\r\n\tFrederick Law Olmsted, who is considered the father of American landscape architecture, often commented that landscape architecture is a profession that creates natural design approaches in human interaction with the environment. \r\n\tLandscape design answers the people's need for parks and gardens in urban areas.
\r\n
\r\n\tIt is a necessary design tool to understand to make more livable places in our living environment. Landscape Planning gives an option to protect our natural resources and natural environment. It draws a balance between the use and protection of natural areas. Urban Design is one of the important issues for landscape architecture. It is a common point of architecture, planning, and the landscape architecture profession. It helps to create sustainable, healthy spaces in urban development. Rural planning and development is the core issue of landscape architecture that makes a balance between natural and human interactions.
\r\n
\r\n\tThis book project welcomes research papers related to all Landscape Architecture topics. We hope to provide an opportunity for Landscape Architecture researchers to distribute information to the academic community.
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Mustafa Ergen and Associate Prof. Yasar Bahri Ergen",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10766.jpg",keywords:"Landscape Design, Sustainable Development, Healthy Spaces, Livable Spaces, Protection of Nature, Landscape Planning, Urban Design, Rural Planning, Rural Development, Landscape Architecture, Townscapes, Vegetation Conservation",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"March 15th 2021",dateEndSecondStepPublish:"April 12th 2021",dateEndThirdStepPublish:"June 11th 2021",dateEndFourthStepPublish:"August 30th 2021",dateEndFifthStepPublish:"October 29th 2021",remainingDaysToSecondStep:"7 days",secondStepPassed:!0,currentStepOfPublishingProcess:3,editedByType:null,kuFlag:!1,biosketch:"Dr. Mustafa Ergen received his Dr.-Eng. degree at the Technical University of Dortmund, Germany. His studies focus on cityscapes, urban and landscape planning, design, and Geographic Information Systems, and Remote Sensing.",coeditorOneBiosketch:"Yasar Bahri Ergen received a Doctoral Degree from Gazi University, Turkey. Currently, he is the chair of the City and Regional Planning Department at Siirt University.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"166961",title:"Dr.Ing.",name:"Mustafa",middleName:null,surname:"Ergen",slug:"mustafa-ergen",fullName:"Mustafa Ergen",profilePictureURL:"https://mts.intechopen.com/storage/users/166961/images/system/166961.jpg",biography:"Mustafa Ergen graduated from Abant İzzet Baysal University’s Department of Landscape Architecture in 2000 in Turkey. In 2005, he completed his first master’s degree at the Gebze Institute of Technology in Urban and Regional Planning in Turkey and his second master’s degree at Anhalt University of Applied Sciences in Landscape Architecture, Germany, in 2006. He studied Geographic Information Systems and Remote Sensing at the Mediterranean Agronomic Institute of Chania between 2007 and 2008 in Greece and was granted a specialization diploma in Environmental Management. He received the Dr.-Ing. degree at the Technical University of Dortmund, Germany, in 2013. In 2005, he started his professional life at the Yüzüncü Yıl University, Turkey, as a Research Assistant. Between 2010 and 2016, he worked in the Department of Urban Design and Landscape Architecture at Amasya University in Turkey. Currently, He works in the Department of Architecture at Siirt University in Turkey. He speaks English fluently, he has an intermediate level of the German language, and his mother tongue is Turkish. 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He received the Doctoral Degree from Gazi University in 1987. His academic life started in 1982 and took turn from the US to Gazi University. After 1995, he established Departments of Architecture and City and Regional Planning at Bozok University and Amasya University. In 2017, he established Faculty of Fine Arts and Design and the Faculty Departments at Siirt University. 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1. Introduction
In the whole chapter, the notations R and C are reserved for fields of the real and complex numbers, respectively. Cm×n stands for the set of all m×n matrices over C. Crm×n determines its subset of matrices with a rank r. For A∈Cm×n, the symbols A∗ and rkA specify the conjugate transpose and the rank of A, respectively, ∣A∣ or detA stands for its determinant. A matrix A∈Cn×n is Hermitian if A∗=A.
A† means the Moore-Penrose inverse of A∈Cn×m, i.e., the exclusive matrix X satisfying the following four equations:
AXA=AE1
XAX=XE2
AX∗=AXE3
XA∗=XAE4
For A∈Cn×n with index IndA=k, i.e., the smallest positive number such that rkAk+1=rkAk, the Drazin inverse of A, denoted by Ad, is called the unique matrix X that satisfies Eq. (2) and the following equations,
AX=XA;E5
XAk+1=AkE6
Ak+1X=Ak.E7
In particular, if IndA=1, then the matrix X is called the group inverse, and it is denoted by X=A#. If IndA=0, then A is nonsingular and Ad=A†=A−1.
It is evident that if the condition (5) is fulfilled, then (6) and (7) are equivalent. We put both these conditions because they will be used below independently of each other and without the obligatory fulfillment of (5).
A matrix A satisfying the conditions i,j,… is called an ij…-inverse of A, and is denoted by Aij…. The set of matrices Aij… is denoted Aij…. In particular, A1 is called the inner inverse, A2 is called the outer inverse, A12 is called the reflexive inverse, A1,2,3,4 is the Moore-Penrose inverse, etc.
For an arbitrary matrix A∈Cm×n, we denote by
NA=x∈Hn×1:Ax=0, the kernel (or the null space) of A;
CA=y∈Hm×1:y=Axx∈Hn×1, the column space (or the range space) of A; and
RA=y∈H1×n:y=xAx∈H1×m, the row space of A.
PA≔AA† and QA≔A†A are the orthogonal projectors onto the range of A and the range of A∗, respectively.
The core inverse was introduced by Baksalary and Trenkler in [1]. Later, it was investigated by S. Malik in [2] and S.Z. Xu et al. in [3], among others.
