The approach to solving the problem of complex relaxation spectra is presented.
Presentation for the XI International Conference on Defect interaction and anelastic phenomena in solids. Tula, 2007.
system of algebraic equation by Iteration methodAkhtar Kamal
solve the system of algebraic equation by Iteration method
classification of Iteration method:-
(1) Jacobi's method
(2) Gauss-Seidel method
each problem
Identification of unknown parameters and prediction of missing values. Compar...Alexander Litvinenko
H-matrix approximation of large Mat\'{e}rn covariance matrices, Gaussian log-likelihoods.
Identifying unknown parameters and making predictions
Comparison with machine learning methods.
kNN is easy to implement and shows promising results.
system of algebraic equation by Iteration methodAkhtar Kamal
solve the system of algebraic equation by Iteration method
classification of Iteration method:-
(1) Jacobi's method
(2) Gauss-Seidel method
each problem
Identification of unknown parameters and prediction of missing values. Compar...Alexander Litvinenko
H-matrix approximation of large Mat\'{e}rn covariance matrices, Gaussian log-likelihoods.
Identifying unknown parameters and making predictions
Comparison with machine learning methods.
kNN is easy to implement and shows promising results.
Representation of of Stochastic Processes in Stochastic Processes in Real and Spectral Domains Real and Spectral Domains and and Monte Monte-Carlo sampling
Introduction to Mathematical ProbabilitySolo Hermelin
This is a lecture I've put together summarizing the topics of mathematical probability.
The presentation is at a Undergraduate in Science (Math, Physics, Engineering) level..
In the Upload Process a part of Figures and Equations are missing. For a better version of this presentation please visit my website at http://solohermelin.com at Math Folder and open Probability presentation.
Please feel free to comment and suggest improvements to solo.hermelin@gmail.com.Thanks!
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
Hierarchical matrix techniques for maximum likelihood covariance estimationAlexander Litvinenko
1. We apply hierarchical matrix techniques (HLIB, hlibpro) to approximate huge covariance matrices. We are able to work with 250K-350K non-regular grid nodes.
2. We maximize a non-linear, non-convex Gaussian log-likelihood function to identify hyper-parameters of covariance.
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko
We apply low-rank Tensor Train format to solve PDEs with uncertain coefficients. First, we approximate uncertain permeability coefficient in TT format, then the operator and then apply iterations to solve stochastic Galerkin system.
On Optimization of Manufacturing of a Sense-amplifier Based Flip-flopBRNSS Publication Hub
The paper describes an approach for increasing of density of field-effect heterotransistors in a sense-amplifier based flip-flop. To illustrate the approach, we consider manufacturing of an amplifier of power in a heterostructure with specific configuration. One shall dope some specific areas of the heterostructure by diffusion or ion implantation. After that, it should be done optimized annealing of radiation defects and/or dopant. We introduce an approach for decreasing of stress between layers of heterostructure. Furthermore, it has been considered an analytical approach for prognosis of heat and mass transport in heterostructures, which can be take into account mismatch-induced stress.
Основні результати опубліковані в:
http://scipeople.ru/publication/105074 (Вища освіта України. Тематичний випуск "Вища освіта України в контексті інтеграції до Європейського освітнього простору" / Ред. Кремень В.Г., Савченко О.Я., Маноха О.П. та ін. – 2011. - Додаток 2 до № 3. Т. 3 (28) – С. 29 – 35.);
http://scipeople.ru/publication/111134 (Higher Education in Ukraine: Internationalization, Reform, Innovation. International Conference. April 20-21, 2012. Kyiv, Ukraine);
http://www.anvsu.org.ua/index.files/Articles/Bakhruschin.htm
Representation of of Stochastic Processes in Stochastic Processes in Real and Spectral Domains Real and Spectral Domains and and Monte Monte-Carlo sampling
Introduction to Mathematical ProbabilitySolo Hermelin
This is a lecture I've put together summarizing the topics of mathematical probability.
The presentation is at a Undergraduate in Science (Math, Physics, Engineering) level..
In the Upload Process a part of Figures and Equations are missing. For a better version of this presentation please visit my website at http://solohermelin.com at Math Folder and open Probability presentation.
Please feel free to comment and suggest improvements to solo.hermelin@gmail.com.Thanks!
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
Hierarchical matrix techniques for maximum likelihood covariance estimationAlexander Litvinenko
1. We apply hierarchical matrix techniques (HLIB, hlibpro) to approximate huge covariance matrices. We are able to work with 250K-350K non-regular grid nodes.
