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...dies on several topics of Geometry and Number theory. The material is divided into ve chapters: Geometric theorems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrationality results. Chapter 1 deals essentially with geometric inequalities for the remarkable elements of triangles or tetrahedrons. Other themes have ...

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...sef Sándor GEOMETRIC THEOREMS, DIOPHANTINE EQUATIONS, AND ARITHMETIC FUNCTIONS ************************************* AB/AC=(... ...A, ROMANIA Geometric Theorems, Diophantine Equations, and Arithmetic Functions Am... ...into ve chapters: Geometric the- orems; Diophantine equations; Arithmetic functions; Divisibility properties of numbers and functions; and Some irrat... ...le 20 may be considered also as an introduction to Chapter 3 on Arithmetic functions. Here many papers study the famous Smarandache function, the sour... ...he author has discovered various generalizations, extensions, or analogues functions. Other topics are connected to the com- position of arithmetic fu... ...ave remarked, r is always integer, since r = p = 3y = t 3 =u (in other words, in (10)u n represents the inradius of thenth CH-triangle). If one de... ... 2 (where x is the height corresponding to the base a), so q = x. In other words, if an isosceles triangles ABC is H-triangle, then its height x must ... ...+ 1 = bc. Let b = B with B > 0 and let c> 0. We get:a 2 +1 =Bc. In other words we must solve in positive integers the equation xy =z 2 + 1. The foll... ...(y x 2u) = u 2 + 1. By u = z x it results y x 2u = y +x 2z. In other words, (x 0 ;x 0 +y 0 2z 0 ;z 0 x 0 ) satis es the equation xy =z 2 + 1...

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...i-theses, anti- phrases, antagonism, non-conformism, the paradoxes in other words of anything (in literature, art, science), while futurism, cubism, ... ... - style of the non-style; - poems without verse (because poems don’t mean words) - dumb poems with loud voice; - poems without poems (because the ... ...uistic verse (only!): graphics, lyrical portraits, drawings, drafts… - non-words and non-sentence poems; - very upset free verse and trivial hermeti... ...s, the contradictions, antagonism, non-conformism, the paradoxes in other words of anything (in literature, art, science), while futurism, cubism,... ...in Craiova), în ordine cronologic ă: 1) An Introduction tu the Smarandache Function, de Charles Ashbacher, Vail, 1995, 60 p. 2) Collection of Probl... ...nd Problems, de Kenichiro Kashihara, Vail, 1996, 46 p. 4) The Smarandache Function, de C. Dumitrescu şi V. Seleacu, Vail, 1996, 134 p. 5) Surfing ... ...pton, 1998, 87 p. 8) Pluckings from the Tree of Smarandache Sequences and Functions, de Charles Ashbacher, Lupton, 1999, 87. 9) On Some of the Sma... ... de Krassimir Atanassov, Lupton, 1999, 88 p. 10) A Set of New Smarandache Functions, Sequences and Conjectures in Number Theory, de Felice Russo, L... ... ă cu American Research Press, sub forma unei reviste anuale: “Smarandache Function Journal”, ISSN 1053-4792, vol. 1-6 (1990-1995) şi “Smarandache N...

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...nd b and it is denoted by l.c.m or [a, b]. 8 DEFINITION 1.1.3: Any function whose domain is some subset of set of integers is called an arith... ...on whose domain is some subset of set of integers is called an arithmetic function. DEFINITION 1.1.4: An arithmetic function f(n) is said to be a ... ...INITION 1.1.4: An arithmetic function f(n) is said to be a multiplicative function if f(mn) = f(m) f(n) whenever (m, n) = 1. Notation: If x ∈ R (R... ...s of loops with a natural simple operation. As to introduce loops several functions or maps are defined satisfying some desired conditions we felt t... ...s on L as in the above theorem. Now construction of π can be put in other words as follows. Let the number of objects be 2n say 1, 2, … , 2n. From th... ... left, 20 right, 20 Anti-symmetric, 11 ARIF-loop, 50, 72 Arithmetic function, 8 Associative loop, 22, 35 Associative triple, 26, 70 Associa...