Definition 1.1. [1] A matrix X∈Cn×n is called the core inverse of A∈Cn×n if it satisfies the conditions
AX=PA,andCX=CA.
When such matrix X exists, it is denoted as A○#.
In 2014, the core inverse was extended to the core-EP inverse defined by K. Manjunatha Prasad and K.S. Mohana [4]. Other generalizations of the core inverse were recently introduced for n×n complex matrices, namely BT inverses [5], DMP inverses [2], CMP inverses [6], etc. The characterizations, computing methods, and some applications of the core inverse and its generalizations were recently investigated in complex matrices and rings (see, e.g., [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]).
In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, for generalized inverse matrices, there exist different determinantal representations as a result of the search of their more applicable explicit expressions (see, e.g. [19, 20, 21, 22, 23, 24, 25]). In this chapter, we get new determinantal representations of the core inverse and its generalizations using recently obtained by the author determinantal representations of the Moore-Penrose inverse and the Drazin inverse over the quaternion skew field, and over the field of complex numbers as a special case [26, 27, 28, 29, 30, 31, 32, 33, 34]. Note that a determinantal representation of the core-EP generalized inverse in complex matrices has been derived in [4], based on the determinantal representation of an reflexive inverse obtained in [19, 20].
The chapter is organized as follows: in Section 2, we start with preliminary introduction of determinantal representations of the Moore-Penrose inverse and the Drazin inverse. In Section 3, we give determinantal representations of the core inverse and its generalizations, namely the right and left core inverses are established in Section 3.1, the core-EP inverses in Section 3.2, the core DMP inverse and its dual in Section 3.3, and finally the CMP inverse in Section 3.4. A numerical example to illustrate the main results is considered in Section 4. Finally, in Section 5, the conclusions are drawn.
2. Preliminaries
Let α≔α1…αk⊆1…m and β≔β1…βk⊆1…n be subsets with 1≤k≤minmn. By Aβα, we denote a submatrix of A∈Hm×n with rows and columns indexed by α and β, respectively. Then, Aαα is a principal submatrix of A with rows and columns indexed by α, and Aαα is the corresponding principal minor of the determinant ∣A∣. Suppose that
Lk,n≔α:α=α1…αk1≤α1<⋯<αk≤n
stands for the collection of strictly increasing sequences of 1≤k≤n integers chosen from 1…n. For fixed i∈α and j∈β, put Ir,mi≔α:α∈Lr,mi∈α and Jr,nj≔β:β∈Lr,nj∈β.
The jth columns and the ith rows of A and A∗ denote a.j and a.j∗ and ai. and ai.∗, respectively. By Ai.b and A.jc, we denote the matrices obtained from A by replacing its ith row with the row b, and its jth column with the column c.
Theorem 2.1. [28] IfA∈Hrm×n, then the Moore-Penrose inverseA†=aij†∈Cn×mpossesses the determinantal representations
aij†=∑β∈Jr,niA∗A.ia.j∗ββ∑β∈Jr,nA∗Aββ=E8
=∑α∈Ir,mjAA∗j.ai.∗αα∑α∈Ir,mAA∗αα.E9
Remark 2.2. For an arbitrary full-rank matrix A∈Crm×n, a row vector b∈H1×m, and a column-vector c∈Hn×1, we put, respectively,
3. Determinantal representations of the core inverse and its generalizations
3.1 Determinantal representations of the core inverses
Together with the core inverse in [35], the dual core inverse was to be introduced. Since the both these core inverses are equipollent and they are different only in the position relative to the inducting matrix A, we propose called them as the right and left core inverses regarding to their positions. So, from [1], we have the following definition that is equivalent to Definition 1.1.
Definition 3.1. A matrix X∈Cn×n is said to be the right core inverse of A∈Cn×n if it satisfies the conditions
AX=PA,andCX=CA.
When such matrix X exists, it is denoted as A○#.
The following definition of the left core inverse can be given that is equivalent to the introduced dual core inverse [35].
Definition 3.2 A matrix X∈Cn×n is said to be the left core inverse of A∈Cn×n if it satisfies the conditions
XA=QA,andRX=RA.E15
When such matrix X exists, it is denoted as A○#.
Remark 3.3. In [35], the conditions of the dual core inverse are given as follows:
A○#A=PA∗,andCA○#⊆CA∗.
Since PA∗=A∗A∗†=A†A∗=A†A=QA, and RA=CA∗, then these conditions and (15) are analogous.
Due to [1], we introduce the following sets of quaternion matrices
CnCM=A∈Cn×n:rkA2=rkA,CnEP=A∈Cn×n:A†A=AA†=CA=CA∗.
The matrices from CnCM are called group matrices or core matrices. If A∈CnEP, then clearly A†=A#. It is known that the core inverses of A∈Cn×n exist if and only if A∈CnCM or IndA=1. Moreover, if A is nonsingular, IndA=0, then its core inverses are the usual inverse. Due to [1], we have the following representations of the right and left core inverses.
Remark 3.5. In Theorems 3.6 and 3.7, we will suppose that A∈CnCM but A∉CnEP. Because, if A∈CnCM and A∈CnEP (in particular, A is Hermitian), then from Lemma 3.4 and the definitions of the Moore-Penrose and group inverses, it follows that A○#=A○#=A#=A†.
Theorem 3.6.LetA∈CnCMandrkA2=rkA=s. Then, its right core inverse has the following determinantal representations
where e.l and el. are the unit column and row vectors, respectively, such that all their components are 0, except the lth components which are 1; a˜lf is the (lf)th element of the matrix A˜≔A2A∗.
Here a¯.f and a¯l. are the fth column and lth row of A¯≔A∗A2.
3.2 Determinantal representations of the core-EP inverses
Similar as in [4], we introduce two core-EP inverses.
