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 Book Id: WPLBN0002097645 Subjects: Non-Fiction, Education, Smarandache Collections Collections: Literature, Education, Authors Community, Mathematics ► Abstract Full Text Search Details...be a multiplicative function if f(mn) = f(m) f(n) whenever (m, n) = 1. Notation: If x ∈ R (R the set of reals), Then [x] denotes the largest integ... ... is left for the reader as an exercise to verify L is non-associative. Notation: Let L be a loop. The number of elements in L denoted by o(L) or |... ... Hence 2.4 = 3. n e 2 1 n - 1 1 6 7 5 2 3 4 e 33 Notation: Let L n denote the class of loops. L n (m) for fixed n and vario... ...A ⊂ L). L s = 〈{x ∈ A / x = (y, z) for some y, z ∈ A}〉 with the usual notation and if L has no S- subloops but L is a S-loop we replace A by L. ... ...L) i.e. L A = 〈{x ∈ A/ x = (a,b,c) for some a, b, c ∈ L}〉 with the usual notation 〈B〉 denotes the loop generated by the set B. If L has no S-subloo... ... Conf., 33-37 (1995). 76. Vasantha.W.B, Loop algebras of Moufang loops, Mathematical Forum, Vol. XI, 21-25 (1997). 77. Vasantha.W.B, On Quasi l...
 Book Id: WPLBN0002097045 ► Abstract Description Details...nitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matric... Full Text Search Details... by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem un... ...This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs,... ...12 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ n n mm m n aa a aa a aa a ‘ ∪’ is just the notational convenience (symbol) only. The above array is called a m ... ...c bimatrix). Just for the sake of clarity we give some illustrations. Notation: We in this book denote a neutrosophic bimatrix by A N = A 1... ... bimatrix). Just for the sake of clarity we give some illustration. 18 Notation: We denote a fuzzy bimatrix by A F = A 1 ∪ A 2 . Examp... ...We denote the combined FCM adjacency matrix by E = E 1 + E 2 + …+ E p . NOTATION: Suppose A = (a 1 , … , a n ) is a vector which is passed int... ...t will be easy for an expert to work with. He need not be a person with good mathematical background. For an expert may be a doctor or a lay man wh... ... of the NRM model we will proceed on to describe the concepts of it in a more mathematical way. DESCRIPTION OF A NRM: Neutrosophic Cognitive Ma... .... 58. Jacobson, N. Structure of Rings, Colloquium Publications, 37, American Mathematical Society, 1956. 59. Jang, S.R., C.T. Sun and E. Mizutani. ...
 Book Id: WPLBN0002097099 ► Abstract Description Details...This book arose out of a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man who seems to have no end of ideas. Most of the papers were published in Smarandache Notions Journal and there was a great deal of overlap. My intent in ... Full Text Search Details...f a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man w... ...ious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also l... ...f a collection of papers written by Amarnath Murthy. The papers deal with mathematical ideas derived from the work of Florentin Smarandache, a man w... ...ious errors. However, I made no attempt to verify every statement, so the mathematical work is almost exclusively that of Murthy. I would also l... ..., the proposition is also true for r+1. Therefore, by the principle of mathematical induction, the expression is true for all n. Note: If a 1 ,... ... N. If N is a square-free number with n prime factors, we will use the notation F’ ** (N) = F ** (1#N). Examples: ... ...the index equal to 1, hence by induction the proof is complete. In matrix notation if we write [ b 1 , b 2 , . . b n ] 1xn * [p i,j ] ' nxn = [... ... C 15 , . . . , The expressions of the sequences using the n C r notation suggests the following conjecture. Conjecture: The sequence ob... ..., 36, . . . Let Pd(n) denote the product of the digits of N. With this notation, we have 167 Pd(36) = 18, Pd(144) = 9 2916 = 54 2 . Pd(2916)...
 Date: 2013, Volume 9, Number 2 Book Id: WPLBN0002828492 Subjects: Non Fiction, Education, Algebra ► Abstract Description DetailsA short collection of mathematical papers from various authors relating to Florentin Smaradache's theories. Excerpt Details...hat follows, we could define also the additive analogues functions by the use of Euler's gamma function. However, we shall apply the more transparent notation of a factorial of a positive integer....
 Book Id: WPLBN0002097103 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Description Details...As an alternative to the existing logics we propose the Neutrosophic Logic to represent a mathematical model of uncertainty, vagueness, ambiguity, imprecision, undefined, unknown, incompleteness, inconsistency, redundancy, contradiction. It is a non-classical logic. Eksioglu (1999) explains some of them: “Imprecis... Full Text Search Details...veloped the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that rigorously defines the infinitesimals. Infor... ...native to the existing logics we propose the Neutrosophic Logic to represent a mathematical model of uncertainty, vagueness, ambiguity, imprecision, ... ... P., Upper and Lower Probabilities Induced by a Multivalued Mapping, Annals of Mathematical Statistics, 38, 325-339, 1967. [8] Dezert, J., Autonomo... ...alysis, Princeton University Press, Princeton, NJ, 1996. [37] Shafer, Glenn, A Mathematical Theory of Evidence, Princeton University Press, NJ, 1976... ...Western Section Meeting, Santa Barbara, CA, USA, Meeting # 951 of the American Mathematical Society, March 11-12, 2000, http://www.ams.org/amsmtgs/2... ...the information granules and as follows: and for any and , (31) The summation notation must be interpreted as the sum over all such that (interpre ... ...interpreted as the sum over all such that (interpre tation for other summation notations follows directly by analogy). The orthogonal sum is a proper... ...ng these numbers is exactly the Dedekind’s problem. The choice of letter in our notation to represent the hyper power set of is in honour of the great... ... characterized respectively by their bpa , and their cores , . Following Sun’s notation [66], each source of information will be denoted (42) (43) ...
