... ISBN: 1-931233-48-9 Standard Address Number 297-5092 Printed in the United States of America 4 Dedication to a nation with so muc... ...: 1-931233-48-9 Standard Address Number 297-5092 Printed in the United States of America 4 Dedication to a nation with so much relig... ...ational one, is deemed as an element of the space−time. 4.The fundamental law governing spatio-temporality and its elements is the permanent and sto... ...the spatio-temporality. The absence of such a motion would contradict the law of permanent motion in spatio- temporality, so this hypothesis is nece... ...mporality, so this hypothesis is necessarily imposed, ensuring a relative, law-like shifting from the left to the right of any entity. Furthermore, t... ...his great divine work. In the ″History of Ancient Thought″, Athanase Joja states that <.… the theory of reminiscence is purely Platonic. The idea th... ...ews crossed one of the shallow lakes of the area. The Santorin hypothesis states that following the extremely powerful eruption of the volcano beari... ...l necessity of humans of feeling protected against the unknown, and to be united in truth and righteousness with his fellow-beings. Identifying the...

... ISBN: 1-931233-63-2 Standard Address Number: 297-5092 PRINTED IN THE UNITED STATES OF AMERICA 3 CONTENTS Preface 5 1... ... 1-931233-63-2 Standard Address Number: 297-5092 PRINTED IN THE UNITED STATES OF AMERICA 3 CONTENTS Preface 5 1. Gene... ... property). 2. a, b, c ∈ G implies a • (b • c) = (a • b) • c (associative law). 3. There exist an element e ∈ G such that a • e = e • a = a for all ... ...tional condition (L 4 ) x ∩ x = x, x ∪ x = x. (L 1 ) is the commutative law (L 2 ) is the associative law (L 3 ) is the absorption law and (L 4... ...sociative law (L 3 ) is the absorption law and (L 4 ) is the idempotent law. DEFINITION 1.3.7: A partial order relation ≤ on A is called a total... ...of the classical Cayley's theorem for groups to S-loops. Cayley's theorem states "Every group is isomorphic to a subgroup of S n for some appropriat...

... ISBN: 1-931233-62-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Printed in the United States of America C... ...N: 1-931233-62-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Printed in the United States of America CONTENTS... ...ure). 2. For all a, b, c ∈ S we have (a ∗ b) ∗ c = a ∗ (b ∗ c) (associative law), We denote by (S, ∗) the semigroup. DEFINITION 1.1.2: If in a se... ...ure property) 2. a, b, c ∈ G implies a • (b • c) = (a • b) • c (associative law) 3. There exists an element e ∈ G such that a • e = e • a = a for al... ...he additional condition L 4 . x ∩ x = x, x ∪ x = x. L 1 is the commutative law, L 2 is the associative law, L 3 is the absorption law and L 4 is... ...uence simple finite semiring. Using the classical theorem of Cayley which states, "Every group is isomorphic to a subgroup of A(S) for some approp...

...931233-71-3 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface Chapter One: INTRODUC... ...71-3 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface Chapter One: INTRODUCTORY CO... ...3 ∈ N; (n 1 + n 2 ) n 3 = n 1 n 3 + n 2 n 3 (right distributive law). This near-ring will be termed as right near-ring. If n 1 (n 2 ... ...ents of N are left cancelable we say that N fulfils the left cancellation law. Similarly one defines right cancelable. N is an integral domain if N ... ... ∪ A 2 and a function δ : Z × (A 1 ∪ A 2 ) → Z, Z is called the set of states and A = A 1 ∪ A 2 the input alphabet and δ the next state function... ...njoy varied properties. Hence we see in A = A 1 ∪ A 2 , A 1 may take states in a varied way and A 2 in some other distinct way. So when we tak... ...t) cancellable we say that N fulfils the left (right) 196 cancellation law. N is a bi-integral domain if both N 1 and N 1 has no zero-divisors. ...

...ress ISBN 1-879585-50-2 Standard Address Number 297-5092 Printed in the United States of America Preface The previous volume in this series, An ... ...SBN 1-879585-50-2 Standard Address Number 297-5092 Printed in the United States of America Preface The previous volume in this series, An Introdu... ... result. However, the well-known result of Godel's Incompleteness Theorem states that in any system powerful enough to perform arithmetic, there wil...