Definition 3.8. A matrix X∈Cn×n is said to be the right core-EP inverse of A∈Cn×n if it satisfies the conditions
XAX=A,andCX=CX∗=CAd.
It is denoted as A○†.
Definition 3.9. A matrix X∈Cn×n is said to be the left core-EP inverse of A∈Cn×n if it satisfies the conditions
XAX=A,andRX=RX∗=RAd.
It is denoted as A○†.
Remark 3.10. Since CA∗d=RAd, then the left core inverse A○† of A∈Cn×n is similar to the ∗core inverse introduced in [4], and the dual core-EP inverse introduced in [35].
Due to [4], we have the following representations the core-EP inverses of A∈Cn×n,
A○†=A2,3,6aandCA○†⊆CAk,A○†=A2,4,6bandRA○†⊆RAk.
Thanks to [35], the following representations of the core-EP inverses will be used for their determinantal representations.
Lemma 3.11.LetA∈Cn×nandIndA=k. Then
A○†=AkAk+1†,E21
A○†=Ak+1†Ak.E22
Moreover, if IndA=1, then we have the following representations of the right and left core inverses
A○#=AA2†,E23
A○#=A2†A.E24
Theorem 3.12.SupposeA∈Cn×n,IndA=k,rkAk=s, and there existA○†andA○†. ThenA○†=aij○†,randA○†=aij○†,lpossess the determinantal representations, respectively,
where at.k+1∗ is the tth row of Ak+1∗. Since ∑t=1naitkat.k+1∗=âi., then it follows (25).
The determinantal representation (26) can be obtained similarly by integrating (8) for the determinantal representation of Ak+1† in (22).□
Taking into account the representations (23)-(24), we obtain the determinantal representations of the right and left core inverses that have more simpler expressions than they are obtained in Theorems 3.6 and 3.7.
Corollary 3.13.LetA∈Csn×n,IndA=1, and there existA○#andA○#. ThenA○#=aij○#,randA○#=aij○#,lcan be expressed as follows
The name of the DMP inverse is in accordance with the order of using the Drazin inverse (D) and the Moore-Penrose (MP) inverse. In that connection, it would be logical to consider the following definition.
Definition 3.16. Suppose A∈Cn×n and IndA=k. A matrix X∈Cn×n is said to be the MPD inverse of A if it satisfies the conditions
XAX=X,AX=AAd,andXAk=A†Ak.
It is denoted as A†,d.
The matrix A†,d is unique, and it can be represented as
A†,d=A†AAd.E33
Theorem 3.17.LetA∈Csn×n,IndA=k, andrkAk=s1. Then, its MPD inverseA†,d=aij†,dhas the following determinantal representations
Here, âl. and â.f are the lth row and the fth column of Â≔A∗Ak+1.
Proof. The proof is similar to the proof of Theorem 3.15.□
3.4 Determinantal representations of the CMP inverse
Definition 3.18. [6] Suppose A∈Cn×n has the core-nilpotent decomposition A=A1+A2, where IndA1=IndA, A2 is nilpotent, and A1A2=A2A1=0. The CMP inverse of A is called the matrix Ac,†≔A†A1A†.
Lemma 3.19. [6] LetA∈Cn×n. The matrixX=Ac,†is the unique matrix that satisfies the following system of equations:
XAX=X,AXA=A1,AX=A1A†,andXA=A†A1.
Moreover,
Ac,†=QAAdPA.E34
Taking into account (34), it follows the next theorem about determinantal representations of the quaternion CMP inverse.
Theorem 3.20.LetA∈Csn×n,IndA=m, andrkAm=s1. Then, the determinantal representations of its CMP inverseAc,†=aijc,†can be expressed as
Here, ût. is the tth row and û.k is the kth column of Û≔UAA∗, g˜t. is the tth row and g˜.k is the kth column of G˜≔A∗AG, and the matrices U=uij∈Hn×n and G=gij∈Hn×n are such that
In this chapter, we get the direct method to find the core inverse and its generalizations that are based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, the DMP, MPD, and CMP inverses are derived.
\n',keywords:"Moore-Penrose inverse, Drazin inverse, core inverse, core-EP inverse, 2000 AMS subject classifications: 15A15, 16W10",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/69235.pdf",chapterXML:"https://mts.intechopen.com/source/xml/69235.xml",downloadPdfUrl:"/chapter/pdf-download/69235",previewPdfUrl:"/chapter/pdf-preview/69235",totalDownloads:192,totalViews:0,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"May 9th 2019",dateReviewed:"August 26th 2019",datePrePublished:"November 27th 2019",datePublished:"June 17th 2020",dateFinished:"September 25th 2019",readingETA:"0",abstract:"Generalized inverse matrices are important objects in matrix theory. In particular, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/69235",risUrl:"/chapter/ris/69235",book:{slug:"functional-calculus"},signatures:"Ivan I. Kyrchei",authors:[{id:"226980",title:"Dr.",name:"Ivan",middleName:null,surname:"Kyrchei",fullName:"Ivan Kyrchei",slug:"ivan-kyrchei",email:"kyrchei@online.ua",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Preliminaries",level:"1"},{id:"sec_3",title:"3. Determinantal representations of the core inverse and its generalizations",level:"1"},{id:"sec_3_2",title:"3.1 Determinantal representations of the core inverses",level:"2"},{id:"sec_4_2",title:"3.2 Determinantal representations of the core-EP inverses",level:"2"},{id:"sec_5_2",title:"3.3 Determinantal representations of the DMP and MPD inverses",level:"2"},{id:"sec_6_2",title:"3.4 Determinantal representations of the CMP inverse",level:"2"},{id:"sec_8",title:"4. An example",level:"1"},{id:"sec_9",title:"5. Conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'Baksalary OM, Trenkler G. Core inverse of matrices. Linear and Multilinear Algebra. 2010;58:681-697'},{id:"B2",body:'Malik S, Thome N. On a new generalized inverse for matrices of an arbitrary index. Applied Mathematics and Computation. 2014;226:575-580'},{id:"B3",body:'Xu SZ, Chen JL, Zhang XX. New characterizations for core inverses in rings with involution. Frontiers in Mathematics China. 2017;12:231-246'},{id:"B4",body:'Prasad KM, Mohana KS. Core EP inverse. Linear and Multilinear Algebra. 2014;62(3):792-802'},{id:"B5",body:'Baksalary OM, Trenkler G. On a generalized core inverse. Applied Mathematics and Computation. 2014;236:450-457'},{id:"B6",body:'Mehdipour M, Salemi A. On a new generalized inverse of matrices. Linear and Multilinear Algebra. 2018;66(5):1046-1053'},{id:"B7",body:'Chen JL, Zhu HH, Patrićio P, Zhang YL. Characterizations and representations of core and dual core inverses. Canadian Mathematical Bulletin. 2017;60:269-282'},{id:"B8",body:'Gao YF, Chen JL. Pseudo core inverses in rings with involution. Communications in Algebra. 2018;46:38-50'},{id:"B9",body:'Guterman A, Herrero A, Thome N. New matrix partial order based on spectrally orthogonal matrix decomposition. Linear and Multilinear Algebra. 2016;64(3):362-374'},{id:"B10",body:'Ferreyra DE, Levis FE, Thome N. Maximal classes of matrices determining generalized inverses. Applied Mathematics and Computation. 