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...ition to give a class of loops which are as concrete as the groups of the form S n , D 2n etc. which makes the study of these non- associative struc... ... have just given the definition of lattices and its properties to see the form of the collection of subgroups in loops, subloops in loops and normal... ...ps in loops. The subgroups in case of Smarandache loops in general do not form a modular lattice. Almost all the proofs of the theorem are given ... ... the reader. DEFINITION 1.2.1: A non-empty set of elements G is said to form a group if on G is defined a binary operation, called the product and... ... commutative if every proper subloop of L is commutative but they are not cyclic groups. DEFINITION 2.4.9: Let L be a loop, L is said to be semi r... ...such that i • i = e for all i ∈ L n (m) that is every element generates a cyclic group of order two; now it is left for the reader to prove that no ... ...struction of L n (m) we see L n (m) is a S-loop every element generates a cyclic group of order 2. Thus we have a class of loops L n which are S- su... ...). Clearly L 11 (3) is a S-Cauchy loop as every subgroup of L 11 (3) is a cyclic group of order two and 2/|L 11 (3)| as the order of L 11 (3) is 12.... ... loop etc. and study them. Further we introduce concepts like Smarandache cyclic loop, Smarandache pseudo commutative loops and characterize them. W...

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...s poetry is his biography. But the latter assertion comes to eliminate the former and, the problem of choice is no longer a problem. However, when wou... ...Paradoxist Movement". The editorial debut was made by the volume of poems "Formule pentru spirit" (" Formulae for the Spirit" ), Bucharest, Editura " ... ...roon. Radiographing his volumes of poetry, suggestively entitled Nonpoems, Formulae for the Spirit, The Sense of the non­sense, and others, we discove... ...The haiku micropoem, of Japanese origin, is a species of poetry with fixed form, which is characterized by three groups of rules: Prozodical, structur... ...ike the emptiness of a container that obliges the matter to take its inner form, or like the space between the spokes of a wheel, that makes the wheel... ...des and the acids of the words used by Alecsandri and working with modern, cyclic complexes of the language, so coming without any difficulty to the e...

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... I allowed any boldness: - anti-literature and its literature; - flexible forms fixed, or the alive face of the death! - style of the non-style; -... ...r used, but very concrete at the same time: non-restrictive verse from any form or content. It takes advantage of cliche against itself. 10 EVER... ...florescendo! E 12 nós fazemos escárnio da criação, do senso inverso, de forma sincrética. Ent āo o paradoxismo nascia. As piadas do povo, uma... ...O “n āo” e “anti” do meu manifesto paradoxista tem um caráter criativo, de forma alguma niilista (C. M. Popa). A passagem dos paradoxos para o parado... ...mas. Eu permiti qualquer adácia: - anti-literatura e sua literatura; - formas fixas flexiveis, ou a viva face da morte! - estilo do não-estilis... .... And again they come back to life with a “neo” prefix art, because art is cyclic. Let`s catch the paradoxism in art – an avante-garde movement I ...

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...ind many epigrams of this period—some by Callimachus—directed against the “cyclic” poets, by whom were meant at that time those who were always draggi... ...re well summarized by Professor Robinson Ellis as fol lows: “Precision in form and metre, refinement in dic tion, a learning often degenerating into... ...r among strangers he might lose his home return. (ll. 18 22) The ship, as former bards relate, Argus wrought by the guidance of Athena. But now I wil... ...e sang how the earth, the heaven and the sea, once mingled together in one form, after deadly strife were separated each from other; and how the stars... ... should escape. And fishermen dragged him to shore at the island of Oenoe, formerly Oenoe, but afterwards called Sicinus from Sicinus, whom the water ... ...behold the various paths of that sea; and they brought their ship from its former anchorage to the harbour, Chytus; and the path they trod is named th...

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... I allowed any boldness: - anti-literature and its literature; - flexible forms fixed, or the alive face of the death! - style of the non-style; -... ...r used, but very concrete at the same time: non-restrictive verse from any form or content. It takes advantage of cliche against itself. 10 EVER... ...florescendo! E 12 nós fazemos escárnio da criação, do senso inverso, de forma sincrética. Ent āo o paradoxismo nascia. As piadas do povo, uma... ...O “n āo” e “anti” do meu manifesto paradoxista tem um caráter criativo, de forma alguma niilista (C. M. Popa). A passagem dos paradoxos para o parado... ...mas. Eu permiti qualquer adácia: - anti-literatura e sua literatura; - formas fixas flexiveis, ou a viva face da morte! - estilo do não-estilis... .... And again they come back to life with a “neo” prefix art, because art is cyclic. Let`s catch the paradoxism in art – an avante-garde movement I ...

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...he concept of a mapping precise for us but we shall almost never use it in this form. Instead we do prefer to think of a mapping as a rule which ass... ...d on to define a group. DEFINITION: A non empty set of elements G is said to form a group if in G there is defined a binary operation, called the... ...of a group G is said to be a subgroup of G if, under the product in G, H itself forms a group. The following remark is clear; if H is a subgroup o... ... conjugate class of a in G; it contains the set of all distinct elements of the form y -1 ay as y ranges over G. DEFINITION: If a ∈ G, then N(a), ... ...s o(a). The element a itself furnishes us with this subgroup by considering the cyclic subgroup generated by a that is, (a) of G; (a) consists of e,... ... yet 0 < j − i < o(a) which would contradict the very meaning of o(a). Thus the cyclic subgroup generated by a has o(a) elements, whence, by Lagrang... ...for groups having a single element. If G has no subgroups H ≠ (e), G, must be cyclic of prime order. This prime must be p, and G certainly has p-1... ...he commutative semigroup, Smarandache weakly commutative semigroup, Smarandache cyclic and weakly cyclic semigroups. DEFINITION: The Smarandache... ...TION: Let S be S-semigroup if every proper subset A of S which is a subgroup is cyclic then we say S is a Smarandache cyclic semigroup. DEFINITIO...

