Q r Q r→ ∈ − × − . 0, , ,n N= [ ],0Q r− – -
[ ],0r− -
. : [ ]
[ ]
,0
,0
sup ( )Q r
s r
g g s−
∈ −
= [ ],0 .g Q r∈ − ,
ijΦ [ ],0Q r− .
[4, 5], , -
p
τ v, w, C1, C2,
0, , ,n N= 0, ,k M= [ ],k kt t r t∈ −
1 2( ) ( , ) max ( , ) ,k m p
n n n n m
k L m k
g t v x t C v v x t C τ
− ≤ ≤
− ≤ − +
1 2( ) ( , ) max ( , ) .k m p
n n n n m
k L m k
h t w x t C w w x t C τ
− ≤ ≤
− ≤ − +
1
1 1 1 1
11 122 2
2 2
,
m m m m m m m m
m mn n n n n n n n
n n
v v v v v w w w
g h
h hτ
+
+ − + −− − + − +
= − + Φ + Φ (16)
1 1
21 222
2
0 ,
m m m
m m m mn n n
n n n n
w w w
v w g h
h
β + −− +
= + + + Φ + Φ (17)
1, , 1,n N= − 0, , 1m M= − (16) 0, ,m M= (17),
0 0
( ,0), ( ,0), 0, , ,n n n nv x w x n Nϕ ψ= = = (18)
[ ]0 0
( ) ( , ), ( ) ( , ), 0, , , ,0n n n ng t x t h t x t n N t rϕ ψ= = = ∈ − , (19)
0 0 0, 0, , .m m m m
N Nv v w w m M= = = = = (20)
[2, 3] (1) – (4)
ϕ ψ
21 220 ( ,0) ( ) ( ,0) ( , ) ( , ), (0, ).x x x x xϕ β ψ ϕ ψ π= + + Δ + Φ ⋅ + Φ ⋅ ∈
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. .
2013, 5, 2 49
, .
(16), (17) ( , )m m m
n n nψ ξ η= ,
1 1 1
2
1 1
11 122
( , ) ( , ) ( , ) 2 ( , ) ( , )
( , ) 2 ( , ) ( , )
( , ) ( , ),m m
m n m n m n m n m n m
n
t tn m n m n m
n n
v x t v x t v x t v x t v x t
h
w x t w x t w x t
v x w x
h
ξ
τ
+ + −
+ −
− − +
= − +
− +
+ − Φ ⋅ − Φ ⋅
1 1
21 222
( , ) 2 ( , ) ( , )
( , ) ( , ) ( , ) ( , ).m mt tm n m n m n m
n n m n m n n
w x t w x t w x t
v x t w x t v x w x
h
η β + −− +
= + + + Φ ⋅ + Φ ⋅
, 4 [5], 1
(11) , (12), 1 3 [5] .
3. 0β < , 2
hτ ≤ , [ ]2 2
: ,0L
I R Q r+
→ −
0p
τ , m
nψ 1 2p p
hτ + .
(1)–(4) { }0 1 2min ,p p p
hτ + .
, ijΦ
, .
4.
( 1)ij ijg a gΦ = − [ ]1,0 ,g Q∈ − i, j = 1, 2. , (1)–(4)
[ ]11 12( , ) ( , ) ( , ) ( , 1) ( , 1), ( , ) [0, ] 0,tv x t v x t w x t a v x t a w x t x t Tπ= Δ − Δ + − + − ∈ × , (21)
[ ]21 220 ( , ) ( ) ( , ) ( , 1) ( , 1), ( , ) [0, ] 0,v x t w x t a v x t a w x t x t Tβ π= + + Δ + − + − ∈ × , (22)
(3), (4). (16), (17)
, , - -
1
1 1 1 1
11 122 2
2 2
,
m m m m m m m m
m L m Ln n n n n n n n
n n
v v v v v w w w
a v a w
h hτ
+
− −+ − + −− − + − +
= − + + (23)
1 1
21 222
2
0 ,
m m m
m m m L m Ln n n
n n n n
w w w
v w a v a w
h
β − −+ −− +
= + + + + (24)
1, , 1, 0, , 1n N m M= − = − (23) 0, ,m M= (24), (18), (19)
(20). -
τ , - – 2
τ [9].
1, .
