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...ur SUPER INTERVAL SEMILINEAR ALGEBRAS 137 Chapter Five SUPER FUZZY INTERVAL MATRICES 203 5.1 Super Fuzzy Interval Matrices 203 5.2 Special Fuzzy Linear Algebras Using Super Fuzzy Interval Matrices 246 Chapter Six APPLICATION OF SUPER INTERVAL MATRICES AND SET LINEAR ALGEBRAS BUILT USING SUPER INTERVAL MATRICES 255 Chapter Seven SUGGESTED PROBLEMS 257 FURT...

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...super FAMs etc. In chapter five we define interval matrix and matrix interval using the natural class of intervals, that is intervals of the form [a, b] where a b or a and b not comparable. In the final chapter we have defined the new notion of bimatrices and n-matrices. Using these new concepts we have constructed linear algebra of type I and linear algebra of type II. U...

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...e basics. The second chapter deals with different types of neutrosophic bilinear algebras and bivector spaces and proves several results analogous to linear bialgebra. In chapter three the authors introduce the notion of n-linear algebras and prove several theorems related to them. Many of the classical theorems for neutrosophic algebras are proved with appropriate modific...

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...1 Introduction to Bilinear Algebras and their Generalizations 9 1.2 Introduction to Neutrosophic Algebraic Structures 11 Chapter Two NEUTROSOPHIC LINEAR ALGEBRA 15 2.1 Neutrosophic Bivector Spaces 15 2.2 Strong Neutrosophic Bivector Spaces 32 2.3 Neutrosophic Bivector Spaces of Type II 50 2.4 Neutrosophic Biinner Product Bivector Space 198 Chapter Three NEUTROSOP...

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...easily understood by a beginner. Binary symmetric channel is an illustration of a model for a transmission channel. Now we will proceed onto define a linear code algebraically....

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...Preface 5 Chapter One BASIC CONCEPTS 7 Chapter Two ALGEBRAIC LINEAR CODES AND THEIR PROPERTIES 29 Chapter Three ERASURE DECODING OF MAXIMUM RANK DISTANCE CODES 49 3.1 Introduction 49 3.2 Maximum Rank Distance Codes 51 3.3 Erasure Decoding of MRD Codes 54 Chapter Four MRD CODES ...

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... in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research...

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... chemical industries using fuzzy neutral networks. Chapter five gives the method of minimization of waste gas flow in chemical industries using fuzzy linear programming. The final chapter suggests when in these studies indeterminancy is an attribute or concept involved, the notion of neutrosophic methods can be adopted. The authors feel that the reader should be well verse...

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...Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector space...

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...Matrices provide a very powerful tool for dealing with linear models. Bimatrices which we are going to define in this chapter are still a powerful and an advanced tool which can handle over one linear model at a time. Bimatrices will be useful when time bound comparisons are ne...

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... 1 INTRODUCTION TO BIMATRICES W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Tec... ..., Georgia State University, Atlanta, GA 30302, USA Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache and K. Ilanthenral La... ...EIR PROPERTIES 2.1 Introduction to Bivector Spaces 79 2.2. Systems of linear biequations 87 2.3 Bitransformation in bivector spaces 96 2.4 ... ...paces 105 2.5 Bieigen values and Bieigen vectors 117 2.6 Bidiagonalizable linear bioperator and its properties 124 4 Chapter Three ... ...zy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices ... ...odels. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of eac... ...vention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are ... ...my for cover designing and Kama Kandasamy for formatting this book. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K. ILANTHENRAL 7 Ch... ...and define some of basic operation on it. DEFINITION 1.1.1: A bimatrix A B is defined as the union of two rectangular away of numbers A 1 and A...

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...namics of the Universe and Life Chapter 3: Hominids Pgs 132-187 A: How we evolved into Humans Pgs 188-222 B: Summary of Hominid-Human Development Chapter 4: Modern Human Dynamics Pgs 223-266 Human Psycho-biologic Totality. Chapter 5: Modern Humans: Pgs 267-299 The Transition from Hunter-gatherers to S...

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... A: How we evolved into Humans Pgs 188-222 B: Summary of Hominid-Human Development ... ...03 A: The Beginnings of Civilization Pgs 704-1469 B: The Effect of Civilization on Humans Pgs 1470-1868 ... ...ollywood and Perfection Pg 527 Tool Principles vs. Human principles Pg 529 Linear Functionality Pg 540 Tool-functioning vs. Human Functioning P... ...Unclearness of History Pg 694 Wisdom Pg 697 Incrementalism Chapter Six ‘B’: Civilization The Effect of Civilization on Humans Pg 706 Incre... ... series of Three Basic Causal Splits What is a Dimension? A Dimension is a Linear Condition which is oriented 90 degrees to all other Dimensions.... ...o around. Impetus does not just change stillness into an absolutely straight linear path of motion, it also changes stillness into a linear path th... ...all the Impetus of all the Energy in the Universe not only goes in a straight linear direction…. it also goes around. Science has tried to separate... ...petus is not only the basic force that makes things move in our Universe in a Linear path… it is also what makes things go around. The Basic Condit... ...Space: there have to be three energies: not two. In other words: A - c - B: is a balanced dynamic of Triality. Not: A - B There has to be...