2018;333:42-52'},{id:"B11",body:'Ferreyra DE, Levis FE, Thome N. Revisiting the core EP inverse and its extension to rectangular matrices. Quaestiones Mathematicae. 2018;41(2):265-281'},{id:"B12",body:'Liu X, Cai N. High-order iterative methods for the DMP inverse. Journal of Mathematics. 2018;8175935:6'},{id:"B13",body:'Ma H, Stanimirović PS. Characterizations, approximation and perturbations of the core-EP inverse. Applied Mathematics and Computation. 2019;359:404-417'},{id:"B14",body:'Mielniczuk J. Note on the core matrix partial ordering. Discussiones Mathematicae Probability and Statistics 2011;31:71-75'},{id:"B15",body:'Mosić D, Deng C, Ma H. On a weighted core inverse in a ring with involution. Communications in Algebra. 2018;46(6):2332-2345'},{id:"B16",body:'Prasad KM, Raj MD. Bordering method to compute core-EP inverse. Special Matrices. 2018;6:193-200'},{id:"B17",body:'Rakić DS, Dinčić ČN, Djordjević DS. Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra and its Applications. 2014;463:115-133'},{id:"B18",body:'Wang HX. Core-EP decomposition and its applications. Linear Algebra and its Applications. 2016;508:289-300'},{id:"B19",body:'Bapat RB, Bhaskara Rao KPS, Prasad KM. Generalized inverses over integral domains. Linear Algebra and its Applications. 1990;140:181-196'},{id:"B20",body:'Bhaskara Rao KPS. Generalized inverses of matrices over integral domains. Linear Algebra and its Applications. 1983;49:179-189'},{id:"B21",body:'Kyrchei I. Analogs of the adjoint matrix for generalized inverses and corresponding Cramer rules. Linear and Multilinear Algebra. 2008;56(4):453-469'},{id:"B22",body:'Kyrchei I. Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations. Applied Mathematics and Computation. 2013;219:7632-7644'},{id:"B23",body:'Kyrchei I. Cramer’s rule for generalized inverse solutions. In: Kyrchei I, editor. Advances in Linear Algebra Research. New York: Nova Science Publ; 2015. pp. 79-132'},{id:"B24",body:'Stanimirović PS. General determinantal representation of pseudoinverses of matrices. Matematichki Vesnik. 1996;48:1-9'},{id:"B25",body:'Stanimirović PS, Djordjevic DS. Full-rank and determinantal representation of the Drazin inverse. Linear Algebra and its Applications. 2000;311:131-151'},{id:"B26",body:'Kyrchei I. Determinantal representations of the Moore-Penrose inverse over the quaternion skew field. Journal of Mathematical Sciences. 2012;180(1):23-33'},{id:"B27",body:'Kyrchei I. Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer’s rules. Linear and Multilinear Algebra. 2011;59(4):413-431'},{id:"B28",body:'Kyrchei I. Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Applied Mathematics and Computation. 2014;238:193-207'},{id:"B29",body:'Kyrchei I. Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field. Applied Mathematics and Computation. 2015;264:453-465'},{id:"B30",body:'Kyrchei I. Explicit determinantal representation formulas of W-weighted Drazin inverse solutions of some matrix equations over the quaternion skew field. Mathematical Problems in Engineering. 2016;8673809:13'},{id:"B31",body:'Kyrchei I. Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation. Journal of Applied Mathematics and Computing. 2018;58(1-2):335-365'},{id:"B32",body:'Kyrchei I. Determinantal representations of the Drazin and W-weighted Drazin inverses over the quaternion skew field with applications. In: Griffin S, editor. Quaternions: Theory and Applications. New York: Nova Sci. Publ.; 2017. pp. 201-275'},{id:"B33",body:'Kyrchei I. Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore-Penrose inverse. Applied Mathematics and Computation. 2017;309:1-16'},{id:"B34",body:'Kyrchei I. Determinantal representations of the quaternion weighted Moore-Penrose inverse and its applications. In: Baswell AR, editor. Advances in Mathematics Research 23. New York: Nova Science Publ; 2017. pp. 35-96'},{id:"B35",body:'Zhou M, Chen J, Li T, Wang D. Three limit representations of the core-EP inverse. Univerzitet u Nišu. 2018;32:5887-5894'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Ivan I. Kyrchei",address:"kyrchei@online.ua",affiliation:'
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv, Ukraine
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1. Introduction
With the formulation of the theory of relativity, we gained a new tool with which to explain the world. It appears that the laws of quantum mechanics explain the processes governing the deepest layers of reality, operating at the levels of the smallest particles of our world. Through experiments and analyses, we can assume that these laws explain phenomena both on the micro- and macroscale. Rules different from classical physics explain the heretofore unexplained and, crucially, allow us to design new experiments within a world entirely unavailable to our senses. The analyses of the visible outcomes of interactions between components—that is to say, events in the real world—return little information about the structure of the observed reality. By interpreting outcomes (events), instead of images of detailed relations, we project our expectations; the mathematical structures we use to explain outcomes are merely an attempt to fit our model to reality, and we can only determine the model’s applicability when, and insofar as, the observable reality confirms our reasoning. Nevertheless, the fact remains that mathematics allows us to draw conclusions as to the rules governing the functioning of the world. The more appropriate the model we use, the better will be our results. If reality is like music, then the tools of analysis are our music score [1]. The notes we use will be the substance of music, but not the music itself. Similarly with analyses, a model of reality used for analysis is the score whose reality is created with many unspecified or loosely defined components. This may lead us to conclude that reality can only perform a score once it has been written. Such an anthropocentric approach leads us to believe that we can influence and shape events. This is especially clear in analysing risk. We usurp the right to assess risk and event probability and expect that reality will perform our freshly composed score. We must, however, allow that mathematical analysis may try to impose its assumptions on reality, which may or may not succeed (e.g. in management methods used in banks and insurance companies).