2. v, w (21), (22), (3), (4) , v
t, v, w
x. (23), (24) 2
hτ + .
3
1. 0β < , 2
hτ ≤ , - - .
(1)–(4) 2
hτ + .
. 1 (v,w) β = –0,75, T = 10, M = 1000, N = 16
c ( ,0) sin , [0, ]v x x x π= ∈ (7)–(10).
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. .
2013, 5, 2 51
NUMERICAL SOLUTION OF DELAYED LINEARIZED QUASISTATIONARY
PHASE-FIELD SYSTEM OF EQUATIONS
E.A. Omelchenko
1
, M.V. Plekhanova
2
, P.N. Davydov
3
For delayed linearized quasistationary phase-field system of equations the numerical method of so-
lution was proposed. The convergence of explicit difference scheme that takes account of delay in the
system under investigation was thoroughly studied. On the basis of the results obtained the implementa-
tion of the method was realized.
Keywords: Sobolev type equation, quasistationary phase-field system of equations, difference
scheme.
References
1. Plotnikov P.I., Klepacheva A.V. Siberian Mathematical Journal. 2001. Vol. 42, no. 3. pp. 551–
567.
2. Fedorov V.E., Omelchenko E.A. On solvability of some classes of Sobolev type equations with
delay. Functional Differential Equations. 2011. Vol. 18, no. 3–4. pp. 187–199.
3. Fedorov V.E., Omel’chenko E.A. Siberian Mathematical Journal. 2012. Vol. 53, no. 2. pp. 335–
344.
4. Lekomtsev A.V., Pimenov V.G. Russian Mathematics (Izvestiya VUZ. Matematika). 2009.
Vol. 53, no. 5. pp. 54–58.
5. Pimenov V.G. Trudy Inst. Mat. i Mekh. UrO RAN. 2010. Vol. 16, no. 5. pp. 151–158. (in Russ.).
6. Pimenov V.G., Lozhnikov A.B. Proceedings of the Steklov Institute of Mathematics (Supplemen-
tary issues). 2011. Vol. 275. Suppl. 1. pp. 137–148.
7. Fedorov V.E., Urazaeva A.V. Trudy Voronezhskoy zimney matematicheskoy shkoly (Proc. of the
Voronezh Winter Mathematical School). Voronezh: VGU. 2004. pp. 161–172.
8. Richtmyer R.D., Morton K.W. Difference Methods for Initial-Value Problems. Interscience, New
York, 1967. 405 p.
9. Kim A.V., Pimenov V.G. i-Gladkiy analiz i chislennye metody resheniya funktsional'no-
differentsial'nykh uravneniy (i-differential analysis and numerical methods of solutions of functional and
differential equations). Izhevsk: RKhD, 2004. 256 p. (in Russ.).
15 2013 .
1
Omelchenko Ekaterina Aleksandrovna is Senior Lecturer, Department of Humanities and socio-economic disciplines, Ural Branch of Russian
Academy of Justice.
E-mail: omelchenko_ea@mail.ru
2
Plekhanova Marina Vasilyevna is Associate Professor, Differential and Stochastic Equations Department, South Ural State University.
E-mail: mariner79@mail.ru
3
Davydov Pavel Nikolaevich is Post-graduate student, Mathematical Analysis Department, Chelyabinsk State University.
E-mail: davydov@csu.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
53. . .,
. .