2. We maximize a non-linear, non-convex Gaussian log-likelihood function to identify hyper-parameters of covariance.
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
Response Surface in Tensor Train format for Uncertainty QuantificationAlexander Litvinenko
We apply low-rank Tensor Train format to solve PDEs with uncertain coefficients. First, we approximate uncertain permeability coefficient in TT format, then the operator and then apply iterations to solve stochastic Galerkin system.
On Optimization of Manufacturing of a Sense-amplifier Based Flip-flopBRNSS Publication Hub
The paper describes an approach for increasing of density of field-effect heterotransistors in a sense-amplifier based flip-flop. To illustrate the approach, we consider manufacturing of an amplifier of power in a heterostructure with specific configuration. One shall dope some specific areas of the heterostructure by diffusion or ion implantation. After that, it should be done optimized annealing of radiation defects and/or dopant. We introduce an approach for decreasing of stress between layers of heterostructure. Furthermore, it has been considered an analytical approach for prognosis of heat and mass transport in heterostructures, which can be take into account mismatch-induced stress.
Основні результати опубліковані в:
http://scipeople.ru/publication/105074 (Вища освіта України. Тематичний випуск "Вища освіта України в контексті інтеграції до Європейського освітнього простору" / Ред. Кремень В.Г., Савченко О.Я., Маноха О.П. та ін. – 2011. - Додаток 2 до № 3. Т. 3 (28) – С. 29 – 35.);
http://scipeople.ru/publication/111134 (Higher Education in Ukraine: Internationalization, Reform, Innovation. International Conference. April 20-21, 2012. Kyiv, Ukraine);
http://www.anvsu.org.ua/index.files/Articles/Bakhruschin.htm
Робота з файлами даних в R, блоки виразів, цикли, функціїVladimir Bakhrushin
Приклади зчитування інформації з файлів даних та запису до файлів в R, списки, таблиці даних, блоки виразів, організація умовних переходів та циклів, створення функцій
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
The Queue Length of a GI M 1 Queue with Set Up Period and Bernoulli Working V...YogeshIJTSRD
Consider a GI M 1 queue with set up period and working vacations. During the working vacation period, customers can be served at a lower rate, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to a set up period with probability or continue the working vacation with probability , and when the set up period ends, the server will switch to the normal working level. Using the matrix analytic method, we obtain the steady state distributions for the queue length at arrival epochs. Li Tao "The Queue Length of a GI/M/1 Queue with Set-Up Period and Bernoulli Working Vacation Interruption" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-5 , August 2021, URL: https://www.ijtsrd.com/papers/ijtsrd43743.pdf Paper URL: https://www.ijtsrd.com/mathemetics/other/43743/the-queue-length-of-a-gim1-queue-with-setup-period-and-bernoulli-working-vacation-interruption/li-tao
We combined: low-rank tensor techniques and FFT to compute kriging, estimate variance, compute conditional covariance. We are able to solve 3D problems with very high resolution
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russia)
TITLE: Dynamical Groups, Coherent States and Some of their Applications in Quantum Optics and Molecular Spectroscopy
Decision-making on assessment of higher education institutions under uncertaintyVladimir Bakhrushin
Presentation for XХXII International Conference Problems of Decision Making under Uncertainties (PDMU-2018), August 27-31, 2018, Prague, Czech Republic
DOI: 10.13140/RG.2.2.27143.44966
Порівняння розуміння, мети та принципів освіти в проектах Закону України "Про освіту", підгтовлених робочою групою Комітету Верховної Ради з питань науки та освіти і Міністерством освіти і науки. Маємо змогу побачити у чому полягають основні розбіжності.
Окремі аспекти реформування освіти України з погляду системного підходуVladimir Bakhrushin
З погляду системного підходу розглянуто окремі аспекти реформування освіти України, зокрема: відображення входів та виходів системи освіти; групи інтересів та необхідність пошуку балансу їх інтересів; багатовимірні оцінки в освіті; обмеження при прийнятті рішень.
Some problems of decision-making in education (raw data, multicriteriality, uncertainty, interest groups) are considered. There are given examples of erroneous decisions, assessment of universities, the applicants selection etc. Also certain requirements for the new Law of Ukraine on education are formulated.