The job of the risk assessor at such institutions is to draw conclusions about the structure of the world based on mathematical structures. In this context, the ideas of quantum physics—in particular, the concept of the state of an object in a Hilbert space [2], the phase space and the quantum system—may help our analyses, allowing a fuller understanding of reality.
In light of the present investigation, we can conclude that quantum mechanics does not apply to individual events but is a theory of interactions between groups of events (composition series) whose behaviour observably conforms to the laws of statistics. All measurement attempts made within a quantum system will, in essence, be performed on groups of identically prepared objects. These objects may realise every possible state. The results of these measurement attempts come in the form of probability distribution of all possible measurement results. In line with this interpretation, we can focus strictly on looking for the probability distribution and ignore individual events.
Risk assessors have been using this interpretation for some time now. The theory of inertia [3], as an example, asserts that the probabilities of a given state occurring and not occurring are equal. This means that at any given moment a given event may occur or not. If we assign a value of 1 to the state occurring, and a value of 0 to the state not occurring, the distribution will result in a mean of 0 and standard deviation of 1. The potential of the event occurring has equal probability, which means that at any given moment we can expect a given state to occur and not occur—just as described by Schrödinger in his famous 1935 experiment [4]. Each object interacts with its environment, and at the quantum level this interaction can be described as consistent with the second law of thermodynamics, i.e. with quantum decoherence, where every system moves towards increased entropy if it is devoid of energy needed to preserve its current state. The identified risk potential is an expression of quantum entanglement and exhibits a tendency to equalisation (entanglement reduction), towards irreversible change of interference between the system and the environment. Risk, therefore, should be understood as the measurement of loss of information about a given system, as a result of its interaction with the environment. In this context, it is imperative to attempt a quantum-mechanical analysis of risk.
2. Quantum mechanics in interpretation of phenomena
The desire to understand the world and to describe it in terms of mathematical formulae is as old as the human desire to dominate it. Each age has tried to explain observable correlations as causes and effects, in a manner peculiar to itself. In ancient Greece, atomistic theories of the likes of Democritus, according to which matter consisted of final, eternal, unchanging and indivisible atoms, clashed with continuous theories of Aristotle and others, who believed that matter was fluid and ever-changing. Such theoretical clashes across the ages have always encouraged further investigations into our questions about the world, leading to new questions, new theories and new clashes. This progress of human knowledge sped up with the industrial revolution and the creation of more efficient tools of observation. When electrons were discovered, the structure of previously indivisible atoms was called into question. The discovery of the atomic nucleus led to the formulation of the planetary model of the atom, according to which nearly all the mass and positive charge are concentrated in the nucleus, around which negatively charged electrons orbit. During his work on black-body radiation, Max Plank formulated the hypothesis of quanta of energy, when existing analyses based on classical physics proved ineffectual. Further discoveries and ideas followed; one of which was the wave-particle duality, which posits that the entire universe behaves consistently with laws governing either the behaviour of waves or that of particles. This idea, and the results of many experiments which it enabled (including the work of Wheeler [5]), suggests that the observed structures change as the result of observation. So, it is probable that a given event’s result, which we wish to observe, will occur depending on whether and how we observe the event. The manner in which we choose to measure the “present moment” will influence that which caused the present event. Ideas like these are far removed from those of Aristotle and Democritus and require a completely fresh gaze. The premises of quantum physics appear to be verifiable both on micro and macro level, which begs the question: if quantum physics applies to the level of elementary particles, will they apply to modelling real events? If the answer is yes, there should be no objection to the use of analytical tools proper to quantum physics in analysing events such as those which risk analysts are concerned with. The language of mathematics seems to be the only tool precise enough to describe the subtle and rich structures of reality. Structures transparent to human senses are revealed in mathematics, and numbers allow the identification of their states and properties (e.g. their minima, maxima or functions). Let us attempt a certain simplification. The classical, Newtonian understanding of phenomena is determined by the notions of motion, point particle and rigid body. Classical mechanics relies on the premise that there exist objective, quantifiable objects, in motion along specific trajectories, possessing other specific properties, such as position, mass or charge. Elements of a system interact, as do point particles, in strictly defined ways. Contacts and collisions occur, whether directly or indirectly through fields (e.g. of energy, temperature, etc.). These observations lead to the conclusion that the properties of a physical situation can be defined by absolute terms and numbers. We identify laws of cause and effect and conjecture that, under identical conditions, objects will behave identically. We conclude that all objects in the world are determinate and their behaviour is strictly defined and uniform. In the standard (classical) probability measure of a given state occurring, a family of events is characterised as follows:
There exists a sample space Ω and a family Z of subsets of the sample space Ω, called events. The following premises are true:
∅ (empty set) and Ω (sample space) are events.