2013, 5, 2 53
, [6, 7], Q
: ( ( , ))Q Q x tψ= , ( , ) constx tψ = – -
( , )Q x t . (3) (4)
''' 3 '' '
( 3 ) ( ) 0x x t x xx xt xxx xkQ Q k Q k qψ ψ ψ ψ ψ ψ ψ ψ+ − + + − + + = , (5)
''' '3 3 '' '
( ) ( 3 ) ( ) 0x x t x xx xt xxx xkQ Q Q k Q k qα ψ ψ ψ ψ ψ ψ ψ ψ− + − + + − + + = . (6)
( )' ψ . (5) (6), -
, ( , )x tψ , ,
0xψ ≠ [6, 7]
12
3
( ),t xx
x
k
f
ψ ψ
ψ
ψ
− +
= 23
( )xt xxx x
x
k q
f
ψ ψ ψ
ψ
ψ
− + +
= . (7)
1( ),f ψ 2 ( )f ψ – . , (7)
. 2
1( )/(3 ).xx t xf kψ ψ ψ= + -
x xxxψ -
. xtψ ,
2
1
,
3
t x
xx
f
k
ψ ψ
ψ
+
= 3
3 1
3 1
2 3
xt x x x tq f f
k
ψ ψ ψ ψ ψ= + + , ' 2
3 1 1 2
1 1 3
2 3 2
f f f f
k
= + − . (8)
(8) ,
( xxt xtxψ ψ= ). ,
2 2 4
1 3 4
3 1
3 ,
2 3
tt t t t x xq f f f
k
ψ ψ ψ ψ ψ ψ= + + + '
4 1 3 32 3f f f kf= + . (9)
xtψ ,
. , xtt ttxψ ψ= . -
,
2 2 ' 4
4 3 4 1 46 6 3 ( / ) 0t x t xf f k f f f kψ ψ ψ ψ+ + + = . (10)
, 3 0f = , (10) . -
:
1. (7) , ( , )x tψ -
(8), (9) ' 2
2 1 1/3 2 /(9 )f f f k= + , 1( )f ψ – .
(5), (6)
''' '' ' ' 2
1 1 1[ /3 2 /(9 )] 0kQ Q f Q f f k+ + + = ; ''' '3 '' ' ' 2
1 1 1( ) [ /3 2 /(9 )] 0kQ Q Q f Q f f kα− + + + = .
3 0f ≠ , , (10) -
(8), (9), , 1( )x gψ ψ= , 2 ( ).t gψ ψ= -
, 2 1( ) ( ), const.g Cg Cψ ψ= = -
2. 3 0f ≠ , ( ),ax btψ ψ= + consta = , constb = . ,
'
( ) ( )x xu Q Q ψ ψ ψ= = , ( )u u ax bt= + (1) (2)
2
0,z zzbu ka u qu− + + = 2 3
0,z zzbu ka u qu uα− + + − = .z ax bt= +
, 1 constf = 2
2 12 /(9 ) constf f k= = . (10) -
. (9) tψ , , x – . ,
0tψ ≠ , (9) 1(1/ ) 1,5 (1/ ) /(3 )t t tq f kψ ψ+ = . -
, 11,5 ( )exp(3 / 2)/[1 ( )exp(3 )/(3 )]t qC x qt f C x qt kψ = − . (8)
xψ . ,
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1exp(3 / 2)/[1 ( )exp(3 / 2)/(3 )].x xC qt f C x qt kψ = −
(8), -
( )C x : /(2 )xxC qC k= . 1 2( ) exp[ /(2 )] exp[ /(2 )].C x A x q k A x q k= ± +
1 const,A = 2 constA = . , 1 1A = , 2 0A = -
tψ xψ
1
3 exp[(3 / 2) /(2 )]/ 2
,
1 exp[(3 / 2) /(2 )]/(3 )
t
q qt x q k
f qt x q k k
ψ
±
=
− ± 1
/(2 ) exp[(3 / 2) /(2 )]
1 exp[(3 / 2) /(2 )]/(3 )
x
q k qt x q k
f qt x q k k
ψ
± ±
− ±
.
xψ
. , ,
1 13 ln{1 exp[(3 / 2) /(2 )]/(3 )}/k f qt x q k k fψ = − − ± . (11)
2 0f = . 1( ) 3 /(2 )of kψ ψ ψ= + , constoψ = ( . 1). ,
( , )x tψ (8), (9).
(9) (8) , x – , ,
2 [ ( ) 3 2 / 2],t o oM x qψ ψ ψ ψ ψ= + + + ( ) 2 exp(3 / 2)x oN x qtψ ψ ψ= + .
(8),
2
2
2
2 3
exp ,
2 2 2 3 2
ok q q M
qt x
k k kq
ψ
ψ = ± ± − − const,M = constoψ = . (12)
, .
2 0f = , constt xxk q Aψ ψ ψ− − = = . ,
2 2
4 /(9 )o M qψ = , ( , )x tψ (2) (1) 0A = . ,
1. 2 2 2
( , ) (2 / ){ /(2 ) exp[3 / 2 /(2 )] /(3 )} / 2ou x t k q q k qt x q k M k ψ= ± ± − −
t xxu ku qu C= + + , constM = , constoψ = ,
( , , )oC C q M ψ= , (2).