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...hat a semigroup S is called an rpp semigroup if all its principal right ideals aS1, regarded as right S1-systems, are projective. According to J.B. Fountain[5], a semigroup S is rpp if and only if, for any a 2 S, the set Ma=fe 2 E j S1a µ Se and for all x; y 2 S1, ax = ay ) ex = eyg is a non-empty set, where E is the set of all idempotents of S. An rpp semigroup S is ...

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...X. Pan and B. Liu : On the irrational root sieve sequence 1 H. Liu : Sub-self-conformal sets 4 K. Liu : On mean values of an arithmetic function 12 L. Wang and Y. Liu : On an infinite series related to Hexagon-numbers 16 X. Ren, etc...

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...omic clocks: A rocket travels at a constant speed v with respect to the earth. In the rocket, a light pulse is emitted by a source from A to a mirror B that reflects it back to A where it is detected. The rocket’s movement and the light pulse’s movement are orthogonal. There is an observer in the rocket (the astronaut) and an observer on the earth. The trajectory of light ...

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...Relativistic Paradoxes in ATR: 55 3.4. Removing Lorentz Factor in ATR: 56 3.5. Physics Laws might not be the same in all Inertial Systems: 58 3.6. Linear Trajectories for Both Observers: 59 3.6.1. Orthogonal Trajectory Vectors: 54 3.6.2. Non-Orthogonal Trajectory Vectors and C as Ultimate Velocity: 59 3.6.3. Non-Orthogonal Trajectory Vectors and Arbitrary Ultimate Sp...

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...4 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of... ...alia. Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania. Dr. B.S. Kirangi, Department of Mathematics and Computer Science, Univers... ...r Science, University of Mysore, Karnataka, India. Copyright 2004 by W. B. Vasantha Kandasamy and Florentin Smarandache Layout by Kama K... ... resolution based on Archimedean triangular norms 40 1.6 Solving non-linear optimization problem with FRE constraints 48 1.7 Method... ...8 4 1.19 A FRE in dynamic fuzzy systems 113 1.20 Solving FRE with a linear objective function 117 1.21 Some properties of minimal sol... ...h deneutrofication algorithm for largest solution 268 4.9 Solving NRE with a linear objective function 269 4.10 Some properties of minimal soluti... ...ing of the book. Without his help this book would not have been possible. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE 9 Chapter One... ...ments of X are comparable by the ordering such an ordering is usually called a linear ordering, some alternative names used in the literature are t... ...A fuzzy weak ordering R is an ordering satisfying all the properties of a fuzzy linear ordering except anti-symmetry. Alternatively it can be though...

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...avirent in tcrram, ita terra viciffim ill car pora zquali tcr gravi tat ; b wavitaeis enim a&ionem ci‘li; mutuam & utrinquc t~qualem, iic oltenditur... ...s in- tervallis profrAi. feruntwr in riciniam folk9 & nonnuhquam adeo ad b ipfLlm proxime accedunt ut globum ejus, in periheliis fuis verfantes, ta... ...i ita vifiun fuilEt. Rationi enim confonum efi, ut a plu- ribus caufis, ;~b igvicem nonnihil diverfis, idem pofrit eEeAus pro- iicii‘ci: hzc autem v... ...m hi mote debent effk confcXmcs dPlatornm corporum motibus, q!li fimt fun-b me regulares, & peraguntur in fetilionihus conicis nunc valde eccetz- t... ... G ; dein accedant puntta 23, B, 6, ad punfia 4 d,g, iitque J interfc&io linearurn .L? G, AG ultimo faAa hi p~:lh Da B accedunt ufque ad A. Manifk... ...erit; 8r: prop- terea figura erit para- bola axem habens SH parallelurn linear T K, & inde dabitur. Quad ii corpus majori adhuc cum velocitate d... ...gem figura erit Q?T, perbola axem habens principalem Equalem differentiz linearumJ@F & TAT, & inde dabitur. Nam ii corpus in his cafibus ,~~voI$$r;... ...do cafu abeunte in infinitum ; in terrio ca- dente ad contrariam partem linear G F. Nam ii demittantur ad G FE perpendicula C f, fD I(; ” erit I... ...b duabus aliis T 5’ x TT, fe&io conica evadet cir- culus . Idem fret, ii linear quatuor ducantur in anguhs quibufvis, & re&anguIum fub duabus duei...

