...rom Japan, proposed the concept of a paper defining G-Dimensional theory in relation to the Smarandache quantum paradoxes. After reviewing a draft, ... ...(FTL) Hypothesis as well. So it is hoped that a determination of parametric relationships between the quantum paradoxes and GDT proves helpful in adv... ...e greater the similarity between set B and set A, the more logical is their relationship. We see then, (somewhat obviously), that these statements a... ...roportional focal length" (f), defines the general kinetic vector geometric relationship with elliptic eccentricity established in terms of (a/b); f... ...ations The analog to Lorentz transformations is inferred from the geometric relation to the vector. The elliptic focus positive with the vector inte... ...two opposing Doppler shifts longitudinal with the vector in k' translate in congruence with SRT/GRT [14] i.e. for, Φ = [{0}, {π}] ; 12 λ = λ ο ... ...ec(θ) [{1 - v}, {1 + v}] . (7.1.1) These are the only angles of congruence required of a theory to have experimental congruence with GRT... ...esis of gravitational repulsion between nucleons and electrons has apparent congruence with the physical world and may be impossible to experimental... ...e unconditionally, some reluctantly), so that mere approximately equivalent congruence to the MT model might likely fail to persuade, but that the i...

...velopment and dissemination of G-dimensional theory (GDT). K. Toshihara, from Japan, proposed the concept of a paper defining G-Dimensional theory in relation to the Smarandache quantum paradoxes. After reviewing a draft, Dr. M. L. Perez, Editor of the SNJ, suggested the inclusion of the Smarandache (FTL) Hypothesis as well. So it is hoped that a determination of parametri...

...comparing opposites — hot vs. cold, black vs. white. We recognize one in relation to the other. if everything were white, we wouldn’t detect anythi... ... oday the ego has evolved to such a degree that we pre- fer a more virtual relationship with others. it is not that we do or do not want others. but w... ...eek deeper contact with others, so for the time being we turn to virtual relationships. A Social Network on the Spiritual Level 3 1 in the next ph... .... A Social Network on the Spiritual Level 3 1 in the next phase, virtual relationships will not satisfy us and we will want an even deeper connecti... ... root of cre- ation. a man and a woman, or groom and bride, represent the relationship between the creator and the creature—the creature being a wom... ...,” a force that leads us back into the general system. this force acts in congruence with the power of our will—the extent to which we awaken, reque...

...e first section we just give the basic concepts or notion like equivalence relation greatest common divisor etc. Second section is devoted to giving ... ...ain aim of this section is to introduce the basic concepts of equivalence relation, equivalence class and introduce some number theoretic results use... ...rgest integer that does not exceed x. Result 1: If d = (a, c) then the congruence ax ≡ b (mod c) has no solution if d / b and it has d mutually i... ...= c -1 ac. We denote a conjugate to b by a ~ b and we shall refer to this relation as conjugacy relation on G. 9 DEFINITION 1.2.4: Let G be a... ...properties of them. DEFINITION 1.3.1: Let A and B be non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to B are cal... ...tion R from A to B is a subset of A × B. Relations from A to B are called relations on A, for short. If (a, b) ∈ R then we write aRb and say that 'a...

... of exis- tence. But at the human level, we ourselves must build this kind of relationship. Nature has left it for us so that we can elevate ourselv... ...l other creatures. In this book, we will discuss how to implement al- truistic relationships, since it is no small task to change human Nature. We we... ...intensifica- tion of the ego is the decline of the family institution. Family relationships in general, and particularly between husband and wife, a... ...ion of cells, and move- ment toward a certain location in the body, unfold in congruence with the body’s needs. CONNECTEDNESS CREATES LIFE IN A NEW... .... One such example is the yucca plant, which has a symbiotic (interdependent) relationship with the yucca butterfly. The female butterfly helps fert... ...the plant to allow the continuation of the plant. By sustaining this kind of relationship, both plant and butterfly ensure the continuation of thei... ...econd Temple were in a state of spiritual attainment. At that time, there was congruence between the spiritual degree of the people of Israel and it... ...of the people to the land of Israel preceded their spiritu- al return, but the congruence between the spiritual root and the corporeal branch must be... ...ces simply do not allow one to be at rest in that land, without the spiritual congruence. To encourage the residents of the physical land of Israel ...