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... the author will be part of your experience. For the reading ofa book is a form of mind- joining, where the author tries to create the opportunity for... ...equence: 12,1342.135642, 13578642, 1357910864~ 135791112108642,. Or as a formula 5 SPS(n) = 135 (2n-3)(2n-l )(2n)(2n-2)(2n-4) 642 where SPS is a... ... k = SPS(n) With the hint The last digit must be 2 for exponents of the form 4k + I or 8 for exponents of the form 4k + 3. And the final statemen... ...c. 9 1 modulo 10 = 9 9 2 modulo 10 = I 9 3 modulo 10 = 9 etc. This cyclic behavior is typical and provides a powerful proof mechanism when the... ...he equation m k = SPS(n) 7 are when m terminates in 2 and k is of the form 4k + I or m terminates in 8 and k is of the form 4k -'- 3. The searc... ...onclusion of the theorem, assume that such an M exists Choose Pk > M + 1. Form the square number created by the product of all primes less than or e...

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... Dr. A. R. T. Solarin, Dept. of Mathematics, Obafemi Awolowo University, (formerly University of Ife), Ile-Ife, Nigeria. Copyright 2... ...union of two subgroups, or two subrings, or two subsemigroups etc. do not form any algebraic structure but all of them find a nice bialgebraic struc... ... and future reference. DEFINITION 1.1.1: A non-empty set G, is said to form a group if in G there is defined a binary operation, called the produ... ...-1 , ab, ab 2 , …, ab n-1 }. |D 2n | = 8 2n. G = 〈g |g n = 1 〉 is the cyclic group of order n; i.e. G = {1, g, g 2 , …, g n-1 }. A n will denot... ...r properties about them. DEFINITION 1.1.2: A non-empty set L is said to form a loop if in L is defined a binary operation called product and denot... ...ative (S-weakly commutative). If every proper subset which is subgroup is cyclic then we call S a Smarandache cyclic semigroup (S-cyclic semigroup).... ... cyclic semigroup (S-cyclic semigroup). If atleast one of the subgroup is cyclic we call the semigroup Smarandache weakly cyclic (S-weakly cyclic). ... ... a S-pseudo ideal I related to a field A, A ⊂ R is said to be Smarandache cyclic pseudo ideal (S-cyclic pseudo ideal) related to A if I can be gener... ... also their Smarandache analogue. The concept of Smarandache commutative, cyclic, weakly commutative and weakly cyclic are defined in the near-ring....

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...ortly say as semirings and its generalizations), which are not found in the form of text. Secondly, to define Smarandache semirings, semifields and ... ...elf. The set S n together with the composition of mappings as an operation forms a non-commutative group. This group will be addressed in this book ... ...oup the group ring RG of the group G over the ring R consists of all finite formal sums of the form ∑ α i i i g (i runs over finite number) where α... ... Example 1.5.1: Q be the field of rationals and G = 〈g / g 2 = 1 〉 be the cyclic group. The group ring QG = {a + bg | a, b ∈ Q and g ∈ G} is a com... ...2, 3, 4} be the prime field of characteristic 5. G = 〈g | g 12 =1 〉 be the cyclic group of order 12. The group ring Z 5 G is a commutative ring wit... ...4, 1989. 14. Vasantha Kandasamy, W.B. Idempotents in the group ring of a cyclic group. Vikram Math. Journal, Vol. X, 59-73, 1990. 15. Vasantha... ... indeterminate. The polynomial semiring S[x] consists of all symbols of the form s o + s 1 x + …+ s n x n where n can be any non-negative integer ... ...iring. Example 2.5.2: Let Q o be the semiring. G = 〈g/ g n = 1 〉 be the cyclic group of order n. Clearly Q o G is the group semiring of infinite... ...h it. Example 2.5.4: Let S be the semiring C 3 . G = 〈g 〉 be the infinite cyclic group. SG is an infinite commutative semiring and has no characte...

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... by Abuse Victim reaction to Abuse By Narcissists and Psychopaths The Three Forms of Closure Surviving the Narcissist Mourning the Narcissist The Inv... ...tic Personality Disorder (NPD)? The Narcissistic Personality Disorder (NPD) (formerly known as megalomania or, colloquially, as egotism) is a form o... ...ch as under constant public scrutiny and exposure, a transient and reactive form of the Narcissistic Personality Disorder (NPD) has been observed b... ...e of self-worth by extracting from his environment Narcissistic Supply (any form of attention, both positive and negative). There is a whole range o... ...andon their efforts in mid-stream, give up, vanish, lose interest, devalue former pursuits, fail, or slump. Why is that? Narcissists are prone to se... ...seeable future. I tend to think of these episodes with narcissists as being cyclic. You will likely need to cut loose for a while when your child is...

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...nderstand this new subject. The authors are grateful to Ilanthenral for formatting the book and drawing all the graphs and diagrams in this book... .... A study of these basis and its relations happens to be an interesting form of research. DEFINITION 1.2.6: Let G(I) be a neutrosophic vector ... ...in the representation of Binary relations on a single set. Thus one of the forms of representation of a fuzzy relation R(X, X) is represented by th... ...edge of G incident to such a vertex of G is a pendant edge of G. A sequence formed by the degrees of vertices of G is called a degree sequence of G.... ...at have one end vertex in S and the other in S . A set of edges of G of the form {S, S } is called an edge cut of G. An edge e is a cut edge of G, i... ... a connected simple graph containing at least two disjoint cycles. Then the cyclical edge connectivity of G is defined to be the minimum number of e... ...rom 3 to n. G is called vertex pancyclic if each vertex υ of G belongs to a cyclic of every length l, 3 ≤ l ≤ n. The following graph is an examp... ...e graph with a cut-edge then χ' (G) = 4. DEFINITION 2.7.10: A snark is a cyclically 4-edge connected cubic graph of girth at least 5 and has chro... ...ent of a graph, 26-7 Connected graph, 19, 34 Covering of G, 40 Cycle, 19 Cyclical edge connectivity, 37 D Degree sequence of a graph, 25-6 ...