Останнім часом активізувалися дискусії про стан системи освіти України, її актуальні проблеми, можливі шляхи їх вирішення. У Комітеті Верховної Ради України з питань науки і освіти на весну заплановані обговорення проекту Концепції нової редакції Закону України “Про освіту” у березні та стану підготовки відповідного законопроекту у квітні. Аналіз окремих проблем, які потрібно вирішити у новому Законі, а також пропозиції до Закону містяться у багатьох публікаціях останнього часу. Зокрема, це статті О. Єльникової, І. Лікарчука, В. Огнев’юка, Ю. Шукевича та інших відомих фахівців на порталі Освітня політика. Учасники дискусій, що відбуваються, висловлюють різні, нерідко протилежні, погляди на майбутній закон. Тому на цьому етапі доцільно обговорити деякі передумови його прийняття, виходячи із загальних принципів теорії систем, теорії управління і теорії прийняття рішень.
http://education-ua.org/ua/draft-regulations/382-zakon-pro-osvitu-deyaki-peredumovi
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
• How SAP Build Code includes SAP Fiori tools and other generative artificial intelligence capabilities
• How SAP Fiori paves the way for using AI in SAP apps
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
Identification of the Mathematical Models of Complex Relaxation Processes in Solids
1. Identification of theIdentification of the
Mathematical Models ofMathematical Models of
Complex Relaxation ProcessesComplex Relaxation Processes
in Solidsin Solids
Bakhrushin V.E.Bakhrushin V.E.
University of HumanitiesUniversity of Humanities
“ZISMG”, Zaporozhye, Ukraine“ZISMG”, Zaporozhye, Ukraine
2. Relaxation processes:
- internal friction;
- dispersion of modulus;
- stress relaxation;
- elastic aftereffect.
Parameters:
- interstitial concentrations for different states;
- interstitial solubility;
- local diffusion coefficients;
- activation energies for jumps.
3. Identification tasks
1. To choose the type of mathematical model: ideal
Debay peak (model of the standard linear body); the
sum of ideal peaks (processes); enhanced Debay
peak; the sum of enhanced peaks; the sum of peaks +
background.
2. To determine the quantity of relaxation processes
3. To determine the parameters of relaxation processes
4. ( )
n
1 1 1 1 i
0 0i
i 1 0i
E E 1 1
Q T Q exp Q cosh
RT R T T
− − − −
=
= − + − ÷ ÷
∑
0i
i 0i
kT
E RT ln
hf
=
Model of spectrum at Snoeck relaxation area:
1
0Q ,E−
– background intensity and activation energy;
1
0i 0iQ ,T−
– i-th peak height and temperature
– i-th peak activation energy
f – sample vibration frequency
5. Parameters, which must be determined are:
1
0i 0in,Q ,T−
or
1
0i in,Q ,E−
( )0,14139 0,003245
0T 12,89967 2,706674f 0,04547 0,04929f E,= + + − +
An error for 300 – 800 К interval at f = 20 – 60 Hz
is not more, then 1 %
From Wert & Marx formula such approximation may
be obtained:
( )( )
m 2
1 1
j j
j 1
S Q Q T min− −
=
= − →∑
( )( )
2
1 1
m
j j
1 2
j 1 j
Q Q T
S min
− −
=
−
= →
σ
∑
6. ( )
( )
2
exp z/
z
− β
ψ =
β π
( )mz ln /= τ τ z2β = σ
Peak enhance may be taken into account by the model
of log-normal distribution of relaxation time:
7. max
max id
δ
ρ =
δ
2
0,0853 0,197 0,970ρ = ϕ + ϕ +
n
1 1 1 i
0i
i 1 0i
E 1 1
Q Q cosh
R T T
− − −
=
= − ÷
ρ
∑
In this case:
( )mlnϕ = ωτ
For ρ value from N.P. Kushnareva & V.S. Petchersky
data such approximation may be obtained:
10. Linear least-squares method:
( )
n
1
j i i j i
i 1
x A cosh [B (x c )]−
=
ϕ = −∑
1
i 0i j j i 0iA Q , x 1/T , c 1/T−
= = =
i
i i
1 k
B ln
c hfc
=
11. ( )
( )
( )
( )
( )
( )
n
j
j j
j 1 1
n
j
j j
j 1 2
n
j
j j
j 1 n
x
y x 0;
A
x
y x 0;
A
x
y x 0,
A
=
=
=
∂ϕ
− ϕ = ∂
∂ϕ
− ϕ = ∂
∂ϕ
− ϕ = ∂
∑
∑
∑
LLLLLLLLLLLL
12. ( )j 1
i j i ij
i
x
cosh [B (x c )] F
A
−
∂ϕ
= − =
∂
n n n n
2
1 1j 2 1j 2 j n 1j nj j 1j
j 1 j 1 j 1 j 1
n n n n
2
1 2 j 1j 2 2 j n 2 j nj j 2 j
j 1 j 1 j 1 j 1
1 nj 1j 2 nj
A F A F F A F F y F ;
A F F A F A F F y F ;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A F F A F
= = = =
= = = =
+ + + =
+ + + =
+
∑ ∑ ∑ ∑
∑ ∑ ∑ ∑
L
L
n n n n
2
2 j n nj j nj
j 1 j 1 j 1 j 1
F A F y F
= = = =
+ + =∑ ∑ ∑ ∑L
13. 1 11 2 12 n 1n 1
1 21 2 22 n 2n 21
1 n1 2 n2 n nn n
A W A W ... A W Z ;
A W A W ... A W Z ;
. . . . . . . . . . . . . . . . . . . . . . . . . .
