International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
Or: Invariant theory, magnetism, subfactors and skein
This talk is a summary of the history of the Temperley-Lieb calculus, starting with famous works of Rumer-Teller-Weyl and Temperley-Lieb, over subfactors to skein theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Slides: dtubbenhauer.com/talks.html
Presentation about chapter 1 of electrical circuit analysis. standard prefixes. basic terminology power,current,voltage,resistance.How power is absorbed by the circuit and its calculation with passive sign convention.
Or: Invariant theory, magnetism, subfactors and skein
This talk is a summary of the history of the Temperley-Lieb calculus, starting with famous works of Rumer-Teller-Weyl and Temperley-Lieb, over subfactors to skein theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Slides: dtubbenhauer.com/talks.html
Or: From ice to R-matrices
This talk is a summary of the history of quantum groups, describe how they arose from questions in statistical mechanics. The keyword is the Yang-Baxter equation, which was crucial for the development of the field.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Here the concept of "TRUE" is defined according to Alfred Tarski, and the concept "OCCURING EVENT" is derived from this definition.
From here, we obtain operations on the events and properties of these operations and derive the main properties of the CLASSICAL PROB-ABILITY. PHYSICAL EVENTS are defined as the results of applying these operations to DOT EVENTS.
Next, the 3 + 1 vector of the PROBABILITY CURRENT and the EVENT STATE VECTOR are determined.
The presence in our universe of Planck's constant gives reason to\linebreak presume that our world is in a CONFINED SPACE. In such spaces, functions are presented by Fourier series. These presentations allow formulating the ENTANGLEMENT phenomenon.
Global Journal of Science Frontier Research: FMathematics and Decision Sciences Volume 18 Issue 2 Version 1.0 Year 2018
This presentation is about electromagnetic fields, history of this theory and personalities contributing to this theory. Applications of electromagnetism. Vector Analysis and coordinate systems.
ALL-SCALE cosmological perturbations and SCREENING OF GRAVITY in inhomogeneou...Maxim Eingorn
M. Eingorn, First-order cosmological perturbations engendered by point-like masses, ApJ 825 (2016) 84: http://iopscience.iop.org/article/10.3847/0004-637X/825/2/84
In the framework of the concordance cosmological model, the first-order scalar and vector perturbations of the homogeneous background are derived in the weak gravitational field limit without any supplementary approximations. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The expressions found for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge at all points except at the locations of the sources. The average values of these metric corrections are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant, this part represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggested connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
To give the complete list of Uq(sl2)-actions of the quantum plane, we first obtain the structure of quantum plane automorphisms. Then we introduce some special symbolic matrices to classify the series of actions using the weights. There are uncountably many isomorphism classes of the symmetries. We give the classical limit of the above actions.
Differential Geometry for Machine LearningSEMINARGROOT
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
Or: From ice to R-matrices
This talk is a summary of the history of quantum groups, describe how they arose from questions in statistical mechanics. The keyword is the Yang-Baxter equation, which was crucial for the development of the field.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Here the concept of "TRUE" is defined according to Alfred Tarski, and the concept "OCCURING EVENT" is derived from this definition.
From here, we obtain operations on the events and properties of these operations and derive the main properties of the CLASSICAL PROB-ABILITY. PHYSICAL EVENTS are defined as the results of applying these operations to DOT EVENTS.
Next, the 3 + 1 vector of the PROBABILITY CURRENT and the EVENT STATE VECTOR are determined.
The presence in our universe of Planck's constant gives reason to\linebreak presume that our world is in a CONFINED SPACE. In such spaces, functions are presented by Fourier series. These presentations allow formulating the ENTANGLEMENT phenomenon.
Global Journal of Science Frontier Research: FMathematics and Decision Sciences Volume 18 Issue 2 Version 1.0 Year 2018
This presentation is about electromagnetic fields, history of this theory and personalities contributing to this theory. Applications of electromagnetism. Vector Analysis and coordinate systems.
ALL-SCALE cosmological perturbations and SCREENING OF GRAVITY in inhomogeneou...Maxim Eingorn
M. Eingorn, First-order cosmological perturbations engendered by point-like masses, ApJ 825 (2016) 84: http://iopscience.iop.org/article/10.3847/0004-637X/825/2/84
In the framework of the concordance cosmological model, the first-order scalar and vector perturbations of the homogeneous background are derived in the weak gravitational field limit without any supplementary approximations. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The expressions found for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge at all points except at the locations of the sources. The average values of these metric corrections are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant, this part represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggested connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
Text Book: An Introduction to Mechanics by Kleppner and Kolenkow
Chapter 1: Vectors and Kinematics
-Explain the concept of vectors.
-Explain the concepts of position, velocity and acceleration for different kinds of motion.