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...tands— With all the music in her tone, 10 A hollow echo of my own,— A hollow form with empty hands.’ And shall I take a thing so blind, Embrace her as... ...way. 20 14 In Memoriam XIII. TEARS of the widower, when he sees A late lost form that sleep reveals, And moves his doubtful arms, and feels Her place... ... take 5 The touch of change in calm or storm; But knows no more of transient form In her deep self, than some dead lake That holds the shadow of a lar... ...ife that almost dies in me; That dies not, but endures with pain, And slowly forms the firmer mind, Treasuring the look it cannot find, The words that a... ... our fair companionship, And spread his mantle dark and cold, And wrapt thee formless in the fold, 15 And dull’d the murmur on thy lip, And bore thee ... ...f fluent heat began, And grew to seeming random forms, 10 The seeming prey of cyclic storms, Till at the last arose the man; Who throve and branch’d fr...

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...o very different handling from that which the in- dulgence or credulity of former ages could allow. Mere state- 4 The Iliad of Homer ments are jealou... ... Iliad of Homer ments are jealously watched, and the motives of the writer form as important an ingredient in the analysis of his his- tory, as the fa... ...ich human experience, whether actual or traditionary, has fur- nished. T o form correct views of individuals we must regard them as forming parts of a... ...ecame ac- quainted with the legends respecting Ulysses, which after- wards formed the subject of the Odyssey. The inhabitants of Ithaca assert, that i... ... had an opportunity of listening to the improvisation of Italy, can easily form an idea of Demodocus and Phemius.”— Ancient Greece, p. 94.] But povert... ...e by Alcmena, the wife of Amphitryon. 111 Ægiale daughter of Adrastus. The Cyclic poets (See Anthon’s Lempriere, s. v.) assert V enus incited her to i...

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...f there are as many religious creeds as there are adherents. It is true that thevast majority of Westerners continue to list themselves on the census forms as Christian or Jewish, but for many...

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... the vast majority of Westerners continue to list themselves on the census forms as Christian or Jewish, but for many people these terms have come to ... ...was reared in a fundamentalist Christian family which had little by way of formal affiliation with any particular church but which made a practice of ... ...rship. It had no budget. It had no teachings. It continued the traditional form of Quaker Meetings for Worship, 2 which consists of silent meditation ... ...tude” has a great deal to do with their suffering, and the finding of some form of “religious attitude” has much to do with their discovery of some ki... ...stablishing psychology as a study in its own right, especially the various forms of Depth Psychology. Pierre Janet and Sigmund Freud were already publ... ...e are eternal. After we die we eventually enter new bodies and the endless cyclic sequence of birth, life, and death continues on through all eternity... ... Western and especially the Christian view. For Native Americans, time was cyclical, based on the recurring sacred cycles of the seasons, and these cy...

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... Dr. A. R. T. Solarin, Dept. of Mathematics, Obafemi Awolowo University, (formerly University of Ife), Ile-Ife, Nigeria. Copyright 200... ...eface 5 1. BASIC CONCEPTS 1.1 Basics of vector space and bilinear forms 7 1.2 Smarandache vector spaces 8 1.3 Basic definitions of other al... ...ection mainly describes the specific notions of vector space and bilinear forms. In section two we develop the definition of Smarandache K-vectorial... ...oups, S-groupoids and loops. 1.1 Basics of Vector space and bilinear forms In this section we just recall the concept of vector space, biline... ... S-commutative. DEFINITION 2.1.29: Let L be a loop. L is a Smarandache cyclic loop (S-cyclic loop) if L contains atleast a proper subset A which ... ...c loop (S-cyclic loop) if L contains atleast a proper subset A which is a cyclic group. We say L is 19 Smarandache strongly cyclic (S-strongly cy... ...ongly cyclic (S-strongly cyclic) if every subset which is a subgroup is a cyclic group. THEOREM [72]: Let n be a prime, every loop in the class o... ...]: Let n be a prime, every loop in the class of loops L n are S-strongly cyclic loops. Proof: Follows from the fact when n is a prime, L n (m) f... ... the fact when n is a prime, L n (m) for any m have only n subgroups each cyclic of order 2. THEOREM [72]: Let L n be a class of loops, n > 3, i...

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...t is important to mention that unlike in rings where the two sided ideals form a modular lattice, we see in case of Smarandache rings the two sided ... ...e see in case of Smarandache rings the two sided ideals in general do not form a modular lattice which is described in the cover page of this book. T... ...e problems, specially the zero divisor conjecture find several equivalent formulations. We have given many equivalent formulations, for this conject... ...ena and Kama for single- handedly helping me by spending all their time in formatting and correcting this book. I dedicate this book to be my bel... ...xed element of G and n is an arbitrary integer. In this case G is called a cyclic group and the element a is said to generate G. Example 1.1.1: L... ...lo p. Z p \{0} is a group under multiplication modulo p. This is a finite cyclic group of order p-1. DEFINITION 1.1.2: Let G be a group. If a . ... ...group if (H, o) is a group. Example 1.1.4: Let G = 〈g / g 8 = 1〉 be a cyclic group of order 8. H={g 2 , g 4 , g 6 , 1} is subgroup of G. Exa... ... -1 ha / for every a in G and every h ∈ H}. If G is an abelian group or a cyclic group then all of its subgroups are normal in G. Example 1.1.6:... ...groups or normal subgroups. Example 1.1.8: Let G= 〈g / g 12 =1〉 be the cyclic group of order 12. The series {1} ⊆ {g 6 , 1} ⊆ {1, g 3 , g 6 , g 9...

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...ss Rehoboth 2002 3 This book can be ordered in microfilm format from: Bell and Howell Co. (University of Microfilm Internati... ...is a mere soul>.- According to this thesis of Socrates, matter is another form of existence of the spirit – a conception agreeing with the paradigm ... ...aneous spiritalization) and by the same will, the entity repenetrates the former material support, resuming the initial code. Such a scenario would ... ...itualization cycles, of the XY area, would ″arouse″ subconscious memories form a previous ″life″ − which would naturally explain the exceptional perf... ...he outlooks of the great philosopers Socrates and Plato. By admitting the cyclic character of the materialization and spiritualization process, one c... ...d if the rational entities had the same monadic consistency, during their cyclic evolution they would materialize each time within the same area 1... ...ding way, compatible with and apt to a permanent and unlimited evolution, cyclically controllable by the divine will, while the extraterrestrial wor... ...) The MESER concept asserts, on scientific grounds, the progress of the cyclic process of materialization and spiritualization, characteristic of ... ...the human being, besides the bygone genesis, has also a permanentness and cyclicity character; it is not a completed process. According to the sign...