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...to: M. Popescu, P. Popescu, V. Seleacu, "About the behaviour of some new functions in the number theory" (to appear this year) . "Collection of Pr... ...ce The previous volume in this series, An Introduction to the Smarandache Function, also by Erhus Cniversity Press, dealt almost exclusively with so... ...ealt almost exclusively with some "basic" consequences of the Smarandache function. In this one, the universe of discourse has been expanded to incl... ...e Sequences In the time since it was first published[ 1], the Smarandache function SCm) = n, where n is the smallest number such that m divides n'. ... ...anches of mathematics. A previous volume in this series[2] introduced the function and explored some of the problems derived from the definition. I... ...ive integers M and k such that ~ SSC(x) - SSC(y). ::; Mx - yi k In other words, SSC(x) does not satisfY the Lipschitz condition for any exponent k ... ...tive integers M and k such that • SCC(x) - SCC(y) , :s; Mx _ yk In other words, SCC(x) does not satisfy the Lipschitz condition for any exponent k. ... ...numbers that can be constructed using steps I and 2 are in NCS2. In other words, NCS2(n+ 1) is the smallest number strictly greater than NCS2(n) that... .... 4) Only numbers formed by steps (1), (2) or (3) are in SS122. In other words, SS 122(n-:-l) is the smallest number strictly greater than SS 122(n)...

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...n act that requires several preconditions. I) An interesting and worthwhile subject; 2) A fair, yet demanding editor.; 3) Someone willing to put the words on paper. Given the existence of these three items, the leap to making the book a good one becomes the responsibility of the author. The Smarandache function is simultaneously a logical extension to earlier functions i...

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...amily of estimators of population mean using multiauxiliary information. This error minimization is vital in financial modeling whereby the objective function lies upon minimizing over-shooting and undershooting. This book has been designed for graduate students and researchers who are active in the area of estimation and data sampling applied in financial survey modeling ...

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.... This error minimization is vital in financial modeling whereby the objective function lies upon minimizing over-shooting and undershooting. Th... ...dified class of shrinkage estimators is proposed with its properties. Key Words & Phrases: Two-parameter Weibull distribution, Shape paramete... ...ample of size n from a two-parameter Weibull distribution, probability density function of which is given by : () () {} fx x x x ;, e x p / ; , , ... ...the log-Weibull distribution [e.g. White(1969)], cumulative distribution function of which is given by : () Fy yu b yu b =− − −        ... .../ (/) , (2.4) lies between 0 and 1, {i.e., 0 < w(p) ≤ 1} provided gamma functions exist, i.e., ) 2 / ( h p − > . Substituting (2.3) in (2.1) ... ... 11 The convex nature of the proposed statistic and the condition that gamma functions contained in w(p) exist, provides the criterion of choosing...

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...ibution 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 used in the geographical context, the term ‘migration’ refers to the ‘permanent or semipermanent change in the residence of an individual person or group of people.’ Traditional s...

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...fe-styles, employment, socioeconomic and political stability; or in other words, it influences the entire society. When used in the geographical con... .... This perceived and heightened sense of masculinity is garnered from the words the AIDS affected migrant workers use to describe the reasons for vi... ...ugh NGO intervention has given CSWs responsibility and alacrity in their functioning, the men from rural areas visiting them do not adopt safe-sex ... ...tudy it has become inevitable for us to use fuzzy theory, as these models function mainly based on the working of the neurons of the brain. The d... ...concept C j measures how much C i causes C j . The time varying concept function C i (t) measures the non-negative occurrence of some fuzzy event,... ... some disadvantages. The main advantage of this method: it is simple. It functions on expert’s opinion. When the data happens to be an unsupervised... ...r an ideal in life then he terms them as person for whom only the body functions and they have no work for the brain. P 11 – CSW. 59 P ... ... His wife was very angry, she does not talk to him and uses disrespectful words and ill-treats him. If the government helps his children it is suffi... ...to his wife. He was found to have HIV in Pattukottai Hospital. In his own words, he had “enjoyed” life in Mumbai. His friends are well and unaffec...