... ISBN: 1-931233-59-4 Standard Address Number: 297-5092 Printed in the United States of America 3 ... ... ISBN: 1-931233-59-4 Standard Address Number: 297-5092 Printed in the United States of America 3 Prefac... ...τ o δ) that is the composition of mappings in general satisfies the associative law. Further if σ, τ : S → S is one to one then σ o τ and τ o σ ar... ...  b ∈ G (closed) 2. a, b, c ∈ G implies a  (b  c) = (a  b)  c (associative law) 3. There exists an element e ∈ G such that a  e = e  a = a fo... ... may be several such b's). Thus b  (a  x) = b  (a  y) using the associative law this leads to x = e  x = (b  a)  x = b  (a  x) = b  (a  y... ...Z(G), the center of G, then N(a) ≠ G. Recall, too, that the class equation of G states that ∑ = )) a ( N ( o ) G ( o ) G ( o where the sum runs ove... ...semigroups. 5.1 Lagrange's Theorem for S-semigroups Lagrange's Theorem states that if G is a finite group having a subgroup H then o(H)/o(G...

... ISBN: 1-931233-46-2 Standard Address Number 297-5092 Printed in the United States of America 4 G-DIMENSIONAL THEORY & THE SMARA... ...BN: 1-931233-46-2 Standard Address Number 297-5092 Printed in the United States of America 4 G-DIMENSIONAL THEORY & THE SMARANDACHE ... ...rojection. Each bit in the computer's logic system has two possible quantum states (on/off); yet the computer as a whole has a near-infinite number ... ...on/off); yet the computer as a whole has a near-infinite number of possible states. This then serves an example of apparent non- projection of quantu... ...d thus has conjunctive motion with the body. We now suspend Newton's first law of motion, Principia, (1687), as a given principle and treat it inst... ... o = 0: - ϒ = mvsec(θ) /t 2 . (15) From Newton's second law of motion, the resultant acceleration (A), of the body is then, A ... ...s in that quanta. This definition is in complete accord with Newton's first law of motion and moreover provides a causal definition for inertial moti... ...omic and Molecular Processes in Plasmas," funding request: DOE Grant 01-24, United States Department of Energy, Washington D.C., (7-13- 01). [51] On...

... Manifesto of the PARADOXISM is especially a revolt of the emigrant to the United States who doesn’t speak English, against the language - an anti- l... ...sto of the PARADOXISM is especially a revolt of the emigrant to the United States who doesn’t speak English, against the language - an anti- language... ...poems; - poems in Pirissanorench (language spoken in the South-West of the United States by a single person); - super-poems; - graphic poems; - u... ... - poems in Pirissanorench (language spoken in the South-West of the United States by a single person); - super-poems; - graphic poems; - upset-po... ...ublished Quelques Paradoxes {Some Paradoxes), Canada, 1993. Emigrating to United States in March 1990, I continued to spread the paradoxist movemen... ...d Quelques Paradoxes {Some Paradoxes), Canada, 1993. Emigrating to United States in March 1990, I continued to spread the paradoxist movement to En...

... ISBN: 1-931233-64-0 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 ... ... 1-931233-64-0 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 1. P... ...correcting this book. I dedicate this book to be my beloved mother-in-law Mrs. Salagramam Alamelu Ammal, whose only son, an activist-writer and ... ...e concept of lattices as we have a well known result in ring theory which states that “the set of all two sided ideals of a ring form a modular latt... ... ∪ (x ∩ x)) = x lead to x ∩ x = x and x ∪ x = x. L 1 is the commutative law, L 2 is the associative law, L 3 is the absorption law, and L 4 is ... ...e associative law, L 3 is the absorption law, and L 4 is the idempotent law. 16 DEFINITION 1.3.7: A lattice L is called modular if for all ...