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...te that the vertices of the bigraphs i.e. G = G 1 ∪ G 2 in general need not form disjoint subsets of G such that 12 GG ∩ =φ . In the above exa... ...will be called as the mixed trimatrix. Example 2.1.37: The trimatrix of the form A = 30 01 20 11 01 00 − ⎡⎤⎡⎤⎡⎤ ∪∪ ⎢⎥⎢⎥⎢⎥ ⎣⎦⎣⎦⎣⎦ will be c... ...l be called as a 2 × 2 square trimatrix. Example 2.1.38: A trimatrix of the form A = 31 11 10 01 00 10 11 00 10 ⎡⎤⎡⎤⎡⎤ ⎢⎥⎢⎥⎢⎥ ∪∪ ⎢⎥⎢⎥⎢⎥ ⎢⎥⎢⎥⎢⎥... ...C 3 2 j i 4 3 C C , , C C … be the edges of the FCM (i ≠ j). Then the edges form a directed cycle. An FCM is said to be cyclic if it possesses a d... ... the FCM (i ≠ j). Then the edges form a directed cycle. An FCM is said to be cyclic if it possesses a directed cycle. An FCM is said to be acyclic ... ...and G 3 connected by the vertex P 4 such connected trigraphs are called as cyclically connected trigraphs. Now we just give the separated dire... ... Let the edges of the bigraph form a directed bicycle. An FRBM is said to be cyclic if it possesses a directed cycle. An FRBM is said to be acyclic... ...edges of the NCM. Then the edges form a directed cycle. An NCM is said to be cyclic if it possesses a directed cyclic. An NCM is said to be acyclic... ...edge of the NCBM. Then the edges form a directed cycle An NCBM is said to be cyclic if it possesses a directed cycle. An NCBM is said to be acyc...

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....9 Smarandache Factor ……………….. 50 5 Partitions Of A Typical Canonical Form 1.10 Length/Extent of Smarandache Factor Partitions ……………….. 55 1... ...sets of n natural numbers such that the sum of the reciprocals is 1. More formally, ... ...is SRRPS(n) where the elements of each set of size n must be unique. More formally, ... ... a 2 . . . a n-3 + 1 a n-1 = a 1 a 2 . . . a n-2 then the numbers form a set of (n-1) distinct natural numbers such that the sum of the reci... ...DP (n) – f DP (n-1) ≥ f DP (n-1) + n – 3. Replacing n in the last formula by n-1, n-2 and so forth, we get f DP (n-1) – f DP (n-2) ≥ f ... ...on 12 Smarandache Determinant Sequences Definition: The Smarandache Cyclic Determinant Natural Sequence is defined as the determinants of the ... ... a the first term and common difference d. We will define the Smarandache Cyclic Arithmetic determinant sequence in the following way: a ... ...e Comprehensive Geometric Partition Function (SCGP) 206-207 Smarandache Cyclic Determinant Natural Sequence 179-181 Smarandache Distinct Recipro...

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... need Mother, I got what I needed from her, her love. Well, at least a semblance of love, in the form of approval and encouragement. Naturally, I tr... ... love, bonding, coupling? Do we simply get used to other human beings, pets, or objects? Habit forming in humans is reflexive. We change ourselves ... ...n maximum comfort and well being. It is the effort that goes into these adaptive processes that forms a habit. The habit is intended to prevent us f... ...of our acts, intentions, emotions and reactions. It is a mirror reflecting that part in us which formed the habit in the first place. Hence, the fee... ...of socialization, the acts and imprints of our environment and nurture on our psyche during the formative years (0-6 years and in adolescence). Ou... ... for you for the foreseeable future. I tend to think of these episodes with narcissists as being cyclic. You will likely need to cut loose for a whi...

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............ 10 The Impetus of Splitness……………………………………………………………………..……….11 Formlessness………………………………………………………………………………………….14 The First Major Spli... ...s Pg 1385 Disease Pg 1388 Exploration of the Subconscious Pg 1393 Social Form and Function Pg 1395 The Power of Myth Pg 1397 History and Myt... ...l the light rays, all the dust particles; and turns it into a more compressed form of energy. It then spews this intense form of compressed energy ... ... is so simple, so taken for granted: that nobody ever has thought to state it formally as a Universal Law, not just as a Law of Physics for moving p... ... thing about the Universe is that the Dynamic of Permutations grows out of the cyclical nature of its patterns and basic structures. All cyclic patt... ...alities which are involved in these basic patterns and structures. These cyclic patterns act as Totalities, but they are also Dualistic and Tria... ...es, but they are also Dualistic and Trialistic in nature… merging into larger cyclic patterns in overlapping series of Totalities... which in turn, b... ...rns act as highways for energy to continually flow-through these cycles. The cyclical patterns may persist generally, but they themselves are also c... ...4. The atomic weight of 24 or 12-12 obviously should be the totality of the cyclic process for Fission- Fusion occurring within Stars. This appare...

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...c structures. 1.1 Fuzzy Subsets In 1965 Zadeh [144] mathematically formulated the fuzzy subset concept. He defined fuzzy subset of a non-empt... ...group G if µ (xy) = µ (yx) for every x, y ∈ G. This is just an equivalent formation of the normal fuzzy subgroup. Let µ be a fuzzy normal subgroup of... ...Hamiltonian if every subgroup of G is normal) THEOREM 1.2.8: Let G be a cyclic group of prime order. Then there exists a fuzzy subgroup A of G suc... ...ubgroups of A. Now we give more properties about fuzzy subgroups of a cyclic group. THEOREM [16]: Let G be a group of prime power order. Then ... ...c group. THEOREM [16]: Let G be a group of prime power order. Then G is cyclic if and only if there exists a fuzzy subgroup A of G such that for x... ... = {t 0 , t 1 , … , t r }, t 0 > t 1 > … > t r and the level subgroups form a composition chain. THEOREM [16]: Suppose that G is a finite group... ...composition chain 〈e 〉 = A 0 ⊂ A 1 ⊂ …⊂ A r = G where A i / A i–1 is cyclic of prime order, i =1, 2, … , r. Then there exists a composition cha... ...subgroup of µ such that µ K ⊆ µ (µ K ≠µ ) then µ K takes the following form: µ K (x) =    − = α − = 1 , 1 x if i , i x if 0 i with 0 ≤ ... ...f A is bounded over B then A t / B t is a direct product of cyclic groups for all t ∈ [0, A(e)]. DEFINITION 2.2.13: Let G be a S-...