A W A W ... A W Z ,
+ + + =
+ + + =
+ + + =
n
1 1
ik i k
j 1 j i j k
1 1 1 1
W cosh B cosh B
x c x c
− −
=
= − − ÷ ÷ ÷ ÷ ÷ ÷
∑
n
1
i i i
j 1 j i
1 1
Z y cosh B
x c
−
=
= − ÷ ÷
∑
14. The advantages of linear least-squares
method:
- simple realization;
- sufficient accuracy (up to 10 % for the main peaks
heights)
Method disadvantages:
- linearization error;
- necessity of peak temperatures preliminary definition;
- possibility to obtain an ill-condition system;
- supposition of uniformly precise of the data;
- supposition of absolute accuracy of temperature
measurements;
- possibility of obtaining the negative values of peak
heights.
15. The method of gradient descent
(linearized least square method)
Main differences:
- an expression of ideal peak is linearized by Taylor
series expansion in the neighborhood of some point
(initial estimate) with abandonment of only linear
terms;
- the possibility to choose the different types of
objective function (cancellation of supposition that the
data have the same errors)
[L. Crer et. al, 1969; M.S. Ahmad et. al, 1971;
O.N. Razumov et. al., 1974; A.I. Efimov et. al., 1982.]
16. ( )( ) ( )
1 1 1m
j j
2
j 1 j k
Q Q T Q T
0, k 1,2,...q
a
− − −
=
− ∂
= =
σ ∂
∑
{ }1 1 1
0 01 0n 1 n 01 0na E,Q ,Q ,...,Q ,E ,...,E ,T ,...,T− − −
=
q 3n 2= +
( )( )
2
1 1
m
j j
1 2
j 1 j
Q Q T
S min
− −
=
−
= →
σ
∑
From
we can obtain:
at general case, q 3n= without background,
q 2n= for ideal Debay peaks.
17. After linearization we obtain:
( ) ( )
m
0 1
k k k mkm
1
a a a M Z ,−
=
∆ = − = ∑l
( ) ( )1 1m
k 2
j 1 j k
Q T Q T1
M ,
a a
− −
=
∂ ∂
= ÷ ÷σ ∂ ∂
∑l
l
( )( ) ( )
1 1 1m
j j
2
j 1 j
Q Q T Q T
Z ,
a
− − −
=
− ∂
=
σ ∂
∑l
l
where:
derivatives are determined in 0
ka
18. Adjusted values:
0
k k ka a a , 0 1.= + γ∆ < γ ≤
From the definition of ka∆ follows, that it corresponds
with the general formula of gradient search methods:
0
k k 1a a gradS .= −β
Gradient methods realize an iteration procedure, in
which such stopping conditions may be used:
p p 1
k ka a ;−
− < ε p p 1
1 1S S ;−
− < δ 1gradS ;< ξ , , 0.ε δ ξ >
19. Problems and disadvantages:
- poor convergence at the case of large number of
peaks;
- possibility of iteration stopping at the critical point,
which is not the point of minimum;
- possibility of getting into a loop, when the objective
functional S is ravine;
- absence of realization at standard libraries of the most
popular software packages;
- М matrix must be positively defined at the every step
of iterations
20. ( ) ( )(k 1) (k) 1 (k) (k)
H ,+ −
= −X X X G X
Quasi-Newton algorithm
{ }1 1 1
0 01 0n 1 n 01 0na E,Q ,Q ,...,Q ,E ,...,E ,T ,...,T− − −
=
2 2 2
2
1 1 2 1 q
2 2 2
2
2 1 2 2 q
2 2 2
2
q 1 q 2 q
S S S
...
a a a a a
S S S
...
a a a a aH
... ... ... ...
S S S
...
a a a a a
∂ ∂ ∂
÷
∂ ∂ ∂ ∂ ∂ ÷
÷∂ ∂ ∂
÷
∂ ∂ ∂ ∂ ∂= ÷
÷
÷
÷∂ ∂ ∂
÷
∂ ∂ ∂ ∂ ∂
1
2
q
S
a
S
aG
...