... ISBN: 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 PART ONE 1. ... ... 1-931233-74-8 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 PART ONE 1. SOME RE... ... a –1 )(x) = µ (a –1 xa) for every x ∈ G. By using earlier theorems which states o(aµ a –1 ) = o(µ ) for any a ∈ G. For the sake of simplicity an... ...t (right) cancelable we say that N fulfills the left (right) cancellation law. N is a bi-integral domain if both N 1 and N 2 has no zero divisors. ... ...erly contained in 〈S 〉; as in 〈S 〉 we may or may not have the associative law to be true. Remark: Even (ab)c ≠ a(bc) in general for all a, b, c ∈ ... ... - empty sets Z and A and a function δ: Z × A → Z, Z is called the set of states, A the input alphabet and δ the "next state function" of Y. 4...

... ISBN: 1-931233-61-6 Standard Address Number: 297-5092 Printed in the United States of America 3 Pre... ...SBN: 1-931233-61-6 Standard Address Number: 297-5092 Printed in the United States of America 3 Preface 5... ...osure). 2. For all a, b, c ∈ S we have a  (b  c) = (a  b)  c (associative law). (S, ) is a semigroup. DEFINITION: If in a semigroup (S, ) w... ...s properly contained in <S>; as in <S> we may or may not have the associative law to be true. Remark: Even (ab) c ≠ a (bc) in general for all a, b... ... non - empty sets Z and A and a function δ: Z × A → Z, Z is called the set of states, A the input alphabet and δ the "next state function" of Y. ... ... ) = z r . This graph is called the state graph. Example 6.1.1: Z – set of states and A – input alphabet. Let Z = {0, 1, 2} and A = {0, 1}. The f... ... Alternative, 17, 20, 40-44, 56, 62, 65, 91, 92, 106-108 Associative Law, 11, 93 Automaton, 88, 93-96 Binary operation, 8, 11, 16, 25, 3...

... ISBN: 1-931233-51-9 Standard Address Number 297-5092 Printed in the United States of America "...It is just this, which gives the higher ar... ...: 1-931233-51-9 Standard Address Number 297-5092 Printed in the United States of America "...It is just this, which gives the higher arithmeti... ...podaire triangle A 0 B 0 C 0 . Smarandache’s Podaire theorem [2] (see [1]) states that X a 0 b 0 1 4 X a 2 (1) wherea;b;c are the sides of the trian... ...lt a +b +c 3 p 3R is another source for inequalities. For example, by the law of sinus, a = 2R sinA, etc., we get the classical trigonometric inequal... ...rprising properties do appear. For example, the well-known Viviani theorem states that if an interior point P is projected to the sides BC;AC;AB in P ...

...-931233-69-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 1. BAS... ...-69-1 Standard Address Number: 297-5092 Printed in the United States of America 3 CONTENTS Preface 5 1. BASIC CONC... ... (closure axiom) 2. a. (b.c) = (a.b).c for all a, b, c ∈ L. (associative law) (S, .) is called the semigroup. The semigroup in general may not ha... ...b) . c for some a, b, c ∈ R. 3. a. (b + c) = a.b + a.c (The distributive law are true) for all a, b, c, ∈ R. Let R be a non-assoiciative ring. R ... ...is said to be associative if its multiplication satisfies the associative law that is (xy) z = x (yz). 92 A non-associative algebra L is a Lie ...

... ISBN: 1-931233-60-8 Standard Address Number: 297-5092 Printed in the United States of America 3 Foreword As someone who works heav... ... 1-931233-60-8 Standard Address Number: 297-5092 Printed in the United States of America 3 Foreword As someone who works heavily in ... ...g else” it is then possible to avoid the restrictions forced on us by the law of the excluded middle. Chapter three describes the rules of fuzzy log... ...apter 1 Classical Logic 7 Section 1 Propositions Section 2 The Law of the Excluded Middle Section 3 Logical Equivalence Section ... ... as well as the negations of all the tautologies. Section 2 The Law of the Excluded Middle While propositions can be used in many diff... ...in the modal frame. Example: The major league baseball teams in the United States are split up into two leagues, the National and American. At... ...modal frame. Example: The major league baseball teams in the United States are split up into two leagues, the National and American. At the en... ... for tiny particles such as electrons to possess one of two possible spin states, “up” and “down.” They are placed in quotes because the spin proper...