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... Chapter II deals with interesting alterating iterations of the Smarandache function and the Euler φ-function (the number of natural numbers less tha... ... divisor in common with n) An important question concerning the Smarandache function is resolved and an important link to the famous Fermat numbers ... ... of chapter II. It deals with the alternating iteration of the Smarandache function and the sum of divisors function (σ-function). Some light is th... ...ermany. Chapter VII. A k-k additive relationship involves the Smarandache function S(n) which is defined as the smallest integer such that S(n)! ... ...ed as the smallest integer such that S(n)! is divisible by n. A sequence of function values S(n), S(n+1)+ … +S(n+2k-1) satisfies a k-k additive rela... ...re rare – only 6 were found. Chapter IX. Concatenation is a sophisticated word for putting two words together to form one. The words book and mark... ... together to form one. The words book and mark are concatenated to form the word bookmark. Identical words like “abcd” are concatenated to form infin... ...mple like this abcdabcdabcdabcdabcd… and the properties of the extracted worddabcdabcda is then studied. The analysis of concatenations is appl... ...ven it is seen from (10) that α 1 , α 2 , … α j must all be even. In other words the smallest positive integer which can be represented as a sum of ...

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... A FRE in dynamic fuzzy systems 113 1.20 Solving FRE with a linear objective function 117 1.21 Some properties of minimal solution for a FRE 1... ...n algorithm for largest solution 268 4.9 Solving NRE with a linear objective function 269 4.10 Some properties of minimal solution for NRE 270 4... ...ies It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from X to Y, binary relations R(X, ... ...assign to each element of X two or more elements of Y. Some basic operations on functions such as the inverse and composition are applicable to bina... ...fuzzy relation R(X, Y), its domain is a fuzzy set on X, dom R, whose membership function is defined by dom (R(x)) = Y y∈ max R (x, y) for each... ... ∈ R, whenever both (x, y) ∈ R and (y, z) ∈ R for at least one y ∈ X. In other words the relation of x to y and of y to z imply the relation x to z... ... R ≥ [x] and is defined by R ≥[x] (y) = R (x, y) where y ∈ X. In other words the dominating class of x contains the members of X to the deg... ...x 1 , x 2 ) ∈ R implies (h(x 1 ), h(x 2 )) ∈ Q for all x 1 , x 2 ∈ X. In other words a homomorphism implies that for every two elements of the set ... ... 1 2 ) J ˆ = {j ∈ J | i x ˆ , a ij = b j , ∀ i ∈ I ˆ }. (13) In other words, J ˆ is a set of indices of constraints which can be satisfie...

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... A FRE in dynamic fuzzy systems 113 1.20 Solving FRE with a linear objective function 117 1.21 Some properties of minimal solution for a FRE 1... ...n algorithm for largest solution 268 4.9 Solving NRE with a linear objective function 269 4.10 Some properties of minimal solution for NRE 270 4... ...ies It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from X to Y, binary relations R(X, ... ...assign to each element of X two or more elements of Y. Some basic operations on functions such as the inverse and composition are applicable to bina... ...fuzzy relation R(X, Y), its domain is a fuzzy set on X, dom R, whose membership function is defined by dom (R(x)) = Y y∈ max R (x, y) for each... ... ∈ R, whenever both (x, y) ∈ R and (y, z) ∈ R for at least one y ∈ X. In other words the relation of x to y and of y to z imply the relation x to z... ... R ≥ [x] and is defined by R ≥[x] (y) = R (x, y) where y ∈ X. In other words the dominating class of x contains the members of X to the deg... ...x 1 , x 2 ) ∈ R implies (h(x 1 ), h(x 2 )) ∈ Q for all x 1 , x 2 ∈ X. In other words a homomorphism implies that for every two elements of the set ... ... 1 2 ) J ˆ = {j ∈ J | i x ˆ , a ij = b j , ∀ i ∈ I ˆ }. (13) In other words, J ˆ is a set of indices of constraints which can be satisfie...