 Book Id: WPLBN0002097643 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Description Details.... Except for the introduction of smarandacheian notions by Prof. Florentin Smarandache such types of studies would have been completely absent in the mathematical world. Thus, Smarandache groupoids, which are groupoids with a proper subset, which is a semigroup, has several interesting properties, which are defined and studied in this book in a sequential way. This book as... Full Text Search Details...in Smarandache such types of studies would have been completely absent in the mathematical world. Thus, Smarandache groupoids, which are groupoids... ...d a good foundation in number theory. In Chapter 1 we just recall the basic notations and some important definitions used in this book. In Chapter... ... enumerate the properties, which will be used in this book. As concerned with notations, the familiar symbols a > b, a ≥ b, |a|, a / b, b / a occur w... ...ivisible by a and b and it is denoted by l.c.m (a, b) or [a, b]. 8 Notation: 1. Z + is the set of positive integers. 2. Z + ∪ {0}... ... 3 + 6 ≡ 0 (mod 9), 3.3 ≡ 0 (mod 9), 2.8 ≡ 7 (mod 9), 6.2 ≡ 3 (mod 9). This notation will be used and Z n will denote the set of finite integers m... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 45 CHA... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 69 CHA... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol 1, 135-143, (2001). 4. W. B. Vasantha Kan... ... Kandasamy, New classes of finite groupoids using Z n , Varahmihir Journal of Mathematical Sciences, Vol. 1, 135-143, (2001). 4. W. B. Vasantha Ka...
 Book Id: WPLBN0002097104 ► Abstract Excerpt Details...Migration acquires great significance in the study of peoples and populations, for it not only involves the merely mathematical spatial redistribution of people, but also because it has enormous impact on livelihood, life-styles, employment, socioeconomic and political stability; or in other words, it influences the entire society. When u... Full Text Search Details...c logic grew as an alternative to the existing topics and it represents a mathematical model of uncertainty, vagueness, ambiguity, imprecision, unde... ...oor uneducated labourers from rural Tamil Nadu. This book is an extended mathematical analysis of the book Love.Life.Lust.Loss: 101 Real Life Stori... ...the study of peoples and populations, for it not only involves the merely mathematical spatial redistribution of people, but also because it has eno... ...f the literature on the said topic. We like to point out that most of the mathematical research into migration has dealt with it on purely statistic... ...tions simple and easy. 14 It is important to mention that the relevant mathematical notions used in each of these chapters are described in the ... ...enote the combined FCM adjacency matrix by E = E 1 + E 2 + … + E p . NOTATION: Suppose A = (a 1 , … , a n ) is a vector which is passed into a ... ...n which at least one edge is an indeterminacy denoted by dotted lines. NOTATION: The indeterminacy of an edge between two vertices will always be... ... adjacency neutrosophic matrix by N(E) = N(E 1 ) + N(E 2 )+…+ N(E p ). NOTATION: Let (a 1 , a 2 , … , a n ) and (a' 1 , a' 2 , … , a' n ) be two ...
 Date: 2013, Volume 1 Book Id: WPLBN0002828224 ► Abstract Table of Contents Details...ints . . . 68 4.4.8 Example 7 : hybrid DSm model with a mixed constraint . . . 69 4.5 DSm rule of combination for hybrid DSm models . . . 70 4.5.1 Notations . . . . . 70 4.5.2 Programming of the u (X ) function . . . 71 4.5.3 The hybrid DSm rule of combination for 2 sources . . . 73 4.5.4 The hybrid DSm rule of combination for k _ 2 sources . . . 74 4.5.5 On the as...
 Date: 2013, Volume 3 Book Id: WPLBN0002828228 ► Abstract Description Details... this third Volume, DSmT is applied to the entire spectrum of the Information Fusion that would interest any reader in data, sensor, information, and mathematical fusion topics. Highlighted in Figure 1 are the contemporary issues that include the links between (1) data conditioning and information management, (2) combined situation and impact assessment, and (2) knowledge ... Table of Contents Details... . . . . . . . 669 24.3.3 Ontologicmodel . . . . . . . . 670 24.4 VS-IMMwith road constraints (VS-IMMC) . . . . . 670 24.4.1 Track definitions and notations . . . . . . 670 24.4.2 IMM with only one road segment constraint . . . 671 24.4.3 Variation of the set of constrained motion models . . . 673 24.4.4 VS-IMMC within the SB-MHT . . . . . . 674 24.4.5 OOSMalgorithm...
 Book Id: WPLBN0002097121 Subjects: Non Fiction, Education, Smarandache Collections Collections: Technology, Literature, Education, Authors Community, Mathematics ► Abstract Full Text Search Details...ty related to function ¼(x) 114 A6. An arithmetic function 124 References 127 Chapter 1 On Some Smarandache's problems In the text below the followin... ...011111::: ::: 1111001 001111::: ::: 1110011100111::: ::: 1100111110011::: ::: 1 001111111001 ::: ::: 1100111110011::: ... ... Further, we will keep th... ...written in Problem 103, all of its elements are equal to 1. Hence, the values of the elements of the subsequent (second) square will be (using also th... ...function (¡1) !(s) uni- tary analogue of the MÄ obius function ¹(s) and denote this function by ¹ ¤ (s) (see [11, 19]). So, if we agree to use the las... ... Univ. Press, Glendale, 1994. [14] Grauert H., Lieb I., Fischer W, Di®erential- und Integralrech- nung, Springer-Verlag, Berlin, 1967. [15] Handbook o...