2. 2 2 2
( , ) (2 / ){ /(2 ) exp[3 / 2 /(2 )] /(3 )} / 2ou x t k q q k qt x q k M k ψ= ± ± − −
(1) (2),
2 2
4 /(9 )o M qψ = .
, '2
2 ( )f Qα ψ= , 1 (7) -
,
' 2 '2
1 1 2/3 2 /(9 )f f k f Qα+ = = . (13)
, (6) ''' ''
1 0kQ Q f+ = . -
1f (13),
'''' '' '''2 ''2 '2
5 /3 3 / 0Q Q Q Q Q kα− + = .
, '
02 / (3 )Q k α ψ ψ= + , 0 constψ = . -
(13) , 1 06 /(3 )f k ψ ψ= − . , (8), (9).
,
3
01 2 3
exp ,
3 3 2 2 3
q
qt x c
q k
ψ
ψ = ± − − const.c = (14)
( )Q ψ ( 1 constf = )
''' '' ' ' 2
1 1 1[ /3 2 /(9 )] 0kQ Q f Q f f k+ + + = ; ''' '3 '' ' ' 2
1 1 1( ) [ /3 2 /(9 )] 0kQ Q Q f Q f f kα− + + + = .
( )Q ψ , ,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
55. . .,
. .
2013, 5, 2 55
'
1 1 2 1exp[ /(3 )] exp[ 2 /(3 )]Q C f k C f kψ ψ= − + − , 1 const,C = 2 constC = .
, (1) '
x xu Q Qψ= = ,
1
/(2 ) exp[3 / 2 /(2 )]
1 exp[3 / 2 /(2 )]/(3 )
q k qt x q k
u
f qt x q k k
± ±
=
− ±
1 1 2 1{ exp[ /(3 )] exp[ 2 /(3 )]}.C f k C f kψ ψ− + −
ψ (11),
1
1 2
3 3
exp 1 exp .
2 2 2 3 2 2
fq q q q q
u t x C C t x
k k k k
= ± ± + − ±
( )Q ψ . , '
( )Q p ψ= , ,
'
( )p r p= . 3 2
1 12 /(9 )pkrr f r p f p kα+ = − .
2
r ap bp= + , const,a = constb = . ,
/(2 )a kα= ± , 1 /(3 )b f k= . ,
' 1
1
/(3 )
,
/(2 ) exp[ /(3 )]
f k
Q
k C f kα ψ
= −
± −
const.C =
, (2) '
x xu Q Qψ= = , -
(2)
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α± ±
=
− − ±
0 .
2
C C
k
α
= ± (15)
, -
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α +
=
− − ±
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α− +
=
− − ±
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
,
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α −
=
− − ±
1
0 1
[ /(3 )] / exp[3 / 2 /(2 )]
.
1 [ /(3 )]exp[3 / 2 /(2 )]
f k q qt x q k
u
C f k qt x q k
α− −
=
− − ±
(16)
, 2 0f = (5) ''' ''
3 /(2 ) 0.oQ Q ψ ψ+ + = -
, , '
1 2 / 2 oQ C C ψ ψ= − + , 1 const,C = 2 constC = . -
, (12),
1 2
2 3 3 3
exp exp exp .
2 2 2 2 2 3 2 2
k q q q M q
u C qt x qt x C qt x
q k k k k k
= ± ± ± − − ±
2 0f = , (6) ''' '3 ''
( / ) 3 /(2 ) 0.oQ Q k Qα ψ ψ− + + = -
: '
2 / [1/(2 )]oQ k α ψ ψ= ± + .
(6)
1
2 3 3
exp exp .
2 2 2 2 2 2
q k q q q M
u qt x qt x
k k k k kα
−
= ± ± ± − (17)
(14), (2)
1
/ exp[3 / 2 /(2 )]{exp[3 / 2 /(2 )] } ,u q qt x q k qt x q k cα −
= ± ± ± − const.c = (18)
u
( , , ).
, (1) (2) constt = .
, (2) ( , )u x t
( , )t u x [8].