S
a
∂
÷∂
÷
∂ ÷
÷∂=
÷
÷
÷∂
÷
÷∂
21. ij
ij
h , i j;
c
0, i j.
=
=
≠
n
1 T
1 k k k
k 1
P z ,
−
=
= ∑ v v
° 1 1 1
1H C PC ,
− − −
=
zk are eigenvalues and vk are eigenvectors of matrix:
Grinshtadt technique:
1 1
P C HC ,− −
=
It is necessary to provide the positive definiteness of
Hesse matrix or to find an approximation of Н-1
22. and
F is a Fisher criterion value for the corresponding
numbers of degrees of freedom and significance level,
∆2
- sum of errors squares (relative errors) of
experimental points.
Adequacy criteria for spectrum models:
2
S
F≤
∆
2
F
S
∆
≤
( )0,052
S
F F 2,0...2,5> ≈ ⇒
∆
number of model parameters must be increased;
2
F
S
∆
> ⇒
number of model parameters must be decreased.
23. Quasi-unimodelity (an absence of physically
different minimums) of objective functional, that is all
minimums of objective functional correspond to the
same physical model of a spectrum.
Deviation from quasi-unimodality may be caused
with:
- the presence of excess peaks in the model;
- absence of some essential peak in the model;
- presence at the real spectrum of some collateral
peak, which height is close to measurement error.
24. Absence of model residuals serial correlation
(Darbin & Watson criterion):
( )
m 2
j j 1
j 2
m
2
t
j 1
e e
d ,
e
−
=
=
−
=
∑
∑
( )1 1
j j je Q Q T− −
= − - model residuals.
d 2≈ - serial correlation is absent;
d 0→
d 4≈
- positive serial correlation;
- negative serial correlation
(there are excess peaks).
66. Nb – 12 at.% W Nb – 6
at.%
W
4
peaks
5 peaks 4
peaks1 set 2 set
E1, kJ/mol 102,2 86
E2, kJ/mol 109,5 111,5 110,1 109,8
E3, kJ/mol 116,3 116,9 116,5 116,5
E4, kJ/mol 128,9 129,6 129,1 128,3
E5, kJ/mol 1456 145,9 145,9 145,9
67. 0
0,3
0,6
0,9
500 600 700 800
Nb – 2 at.% Hf – 0,32 at.% N (3 peaks):
E = 1,47; 1,61; 1,76 kJ/mol;
d = 0,90; F = 1,73.
68. 0
0,3
0,6
0,9
500 600 700 800
Nb – 2 at.% Hf – 0,32 at.% N (4 peaks):
E = 1,29; 1,48; 1,62; 1,79 kJ/mol;
d = 1,26; F = 2,64.
69. 0
0,3
0,6
0,9
500 600 700 800
Nb – 2 at.% Hf – 0,32 at.% N (5 peaks, 1 result):
E = 1,44; 1,54; 1,63; 1,77; 1,91 kJ/mol;
d = 1,06; F = 3,21.
70. 0
0,3
0,6
0,9
500 600 700 800
Nb – 2 at.% Hf – 0,32 at.% N (5 peaks, 2 result):
E = 1,26; 1,46; 1,57; 1,64; 1,80 kJ/mol;
d = 1,55; F = 3,58.
71. n
N i
i 1
0i
i 2 2 2
i
i
0i
i
M(T) M M (T);
M
M (T) ;
1 4 f
E 1 1
exp
R T T
.
2 f
=
= − ∆
∆
∆ =
+ π τ
− ÷
τ = π
∑
The temperature dependence of dynamic elastic
modulus in a case of n processes, which satisfy the
model of standard linear body, may be determined
from a system:
MN – non-relaxed modulus.
72. Model parameters, which must be identified, are:
0i 0iM ,T .∆
( ) ( )
m 2
1 exp j j
j 1
S M T M T min,
=
= − → ∑
We are to solve such minimization problem:
( )exp jM Twhere are experimental data for modulus at Tj.
(*)
73. Functional (*) has a great number of minimums,
so the result of minimization strongly depends on initial
assumption.
Adequate model may be obtained by using as T0i
initial values the results of relaxation spectrum
decomposition and setting initial values as0iM∆
1
0i N 0iM 2M Q .−
∆ =
T0i values after minimization are very close with the
initial ones, and values change essentially. But
there is a correlation (r = 0,90 – 0,97) between partial
Snoek peaks heights and results for :
0iM∆
0i
1
0i H
M
2,00 0,15.
Q M−
∆
= ±
0iM∆
74. Nb – 12 at.% W – N
[N], at.%: - 0,11;
■ – 0,16;
▲ – 0,22;
♦ - 0,31