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...Research Press Rehoboth 2002 This book can be ordered in microfilm format from: Books on Demand ... ...e the sides of the triangleABC. Our aim is to improve (1) in the following form: X a 0 b 0 1 3 X a 0 2 1 12 X a 2 1 4 X a 2 : (2) First ... ...f the well-known bisector theorem. LetAM be a cevian of the triangle which forms the anglesu andv with the sidesAB and AC, respectively. Then AB AC = ... ...rue (see Theorem 25 in page 112): Let AD and AE be two cevians (D;E2 (BC)) forming angles ; with the sides AB;AC, respectively. If b A 90 and ... ...D CE AB 2 AC 2 : (2) Indeed, by applying the area resp. trigonometrical formulas of the area of a triangle, we get BD CD = A(ABD) A(ACD) = AB sin ... ...he order of a subgroup divides the order of the group - here one considers cyclic subgroups). Other similar properties of ’ are included in [4]. Bibli...

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...h the Habsburgs: Bosnia to the latter in return for a free hand in Macedonia to the former. The wily Austro-Hungarians regarded the Serbs as canno... ...ainst the Russians and the Turks. In 1885, Bulgaria was at last united - north and formerly Turk-occupied south - under the Kremlin's pressure. The... ...ssociation to a power hungry monstrosity. In 1912 Bulgaria, Serbia, and Greece - former bitter foes - formed the Balkan League to confront an eve... ...s, both active duty and reserve) and with the tacit blessing of Tsar Boris and the forming National Alliance (later renamed the Democratic Alliance... ...oups often do) but staffed by peasant, lumpenproletariat and dwellers of the slums formed by Macedonian refugees all over the Balkans and especiall... ...rt of an eternity, a cycle, which goes on forever of life, death and renewal. This cyclic view of life and the creation is inevitable within any th...

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... 2 C C j i C C C C , , 4 3 … be the edges of the FCM (i ≠ j). Then the edges form a directed cycle. An 12 FCM is said to be cyclic if it possesse... ... FCM (i ≠ j). Then the edges form a directed cycle. An 12 FCM is said to be cyclic if it possesses a directed cycle. An FCM is said to be acyclic ... ...FINITION 1.1.11: If the FCM settles down with a state vector repeating in the form A 1 → A 2 → … → A i → A 1 then this equilibrium is called a l... ... D i ) be the edges of an FRM, j = 1, 2,…, m and i= 1, 2,…, n. Let the edges form a directed cycle. An FRM is said to be a cycle if it posses a di... ...FINITION 1.3.10: If the FRM settles down with a state vector repeating in the form A 1 → A 2 → A 3 → … → A i → A 1 (or B 1 → B 2 → …→ B i ... ... inclusion of the concept of indeterminate situation with fuzzy concepts will form the neutrosophic logic. As in this book the concept of only fuz... ...edges of the NCM. Then the edges form a directed cycle. An NCM is said to be cyclic if it possesses a directed cyclic. An NCM is said to be acyclic...

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...b = c if σ (a, b) = c. DEFINITION: A non empty set of elements G is said to form a groupoid if in G is defined a binary operation called the produ... ...ial subgroups. Proof: It is clear from the fact that every pair {m, e} is a cyclic group of order 2; Hence the claim. THEOREM 3.5.2: Let G = (Z... ...s. Thus two examples give no doubt only a Smarandache strong groupoid but are formulated differently. THEOREM 4.3.7: Every Smarandache strong alt... ...ids. Thus, Smarandache direct products helps us in many ways to get a desired form of SGs. This concept of Smarandache direct product of groupoids wi...

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... in the exquisite autograph “Hunt MS.” of “Julian and Maddalo”, Mr. Buxton Forman, the most conservative of editors, finds it necessary to supplement ... ...tly cared to sing, to the neglect and detriment of the merely external and formal element of his song. Shelley recked little of the jots and tittles o... ...’a great poet is not of an age, but for all time.’ Irregular or antiquated forms such as ‘recieve,’ ‘sacrifize,’ ‘tyger,’ ‘gulph,’ ‘desart,’ ‘falshood... ... his spelling reversed in accordance with modern usage. All weak preterite-forms, whether indicatives or participles, have been printed with “ed” rath... ... as ‘past,’ alone excepted. In the case of ‘leap,’ which has two preterite-forms, both employed by Shelley (See for an example of the longer form, the... ...an be considered only, like the Iliad, as a continuation of that series of cyclic poems, which have already been candidates for bestowing immor- talit...

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...In this section we give the definition of linear algebra and just state the few important theorems like Cayley Hamilton theorem, Cyclic Decomposition Theorem, Generalized Cayley Hamilton Theorem and give some properties about linear algebra....

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...a and just state the few important theorems like Cayley Hamilton theorem, Cyclic Decomposition Theorem, Generalized Cayley Hamilton Theorem and giv... ...nimal polynomial divides the characteristic polynomial for T. THEOREM: (CYCLIC DECOMPOSITION THEOREM): Let T be a linear operator on a finite dim... ... respect to the inner product 〈 〉. We just recall the notion of quadratic form. The quadratic form determined by the inner product is the functio... ...}. Clearly B generates a linear algebra. But all subsets of A do not 18 form a S-basis of V, so A is not a S-strong basis of V but only a S-basis... ...r. Using the new pseudo inner product once again we will have the modified form of spectral theorem. That is, the Smarandache spectral theorem which... ...x is productive if each of its column sums is less than one. Now we will formulate the Smarandache analogue for this, at the outset we will justif... ... G 1 = Q \ {0} with usual multiplication and G 2 = 〈 g | g 2 = I 〉 be a cyclic group of order two. Clearly G is a commutative bigroup. We say th... ...ow the concept of linear bialgebra i.e. bimatrices are used when we define cyclic bicodes. So we define a cyclic bicode as the union of two cyclic c... ...clic bicode as the union of two cyclic codes for we know if C (n, k) is a cyclic code then if (x 1 …x n ) ∈ C (n, k) it implies (x n x 1 …x n–1 ) ...