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...ical extension, the interior of a phenomenon conquers its exterior. In other words, it belongs to itself also what does not belong to it.” F. SM... ... through a deliberate expressiveness, without avoiding (arghezian) the “hard” words, the “mould flowers” of spoken language etc. A genuine Fernand... ...tinomies, the anti- theses, antagonism, nonconformism, the paradoxes in other words of anything ( in literature, art, science), while futurism, cubi... ...ture. The style of non-style. Poems without verses. Poems without poems, non- words...”etc. It is easy to notice that Ion Rotaru (resumed by Gheorghe... ...ilian (alias Ion Barbu) asserted that these are “contiguous with the poetical functions which, approaching disjunctive elements through metaphor, un... ...ematician Florentin Smarandache: publication in USA from 1990 of “Smarandache Function Journal” review (since 1996- “Smarandache Notions Journal”); ... ...m Moldova) he obtained the title of doctor in mathematics with the thesis New functions in the theory of numbers; about the Smarandache function have... ...des between the mathematical notions some of Romanian origin, among that: the function Smarandache , the sequences (41) Smarandache, the constants S... ...n the new mathematician still has had his memory full of formula, algorithms, functions and ... mathematicians. The mentioned creation seems a kind o...

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...ant on many parameters, and therefore they can be considered set-valued vector functions or even operators. The parameters can be: time, space, etc.... ...changes with respect to the observer (subjectivity is another parameter of the functions/operators T, I, F). For example: “John is smart” can be (.3... ...ent sequence, calculating more and more terms, or by calculating the limit of a function successively substituting the argument with values closer an... ...61: "propositions may take many values beyond simple truth and falsity, values functionally determined by the values of their components"; Lukasiewi... ...I, F are subsets, but dynamically the components T, I, F are set-valued vector functions/operators depending on many parameters, such as: time, space... ... elementary (possibly all ) and zero to all possible disjunctions of . In other words there exists a bayesian bpa such that and Bel (24) 3.3 Plausibi... ...formative and can be used to help us to take legitimous final decision. In other words, this new theory can be interpreted as a general and direct exte... ...inimal credibility committed to and as the plausibility committed to . In other words, the Appriou’s transformation model within the DST is the follow... ...ence (i.e. value) committed to some elements of the hyper power set . In other words, the information one has to deal with is the reports: Report fo...

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... Table of Contents …………………. …………………… 4 Chapter 1 Smarandache Partition Functions ………………. 11 1.1 Smarandache Partition Sets, Sequences and Fun... ...Reciprocal Partition Theory ………………. 18 1.4 Generalization of Partition Functions, Introduction of the Smarandache Partition Function ………………. 1... ...rtition Theory ………………. 28 1.6 A General Result on the Smarandache Star Function ………………. 30 1.7 More Results and Applications of the Generalize... ... 1.7 More Results and Applications of the Generalized Smarandache Star Function ………………. 40 1.8 Properties of the Smarandache Star Triangle ……... ... A Few More Sequences ………………… 86 2.6 Some New Smarandache Sequences, Functions and Partitions ………………… 89 6 2.7 Smarandache Reverse Auto C... ... equal to one. He misunderstood the conjecture, perhaps due to inadequate wording. What he actually proved was the proposition that for every n ther... ...p 2 α2 . . . p r αr . . . p n αn and p r is the rth prime. In other words, p 1 = 2, p 2 = 3, and so forth. 2) For the case where N is a sq... ... F’ * (N) = Σ F’(d r ), where d r | N. d|N In other words, F’ * = sum of F’(d r ) over all the divisors of N. Example: N =... ...r those SFPs of N in which no element is a multiple of p 1 p 2 . In other words the SFPs in which the factors of p 1 a1 and p 2 a2 are isolated. I...

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...able will have 2 n rows. The result column of a table defines a Boolean function, named after George Boole, the mathematician who first described ... ...cian who first described many of the rules of logic. Therefore, a Boolean function is a function that assigns values of true and false to a set of v... ...es for every entry in the result column, there are 2 n different Boolean functions for n logical variables. Given the truth values in the colu... ... values in the column above the 5, a proposition that defines the Boolean function is p /\ q \/ (r /\ ¬ s )... ... proposition, it is possible to determine the truth values of the Boolean function it defines. 9 Note: For any Boolean function defined by a t... ...ns, where B is a consequence of the two hypotheses, A → B and A. In other words, if A → B and A are true, we can infer that B is also true. This rul... .... ,a k , b 1 , . . . , b j ├ p , where b I are also wffs. In other words, if we can prove a wff p using the wffs in a set, adding additional w... ... a friend of mine. This also requires an understandable definition of the word friend. 20 Definition 1.8.1: Words such as all, any, every, and ... ...cation theory. Example: In the statement All men are mortal The word “All” of course means that the expression holds for every value for th...