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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2 2 2 3 3
( 2 ) ( ) 0u u xx x u xu x uu ut k t t t t t t t qu u tα+ − + − − = . (19)
(19) ( ) ,u xϕ ξ− = x η= :
2 2 2 2
[ ( 2 ) 2 ( )( ) ( ) ]x x xx x x xt k t t t t t t t t t t t t tξ ξ ηη ξη ξξ ξ ξ η ξ ξη ξξ ξξ η ξϕ ϕ ϕ ϕ ϕ ϕ+ − + − − − − + −
3 3
[ ( ) ( ) ] 0.q tξξ ϕ α ξ ϕ− + − + =
, constξ = – (19), -
tξξ . , 0tη = . ,
constξ = constt = . , (19) ,
2 3 3 3
[ ( ) ( ) ] 0.xxt kt q tξ ξ ξϕ ξ ϕ α ξ ϕ− − + − + =
, , 0α = ( (1)), uξ ϕ+ = , , -
constt =
0 0 1( , ) { ( ) ( )sin[ ( ( )) / ]}/ ,u t x w t c t x c t q k q= + ± +
0 ( ) const,c t = 0 ( ) const,w t = 1( ) const.c t = (20)
(2). , ,
( , )x u t –
2 3 3
( ) 0t u uu ux x kx qu u xα− + − = . (21)
0α ≠ , constt = , -
1
4 2
0 0/(4 ) /(4 )
du
x c
u k qu k w u cα
= ±
− + +
. (22)
, , 1 1 2 2 0 0( ), ( ), ( )c c t c c t w w t= = = ,
(21),
1 1 2 2 0 0( ), ( ), ( )c c t c c t w w t= = = . -
( , )u x t , (22):
4 2 2 2 2
0 0/(4 ) /(4 ) [ /(4 )][ 2 ( ) ( / 2 ( ) ( ))][ 2 ( )u k qu k w u c k u a t u q a t b t u a t uα α α− + + = + − − − −
2
( / 2 ( ) ( ))]q a t b tα− − +
(22) -
, [9]. -
a constt = ( , ) ( )u x t x=℘ ( ( )x℘ – -
). [10], , -
, , , (20)
(1).
(2) , , -
, , (
(15)–(18) ) .
. 1 (1:
t = 0; 2: t = 0,5; 3: t = 1; 4: t = 1,5; 5: t = 2; 6: t = 2,5). . 2
(1: t = 0; 2: t = 0,1; 3: t = 0,5; 4: t = 1; 5: t = 1,5).
(15), (17), (18) , ,
, –
( . . 3, ( , )u x t
c 1t = (1: c = 100; 2: c = 1000; 3: c = 2500;
4: c = 4000; 5: c = 7000).
{ 0, 0}x t= = . , (18) [ / (0,0)]/ (0,0).c q u uα= ± − (0,0) / ,u q α= 0c =
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
57. . .,
. .
2013, 5, 2 57
( , ) const /u x t q α= = . ,
( . 4, ( , )u x t
c , 1: c = 1600; 2: c =
2800; 3: c = 4900). , (15)
, , (15)
( . (16)). (0,0)u
( . 1,
. 2).
0,00 2,00 4,00 6,00 8,00 x
0,00
0,04
0,08
0,12
u
6
5
4
3
2
1
. 1.
12345
30,0
u
20,0
10,0
0,0
–4,00 –2,00 0,00 2,00 x
. 2.
0,00 4,00 8,00 x
0,00
4,00
8,00
u
12,00
1
2 3
4
5
. 3.
0,0
100,0
200,0
u
300,0
1
2
3
7,0 8,0 9,0 x10,0
. 4.
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
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. .
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TOWARDS THE DIFFERENCES IN BEHAVIOUR OF SOLUTIONS OF LINEAR AND
NON-LINEAR HEAT-CONDUCTION EQUATIONS
L.I. Rubina
1
, O.N. Ul’yanov
2
The linear and non-linear heat-conduction equations are analyzed by the previously initiated geo-
metrical method of analyzing linear and non-linear equations in partial derivatives. The reason of the
difference in behavior of solutions of equations under consideration was stated, as well as the reason of
aggravation of the non-linear equation. A class of solutions of linear equations that represents the sur-
faces of the levels of non-linear heat-conduction equations was excluded.
Keywords: non-linear equations in partial derivatives, heat-conduction equations, exact solutions,
surfaces of the level.