 Book Id: WPLBN0002097108 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...their properties It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from X to Y, binary re... ...es of binary relations such that R = P ° Q. We can then write using this matrix notations [r ij ] = [p ik ] ° [q kj ] where r ij = k max mi... ...n are replaced with the min and max operations respectively. We shall give the notational conventions. Consider three fuzzy binary relations P (X, ... ... solving p ° Q = r we need to introduce some additional concepts and convenient notation. First let ℘ denote the set of all possible vectors. p = {... ... 1) n = 2, 3,… where R is a fuzzy relation on X 2 and R (1) = R. Using this notation the reader is expected to prove the following theorem: T... ...ominated criterion vectors is called the non-dominated set. In the absence of a mathematical specification of the decision maker’s utility function,... ...ity to determine outputs for a given set of inputs without using conventional, mathematical models. FLC follows the general strategy of control wor... ... with sup-t composition nor adjoint equation. Therefore, simple control is not “mathematically correct”. However, solutions of fuzzy relational equa... ...of the fluid under study their by changing the flow rates while analyzing as a mathematical model. So in the study of flow rates in chemical plants...
 Book Id: WPLBN0002097105 ► Abstract Full Text Search Details...their properties It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from X to Y, binary re... ...es of binary relations such that R = P ° Q. We can then write using this matrix notations [r ij ] = [p ik ] ° [q kj ] where r ij = k max mi... ...n are replaced with the min and max operations respectively. We shall give the notational conventions. Consider three fuzzy binary relations P (X, ... ... solving p ° Q = r we need to introduce some additional concepts and convenient notation. First let ℘ denote the set of all possible vectors. p = {... ... 1) n = 2, 3,… where R is a fuzzy relation on X 2 and R (1) = R. Using this notation the reader is expected to prove the following theorem: T... ...ominated criterion vectors is called the non-dominated set. In the absence of a mathematical specification of the decision maker’s utility function,... ...ity to determine outputs for a given set of inputs without using conventional, mathematical models. FLC follows the general strategy of control wor... ... with sup-t composition nor adjoint equation. Therefore, simple control is not “mathematically correct”. However, solutions of fuzzy relational equa... ...of the fluid under study their by changing the flow rates while analyzing as a mathematical model. So in the study of flow rates in chemical plants...
 Book Id: WPLBN0002097090 Subjects: Non Fiction, Philosophy, Smarandache Collections ► Abstract Full Text Search Details... p and q are said to be logically equivalent if p ↔ q is a tautology. The notation for this relationship is p q. It is possible to show tha... ...( ¬p \/ ¬q ). The equals sign (=) is often used as an alternate notation for logically equivalent. The statements in theorem 1.3.1 are... ...n infer that B is also true. This rule is commonly abbreviated MP and the notation used is A → B, A . B Def... ... rules of the system to conclude that another statement is also true. The notation for such an inference is Ґ ├ p. Generally, we write a 1 , a 2 ... ...hat the expression holds for every value for the variable men. The formal mathematical term is “for all” and it is known as the universal quantifier... ...ion. Therefore, the expression 21 (Ex) x > z is mathematically equivalent. Once the values of the elements in the unive... ...three-valued logic is an extension of classical logic. In most cases, new mathematical structures are defined so that they are consistent with those... ...= min{ f 1 , f 2 , . . ., f k , f k+1 } Therefore, by the principle of mathematical induction, the formula is true for all n. Corollary: If ... ... A \/ B being an absolute contradiction. Therefore, by the principle of mathematical induction, we have reached the desired conclusion. The t...
 Book Id: WPLBN0002097060 Subjects: Non Fiction, Philosophy, Art ► Abstract Full Text Search Details...same time, the neutrosophic logic value NL(paradox) = (1, i, 1), but this notation cannot be used in determining the fuzzy logic value FL(paradox), ... ...ed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, which rigorously defines the infinitesimals. Informall... ...n's Paradox: Let M be a mathematician who may not be characterized by his mathematical work. a) To be a mathematician, M should have some mathematic... ...ork. 31 b) The reverse judgement: if M may not be characterized by his mathematical work, then M is not a mathematician. Divine Paradox (I): ... ...false", where S(n) is a sentence relating on parameter n. Can we prove by mathematical induction that P(n) is true? d) " is true if and only if <... ...., "A Practical Guide to Cooperative Learning in Collegiate Mathematics", Mathematical Association of America, Washington, D.C., 1995. ] The m... ...oenfinkel, Haskell Curry, 1920s) is a system for reducing the operational notation of logic, mathematics, or functional language to a sequence of mo... ...cture. Temporal Logic is an extension of Predicate Calculus that includes notation for arguing about when (at what time) statements are true, and em... ...inacy-, and false-neutrosophic connectors. Let's reconsider the previous notations. 108 2.12. Neutrosophic Topology. A) Definitions: - Le...
 Book Id: WPLBN0002097093 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...A 2 ;C and A: c 2 A 2 C a AA 2 2 +b 2 A 2 B =A 2 B A 2 C a; with the notation AA 2 =s a , the following formula can be deduced: s 2 a = b 2 c 2 ... ...since f( ) 0, let x = X. Then 1 y = 1 z + 1 X , which is the same, with other notations, as the equation in the title. We shall use, among others the two... ... 1 x 2 + 1 y 2 = 1 z 2 in integers, these Selected papers. 5. Ch.W. Trigg, Mathematical Quickies, New York, 1968. 58 2 On the equation 1 x 2 + 1 y 2 ... ...i surprize ^ n matematic a , Ed. Enc. Rom^ an a, 1975; English edition: Mathematical Quickies, New York, 1968. 4. B. Margit, Problem E:11248, Mat. ...