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...ache. A few of the mentionable characteristics of this mode of thinking are [90-94]: It proposes new philosophical theses, principles, laws, methods, formulas and movements; it reveals that the world is full of indeterminacy; it interprets the uninterpretable; regards, from many different angles, old concepts, systems and proves that an idea which is true in a given refere...

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...[90-94]: It proposes new philosophical theses, principles, laws, methods, formulas and movements; it reveals that the world is full of indeterminacy... ...-A>, there is a continuum-power spectrum of Neutralities. This philosophy forms the basis of Neutrosophic logic. Neutrosophic logic grew as an a... ...ology such as office plants; analysis of business performance assessment; formalism debate and legal rules; creating metabolic and regulatory network... ...nt intrusion detection system; hyper-knowledge representation in strategy formation; female infanticide; depression in terminally ill patients and f... ...book, Kama who patiently drew all the cognitive maps and ensured that the formatting and page making of the book was intact and Meena who collected ... ...e FCM (i ≠ j). Then the edges form a directed cycle. An FCM is said to be cyclic if it possesses a directed cycle. An FCM is said to be acyclic if i... ...es of the NCM. Then the edges form a directed cycle. An NCM is said to be cyclic if it possesses a directed cyclic. An NCM is said to be acyclic if ... ...ined FRM, 94 Combined NCMs, 136 Combined NRM, 171 Connection matrix, 8 Cyclic FRM, 93 Cyclic NCM, 135 Cyclic NRM, 170 D Density equation, ...

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...ed the problem of finding the S-perfect and completely S-perfect numbers, but his proof is not complete. He has proved that the only S-perfect of the form n = p q is n = 6 and there is no S-perfect number of the form n = 2kq where k ¸ 2 is an integer and q is an odd prime. On the other hand, Theorem 2.1 gives all the S-perfect numbers. Again, Sandor only proved that, the o...

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... Smarandache double factorial function and Euler function 8 A. A. K. Majumdar : S-perfect and completely S-perfect numbers 12 B. Zhao and S. Wang: Cyclic dualizing elements in Girard quantales 21 P. Chun and Y. Zhao : On an equation involving the Smarandache function and the Dirichlet divisor function 27 F. Li and Y. Wang : An equation involving the Euler function an...

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...o every connected com- ponent of A is analytic or antianalytic. Now we can formally define a Klein surface. A Klein surface is a Hausdorff, connected, t... ...known that for a given Klein surface S, the set AutS of automorphisms of S forms a group with respect to the composition operation and AutH =PGL(2,R).... ... automorphism of the s-manifold C(T,n). All automorphisms of an s-manifold form a group under the composition operation, called the automorphism group... ...c forembedding ofagraphon orientablesurfaces isfound by Heffter in 1891 and formulated by Edmonds in 1962, states as follows. Theorem 1.2.2([17]) Every... ...X α,β with X =E(Γ) is an automorphism of map M iff ∀v∈V(M), g preserves the cyclic order of v. Proof Assume that ζ ∈AutM is induced by the extend actio... ...,g(x 2 ),···,g(x ρ(u) )) =(y 1 ,y 2 ,···,y ρ(v) ). Whence, g preserves the cyclic order of vertices in the mapM. Chapter 1 Preliminary 12 Ontheotherh... ...v = (x 1 ,x 2 ,···,x ρ(v) )(αx ρ(v) ,···,αx 2 ,αx 1 ). Denote by < v > the cyclic group generated by v. Then we get the following property for the aut... ...en ∀v∈V(M), (i) if ∀g∈G,g is orientation-preserving, then G v <v >, is a cyclic group; (ii) G v <v >×<α>. Proof (i)Let M = (X α,β ,P). since for a... ...sume ξ is the least power of elements in G v . Then G v =<v ξ > <v > is a cyclic group generated by v ξ . (ii)For∀g∈G v , v g =v, i.e., [(x 1 ,x 2 ,·...

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...nted edition) 2002 This book can be ordered in microfilm format from: Books on Demand ProQuest Information & Learning... ...s. In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, that ri... ...e infinitesimals. Informally, an infinitesimal is an infinitely small number. Formally, x is said to be infinitesimal if and only if for all positi... ...e of thinking: - proposes new philosophical theses, principles, laws, methods, formulas, movements; - reveals that world is full of indeterminacy; ... ...equency of the events' occurrence.” “The vagueness which constitutes another form of uncertainty is the character of those which contours or limit... ... attractor. Within this perspective there are three types of attractor: point, cyclic and strange (or chaotic). Simplifying a little, the first assu...

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...I allowed any boldness: - anti-literature and its literature; - flexible forms fixed, or the alive face of the death! - style of the non-style; - ... ...r used, but very concrete at the same time: non-restrictive verse from any form or content. It takes advantage of cliche against itself. EVERYTHING I... ... 1998]. (1999-12-14) 3.4. Neutrosophic Probability: <logic> An extended form of probability based on Neutrosophy, in which a statement is held to b... ...rch Press, 1998]. (1999-07-05) These definitions receive a more general form in Smarandache’s next two papers, and are explained, exemplified, giv... ...sis. In 1960s Abraham Robinson has developed the non-standard analysis, a formalization of analysis and a branch of mathematical logic, which rigoro... ...ween any two categories. N) Evolution of an Idea <A> in the world is not cyclic (as Marx said), but discontinuous, knotted, boundless: <Neut-A> = ... ...s justifying to me an American friend kept by his wife henpecked! A cyclic social development. The woman becomes the center of the societ... ...ple to complex and from homogeneous to heterogeneous, can be updated to a cyclic movement: - from simple to complex and back to simple - since any c...

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...ers should be professional, in good English, containing a brief review of a problem and obtained results. All submissions should be designed in LATEX format using Progress in Physics template. This template can be downloaded from Progress in Physics home page http://www.geocities.com/ptep_online. Abstract and the necessary information about author(s) should be included int...