References:
Halliday, Resnick and Walker
Berkley Physics Volume-1
To give the complete list of Uq(sl2)-actions of the quantum plane, we first obtain the structure of quantum plane automorphisms. Then we introduce some special symbolic matrices to classify the series of actions using the weights. There are uncountably many isomorphism classes of the symmetries. We give the classical limit of the above actions.
Differential Geometry for Machine LearningSEMINARGROOT
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
On Fully Indecomposable Quaternion Doubly Stochastic Matricesijtsrd
In traditional years, fully indecomposable matrices have played an vital part in various research topics. For example, they have been used in establishing a necessary condition for a matrix to have a positive inverse also, in the case of simultaneously row and column scaling sub ordinate to the unitarily invariant norms, the minimal condition number diagonalizable, sub stochastic matrices , Kronecker products is achieved for fully indecomposable matrices. In the existence of diagonal matrices D1 and D2 , with strictly positive diagonal elements, such that D1 AD2 is quaternion doubly stochastic, is established for an nXn non negative fully indecomposable matrix A. In a related scaling for fully indecomposable non negative rectangular matrices is also discussed. Dr. Gunasekaran K. | Mrs. Seethadevi R. "On Fully Indecomposable Quaternion Doubly Stochastic Matrices" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-5 , August 2019, URL: https://www.ijtsrd.com/papers/ijtsrd25351.pdfPaper URL: https://www.ijtsrd.com/mathemetics/other/25351/on-fully-indecomposable-quaternion-doubly-stochastic-matrices/dr-gunasekaran-k
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Information related to Gauss-Jordan elimination.
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CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
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R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
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Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
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Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
block diagram and signal flow graph representation
k - Symmetric Doubly Stochastic, s - Symmetric Doubly Stochastic and s - k - Symmetric Doubly Stochastic Matrices
1. International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org Volume 3 Issue 8 ǁ August 2014 ǁ PP.11-16
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k - Symmetric Doubly Stochastic, s - Symmetric Doubly Stochastic and s – k - Symmetric Doubly Stochastic Matrices 1,Dr. K. Gunasekaran , 2,Mrs. N. Mohana, Associate professor1 Assistant Professor2 1Department of Mathematics, Government Arts College (Autonomous), Kumbakonam, TamilNadu. 2 Department of Mathematics, A.V.C. College (Autonomous), Mannampandal, TamilNadu. ABSTRACT: The basic concepts and theorems of k-symmetric doubly stochastic s-symmetric doubly stochastic and s-k-symmetric doubly stochastic matrices are introduced with examples. KEY WORDS: k-symmetric doubly stochastic matrix, s-symmetric doubly stochastic matrix and s-k-symmetric doubly stochastic matrix. AMS CLASSIFICATIONS: 15A51, 15B99
I. INTRODUCTION
We have already seen the concept of symmetric doubly stochastic matrices. In this paper the symmetric doubly stochastic matrix is developed in real matrices. Recently Hill and Waters[2] have developed a theory of k-real matrices as a generalization of s-real matrices. Ann Lee[1] has initiated the study of secondary symmetric matrices, that is matrices whose entries are symmetric about the secondary diagonal. Ann Lee[1] has shown that the matrix A, the usual transpose AT and secondary transpose AS are related as AS = VATV and AT = VASV where V is a permutation matrix with units in the secondary diagonal.
II. PRELIMINARIES AND NOTATIONS
AT - Transpose of A Let k be a fixed product of disjoint transpositions in Sn and „K‟ be the permutation matrix associated with k. Clearly K satisfies the following properties. K 2 = I, KT = K.