 Book Id: WPLBN0002097101 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...c logic grew as an alternative to the existing logics and it represents a mathematical model of uncertainty, vagueness, ambiguity, imprecision, undef... ...eural networks, and they have powerful and far-reaching consequences as a mathematical tool for modeling complex systems. Prof. Bart Kosko, the guru... ...ct us almost as much as the determined factors. It is a major handicap in mathematical modeling that we are only able to give weightages for known c... ...denote the combined FCM adjacency matrix by E = E 1 + E 2 + …+ E p . NOTATION: Suppose A = (a 1 , … , a n ) is a vector which is passed into a d... .... Before defining the cognitive structure of the relationship, we give notations to the concepts involved in the analysis as below. C 1 - Fr... ...tial version of FCM matrix can be built as in Table 1. The description of notations is given below: • • : Market Share +1 : Positive cause-and... ...ng issues are naturally solved, since Time is explicitly expressed in the mathematical equations used to describe the relations between concepts. Ho... ...Only the modifier is used. However, its meaning can be represented in the mathematical expression as (Q i ∪ ~ Q i ) ∩ M i which contains the qua... ...(r) = {Affects, Requires, Multiple_relationship, Stop-relationship} The notation used for each type of relationship (causal) is shown in the follow...
 Book Id: WPLBN0002097100 ► Abstract Full Text Search Details...s degree 1 or 4. Jordan in 1869 independently discovered trees as a purely mathematical discipline and Sylvester 1882 wrote that Jordan did so with... ...ice. Before defining the cognitive structure of the relationship, we give notations to the concepts involved in the analysis as below. a b c ... ... in which at least one edge is an indeterminacy denoted by dotted lines. NOTATION: The indeterminacy of an edge between two vertices will always ... ... 3.3.1: The graph given in the figure is a strongly neutrosophic graph. NOTATION: V N (G) will denote the vertices of the neutrosophic graph wit... ...atrices of binary relations such that R = P ° Q. We write this using matrix notation [r ij ] = [p ik ] o [q kj ] where r ij = min max k (p ik , q... ...e reader. 3.8 Neutrosophic Models versus Fuzzy models We know in all mathematical analysis of the unsupervised data, not only uncertainty is ... ...s predominant but also the concept of indeterminacy is in abundance. But in mathematical logic no one used the concept of indeterminacy till the yea... ...sophic logic. So whenever we introduce the notion of indeterminacy in our mathematical analysis we name the structure as Neutrosophic structure. V... ...bility). He contributed with proposed problems and solutions to the Student Mathematical Competitions. His latest interest is in information fusion ...
 Book Id: WPLBN0002097651 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details... fuzzy algebraic structures. 1.1 Fuzzy Subsets In 1965 Zadeh [144] mathematically formulated the fuzzy subset concept. He defined fuzzy subse... ... to verify that µ o θ is the smallest fuzzy ideal of R containing µθ . Notation: At times we will make use of this notation also. Let A (µ ) be an... ... Proof: The reader is requested to refer [40]. We will also using the notation for A ∗ and A ∗ obtain some interesting results about fuzzy sub... ... (s + A) = r s + A. Then R/A is a commutative ring and R/A = R/A ∗ . Notation: Let R α be a subring of R and let A be a fuzzy ideal of R. Let A... ... a fuzzy ideal of R. 55 Proof: Left for the reader as an exercise. Notation. II is a non-empty subset of [0, 1] i.e. we use II as the index se... ...φ 2 : φ | R 2 is a map from R 2 to S 2 is a ring homomorphism. We for notational convenience denote by φ = φ 1 ∪ φ 2 though this union ‘∪’ is ... ...e of fuzzy polynomial near-rings cannot say the number of roots in a nice mathematical terminology that is itself fuzzy. 113 A study of these f... ...494 (1999). 88. MCKOY, N.H., Rings and Ideals, Carus Math. Monograms, 18, Mathematical Association of America. 89. MEIYAPPAN D., Studies on fuzzy ...
 Book Id: WPLBN0002097119 ► Abstract Full Text Search Details...is peer reviewed and included in the abstracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sw... ...ard assumptions are incorrect, which I shall demonstrate with the required mathematical rigour. Contrary to the usual practise, one cannot talk about ... ... 0 =0 in (18) purely and simply by its invalid initial assumptions, not by mathematical imperative. It incor- rectly assumes p C n (r)≡R p (r)≡r, then... ...ly, the Kruskal-Szekeres formulation is meanin- gless, both physically and mathematically. In addition, the so-called “Schwarzschild radius” (not due ... ...as long ago as 1917, by T. Levi-Civita [20], on the grounds that, from the mathematical standpoint, it lacks the invariant character actually required... ...ure (−+++), which gives an advantage in certain cases. They also use other notations for tensor indices: in their book space-time indices are Latin, w... ... = 1 c ∂ϕ ∂t +ϕD + ∇ i q i − 1 c 2 F i q i , (44) where we use his notation for chr.inv.-divergence ∇ i q i = ∂q i ∂x i +q i ∂ln √ h ∂x ... ... thex 3 axis. In integrating equations (8.19) and (8.20), we introduce the notation ∂η 1 ∂t ≡x and ∂η 2 ∂t ≡y. Then we have ˙ x−2ωy =0 ˙ y+2ωx=0 ) . (... ...feld equations) and the so called ADM formalism [7] where(3+1)-dimensional notation has led to the defnition of gravitational Hamiltonian and time as ...