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...20sec. Stable states of unmatter can be built on quarks and anti-quarks: applying the unmatter principle here it is obtained a quantum chromodynamics formula that gives many combinations of unmatter built on quarks and anti-quarks....

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...atter by Experiments, More Types of Unmatter, and a Quantum Chromodynamics Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Informa... ...blem and obtained results. All submissions should be designed in L A T E X format using Progress in Physics template. This template can be downloaded ... ...aches it by means of an ad hoc extension in the ludicrous Kruskal-Szekeres formulation. However, the standard assumptions are incorrect, which I shall... ...n 2 θdϕ 2 , (2b) and claims it the most general, which is incorrect. The form of C(r) cannot be pre-empted, and must in fact be rigorously determine... ...2πr, to satisfy its unproven assumptions aboutr. But this equation is only formally the same as the equation of a circle in the Euclidean plane, becau... ... ⇒ x=0 (6) [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0. (7) The last one is the Jacobi cyclic identity. If{e i } is a vector basis inE 2n , the antiscalar product... ...]σ +Γ ρ λ[μ ,Γ λ ν]σ . (119) Furthermore, (anti) commutation relations and cyclic identities (in particular Bianchi’s identities) hold for the Riemann...

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...ers should be professional, in good English, containing a brief review of a problem and obtained results. All submissions should be designed in LATEX format using Progress in Physics template. This template can be downloaded from Progress in Physics home page http://www.geocities.com/ptep_online. Abstract and the necessary information about author(s) should be included int...

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... Quasi-Paradoxes and Quantum Sorites Paradoxes. . . 7 F. Smarandache A New Form of Matter — Unmatter, Composed of Particles and Anti-Particles. . . . ... ...blem and obtained results. All submissions should be designed in L A T E X format using Progress in Physics template. This template can be downloaded ... ...e best? Really, the result- ing d’Alembert equations are derived from that form of the Ricci tensor obtained under the substantial simplifcations of h... ... deduce the exact d’Alembert equations for the force feld, giving an exact formula for the velocity of waves of the force; (2) Experiments to register... ...d of the gravitational inertial force F α =−2c 2 b β a ∙α β∙ in their fnal form. They are 1 c 2 ∂ ∂t F k F k + 1 c 2 F i ∂F i ∂t +D k m ∂F m... ...hese norms are invariant underC-space Lorentz trans- formations due to the cyclic property of the trace operation and RR −1 =1. If one writes the inva... ... † X> s =||X 2 ||, (16) where once again, we made use of the analog of the cyclic property of the trace, <RX † XR −1 > s = <X † X> s . This way of rew...

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...ed rights. (See "Abortion and the Sanctity of Human Life" by Baruch A. Brody). One form of calculus is the utilitarian theory. It calls for the max... ...e of the use made of an ideal. We compare that which exists with a Platonic Idea or Form (which, according to modern thinking, does not REALLY exist... ...icism is the product not of the process of comparison - but of these ideal Ideas or Forms. Since they do not exist - the thing criticized is felt no... ...dels constructed, one of them selected (on empirical or aesthetic grounds), a goal formulated, an experiment performed and a negative (failure) or ... ...r its definition (hence, for its meaning) – how could we derive meaning from it? A cyclical argument ensues. We can never derive meaning from that ... ...ecome subjects? The will to power. the wish to be able to interpellate. It is this cyclical nature of Althusser's teachings (ideologies interpellat... ... first place. How ironic that Althusser, the brilliant thinker, did not grasp the cyclical nature of his own teachings (that ideologies interpella... ...ed into different kinds of vessels and carriers. It can be continuous or segmented, cyclical (periodic) or punctuated, repetitive or new, etc. The ... .... In an inflationary turn of phrase, "moral hazard" is now taken to encompass anti-cyclical measures, such as interest rates cuts. The Fed - and i...

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...ed rights. (See "Abortion and the Sanctity of Human Life" by Baruch A. Brody). One form of calculus is the utilitarian theory. It calls for the max... ...e of the use made of an ideal. We compare that which exists with a Platonic Idea or Form (which, according to modern thinking, does not REALLY exist... ...icism is the product not of the process of comparison - but of these ideal Ideas or Forms. Since they do not exist - the thing criticized is felt no... ...dels constructed, one of them selected (on empirical or aesthetic grounds), a goal formulated, an experiment performed and a negative (failure) or ... ...r its definition (hence, for its meaning) – how could we derive meaning from it? A cyclical argument ensues. We can never derive meaning from that ... ...ecome subjects? The will to power. the wish to be able to interpellate. It is this cyclical nature of Althusser's teachings (ideologies interpellat... ... first place. How ironic that Althusser, the brilliant thinker, did not grasp the cyclical nature of his own teachings (that ideologies interpella... ...ed into different kinds of vessels and carriers. It can be continuous or segmented, cyclical (periodic) or punctuated, repetitive or new, etc. The ... .... In an inflationary turn of phrase, "moral hazard" is now taken to encompass anti-cyclical measures, such as interest rates cuts. The Fed - and i...

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...naire used in interviewing these migrant labourers. The questionnaire was formatted with the help of Dr. K. Kandasamy. Appendix 2 gives the table of... ...ragement. We are indebted to Ms.Kama Kandasamy who had taken all pains to format, verify the equations and design the books. We deeply acknowledge t... ... services of Dr.K.Kandasamy who had done a lot of background work and for formatting such a scientific linguistic questionnaire without which we wou... ...ts are in a drunken state and so on. So we cannot give it any statistical form/data representation for the psychological study for the cause may rem... ...ive peer-group influence (Bad Company), and Quacks. These six broad heads forms the columns of the matrices. The time periods are taken as the age g... ...e FCM (i ≠ j). Then the edges form a directed cycle. An FCM is said to be cyclic if it possesses a directed cycle. An FCM is said to be acyclic if i... ...es of the NCM. Then the edges form a directed cycle. An NCM is said to be cyclic if it possesses a directed cyclic. An NCM is said to be acyclic if ... ...rlap Block NCM (CDBNCM), 13 Combined Overlap Block NRM (CDBNRM), 13, 301 Cyclic, 41-42 Directed cycle, 41-42, 134 Directed graph, 39 Diverse cl...