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...e V. If f is the characteristic polynomial for T, then f(T) = 0, in other words the minimal polynomial divides the characteristic polynomial for T.... ...roperties in case of complex fields. Here we recall the concepts of linear functionals, adjoint, unitary operators and normal operators. DEFINITI... ... field of reals and V be a vector space over F. An inner product on V is a function which assigns to each ordered pair of vectors α , β in V a scal... ...uadratic form. The quadratic form determined by the inner product is the function that assigns to each vector α the scalar || α|| 2 . Thus we cal... ...lt 1.1.9: Let V be a finite dimensional inner product space and f a linear functional on V. Then there exists a unique vector β in V such that f( α)... ...near biform, the biquadratic biform associated with f = f 1 ∪ f 2 is the function q = q 1 ∪ q 2 from V 1 ∪ V 2 onto F ∪ F defined by q ( α) = ... ...11 12 12 − − ∪ xx ρρ for all x = x 1 ∪ x 2 in G = G 1 ∪ G 2 . In other words a birepresentation of the bigroup G = G 1 ∪ G 2 on V = V 1 ∪ V 2... ... be obtained from the k 1 and k 2 messages in such a way that the bicode words x = x 1 ∪ x 2 satisfy the system of linear biequations. i . e ... ...ns. Hx T = TT 11 22 Hx Hx ∪ = (0) ∪ (0). There are 2 3 ∪ 2 4 bicode words given by {(0 0 0 0 0 0), (0 1 1 0 1 1), (1 1 0 1 1 0), (0 0 1 1 ...

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..., with each object being assigned a value between 0 and 1 by a membership function. Fuzzy set theory was guided by the assumption that classical set... ...1.1.1: Let X be a non-empty set. A fuzzy set (subset) µ of the set X is a function µ : X → [0, 1]. DEFINITION 1.1.2: Let µ be a fuzzy subset of a ... ...epts topologically or analytically like continuity, connected, increasing function or decreasing function. Just we proceed on to define when are two... ...unction or decreasing function. Just we proceed on to define when are two functions disjoint and the concept of min max functions. 10 DEFINITION... ...exist x ∈ X such that µ (x) = 1. DEFINITION 1.1.9: Let f : X → Y be a function. For a fuzzy set µ in Y, we define (f –1 (µ )) (x) = µ ... ... ( Y ) . We recall some definitions from [72]. DEFINITION 1.10.6: A word w ∈ X + is called unbordered if no proper non-empty left factor of w... ... if no proper non-empty left factor of w is a right factor of w. In other words, w is unbordered if and only if w ∈ uX + ∩ X + u implies u = 1. ... ...ROPOSITION 1.10.1: Let X be an alphabet with at least two letters for each word u ∈ X + , there exists v ∈ X + such that uv is unbordered. Proof:... .... Proof: Obvious by the very definition. 141 DEFINITION 1.10.9: A word which is not λ-completable for P is of course ) P ( F λ the set ) P ...

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...primary purpose of this conference is to closely examine those most damning words of Einstein's which nearly everyone accepted. Einstein is reporte... ...d by c to preserve the ratio to c nature of the arguments of the hyperbolic functions. Then to get the time coordinate the 1/g term in the t equati... ...els. But the whole theory is very inadequate, degenerating into the density functional method which is mostly an experimental method for finding gro... ...structure of space-forms give an alternative context with which to view the function spaces on which differential operators act, and from which solu... ...es on which differential operators act, and from which solutions are found. Function spaces have shown themselves to be much too large as spaces, so... ...on spaces have shown themselves to be much too large as spaces, so that the function space's spectral properties do not give any organizational stru... ...7 hertz, as the radioactivity's natural frequency. 76 Key words that can be used to express the main idea Coordinates - Used to ... ...itude of change; each view is real in the eye of the beholder.” Einstein's words may not have been exactly that, but it seems Einstein ignored phys... ...y of Sciences in Berlin on 27 Jan 1921, Einstein gave in; and in 1922 these words appeared in an obscure little book by Einstein (p.35):[3] "Sub sp...