 Book Id: WPLBN0002097644 Subjects: Non-Fiction, Education, Smarandache Collections Collections: Math, Literature, Education, Algebra, Authors Community, Mathematics ► Abstract Full Text Search Details... 1.1.9.) It is left for the reader to find two sided ideals of S(3). 9 Notation: Throughout this book S(n) will denote the semigroup of mappings... ...ed the supremum of B and is denoted by sup B. Now with these notions and notations we define a semilattice. DEFINITION 1.2.5: A poset (L, ≤) i... ...ult). Supplementary Reading 1. Birkhoff, G. Lattice Theory. American Mathematical Society, Providence, R.I., 1967. 2. Birkhoff, G. and Ba... ...g of level II we propose the following problems for the reader to solve. Notation: A semiring of all types in level two will shortly be denoted by... ...swer? Supplementary Reading 1. Birkhoff, G. Lattice Theory. American Mathematical Society, Providence, R.I., 1967. 2. Birkhoff, G. and Ba... ...ve one among researchers. Except for Smarandache notions such rich type of mathematical analysis would be completely absent in the realm of mathemat...
 Book Id: WPLBN0002097650 Subjects: Non-Fiction, Education, Smarandache Collections Collections: Technology, Literature, Education, Logic, Authors Community, Mathematics ► Abstract Full Text Search Details...Smarandache loops (S-loops) for the sake of completeness and also for our notational convenience, as we would be using these notions and notations i... ...t a pair of elements a, b ∈ G, then we call G a non-commutative group. Notation: Let X = {x 1 , x 2 , … , x n }, the set of all one to one mapping... ...is by S n called the symmetric group of degree n. We will adhere to this notation and the order of S n is n!. D 2n will denote the dihedral group ... ...deal can be had from any textbook on algebra [22, 26, 27, 30, 31, 32]. Notation: Let X = {a 1 , …, a n } where a i 's are distinct, that is |X| = ... ...unit element if any and from A. These are S-ring I, but by default of notation we just denote it as S-ring. Example 1.4.1: Let F[x] be a polyn... ...c-Graw Hill, New York, (1970). 4. BIRKHOFF, G. Lattice Theory, American Mathematical Society, Providence, R.I., (1967). 5. BRUCK, R. H., A survey... ...ern Limited, (1984). 30. JACOBSON, Nathan, Structure of rings, American Mathematical Society, (1956). 31. JACOBSON, Nathan, Theory of rings, Am... ...tical Society, (1956). 31. JACOBSON, Nathan, Theory of rings, American Mathematical Society, (1943). 32. LANG, S., Algebra, Addison Wesley, (19...
 Book Id: WPLBN0002097647 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details... good knowledge in algebra for the reader, we give only some definitions, notations and results with the main motivation to make the book self- conta... ...        2 1 3 3 2 1 respectively. We would be using mainly this notation. DEFINITION 1.1.4: Let (G, o) be a group. H a non-empty subset... ...4 , p 5 } given in example 1.1.5 is a normal subgroup of S 3 . 10 Notation: Let S n be the symmetric group of degree n. Then for n ≥ 5, each... ...nvertible elements. We give some problems for the reader to solve. Notation: Throughout this book S(n) will denote the set of all mapping of a... ....) be a ring, if (R \ {0}, . ) is an abelian group we call R a field. Notation: Z – denotes the set of integers positive, negative and zero. Q – ... ...e only S-pseudo ideals and no S-ideals I or S- ideals II. 29. Describe mathematically those group rings which are not a. S-simple rings I. b... ..., Vol. 116, 208- 226, (1988). 38. Jacobson.N, Theory of rings, American Mathematical Society, (1943). 39. Jacobson.N, Structure of ring, America... ...thematical Society, (1943). 39. Jacobson.N, Structure of ring, American Mathematical Society, (1956). 40. Jin Xiang Xin, Nil generalized Hamilto...
 Book Id: WPLBN0002097084 Subjects: Non Fiction, Mathematics, Finances ► Abstract Full Text Search Details...e probability of complete financial ruin for the trader but no meaningful mathematical expression can model this case under the assumption of logari... ...so been theoretical advances in the esoteric area of econophysics whereby mathematical analogs have been proposed and substantiated between utility ... ... logic based on behavioral reasoning, supported in most parts by rigorous mathematical exposition or numerical simulation and sensitivity analysis. T... ...isk classification is proposed – neutrosophic logic being a new branch of mathematical logic which allows for a three-way generalization of binary f... ...ogue – options-based portfolio insurance. In this chapter we propose and mathematically prove a theorem which we have named the “theorem of consiste... ...] / [1 - Σ A∩B = φ m 1 (A) m 2 (B)] ... (1.9) The summation notation Σ A∩B=C is necessarily interpreted as the sum over all A, B ⊆ Θ ...
 Book Id: WPLBN0002097038 Subjects: Non Fiction, Education, Smarandache Collections Collections: Math, Literature, Education, Authors Community, Mathematics ► Abstract Full Text Search Details...42 where SPS is an acronym for Smarandache Permutation Sequence. Note on notation: As a general principle, acronyms will be created for all sequence... ...ces and the page number where the sequence first appears. For purposes of notation the acronym will be used to denote the entire set and the acronym... ...es n and pH does not. = 0 if P does not divide n. Definition 17: Use the notation TPp(n) where p is prime to denote the function n TPp(n) = L SEp(... ... time, we will attempt to resolve the apparent conflicts. For purposes of notation, let SPl" denote the set of Sma rand ache Paradoxist 2'.'umbers. ... ...his result does not violate any principles of number theory or any of the notation used here. For example, the statement 68 The set of integers t... ...yor may not have a solution, depending on how the terms are defined. In a mathematical sense, the word assertion is used to refer to a statement that... ...t consequence of Godel's Theorem that there are questions for which it is mathematically impossible to form an answer Therefore, the answer to the f...