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..., there is some number 1 ‡ n such that e < n n Zw ) ( . Proof. Let’s form a product of distinct primes k p p p q K 2 1 = such that e < q 1 whe... ...k k n Zw = ? Problem 20. Solve the problems from 28 up to 35 already formulated for the Zt function also for the Zw function. Problem 21. Study... ...2. For the Smarandache power sequence [2] the following conjecture can be formulated: 4 3 9 1 m g > for all m values Problem 4. For any Sm... ...) ( ) ( k c j c i c „ + for any three distinct terms c(i), c(j) and c(k) forming the sequence. Problem 10. Find some examples of Smarandache nona... ..., ... [6] (ID=A019546) 14) Smarandache alternating series A series of the form - - k k k a 1 ) ( ) 1 ( where a(k) is any Smarandache sequence. As... ...ven by all the terms a(n) that still belong to the initial sequence on any cyclic rotation of their digits. For example let's consider the sequence a(... ... 2, 4, 6, 8, 20, 22, 24, 26, 28,.... because each term remain even for any cyclic rotation of its digits. In the same way we can build the Smarandache...

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...the word Democratic, the word En-Masse. Of physiology from top to toe I sing, Not physiognomy alone nor brain alone is worthy for the Muse, I say the Form complete is worthier far, The Female equally with the Male I sing. Of Life immense in passion, pulse, and power, Cheerful, for freest action form?d under the laws divine, The Modern Man I sing....

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.................495 What Best I See in Thee....................496 Spirit That Form’d This Scene .........496 As I Walk These Broad Majestic Days ......... ...ing, Not physiognomy alone nor brain alone is worthy for the Muse, I say the Form complete is worthier far, The Female equally with the Male I sing. O... ...g. Of Life immense in passion, pulse, and power, Cheerful, for freest action form’d under the laws divine, The Modern Man I sing. Leaves of Grass –Wh... ...y permanent, The body lurking there within thy body, The only purport of the form thou art, the real I myself, An image, an eidolon. Thy ver... ...tudies the first step pleas’d me so much, The mere fact consciousness, these forms, the power of motion, The least insect or animal, the senses, eyesi... ... mountain cliffs, The fields of Nature long prepared and fallow, the silent, cyclic chemistry, The slow and steady ages plodding, the unoccupied surfa...

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...Excerpt: This volume contains practically all that remains of the post- Homeric and pre-academic epic poetry. I have for the most part formed my own text. In the case of Hesiod I have been able to use independent collations of several MSS. by Dr. W.H.D. Rouse; otherwise I have depended on the apparatus criticus of the several editions, especially that of Rza...

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..................................................................... 236 NON-CYCLIC POEMS ATTRIBUTED TO HOMER ............................................. ...f the post- Homeric and pre-academic epic poetry. I have for the most part formed my own text. In the case of Hesiod I have been able to use independe... ...opu- lar, and not (as it became later) an artificial and academic literary form — passed through the usual three phases, of development, of maturity, ... ..., and we are therefore thrown back upon the evidence of analogy from other forms of literature and of inference from the two great epics which have co... ...ontinental Greece (1), on the other hand, but especially in Boeotia, a new form of epic sprang up, which for the romance and pathos of the Ionian Scho... ...niformly effective. 2) On the other hand while tradition steadily puts the Cyclic poets at various dates from 776 B.C. downwards, it is equally consis... ...ore chrono- logical history began, and at the same time assigns the purely Cyclic poems to definite authors who are dated from the first Olympiad (776... ... (776 B.C.) downwards. This tradition can- not be purely arbitrary. 2) The Cyclic poets (as we can see from the abstract of Proclus) were careful not ... ...necessary to deal in full with that hero’s adventures. (12) In a word, the Cyclic poems are ‘written round’ the Iliad and the Odyssey. 3) The general ...

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...gs which came before and after were told by other bards, who between them narrated the whole ?cycle? of the events of the war, and so were called the Cyclic Poets. Of their works none have survived; but the story of what befell between Hector?s funeral and the taking of Troy is told in detail, and well told, in a poem about half as long as the Iliad....

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...narrated the whole “cycle” of the events of the war, and so were called the Cyclic Poets. Of their works none have survived; but the story of what be... ...ght the ring of his music, and he perhaps had before him the works of those Cyclic Poets whose stars had paled before the sun. We have practically no... ...little more than an amplification or remodel ling of the works of these two Cyclic Poets.” This, however, must needs be pure conjecture, as the only ... ...h to justify us in the conclusion that, even if Quintus had the works of the Cyclic poets before him, which is far from certain, his poem was no mere ... ...on disposes by any means of all difficulties. If Quintus did not follow the Cyclic poets, from what source did he draw his materials? The German crit... ...eath, In grace and comeliness fell, for naught of shame Dishonoured her fair form. Face down she lay On the long spear outgasping her last breath, St... ...o and fro, And will to blind night leave earth, sky, and sea, Till Chaos and formless darkness brood o’er all, That Cronos’ Son may also learn what me... ...uake cleaves A chasm for rushing blasts from underground; So fell the goodly form of Aeacus’ son. He glared, a murderous glance, to right, to left, [U... ... wandering to the tent of Peleus’ son, Now cast down all his length, a giant form, On the sea sands; and thus lamented he: “Achilles, shield and sword...

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...ntly in any one place or rigidly disciplined to one set of conditions. The former attachment to the soil ceases to be an advantage. The human spirit h... ...in the high Alps or along the Riviera. England is rap- idly developing the former Irish grievance of an absentee propertied class. It is only now by t... ...orld over large areas; there is abundant water and food, and the commonest form of cover is the house. But the Legion of Frontiersmen is equipped for ... ...s great, laxly scattered, sea-linked association of ancient states and new-formed countries, Oriental nations, and con- tinental colonies? What will e... ...here lies before me now a crumpled card on which are the notes I made of a former discussion of this very issue, a discussion between a number of prom... ...Normal Social Life is as mute as the standing crops; it is as seasonal and cyclic as nature herself, and reaches towards the future only an intimation...