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...as some disadvantages. The main advantage of this method it is simple. It functions on expert's opinion. When the data happens to be an unsupervised... ...concept C j measures how much C i causes C j . The time varying concept function C i (t) measures the non negative occurrence of some fuzzy event, ... ... 1 on the temporal associative memories recall process. Threshold signal functions synchronously update each concept after each pass, through the c... ...s("EVALUATE")) { /* Capture the number of rows in Matrix */ /* Call the function readandevaluate() to evaluate the values */ /* show the calculat... ... output O is again a vector of order 1 × n. Now we implement the threshold function – a binary conversion of the output vector. Thus we have O(x)... ...total effect of concept C 1 to C 7 is positive or negative, or in other words if the increase of the population of the city will increase or decre... ...uantity fuzzy sets (Q i ) and dis-quantity fuzzy sets( ~ Q i ). In other words, we assume that (FCR-2) is equivalent to (FCR-3) in stock market give... ...ely scholarly debate. Should judges render a decision based merely on the words of the text before them or should they go one step further and attem... ...s an uncertain process for it depends on the search engine; number of key words, sensitivity of search, seriousness of search etc. Hence we can see ...

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...yar, Pallar, Chakkiliyar refers to the Dalits. We have retained the original words in the translations. We feel that it was our lifetime duty to ... ...s some disadvantages. The main advantage of this method is, it is simple. It functions on expert's opinion. When the data happens to be an unsuperv... ... to concept C j measures how much C i causes C j . The time varying concept function C i (t) measures the non negative occurrence of some fuzzy ev... ... on the temporal associative memories, recall process. 16 Threshold signal functions synchronously update each concept after each pass, through t... ... to concept C j measures how much C i causes C j . The time varying concept function C i (t) measures the non negative occurrence of some fuzzy ev... ...s postulated as an alternative to Manudharma and the Varnashrama Dharma. This word gained currency when he founded the Self Respect Movement. 56... ...e people and continues to practice cruelties on them. In such conditions, the words ‘Independence’, ‘Swaraj’, ‘rights’ are words that will only giv... ...is conference just to say, “No need of hate; instead say all are equal.” The word ‘unity’ that is often repeated in our country, and the problems a... ...so. [Kudiarasu, 19-11-1936] 180 Untouchables Today, in this annual function, comrades U.P.A. Soundarapandian, N. Sivaraj, S.Gurusamy, T.N...

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... graphs. DEFINITION 2.7.8: An edge coloring of a loop less graph G is a function π: E (G) → S where S is a set of distinct colors: it is proper ... ...acent edges receive the same color. Thus a proper edge coloring π of G is a function π: E (G) → S such that π(e) ≠ π(e') whenever the edges e and e'... ... Graph Theory [5, 23, 25]. Let G be any graph, given a set of λ colors. The function f(G: λ) is defined to be the number of ways of coloring G prope... ...by example. DEFINITION 2.8.1: Let X and Y be two finite sets contrary to functions from X to Y, binary relations R (X, Y) may assign to each elem... ...ssign to each element X two or more elements of Y some basic operations on functions such as composition or inverse may also be applicable to binar... ...is an uncertain process for it depends on the search engine; number of key words, sensitivity of search, seriousness of search etc. Hence we can se...

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...nary bimatrices if the subbimatrices are conformable for addition. In other words the matrices can be added by blocks. Addition of bimatrices A B a... ... is a representation of a bimatrix. An m × n bimatrix over the field F is a function A from the sets of pairs of integers (i 1 , j 1 ), (i 2 , j 2 )... ...1 ∪ T 2 and U = U 1 ∪ U 2 be linear bitransformations of V in to W. The function T + U defined by [(T 1 ∪ T 2 ) + (U 1 ∪ U 2 )] α = [(T ... ...same. Now we proceed on to define invertible linear bitransformations. The function T = T 1 ∪ T 2 from V into W (where V and W are bivector space... ... = V 1 ∪ V 2 and W = W 1 ∪ W 2 ) is called invertible if there exists a function U = U 1 ∪ U 2 from W into V such that UT = U 1 T 1 ∪ U 2 T... ... ∪ U 2 from W into V such that UT = U 1 T 1 ∪ U 2 T 2 is the identity function i.e., UT = I 1 ∪ I 2 , where I 1 is the identity function on ...