 Book Id: WPLBN0002097043 Subjects: Non Fiction, Algebra, Smarandache Collections ► Abstract Full Text Search Details... may be part of a number of distinct vectors. 9 We simply by default of notation just say V a vector space over the field F and call elements of ... ...omic models. 1.3.1 Smarandache Markov chains Suppose a physical or a mathematical system is such that at any moment it can occupy one of a fin... ..., y 5 ) where (y 1 , y 2 , y 3 , y 4, y 5 ) ∈ V 2 . Thus we follow the notation. Notation: Let V = V 1 ∪ V 2 be the bivector space over the... ... last co ordinates are non zero. ii. T = T 1 ∪ T 2 ‘ ∪’ is just only a notational convenience. iii. T( ν) = T 1 ( ν) if ν ∈ V 1 and T ( ν) ... ... Note that in 〈 , 〉 = 〈 , 〉 1 ∪ 〈 , 〉 2 the ‘ ∪’ is just a conventional notation by default. DEFINITION 2.1.6: Let V = V 1 ∪ V 2 be a bivecto... ...ers of a set introduced, we have studied the theory of directed graphs to mathematically model the types of sets and relation. On similar lines we ... ... and relation. On similar lines we can introduce the directed bigraphs to mathematically model the types of sets and relation. Example 3.5.1: The... ...ich can be attached to some or all of the entities of a system of study. A mathematical bicode is a bicode with some mathematical structure. In many... ...de with some mathematical structure. In many cases as in case of codes the mathematical structure of a bicodes refers to the fact that it forms a bi...
 Book Id: WPLBN0002097044 Collections: Math, Literature, Algebra, Authors Community, Mathematics ► Abstract Full Text Search Details... ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ n n mm m n aa a aa a aa a ‘ ∪’ is just the notational convenience (symbol) only. The above array is called a m by... ...m! (n – m)! choices for 11 1 22 2 12 12 , ,..., , , ,..., mm jj jjj j . The notation 12 ... < << rr r m jj j (r = 1, 2) indicates that the sum is t... ...4 , y 5 ) where (y 1 , y 2 , y 3 , y 4, y 5 ) ∈ V 2 . Thus we follow the notation. Notation: Let V = V 1 ∪ V 2 be the bivector space over th... ...e last co ordinates are non zero. ii. T = T 1 ∪ T 2 ‘ ∪’ is just only a notational convenience. iii. T( ν) = T 1 ( ν) if ν ∈ V 1 and T ( ν)... ... Note that in 〈 , 〉 = 〈 , 〉 1 ∪ 〈 , 〉 2 the ‘ ∪’ is just a conventional notation by default. DEFINITION 2.1.5: Let V = V 1 ∪ V 2 be a bivect... ...16. JACOBSON, N., Structure of Rings, Colloquium Publications, 37, American Mathematical Society, 1956. 17. JOHNSON, T., New spectral theorem for v... ...bility). He contributed with proposed problems and solutions to the Student Mathematical Competitions. His latest interest is in information fusion ...
 Book Id: WPLBN0002097118 Subjects: Non Fiction, Education, Smarandache Collection ► Abstract Full Text Search Details...is peer reviewed and included in the abstracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sw... ...d of gravitational force, which provides two important advantages: (1) The mathematical apparatus of chronometric invariants (physical observable quan... ...d its specifc technical problems. 2 The new approach The basis here is the mathematical apparatus of chronometric invariants, created by Zelmanov in t... ... J. Crothers for his editing this paper. References 1. Eddington A. S. The mathematical theory of relativity. Cambridge University Press, Cambridge, 1... ...y supposed the existence of antimatter in 1928 which resulted from Dirac’s mathematical equation, and fnally the antimatter was discovered/produced in... ...n area, volume, hypervolume units or Clifford-bits. In compact multi-index notation X=X M Γ M one de- notes for each one of the components of the targ... ...in their works on general relativity employ a mathematical formulation and notation which is much simpler from that of purely mathematical or mathemat... ...derivative”. In modern, math- ematically oriented literature more explicit notation such as D γμ or ∇ γμ is used. However, such a notation, although m... ...ponents vanish, i. e., when μ=0,S μν =0 andπ μ =0. This justifes using the notation m for mass. This is basically the distinction between the mass in ...
 Book Id: WPLBN0002097122 ► Abstract Full Text Search Details...the problem. What we need is the fact that any equation written in tensor notation, correct in one frame, is also correct in any other frame of re... ...484, Russia heya@aha.ru; ptemp_online@yahoo.com Abstract: Employing the mathematical apparatus of chronometric invariants (physical observable q... ...isometry (metric preserving transformations) subgroup's lattice structure. (Mathematically, space-forms are defined on spaces of constant curvature)... ...ithin which it is contained. New energy sources from alternative uses of mathematical patterns to describe the physical world There is a beli... ...e maximal torus. Fiber - Above each point in a coordinate space, various mathematical structures can be defined. Such structures are called fiber... ...called a metric state. Godel's consistency and completeness theorems - A mathematical theory which says that there are always mathematical patter...