III. DEFINITIONS AND THEOREMS
DEFINITION: 1 A matrix A Rn x n is said to be symmetric doubly stochastic matrix if A = AT and = 1, j = 1, 2, ………n and = 1, i = 1, 2, ………n and all a ij ≥ 0. If A is doubly stochastic and also symmetric then it is called a symmetric doubly stochastic matrix. DEFINITION: 2 A matrix A Rn x n is said to be k-symmetric doubly stochastic matrix if A = K AT K THEOREM: 1 Let A Rn x n is k-symmetric doubly stochastic matrix then A = K AT K. Proof: K AT K = KAK where AT = A = AKK where AK = KA = AK2 = A where K2 = I THEOREM: 2 Let AT Rn x n is k-symmetric doubly stochastic matrix then AT = KAK. Proof: K A K = KATK where A = AT = ATKK where K AT = AT K = AT K2 = AT where K2 = I
2. k - Symmetric Doubly Stochastic, s – Symmetric…
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THEOREM: 3 Let A, B Rn x n is k-symmetric doubly stochastic matrix then (A + B) is k-symmetric doubly stochastic matrix. Proof: Let A and B are k-symmetric doubly stochastic matrix if A = K AT K and B = KBTK. To prove (A + B) is k-symmetric doubly stochastic matrix we will show that (A + B) = K (A + B)T K Now K (A + B)T K = K (AT + B T) K = K (AT + B T) K = (KAT + KB T) K = (KAT K+ KB T K) = (A + B) where K AT K = A and KBTK = B THEOREM: 4 Any k-symmetric doubly stochastic matrix can be represent as sum of k-symmetric doubly stochastic matrix and skew k-symmetric doubly stochastic matrix. Proof: To prove that (A + KATK) and (A - KATK) are k-symmetric doubly stochastic matrices the we will show that (A + KATK) = K (A + KATK)T K and (A - KATK) = K (A - KATK)T K. K (A + KATK)T K = (A + KATK) using theorem 3 and K (A - KATK)T K = (A - KATK). Then (A + KATK) + (A - KATK) = 2A/2 = A. Hence the theorem is proved. THEOREM: 5 If A and B are k-symmetric doubly stochastic matrices then AB is also k-symmetric doubly stochastic matrix. Proof: Let A and B are k-symmetric doubly stochastic matrix if A = K AT K and B = KBTK. Since AT and BT are also k-symmetric doubly stochastic matrices then AT = KAK and BT = KBK. To prove A B is k-symmetric doubly stochastic matrix we will show that AB= K (A B)T K Now K (A B)T K= KBTATK = K(KBK)(KAK)K where AT = KAK and BT = KBK. = K2B K2A K2 = BA where K2 = I = AB where BA = AB THEOREM: 6 If A and B are k-symmetric doubly stochastic matrices and K is the permutation matrix, k = {(1), (2 3)} then KA is also k-symmetric doubly stochastic matrix. Proof: Let A and B are k-symmetric doubly stochastic matrix if A = K AT K and B = KBTK. Since AT and BT are also k-symmetric doubly stochastic matrices then AT = KAK and BT = KBK. To prove K B is K-symmetric doubly stochastic matrix we will show that KA= K (KA)T K Now K (KA)T K= K(ATKT )K = KATKTK = KAT where KT K = I . = KA where KAT = KA RESULT: For A Rn x n is symmetric doubly stochastic matrices for the following are holds.
[1] A = K AT K
[2] KA is symmetric doubly stochastic matrix.
[3] AK is symmetric doubly stochastic matrix.
Example: A = AT = and k = (1) (2 3) =
3. k - Symmetric Doubly Stochastic, s – Symmetric…
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(i) K AT K = = A Similarly KAK = AT (ii) KA = = = (KA)T KA is symmetric doubly stochastic matrix. Similarly KAT is also symmetric doubly stochastic matrix. (iii) AK = = = (AK)T AK is symmetric doubly stochastic matrix. Similarly AT K is also symmetric doubly stochastic matrix. DEFINITION: 3 A matrix A Rn x n is said to be s-symmetric doubly stochastic matrix if AS = V AT V where V is a permutation matrix with units in the secondary diagonal. THEOREM: 7 Let A Rn x n is s-symmetric doubly stochastic matrix then AS = V AT V. Proof: V AT V = V (VAS V) V = V2AS V2 = AS where V2 = I THEOREM: 8 Let AT Rn x n is s-symmetric doubly stochastic matrix then AT = VASV. Proof: V AS V = V (VAT V) V =V2AT V2 = AT where V2 = I THEOREM: 9 Let A, B Rn x n is s-symmetric doubly stochastic matrix then (A + B) is s-symmetric doubly stochastic matrix. Proof: Let A and B are s-symmetric doubly stochastic matrices if AS = V AT V and BS = VBTV. To prove (A + B) is s-symmetric doubly stochastic matrix we will show that (A + B)S = V (A + B)T V Now V (A + B)T V = V (AT + BT ) V = V(AT + BT ) V = (VAT + VBT) V= (VAT V+ VBT V) = (AS + BS) where V AT V = AS and VBTV = BS = (A + B)S THEOREM: 10 If A and B are s-symmetric doubly stochastic matrices then AB is also s-symmetric doubly stochastic matrix. Proof: Let A and B are s-symmetric doubly stochastic matrices if AS = V AT V and BS = VBTV. Since AT and BT are also s-symmetric doubly stochastic matrices then AT = VASV and BT= VBSV. To prove A B is s-symmetric doubly stochastic matrix we will show that (AB)S = V(A B )T V Now V (A B)T V= VBTATV = V(VBS V)(VASV)V where AT = KASK and BT = KBSK. = V2 BSV2AS V2 = BS AS where V2 = I = (AB)S
4. k - Symmetric Doubly Stochastic, s – Symmetric…
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THEOREM: 11 If A is s-symmetric doubly stochastic matrix and V is a permutation matrix with units in the secondary diagonal then VA is also s-symmetric doubly stochastic matrix. Proof: Let A is s-symmetric doubly stochastic matrices if AS = V AT V. Since AT is s-symmetric doubly stochastic matrices then AT = VASV. To prove VA is s-symmetric doubly stochastic matrix we will show that (VA)S= V(VA)T V Now V (VA)T V = V(ATVT )V= V(VASV) V2 = AS V S where V2 = I = (VA)S RESULT: For A Rn x n is s-symmetric doubly stochastic matrix for the following are holds.