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...on is a triple Y = (Z, A, δ) consisting of two non - empty sets Z and A and a function δ: Z × A → Z, Z is called the set of states, A the input alpha... ... Z, Z is called the set of states, A the input alphabet and δ the "next state function" of Y. 94 Let A = {a 1 , ..., a n } and Z = {z 1 , ..... ...– set of states and A – input alphabet. Let Z = {0, 1, 2} and A = {0, 1}. The function δ: Z × A → Z defined by δ (0, 1) = 1 = δ (2, 1) = δ (1, 1), δ ... ... is a non-empty set called the output alphabet and λ: Z × A → B is the output function. If z ∈ Z and a ∈ A, then we interpret δ (z, a) ∈ Z as the... ...quences of elements of the set A including the empty sequence Λ. In other words, in our study of automaton we extend the input set A to the free ... ...A and B A Z : → × λ where B is the free monoid generated by B. We also extend functions δ and λ from Z × A to Z × A by defining z ∈ Z and a 1 , ..., ...

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...e s-lines curve, but only in a direction perpendicular to the surface. In other words, the s-lines are straight within the surface, and they bend onl... ... containing B, the integral must be zero. These are the properties of an impulse function (see [5]), so we will call the measure of the turning angle... ...h x < 0 is found by traveling in the other direction. In terms of this distance function, the real numbers cover the s-line. In the case of a closed... ... will be parallel, if, and only if, the angles equal two right angles. In other words, given a line a and a point P not on a, there is exactly one l... ...other; and the point is called the centre of the circle.” Here, Euclid uses the word line in the sense that we would use curve. We defined the set o... ...m (b, a √3) to (b + 1, a √3) at (b + x, a √3) and at the same angle θ. In other words, the projection intersects this segment in exactly the same wa... ... in R n is a surface S in R n without self- intersections such that there is a function f: M → S that is one-to-one, onto, continuous, and has a co... ...nverse (i.e., f is a homeomorphism). We may refer to both the surface S and the function f as the embedding. It is relatively easy to imagine that th... ...dding, since any sort of stretching and bending is consistent with a continuous function. All of the orientable closed s-manifolds can be embedded t...

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...0 BT Arts, Modern – 20 th century 550 BT Arts, Modern – 21 th century The word “paradoxism” has entered in many languages: - paradoxism [in Roman... ...rface form but still there is some type of order in its spirit. Or in other words there is a method in this madness. 62 B. VENKATESWARA RAO (In... ... absurd. As a Paradoxist Distich is said to be a two-line poem, one content-word in each line is required at the least to form a distich. Just as a... ... Het „paradoxisme“ (een officiële Nederlandse term moet nog worden uitgevonden), is een avantgardistische beweging in de literatuur... ...ertegenwoordigt een „natuurlijk gedicht“, dat niet meer hoefde geschreven te worden, omdat het meer voelbaar en zichtbaar was in om het even welke ... ...Without it, there’s pure chaos. And I haven’t even gotten into the practical functions yet.” The professor laughed; this was the opening portion of ...

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...fraction of light speed. Those few lines contain too many (six) downer words and phrases to list, so they have been underlined instead. It is es... ... gaining popularity is that the distortions of special relativity, in the words of two mid 20th-century Cornell University physicists, F. K. Richtmye... ...alog of rocketship acceleration. For such an analysis in terms of Bessel functions see for example R. W. Landee et al, Electronic Designers' Handb... ...s. "What a blast!" you think. Especially Dr.Smith and the way he weasel- words himself into and then out of some tight situation or other. "Marvelo... ...ma and the Earth can be Huma 3A with the Moon being Huma 3B. That way the words "sun", "earth", "moon", & "solar system" can be used generically; al...