If A is an event, then A’ = Ω - A also is an event.
If A and B are events, the sum of sets A∪B is also an event.
Probability is the function P:Z→01 where:
PΩ=1.
If A and B are events and the product of sets A∩B=∅, then
PA∪B=PA+PB.E1
It follows from premises (1), (2) and (3) that if A and B are events, A∩B is also an event.
This model works in general but fails in particular, detailed analyses where it turns out that it is impossible to define generic behaviours of such elementary particles as electrons.
We would reach similar conclusions in analysing human behaviours. We cannot predict human reactions to specific stimuli; we can only predict the probability that the person will behave in this or that specific way. In creating models of reality, we try to describe complicated reactions which consist both of defined and undefined situations. In effect, individual points of such a system become a blur. They become less sharply defined as discrete entities and tend towards the state which is the result of our analyses. Whilst we can define, as an example, the length of a road tunnel using classical physics, the result of the measurement of the total velocity of the motion of vehicles in the tunnel, or the time needed to evacuate people from the tunnel in an emergency, will depend on the chosen method of measurement. Defining the method of measurement is key in assessing the safety of a tunnel, both for particular tunnels and all tunnels in general. This in turn creates the risk that, in defining an object, one can describe its properties so that the correct (i.e. well defined) response to the question about its state will be elicited only if the object is in the ground state (assumed by the analysis operator). To return to the example of a road tunnel, the very question whether the tunnel is “safe” is, in effect, a question not about the state of the object but the properties of the analysis operator. The response to such a question will be incidental and unreliable. The basic problem here appears to be expressing the measurement numerically. If we do not use a precise measurement method, the result will always be a blur.
It remains a fact that there are such values, or their relations, which can never be specified precisely (expressed as discrete numbers) at the same time, for example, the number of people at risk in a particular road tunnel emergency. We can assume the minimum, maximum or mean (expected) value, but we can only arrive at a probability of the value of our prediction.
It could be argued that an object in a particular state has the unique ability to respond to the demands of its environment and display a particular property. When its state conforms with the state expected by the operator, the object can return an unambiguous response to the operator’s “question”. In the case of the road tunnel, its state usually does not conform with the state expected by the operator (questioning the tunnel’s safety), and as a result the question generates a random response out of a set of the operator’s ground states. Of the many states identified by the analysis operator, the current state of the object may be constructed, but each of these states may reveal itself as the response to the operator’s question, without the possibility of predicting which. In other words, the complete state of the object fractures into multiple ground states of the operator, and the object picks a state haphazardly and returns it to the operator as its response.
These observations are consistent with the tenets of the Copenhagen interpretation of quantum mechanics. Accordingly, we can set forth the following theses:
Every system is fully described by the wave function Y which fully describes the observer’s understanding of the system (Heisenberg).
A description of nature is probabilistic. The probability of an event is the squared modulus of the wave function associated with the event (Max Born).
We cannot know the values of all properties of a system at a given time; imprecise properties may be expressed as probabilities (Heisenberg’s uncertainty principle).
When the size of the system approaches macroscale, the quantum-mechanical description ought to yield results consistent with the results from a classical treatment (Bohr’s and Heisenberg’s correspondence principle).
An attempt to model future events within a given system consistent with the classical probability modelling must result in producing questions inadequate to the possibility of eliciting responses, as we are unable to predict all variants of events and correlations. Classical probability attempts to ignore reality and proceeds against logic, which allows us to accept the instances of “black swans”. Meanwhile, if we accept that the real-life result is a wave function of the relations between the preparation of the system and its measurement and that it has an operational character inextricable from the observer, we can attempt to identify the final states and define the occurrence probability of the elements leading up to these final states.
Let us consider the following model. Let us suppose, in accordance with the above premises, that the probability p is described as the squared modulus of a certain complex number A, which is the amplitude of probability:
p=A2E2
This pattern has been confirmed in multiple experiments, of which the most representative is the one using a quantum gun.
Let us consider an electron gun (although it could equally well be a ball launcher or traffic organisation in a tunnel). Let us launch electrons towards screen E through an obstacle with two apertures S1,2 and define the variable x (probability) along the screen E. The number of apertures may be greater. The screen is positioned at the distance L. The system may be illustrated as shown in Figure 1.
Figure 1.
Aperture experiment.
Let us assume that the passage of an electron through the apertures can happen in two distinct ways, S1 and S2, each of which is described by the probability amplitude AS1 and AS2.
In classical reasoning, the launched electron can reach the screen either through aperture S1 (trajectory S1) or aperture S2 (trajectory S2). This level will result in flares on the screen appropriate to the location of the apertures, which is illustrated below (Figure 2). The position of the flares conforms to Gaussian (normal) distribution.
Figure 2.
Probability distribution for positions of electrons hitting the screen, consistent with the tenets of classical physics.
In practice, such an ideal model cannot occur. This is because it is also probable that the electron will not pass through the aperture, or that events will suffer from mutual interference. In effect, there will be areas of maximum likelihood of the electron hitting the screen, as well as areas which will never be hit by the electron (under constant conditions of the experiment). This situation will result in a probability amplitude which can be formulated as
AS1orS2=AS1+AS2.E3
This observation replaces the practice of adding up probabilities of classical physics: the final result is not a sum of probabilities of the electron passing through the aperture. It can be illustrated as shown in Figure 3 above.