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...ive belief conditioning rules . . . . . 66 1.10 Conclusion . . . . . . . . . . 68 1.11 References . . . . . . . . . . . 68 Chapter 2 DSm field and linear algebra of refined labels 75 by F. Smarandache, J. Dezert and X. Li 2.1 Introduction . . . . . . . . . . 76 2.2 DSm field and linear algebra of refined labels . . . . 77 2.2.1 Qualitative operators on DSm-LARL . ....

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... APPLICATIONS OF BIMATRICES TO SOME FUZZY AND NEUTROSOPHIC MODELS W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of T... ...y, Anchorage, AK 99508, Alaska, USA. Copyright 2005 by Hexis, W. B. Vasantha Kandasamy, Florentin Smarandache and K.Ilanthenral ... ...s book to Prof. Dr. Bart Kosko, the father of Fuzzy Cognitive Maps. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K.ILANTHENRAL 7 ... ... define some of the basic operations on it. DEFINITION 1.1.1: A bimatrix A B is defined as the union of two rectangular array of numbers A 1 an... ...apply the bimatrices and their extensions to these models. 1. Abraham, R. Linear and Multilinear Algebra, W. A. Benjamin Inc., 1966. 2. Adams, ... ...g, USA. http://www.fuzzysys.com/ftaprod.html 43. Gel'fand, I.M. Lectures on linear algebra, Interscience, New York, 1961. 44. Georgopoulous, V.C.... ...Artificial Neural Networks, Espoo, Finland, 1245-1248, 1991. 46. Greub, W.H. Linear Algebra, Fourth Edition, Springer- Verlag, 1974. 47. Hadjiski, ... ...opics in Algebra, John Wiley, New York, 1975. 55. Hoffman, K. and Kunze, R. Linear algebra, Prentice Hall of India, New Delhi, 1991. 56. Hummel,... ...acob Chen and Anil Milal (eds.), 63-68, 1997. 65. Katsumi, N. Fundamentals of Linear Algebra, McGraw Hill, New York, 1966. 66. Kemeni, J. and J. Sn...

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...anifolds, which is what we will call them, follow a long tradition of piecewise linear approaches to, and avoidances of, the problems of the differen... ...e identified with the point Q on EF that is a distance x from E. In particular, B is identified with F. We will say that these disks share the edge ... ...ate them from the lines in the Euclidean plane. An s-line will be any piecewise linear curve that can be constructed from a line segment lying withi... ...he same total change in direction θ, but all of this change occurs at the point B. In this case, we have a curve with zero curvature everywhere exce... ...require that the integral of the curvature over any part of the path containing B must always be θ, and over any part not containing B, the integral... ...iangles around the elliptic vertex O, as in Figure 18. The turning angles at A, B, C, D, and E are all 60°, so the sum of the turning angles is 300°.... ...folds. Figure 3. The projection of s-lines from Figures 1 and 2. Locally linear projections To help visualize situations where s-lines may or... .... Our s-manifolds are a special case of a class of 2-manifolds called piecewise linear 2-manifolds. These correspond to the triangulations of 2-manif... ...e 17, 18, 19 interior band space 76 Lambert quadrilateral 23 lift 79 locally linear projections 73 Möbius band 77, 81, 84 multiply aligned 27 ...

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...n-1) + n –3 f DP (n) ≥ f DP (n-1) + n - 3. Also for each member (b 1 , b 2 , . . . , b n-1 ) of SDRPS(n-1) there exists a member of SD... ...(2, 2b 1 , 2b 2 , . . . ,2b n-1 ) as we can see that 1 = (1/2)(1+ 1/b 1 + 1/b 2 + . . .+ 1/b n-1 ) = 1/2 +1/2b 1 +. . . + 1/2b n-1 ... ... r r = ∑ [ ( П b t ) ( ∑ b s ) ] -1 b 1 b 2 . . . b r k=1 t=1... ... using the following example. Let A = { a 1 , a 2 , . . . , a r } and B = { b 1 , b 2 , . . . , b s } be two sets such that 19 r ... ...t 19 r s ∑ 1/a k = 1 = ∑ 1/b k k=1 k=1 with A ∩ B = ø. The conjectur... ... . . . p r a . 4) To derive a reduction formula for p a q a as a linear combination of p a-r q a-r for r =0 to a-1. Derive simi... ...1) + r * a (1,r) = 0 for r > 2, i.e. a (2,r) can be expressed as a linear combination of two elements of the first row, (other than the one o... ... a (2,r) = 0 for r > 2. Similarly, a (3,r) can be expressed as a linear combination of two elements of the second row of the type a (2,r) ... ...ed in the sense that the sequence accommodates all natural numbers as the linear combination of its terms with a finite set of coefficients.) A...