 Book Id: WPLBN0002097083 Subjects: Non Fiction, Education, Mathematics ► Abstract Full Text Search Details... Zt n = ¥ ﬁ 2 ) ln( )) ( ln( lim Is this limit convergent to some known mathematical constant? Problem 49. Solve the functional equation: . 2 int... ... an infinite number of prime numbers [7] Theorem 1.2.17 We will use the notation FZw(n)=m to denote, as already done for the Zt(n) function, that m... ...k Zw n = ¥ ﬁ 2 ) ln( )) ( ln( lim Is this limit convergent to some known mathematical constant? Problem 24. Solve the functional equation: 2 int )... ...art with some problems related to the Sdf(n) function. Problem 1. Use the notation FSdf(n)=m to denote, as already done for the Zt(n) and Zw(n) funct... ... n k n = ¥ ﬁ 2 ) ln( )) ( ln( lim Is this limit convergent to some known mathematical constant? Problem 25. Solve the functional equation: n n Sd...
 Book Id: WPLBN0002096589 Subjects: Non Fiction, Education, Algebra Collections: Math, Literature, Education, Algebra, Authors Community, Mathematics ► Abstract Full Text Search Details...roups. It is pertinent to mention that Smarandache notions on all algebraic and mathematical structures are interesting to the world of mathematicia... ...s in groups and Smarandache semigroups. In Chapter one, we introduce some basic notation, Binary relations, mappings and the concept of semigroup an... ...of us, for we have been considering mappings from the very earliest days of our mathematical training. DEFINITION: If S and T are nonempty sets, ... ... ∈ M. We shall say that t is the image of s under the mapping. Now for some notation for these things. Let σ be a mapping from S to T; we often ... ...groups are the only algebraic structures with a single binary operation that is mathematically so perfect that an introduction of a richer structure... ... 17 If no such integer exists we say that a is of infinite order. We use the notation o(a) for the order of a. DEFINITION: A subgroup N of a ... ...ch are subgroups of S(3), are given by the following tables using the following notation,         = 3 2 1 3 2 1 1 ,         = 2 3...
 Book Id: WPLBN0002097649 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...op of L, L s = 〈 {x ∈ A / x = (y, x) for some y, z ∈ A} 〉 with the usual notation and if L has no S-subloops but L is a S-loop we replace A by L. ... ... ∪ x)} this will be called as the supermodular identity for simplicity of notation we replace ‘ ∪’ by ‘+’ and ‘ ∩’ by ‘.’; thus the identity reads a... ... to A if xy ∈ I implies x ∈ I or y ∈ I. The study of groupoid rings in mathematical literature is completely absent so the study of these ideals ... ...zing its algebraic aspects and we do not see its properties via any other mathematical structure. All the more we do not even deal it as a matrix str... ...zed class of non-associative algebras does not find its just place in the mathematical world. Alternative algebras has been intensely studied by [24... ...-247 (1987). 28. GOODAIRE, Edgar G., Alternative loop rings, Publications Mathematical Tomus, 30, 31-38 (1983). 29. GOODAIRE, Edgar G., Groups embe...
 Book Id: WPLBN0002097095 ► Abstract Full Text Search Details...neral manifolds, we know that Riemann surfaces have become a source of the mathematical creative power. Many good ideas for the manifolds with higher ... ..., in fact, any solution for one of these problems needs to establish a new mathematical system ﬁrst. But as soon as the system has been established, t... ...withloopsormultipleedgesandgroupsareﬁniteinthecontext. For terminology and notation not deﬁned in this book can be seen in [33],[34] and [35] for grap... ... if its underlying graph is a complete graph. For a connected graph Γ, the notationsE O (Γ),E N (Γ) andE L (Γ) denote the embed- dings of Γ on the ori... ...K.Polthier and M.Schmies, Straightest geodesics on polyhedral surfaces, in Mathematical Visualization(ed. byH.C.HegeandK.Polthier),Springer-Verlag, Be...
 Book Id: WPLBN0002096873 Subjects: Non Fiction, Social Sciences, Technology ► Abstract Full Text Search Details...ills of analysis and matching that are so critical to developing logical, mathematical, and scientific thinking. How phonetic writing came into bei... ...rld trade soaring. Bankers were major beneficiaries, because this new mathematical tool let them dispense with their counting boards. The ne... ...ecried Archimedes‘s failure two millennia earlier to adopt the convenient notation for the powers of ten built into the zero. Gauss asked sorrowfull... ... Then, as a twenty-two-year-old working on his PhD, Kemeny served as a mathematical assistant to Dr. Albert Einstein. (Asked why the famed physici...
 Book Id: WPLBN0002097024 Subjects: Non Fiction, Technology, History of Technology ► Abstract Full Text Search Details...ills of analysis and matching that are so critical to developing logical, mathematical, and scientific thinking. How phonetic writing came into bei... ...world trade soaring. Bankers were major beneficiaries, because this new mathematical tool let them dispense with their counting boards. The new ... ...ecried Archimedes‘s failure two millennia earlier to adopt the convenient notation for the powers of ten built into the zero. Gauss asked sorrowfull... .... Then, as a twenty-two-year-old working on his PhD, Kemeny served as a mathematical assistant to Dr. Albert Einstein. (Asked why the famed physici...
 Book Id: WPLBN0002097117 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details... India. The fourth and concluding chapter gives observations drawn from our mathematical results based on the Fuzzy and Neutrosophic analysis. A... ... A study of this kind is being carried out for the first time. We have used mathematical tools to study this sociological problem because we wante... ...We denote the combined FCM adjacency matrix by E = E 1 + E 2 + …+ E p . Notation: Suppose A = (a 1 , … , a n ) is a vector which is passed int... ...NCMs adjacency neutrosophic matrix by N(E) = N(E 1 ) + N(E 2 )+…+ N(E p ). NOTATION: Let (a 1 , a 2 , … , a n ) and (a' 1 , a' 2 , … , a' n ) be t... ... of the NRM model we will proceed on to describe the concepts of it in a more mathematical way. DESCRIPTION OF A NRM: Neutrosophic Cognitive Ma... ...ps (FCMs) and Neutrosophic Cognitive Maps (NCMs). For the first time, we use mathematical models to analyze untouchability. We specifically use Fuz... ...ave refrained from giving our views since the sole purpose of this study is a mathematical analysis of untouchability. 43 2.1 Analysis of un...