Figure 3.
Probability distribution for locations of electrons hitting the screen, accounting for possible interferences.
The following equation is, therefore, true:
pS1orS2=AS1orS22=AS1+AS22=AS12+AS22E4
The probability depends on the relative phase of the amplitudes AS1 and AS2, whilst in pS1 and pS2 such phases do not occur. Consequently, event probabilities transfer onto the amplitudes, and this results in the occurrence of new phenomena, unaccounted for by classical physics. Let us assume, for the purposes of our argument, that a given phenomenon can occur in two distinct ways, S1 and S2, each of which is described by the probability amplitude AS1 and AS2. The influence of an interference must be considered if we cannot identify the aperture through which the electron will travel. If we define the phenomenon occurrence probability as = p11(x) + p2 (x) + I(x),
where I(x) denotes the influence of interference calculated as:
Ix=2p1xp2xcosφ1x−φ2xE5
and we must define values for the correlation φ1;2 of x, then in the particle behaviour analysis we can assume that the correlation is linear and can be expressed with the equation
Ix=2p1xp2xcosαxE6
where the constant α depends on the mass and energy of the launched particles, distance of screen from apertures and other conditions. To consider questions other than the particles, we need to define the influence of interference as a corrective.
Probability calculations informed by quantum physics can be defined as follows. There exists a sample space Ω and a family Zk of subsets of Ω, called k-events. The following premises are true:
(1) ∅ (empty set) and Ω (sample space) are k-events.
(2) If A is a k-event, then A’ also is a k-event.
(3K) If A and B are k-events and the product of sets A∩B=∅, then A∪B is also a k-event.
Probability if the function P:Zk→01, so.
(4) PQ=1.
(5) If A and B are k-events and the product of sets A∩B=∅, then P(A∩B) = P(A) + P(B).
The difference between the two approaches hinges on exchanging premise (3), considered earlier, with premise (3K), which prevents us from considering alternatives for the k-events whose conjunction is not itself a k-event.
It follows from premises (1), (2) and (3K) that if A and B are k-events, the product A∩B is a k-event if and only if the sum A∪B is a k-event.
This condition is met when considering random variables which are significant in analysing possible real events.
The function X:Ω→R is a random variable if for each interval ab⊂R the set ωϵΩXωϵab is an event.
A random variable will be continuous if the function ∫:R→R is nonnegative and for each interval ab,PωϵΩ:Xωϵab=∫abfxdx .
The function f will be the distribution density of the variable X. So if for each particle ψq is a wave function which we can define as fq=ψq2, that is the distribution density for the position of the particle along a straight line, then ∫abψq2dq defines the probability that the particle will be within the interval ab.
In probability theory, the pair of random variables XY is called a two-dimensional (bivariate) variable. A nonnegative function hxy is called the distribution density of a bivariate random variable XY, if for any numbers a<b and c<d there is equality:
PωϵΩ:(Xωϵabi(Yωϵcd=∫ab∫cdhxydydxE7
We can also demonstrate that
fx=∫−∞+∞hxydyE8
is the distribution density of the variable X and gx=∫−∞+∞hxydx is the distribution density of the variable Y.
If instead of classical probability we employ quantum-mechanical density amplitude probability, we arrive at a correct definition. This leads us to conclusively abandon the “objective realism” which determines classical probability and replace it with quantum probability. To examine our reasoning, we shall consider the following example. A pair of random variables (q, p) are given, with known distribution densities. We need to establish the distribution density of the bivariate random variable (q, p) as a nonnegative function h(q,p) such that
∫−∞+∞hqpdp=ψq2 and ∫−∞+∞hqpdp=ϕp2.
The above relation between ϕ and ψ, as well as the uncertainty principle, is realised.
The expected values of the observable quantum random variables, calculated according to the definition of expected value in probability theory, are equal to the expected values of the same variables, calculated according to quantum operation formalism.
Following Leon Cohen’s interpretation, there exist functions fulfilling conditions (1) and (2), but a function fulfilling all three conditions does not exist. Consequently, we cannot go too far in probabilistic interpretations of quantum mechanics.
Even so, we can treat both p and q as random variables, but if we consider p and q jointly, we are leaving probability theory behind: the pair pq is not a random variable. Instead, we must broaden the applicability of probability theory. None of the above premises contradict our knowledge of real-life phenomena inside road tunnels so it seems justifiable to use quantum probability in risk assessment calculation models. Quantum physics focuses primarily on an object’s state which may, for its final manifestation, depend on the tools of analysis we use. The state of the object at a given moment is represented by the direction (radius) in the Hilbert space. A Hilbert space illustrates the type of phase space in which referring to real numbers alone is incomplete. A given state should be treated as a superposition of all possible states at every moment, and quantum mechanics typically deals with complex vector spaces. The resultant model of reality is very rich and composed of many interconnected structures whose states are synchronised and the probability of any state’s occurrence at any given moment is always the same.
The conclusion to which quantum physics points us is that event occurrence or nonoccurrence probability is always the same at 50%, and the certainty of this result will be the derivative of the set probability distribution.