References
1. Kurdyumov S.P., Malinetskiy G.G., Potapov A.B. Nestatsionarnye struktury, dinamicheskiy
khaos, kletochnye avtomaty. Novoe v sinergetike. Zagadki mira neravnovesnykh struktur (Unsteady
structures, dynamic chaos, cellular automata. New in synergetics. Mysteries of the world of nonequilib-
rium structure). Moscow: Nauka, 1996. 111 p. (in Russ.).
2. Samarskiy A.A., Galaktionov V.A., Kurdyumov S.P., Mikhaylov A.P. Rezhimy s obostreniem v
zadachakh dlya kvazilineynykh parabolicheskikh uravneniy (Blow-up regimes in problems for quasilin-
ear parabolic equations). Moscow: Nauka, 1987. 477 p. (in Russ.).
3. Vazquez J.L., Galaktionov V. A Stability Technique for Evolution Partial Differential Equations.
A Dynamical System Approach. Birkhauser Verlag, 2004. 377 p. (ISBN: 0-8176-4146-7)
4. Berkovich L.M. Vestnik SamGU – Estestvennonauchnaya seriya. 2005. no. 2(36). pp. 32–64. (in
Russ.).
5. Kurkina E.S., Nikol'skiy I.M. O rezhimakh s obostreniem v uravneniyakh
div( grad )tu u u uσ β
= + (About blow-up regimes in equations div( grad )tu u u uσ β
= + ). Different-
sial'nye uravneniya. Funktsional'nye prostranstva. Teoriya priblizheniy: Tezisy dokladov mezhdunarod-
noy konferentsii, posvyashchennoy 100-letiyu so dnya rozhdeniya Sergeya L'vovicha Soboleva. (Ab-
stracts of the International Conference dedicated to the 100th anniversary of the birth of Sobolev “Dif-
ferential Equations. Functional Space. Theory of Approximation”) Novosibirsk, 2008. p. 512.
6. Rubina L.I., Ul’ianov O.N. Trudy Inst. Mat. i Mekh. UrO RAN. 2010. Vol. 16, no. 2. pp. 209–
225. (in Russ.).
7. Rubina L.I., Ul’yanov O.N. Proceedings of the Steklov Institute of Mathematics (Supplementary
issues). 2008, Vol. 261. Suppl. 1. pp. 183–200.
8. Rubina L.I. Journal of Applied Mathematics and Mechanics. 2005. Vol. 69. Issue 5. pp. 829–836.
(in Russ.).
9. Lomkatsi Ts.D. Tablitsy ellipticheskoy funktsii Veyershtrassa. Teoreticheskaya chast' (Weier-
strass elliptic function charts. Theoretical part). Moscow: VTs AN SSSR, 1967. – 88 p. (in Russ.).
10. Gradshteyn I.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedeniy (Table of inte-
grals, sums, series and compositions). Moscow : Fizmatlit, 1962. 1100 p.
17 2012 .
1
Rubina Liudmila Ilinichna is Cand. Sc. (Physics and Mathematics), Senior Staff Scientist, Institute of Mathematics and Mechanics of the
Russian Academy of Sciences (Ural branch).
E-mail: rli@imm.uran.ru
2
Ul’yanov Oleg Nikolaevich is Cand. Sc. (Physics and Mathematics), Senior Staff Scientist, University’s academic secretary, Institute of
Mathematics and Mechanics of the Russian Academy of Sciences (Ural branch).
E-mail: secretary@imm.uran.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
60. . « . . »60
519.8
. . 1
. -
-
, ( ), . -
«
» . .
, ,
. -
.
: , , ,
, .
. , -
[1, 2]:
{1,2} 1 2 1 2 {1,2}{ } ,{ ( , ), ( , ), ( ), ( )}i i
i i i i V i S i iX f x x x x R x R xϕ∈ ∈ . (1)
(1) , i- « » in
i ix X comp∈ ⊆ R ; -
1 2 1 2( , )x x x X X= ∈ × ; 1 2( , )if x x ( {1,2}i∈ ) -
1 2X X× .
i-
1 2 1 2( ) max ( , ) ( , )
i i
i i i
x X
x f x x f x xϕ
∈
= − ( {1,2}i∈ ).
i-
1 2 1 2( ) max min ( , ) min ( , )
k k k ki i
i
V i i i
x X x Xx X
R x f x x f x x
∈ ∈∈
= − ( , {1,2}i k ∈ ,i k≠ ).