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...White(1969)], cumulative distribution function of which is given by : () Fy yu b yu b =− − −             −∞< <∞−∞< <∞ > 10 exp exp ; , , ... ...           −∞< <∞−∞< <∞ > 10 exp exp ; , , (1.2) where b = 1/β and u = ln α are respectively the scale and location parame... ... The inferential procedures of the above model are quite complex. Mann(1967 A,B, 1968 B) suggested the generalised least squares estimator using ... ...size n from Weibull distribution. Bain(1972) defined an unbiased estimator for b as b yy nK u im mn i m ∧ = − =− −           ∑ (,) 1 1 ... ...      −       = − ∑ 1 1 1 E , (1.4) 7 and v yu b i i = − are ordered variables from the extreme value distrib... ...n AFB, AD.653575, ARL-67-0023. MANN, N. R. (1967 B) : Table for obtaining Best Linear Invariant estimates of parameters of Weibull distribution, Tec... ...ored Sample, Jour. Indian Statist. Association, 24, 43-52. RAO, C. R. (1973) : Linear Statistical Inference and its Applications, Sankhya, Ser. B, 39... ...the lower bound of the variance of () g y M ˆ at (2.1) is the variance of the linear regression estimator () ( ) ( ) 1 2 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z Z X...

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...ncy to filter out the dissonance-causing information. (Bem, 1967). The non-linear feedback process could in effect be the force which keeps blowing u... ...perts. That reasoning that we advance has two basic components – a linear feedback process containing no looping and a non-linear feedback pr... ...triggers the rationalization loop and subsequently feeds it. 1.3 Non-linear Feedback Processes Generating a Rationalization Loop In a l... ...linear autoregressive model of order R, a time series y n is modeled as a linear combination of N earlier values in the time series, with an added c... ...bserved time series y n . The minimization process results in a system of linear equations for the coefficients a n , known as the Yule-Walker equat... ... ... (1.4) The autoregressive coefficients a j (j = 0 ... N) and b j, k (j, k = 1 ... N) are fit by minimizing the mean-squared difference... ...onlinear feedback, driven by a noise x n . In principle, the coefficients b j, k describes dynamical features that are not evident in the power spe... ...ures. Although the equations for the autoregressive coefficients a j and b j, k are linear, the estimates of these parameters are often unstable, ... ....): D Θ → [0, 1] may be defined associated with a given body of evidence B that can support paradoxical information as follows: ...

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...ded edge can be viewed as an 1-band which is defined as a topological space B together with a homeomorphism h :I×I →B, where I = [0,1], the unit interv... ...urface of genus q. Proof By a result in [8], AutM ∼ =N T (A)/A, WhereT =<a,b,c|a 2 =b 2 =c 2 = (ba) 2 = (ac) m = (cb) n = 1 >, A ≤ T and T can be real... ... the semi-arc automorphism groups, for example, Aut1 2 K n =S n but Aut1 2 B n =S n [S 2 ] = AutB n . Γ Aut1 2 Γ order K n S n n! K m,n (m =n) S m ×... ...t1 2 Γ order K n S n n! K m,n (m =n) S m ×S n m!n! K n,n S 2 [S n ] 2n! 2 B n S n [S 2 ] 2 n n! Dp n S 2 ×S n 2n! Dp k,l n (k =l) S 2 [S k ]×S n ×S ... ...the embedding sequence of the graph Γ. Define a functionF(x ¯ ,y ¯ ) to bey-linear if it can be represented as the following form F(x ¯ ,y ¯ ) =f(x 1 ,... ...x 2 ,···) Λ∈O Λ(y ¯ ), where, I denotes a subset of index and O a set of linear operators. Notice that f(x 1 ,x 2 ,···) =F(x ¯ ,0 ¯ ), where 0 ¯ = (... ...(Γ) |Aut1 2 Γ| g ¯ (Γ)) =0. Even more, assume the functionF(x ¯ ,y ¯ ) isy-linear and 2ε(Γ) |Aut 1 2 Γ| ,Γ∈H is a constant. If the embedding sequences... ...x ¯ , 2ε(Γ) |Aut1 2 Γ| g ¯ (Γ))= 0. Now assume that F H (x ¯ ,y ¯ ) is a y-linear function and has the following form F H (x ¯ ,y ¯ ) =f(x 1 ,x 2 ,···... ...,x 2 ,···) i∈I y i + l(x 1 ,x 2 ,···) Λ∈O Λ(y ¯ ), where O is a set of linear operators. If F H (x ¯ ,g ¯ (Γ))= 0, that is f(x 1 ,x 2 ,···) + h(x ...