3. Risk identification
If we want to apply mathematical models to risk analysis, we must clarify our premises and definitions. Risk, in popular understanding, measures the possibility of loss of a given state and may be positive (profit) and negative (loss). Most commonly, “risk” is applied in the context of safety. Most people identify safety as a primary need, without which they experience anxiety and insecurity. It is psychological needs like these that cause individuals, societies, states and organisations to act on their environments in order to remove or reduce factors which increase anxiety, fear, uncertainty or insecurity. As a result, no matter how we define safety, it will ultimately remain an individual interpretation of a given phenomenon. For some people dangerous actions which, if successful, will make them a hero seem right, and their evaluation of possible consequences does not stop them from taking such actions, which would cause fear and inaction in another person. In this context, the security of larger organisations should not rely solely on such subjective assessments. State security is not the same as the sum total of individual securities of each of the state’s citizens, and the safety of an organisation is not tantamount to the safety of each of its stakeholders. In the aftermath of the financial crash which bankrupted many companies in 2008, renewed efforts were undertaken to clarify safety for use both in financial management and in other areas of life. In line with the proposed guidelines, safety must be defined as freedom from unacceptable risk. In process safety procedures used in chemical process facilities, “safety” is understood as the absence of unacceptable risk to health, life, property or environment, whereas risk is the product of probability (frequency of occurrence) of a given phenomenon and the scale of losses (size of undesired results) formulated as
risk=probability×resultsE9
The use of this formula is, however, vitiated by the cognitive determinism of the person identifying the probability and the results. With the 2009 ISO 31000 standard, a new understanding of risk has been proposed. Both the standard and the UN Recommendations on the Transport of Dangerous Goods [6] redefine “risk” as the “effect of uncertainty on objectives”. The standard is a collection of frameworks, processes and rules which ought to be complied with during the risk assessment process in every organisation, commercial and otherwise. Building on the earlier considerations, we can reformulate risk as follows:
Risk−∣uncertainty∣objectiveE10
where “uncertainty” is defined as blurred probability of an event, which cannot be foreseen with absolute certainty, and all the related possibilities and probabilities are variable and possible.
In using measurement instruments (in this context, mathematical analysis) to measure a quantum state, a certain aspect of this state must be adjusted to the state of the instrument used. This is called an observable. In accordance with quantum mechanics’ second postulate, each observable is represented by the linear map (vector—Hermitian operator) acting in a Hilbert space, and the eigenvalues of this operator present all possible results of its measurements. The third postulate proposes that the likelihood, that the measurement of the measurable magnitude of observation A will return a k eigenvalue of the Hermitian operator, equals λk2, thus confirming the aptness of the risk assessment method, but does not bring us any closer to calculating the actual risk in a particular situation. Real-life events, even in their vector manifestation, do not allow the prediction of the direction of change. It follows that we should limit ourselves to identifying the possibility of each state occurring, based on their probability as derived from probability distribution.
4. Example application
Risk analyses are typically related to incident analyses. Let us consider the risk of an incident inside a road tunnel. Real data is available: over the last few years, the number of traffic incidents (traffic jams, attempts to reverse, collisions and fires) averages 340 per annum. We can further identify the likely rate of each type of incident within the total number. Let us conduct an analysis of the risk of change to the state of safety, based on the above-discussed postulates.
Step 1. Identify the objective
Accepting the definition of risk as the effect of uncertainty on objectives, we must first identify the objectives. This will be the number of incidents acceptable under the circumstances. Let us assume that the current state is acceptable, which means that we expect 340 incidents in the coming year. Let us take the year to be 365 days. The expected value (objective) will be 381 over the course of the year. Let us now attempt to measure the influence of uncertainty (i.e. the likelihood of a particular distribution) on the objective, i.e. the given annual number of incidents.
Step 2. Identify the event distribution
Let us assume that incidents occur according to normal distribution. Traffic intensity is constant. There are no planned maintenance works.
For a given tunnel, the number of incidents (vehicles stopped) has been calculated, at certain traffic intensity, as 340 per year.
Calculating the uncertainty of 340 incidents per year (Table 1).
Graphic representation of uncertainty of incident estimate as shown in Figure 4 above.
Parameter
Equation/symbol
Value
Comments
Expected mean
λ
340.000
Median
Me≈λ+1/3 + 0.02/λ
340.333
Mode
340.000
Equals the greatest integer lesser than λ
Kurtosis
1/λ
0.003
Cumulative probability
0.514
Probability mass function for the Poisson distribution
0.0216
Skewness
√ λ
18.439
Measures the asymmetry of the distribution about its mean
Table 1.
Uncertainty parameters for the presumed outcome.
Figure 4.
Representation of uncertainty of 340 incidents occurring.
Conclusion: For the analysed tunnel, there is a 50% uncertainty that the number of incidents (vehicles stopped) in the year will be between 319 and 375.
The analysis leads to the following conclusions:
We do not know if the number of incidents will equal to 0.
It is likely that at least 289 incidents will occur.
We must be prepared for no fewer than 319 but ideally 375 incidents.
We do not know if the number of incidents will exceed 401.
The analysis allows conclusions which will enable a better preparation for incidents than would have been the case using the standard method, which assumes that the risk of incident equals the product of its value and the probability of its occurrence.
5. Conclusion
We ought to understand risk management process as actions coordinated towards the achievement of a predefined state of acceptability. Acceptance of a state, that is to say preparation for it, is the key process in ensuring that an organisation may continue functioning. Risk management process comprises the following stages [7]:
Risk recognition, including identification of objectives
Risk identification, including possible consequences
Risk analysis, or risk magnitude estimation, i.e. quantification of uncertainty in attaining objectives
Risk evaluation, i.e. comparison of results with objectives
Risk analysis is, in this process, just one of the stages and ought not to be conducted apart from the other elements. Aside from describing dangers and consequences, the process will result in identification of conditions for decision-taking regarding actions which consider the uncertainties of danger and dangerous events occurring, as well as identification of possibilities for avoiding or limiting losses.
Therefore, the full process ought to comprise calculations of the influence of uncertainty on our objectives and evaluation of results, including the evaluation whether occurrences of motion deviation are counterbalanced by solutions used.
Risk analysis will be most fully realised when we assume that to identify the level of attainment of objectives (i.e. identification of possible risk), we must first identify the probability of each possible state’s occurrence, based on probability distribution.
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