1 2 1 2( ) max ( , ) min max ( , )
i ik k k k
i
S i i i
x Xx X x X
R x x x x xϕ ϕ
∈∈ ∈
= − ( , {1,2}i k ∈ ,i k≠ ).
, ( )i
V iR x ( )i
S iR x
( {1,2}i∈ ) ,
i ix X∈ . 1 2( , )if x x
Xi , i(x1, x2),
( )i
V iR x , ( )i
S iR x ( {1,2}i∈ ) .
(1) i (x1, x2)
( )1 2 1 2( , ), ( , ), ( ), ( )i i
i i V i S if x x x x R x R xϕ , (2)
fi(x1, x2) ( xi ∈ Xi) -
, .
1. . :
(2) ?
-
( ). , -
:
{1,2} 1 2 {1,2}{ } ,{ ( , )}i i i iX f x x∈ ∈ , (3)
i- 1 2( , )x x « »
1 2( , )if x x .
1
– , ,
. E-mail: solnata@pochta.ru
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
61. . .
2013, 5, 2 61
1. (3) 1 2( , )e e
x x -
( ),
1 1 2 1 1 2 1 1
2 1 2 2 1 2 2 2
( , ) ( , ) ,
( , ) ( , ) ,
e e e
e e e
f x x f x x x X
f x x f x x x X
≥ ∀ ∈
≥ ∀ ∈
1 2( , )e e
x x .
, * *
1 2( , ) 0i x xϕ = ( {1,2}i∈ ), * *
1 2( , )x x .
-
.
2. (3) *
1x
,
2 2 2 21 1
1 1 2 1 1 2max min ( , ) min ( , )
x X x Xx X
f x x f x x∗
∈ ∈∈
= .
1
1( ) 0VR x∗
= .
, *
2x ,
1 1 1 12 2
*
2 1 2 2 1 2max min ( , ) min ( , )
x X x Xx X
f x x f x x
∈ ∈∈
= ,
2 *
2( ) 0VR x = .
,
.
3. , {1,2} 1 2{ } , ( , )i iX f x x∈
1 1 2 1 2( , ) ( , )f x x f x x= ,
2 1 2 1 2( , ) ( , )f x x f x x= − 1 2( , )e e
x x ( ).
1) ,
1 2( , ) 0e e
i x xϕ = ( {1,2}i∈ );
2) ,
( ) 0i e
V iR x = ( {1,2}i∈ );
3) 11 21( , )e e
x x 12 22( , )e e
x x – ( ), -
« » ,
11 21 12 22( , ) ( , )e e e e
f x x f x x= .
, ( ) 1 2( , )e e
x x -
( )1 2 1 2( , ), ( , ), ( ), ( )e e e e i e i e
i i V i S if x x x x R x R xϕ
( {1,2}i∈ ) 1 2( , )e e
x x (
2 1 2 1 1 2( , ) ( , )e e e e
f x x f x x= − ).
« » -
.
( )
{ } { } ( )1 1
1 1 1 2 2 2,..., , ,..., ,m n
ijX x x X x x A a= = = (4)
m n× -
11 12 1
21 22 2
1 2
n
n
m m mn
a a a
a a a
A
a a a
= ,
1 1
i
x X∈ i- A, 22
j
x X∈
j- ,
Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
62. . « . . »62
1 2( , )ji
x x ija ( )ijA a= , ,
1 1 1 2 1 12 2 2 2( , ) ( , ) , ( , ) ( , )j j j ji i i i
ij ijf x x f x x a f x x f x x a= = = − = − . (5)
{ } { } ( ) ( ){ } { }
{ }1 2 1 2
1 12 21,2 1,2
, , , ( ) ( ), ( ), ( )j jk i i
A k ij k ij V V S Sk k
X A a R x R x R x R xϕ∈ ∈
Γ = = Φ = ,
{ }1
1 1 1,..., m
X x x= – , { }1
2 2 2,..., n
X x x= – -
, 1 2( , )ji
x x -
.