 Book Id: WPLBN0002118018 Subjects: Non Fiction, Reference, Style Manual ► Abstract Full Text Search Details...ers. In syllabi, and matter of a similar character, the following scheme of notation and in- dention of subdivisions should ordinarily be adhered t... ...s-3: 12. 254. In formulae, and elsewhere, put a thin space on each side of mathematical signs. Between letters forming products, and before superio... ...be set in roman, 52; use of capitals in, 38. "Master," compounds of, 178. Mathematical signs, spacing of, 254. Measures, metric, how to designate,...
 Book Id: WPLBN0002118089 ► Abstract Full Text Search Details...It results in a reduction of the principal balance of the loan by an established mathematical formula. Usually, by the end of the term of the loan,... ... with the original owner and runs to the present owner. Chains of title include notations of deeds, judgments of distribution from estates, certifi...
 Book Id: WPLBN0002096541 Subjects: Non Fiction, Philosophy, Education ► Abstract Full Text Search Details...ence.” Cloux Certainly a bird is an instrument performing according to mathematical laws which are within the capacity of man to understand. How ... ...y day. There was so much to sing about. I played my lira da braccio. Made notations. As a VOICES FROM THE PAST 236 boy, I thought seriously o... ...ndrea dead at fifty-three! Di me se mai...four words...scattered among my mathematical papers, among my drawings: Is anything ever done! I was twen...
 Book Id: WPLBN0000678920 Subjects: Fiction, Literature, Literature & drama ► Abstract Full Text Search Details...d like to know what greater com- pulsion there can be for any thing than a mathematical cer- tainty. I studied math—” “V ery true, Richard,” interrupt... ...hich, on his return, were regularly transferred to the journal with proper notations of the time, manner, and other little particulars. There was, to ...
 Book Id: WPLBN0000678436 Subjects: Fiction, Literature, Literature & drama ► Abstract Full Text Search Details...mparison with the apparatus of modern math- ematical method. And while the mathematical subjects of the quadrivium were taught as science and for thei... ...and minima, as it is now to be able to read and write. This development of mathematical teaching is only another aspect of the neces- sity that is bri... ...eneral scheme that is the real obstacle to any material improvement in our mathematical teaching. Professor Perry, in his open- ing address to the Eng... ...e pretence in teaching geom- etry, that algebraic formulae and the decimal notation are not yet invented, little boys of nine may be got to apply quad... ...irmly into the teachers’ hands. The problem is much simpler in the case of mathematical teaching than in the case of English, because the educational ... ...t and trial I have suggested for the organization of English teaching. The mathematical reformer may begin now at a point the En- glish language refor...
 Book Id: WPLBN0000660472 ► Abstract Full Text Search Details...ng to Froebel, the actuating force is the presentation of symbols, largely mathematical, corresponding to the essential traits of the Absolute. When t... ...to Froebel, the mystic symbolic value of certain objects and acts (largely mathematical) stand for the Absolute Whole which is in process of unfolding... ... has been led to foresee (infer) a future eclipse. No matter how great the mathematical probability, the inference is hypothetical—a matter of probabi... ...onstituent parts of this strange world. Probably the most conspicuous con- notation of the word knowledge for most persons to-day 195 John Dewey is j... ...In practice, overemphasis upon formed material leads to an exaggeration of mathematical qualities, since intellect finds its profit in physical things... ...ases, of geographical study which tend to become specialized and separate. Mathematical or astronomical, physiographic, topographic, political, commer...
 Book Id: WPLBN0000660463 ► Abstract Full Text Search Details...rst proposing, more particularly by and by. I shall here only, and that very readily, allow, that these maxims and mathematical demonstrations are in ... ...ss it be this,—that the mind is capable of under- standing and assenting firmly to such propositions. And thus all mathematical demonstrations, as wel... ...other propositions, in mathematics itself, which are as clear and evident as this; because, in his search of those mathematical truths, he stopped his... ... inadequate. So that all our com- plex ideas of substances are imperfect and inadequate. Which would be so also in mathematical figures, if we were to... ...is the proper original signification of the word, as is evident from the formation of it; essentia, in its primary notation, signifying properly, bein... ...thers there be who extend this abuse yet further, who take so little care to lay by words, which, in their primary notation have scarce any clear and ... ...that name. The names of substances, if they be used in them as they should, can no more disturb moral than they do mathematical discourses; where, if ... ...h makes us presume things to be true, before we know them to be so. Probability is likeliness to be true, the very notation of the word sig- nifying s...
 Book Id: WPLBN0000661154 Subjects: Fiction, Literature, Literature & drama ► Abstract Full Text Search Details...f Homer and Hesiod. NOTE: Greek words at times appear in ALL CAPS. In text notations are in red and surrounded by parentheses and hyperlinked to the c... ... many Achaeans went to Ilium with the sons of Atreus?’ Homer answered in a mathematical problem, thus: ‘There were fifty hearths, and at each hearth w...
 Book Id: WPLBN0000655832 Subjects: Fiction, Literature, Literature & drama ► Abstract Full Text Search Details... German, he enrolled at the University of Bonn in a program for technical, mathematical, and scientific studies. In June 1997, he requested a leave fr... ...apartment assumed as a center for the group, as he would write on them the notation “Dar el Ansar,” or “house of the followers.” 82 In addition to Att...
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