[1] AS = V AT V
[2] VA is symmetric doubly stochastic matrix.
[3] AV is symmetric doubly stochastic matrix.
Example: A = AT = AS = V = (i) V AT V = =AS Similarly VASV = AT (ii) VA = = = (VA)T VA is symmetric doubly stochastic matrix. Similarly VAT is also symmetric doubly stochastic matrix. (iii) AV = = = (AV)T AV is symmetric doubly stochastic matrix. Similarly AT V is also symmetric doubly stochastic matrix. DEFINITION: 4 A matrix A Rn x n is said to be s-k-symmetric doubly stochastic matrix if
[1] A = KVATVK
[2] AT = KVAVK
[3] A = VKATKV
[4] AT = VKAKV
Where V is a permutation matrix with units in the secondary diagonal and K is a permutation matrix and k = {(1) (2 3)}. THEOREM: 12 Let A Rn x n is s-k-symmetric doubly stochastic matrix then
[1] AS = KVATVK
[2] AT = KVASVK
[3] AS = VKATKV
[4] AT = VK AS KV
5. k - Symmetric Doubly Stochastic, s – Symmetric…
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Proof: (i) KVATVK = K(VATV)K = KAS K where VATV = AS =K(AT)T K = K AT K where AT = A = A = AS where KAT K = A and A = AS (ii) KVASVK = K(VAS V)K = K AT K where VASV = AT = A where K AT K = A = AT where A = AT (iii) VKATKV = V(KATK)V = VAV where KAT K = A = VATV where A = AT = AS where VATV = AS (iv) VKASKV = V(KASK)V = V (K(AT)T K)V = VATV where K(AT )TK = AT = AS where V AT V= AS = (AT )T = AT where A = AT THEOREM: 13 Let A, B Rn x n is s-k-symmetric doubly stochastic matrix then (A + B ) is s-k-symmetric doubly stochastic matrix. Proof: Let A and B are s-k-symmetric doubly stochastic matrix if A =KV AT VK and B = KVBTVK. To prove (A + B ) is s-k-symmetric doubly stochastic matrix we will show that (A + B ) = KV (A + B )T VK Now KV (A + B )T VK = K( V (A + B )T V)K = K (A + B )S K using theorem (9) = K (A + B )T K = (A + B ) using theorem (3) THEOREM: 14 If A and B are s-k-symmetric doubly stochastic matrix then AB is also s-k-symmetric doubly stochastic matrix. Proof: Let A and B are s-k-symmetric doubly stochastic matrix if A =KV AT VK and B = KVBTVK. Since AT and BT are also s-k-symmetric doubly stochastic matrices AT = KVAVK and BT = KVBVK. To prove A B is s-k-symmetric doubly stochastic matrix we will show that AB = KV(A B )T VK Now KV(A B )T VK = K(V(A B )T )VK = K(A B )SK using theorem (10) = K(A B )TK = AB using theorem (5) RESULT: For A Rn x n is s-k-symmetric doubly stochastic matrix the following are equivalent. (i) A = KVATVK (ii) AT = KVAVK (iii) A = VKATKV (iv) AT = VKAKV Example: A = K = V = (i) KVATVK = = A (ii) KVAVK = = AT
6. k - Symmetric Doubly Stochastic, s – Symmetric…
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(iii) VKATKV = = A (iv) VKAKV = = AT REFERENCES:
[1] Ann Lec. Secondary symmetric and skew symmetric secondary orthogonal matrices;
i. Period, Math Hungary, 7, 63-70(1976).
[2] Hill, R.D, and Waters, S.R: On k-real and k-hermitian matrices; Lin.Alg.Appl.,169,17-29(1992)
[3] S.Krishnamoorthy, R.Vijayakumar, On s-normal matrices, Journal of Analysis and Computation, Vol5, No2,(2009)
[4] R.Grone, C.R.Johnsn, E.M.Sa,H.Wolkowiez, Normal matrices, Linear Algebra Appl. 87(1987) 213-225.
[5] G. Latouche, V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999.
[6] Hazewinkel, Michiel, ed. (2001), "Symmetric matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
[7] S.Krishnamoorthy, K. Gunasekaran, N. Mohana, “ Characterization and theorems on doubly stochastic matrices” Antartica Journal of Mathematics, 11(5)(2014).