-
. ( )1
1 ijϕΦ = -
:
{ }1 1
1
1 1 1 1 1 12 2 2
1,...,
( , ) max ( , ) ( , ) max
i
j j ji i i
ij ij ij
i mx X
x x f x x f x x a aϕ ϕ
∈∈
= = − = − ,
( )2
2 ijϕΦ =
{ }22
2
2 1 2 1 2 12 2 2
1,...,
( , ) max ( , ) ( , ) min
j
j j ji i i
ij ij ij
j nx X
x x f x x f x x a aϕ ϕ
∈∈
= = − = − .
1
i
x
{ } { } { }1 1 2 22 2
1
1 1 1 1 12 2
1,..., 1,...,1,...,
( ) max min ( , ) min ( , ) max min min
j ji
j ji i i
V ij ij
j n j ni mx X x X x X
R x f x x f x x a a
∈ ∈∈∈ ∈ ∈
= − = − ,
2
j
x
{ } { } { }1 1 1 122
2
2 1 2 12 2 2
1,...,1,..., 1,...,
( ) max min ( , ) min ( , ) max min max
i ij
j j ji i
V ij ij
j ni m i mx X x Xx X
R x f x x f x x a a
∈∈ ∈∈ ∈∈
= − = − .
1
i
x
{ } { } { }1 12 22 2
1 1 1
1 1 1 1 12 2
1,...,1,..., 1,...,
( ) max ( , ) min max ( , ) max min max
ij j
j ji i i
S ij ij
i mj n j nx Xx X x X
R x x x x xϕ ϕ ϕ ϕ
∈∈ ∈∈∈ ∈
= − = − ,
2
j
x
{ } { } { }1 1 1 122
2 2 2
2 1 2 12 2 2
1,...,1,..., 1,...,
( ) max ( , ) min max ( , ) max min max
ji i
j j ji i
S ij ij
j ni m i mx X x Xx X
R x x x x xϕ ϕ ϕ ϕ
∈∈ ∈∈ ∈∈
= − = − .
(4). (4),
1 2( , )ji
x x (5).
( )1 1 22, ji
x x X X
∗∗
∈ × ( ),
1 1 1 22 2( , ) ( , ) ( , )j jk i i t
f x x f x x f x x
∗ ∗∗ ∗
≤ ≤
1 1
k
x X∈ 2 2
t
x X∈ .
kj i j i t
a a a∗ ∗ ∗ ∗≤ ≤
{ }1,..., ,k m∈ { }1,...,t n∈ .
. ( ) ,
. :
7 1 4 1
4 2 3 2
2 2 5 2
4 3 7 2
A
− −
=
− −
.
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63. . .
2013, 5, 2 63
2 2 2 4 3 2 3 4
1 2 1 2 1 2 1 2( , ),( , ),( , ),( , )x x x x x x x x , -
( ) .
:
1 2
1( ),SR x 1 3
1( ),SR x 2 2
2( ),SR x 2 4
2( )SR x .
( )1 1
S ijϕΦ = ( )
:
1
maxij ij ij
i
a aϕ = − .
1
0 3 11 1
3 0 4 0
5 0 2 0
3 5 0 4
SΦ = .
( )2 2
S ijϕΦ = :
2
max( ) ( ) minij ij ij ij ij
jj
a a a aϕ = − − − = − ,
, 2 1 2( , )ji
ijf x x a= − .
2
11 3 0 5
2 0 1 0
0 0 3 0
7 0 10 1
SΦ = .
:
1
min max 4,ij
i j
ϕ = 2
min max 3ij
j i
ϕ = .
1 1 1
1( ) max min maxi
S ij ij
ij j
R x ϕ ϕ= − 2 2 2
2( ) max min maxj
S ij ij
ji i
R x ϕ ϕ= − ,
:
1 2
1( ) 0,SR x = 1 3
1( ) 1,SR x = 2 2
2( ) 0,SR x = 2 4
2( ) 2SR x = .
, 2 2
1 2( , )x x ,
.
2. S-
(1) :
1 1
1 1 1 1,{ ( ), ( )}e
V SX R x R xΓ = − − ,
1
e
X – ,
2 2
2 2 2 2,{ ( ), ( )}e
V SX R x R xΓ = − − ,
2
e
X – .
1. 1 1
e e
x X∈ 2 2
e e
x X∈ (1)
s- , -
1Γ 2Γ .
1. (1) :
1) 1X 2X ;
2) 1 1 2( , )f x x 1 2X X× 1x
2 2x X∈ ;
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