...mits, are to fake both personalities and papers. He is a predator and he hunts for congruence, cohesion and meaning. He is in constant search of a ... ...ut issue of the intermingling of life and the media. Examples for such incestuous relationships abound: Ronald Reagan, the cinematic president wa... ...bitrary environment. In contrast, a model of reality must have a direct and strong relationship to the world. It must obey the rules of physics and... ...o the world. It must obey the rules of physics and of logic. The absence of such a relationship renders it meaningless. A flight simulator is not m... ...e divorced? It is because we cannot believe in an a-causal world. Causality is a relationship (mostly between two things, or, rather, events, the... ...aling with mundane, routine, causal statements because they do not reveal an OVERT relation between the two events discussed. Moreover, in daily us...

...food, but also from knowing who is giving it to me, knowing with whom I have a relationship. The smallest portion of food I receive from a great figur... ...llives,wehavebecomeusedtonotthinking at all about ourselves, our interpersonal relationships, our families, our work, in all of the deeds we perform i... ...elievetoexistoutsideofus,infactexistonlyinrelationtous. That is, they exist in relation to one who perceives reality in this particular manner. Ifwedo... ...n ourselves, we should try to cultivate both a sense of our own inferiority in relation to the Creator, and a sense of pride in the fact that as human... ...coincide. Then, our degree of perception will be proportional to the degree of congruence between our qualities and those of the spiritual. We can per... ...ification of the Creator with the created is only possible when there exists a congruence of desires. "A blessing" is defined as an outpouring of the ... ...uregoisticdesiresarepresent,wecannot perceive the Light, due to the law of the congruence of qualities, the law of likeness. Twoobjectscanperceiveeach... ...oistic nature? First, we must understand why the Creator instituted the law of congruence. As a result of this, though He fills everything, we are una... ...ward for resisting our egos?" is as follows: The Creator instituted the Law of Congruence. This enables us to perceive only those objects on our own s...

...like the whole of the Torah, speaks exclusively of man (creation) and his relationship with the Creator. The Torah attaches worldly appellations to ... ...Ima (mother), for the subsequent transfer to the children (ZON). The same relationships of ABA (see below) and PBP also transpire between their child... ...o Partzuf Ima (Bina), and if Ima also does not want to receive it, such a relationship between them is called back to back. The same relationship ca... ...HSUT is ZAT de Bina and the Light of Hochma is found only in YESHSUT, the relationship between Malchut and ZA works along the same principle, and th... ...hundreds are in Malchut. Such inverse dependence is caused by the inverse relationship between the Lights and the Kelim: the lowest Light enters the... ...ll. And its Nukva assumes (in Arka) the form (properties) of an eagle, in congruence with her goal and action of Linshor (to fall out)—to bring abou... ...no perfection because the Light of Wisdom can illuminate only if there is congruence in qualities between the Light received and the recipient of th... ... qualities between the Light received and the recipient of the Light. The congruence results in Ohr Hassadim, which is found in the right line. Spir...

... lished separately by the authors, but it makes sense to collate them here so that they can be better seen in perspective as a whole, par- ticularly i... ... 2m ¤ : (11) Then g (m) (k) is a multiplicative function with respect to k,i.e., g (m) (1)=1andforeverytwonaturalnumbersa and b, such that (a;b)=1, th... ...nd b are two di®erent multipliers of (p¡3)! because p 2 <p¡3: Therefore, the number a:b = p divides (p¡3)!, i.e., (p¡3)!´0(modp): Hence in case (a) th... ...ltipliersof (p¡3)!. Therefore, the number q:q k¡1 =q k =p divides (p¡3)!, i.e., (p¡3)!´0(modp): 80 On Some Smarandache's problems Henceincase(b 1 ) th... ... ¡3¸2q: Hence q and 2q are two di®erent multipliers of (p¡3)!. Therefore, the number q 2 =p divides (p¡3)!, i.e., (p¡3)!´0(modp): Henceincase(b 2 ) th... ...(modp): Henceincase(b 2 ) the congruence in the right hand-side of (1') is also impossible. Thus we conclude that ifp>1isanoddcompositenumber, then th... ...conclude that ifp>1isanoddcompositenumber, then the congruence (p¡3)!´ p¡1 2 (modp) is impossible. Let p ¸ 3beprime.Inthiscasew eshallpro v etheabo v ... ...ncides with (3), which proves Theorem 2 in this case. Letn>4 be odd. Then r(k;n)=H(k;n¡1); (5) since [ n 2 ]= n¡1 2 and 2:[ n 2 ]=n¡1: We have also th...

...blished separately by the authors, but it makes sense to collate them here so that they can be better seen in perspective as a whole, particularly in relation to the problems elucidated in Chapter one. Many of the problems, and more especially the techniques employed in their solution, have wider applicability than just the Smarandache problems, and so they should be of mo...

... Chapter 2: A result obtained using the Smarandache Function Chapter 3: A Congruence with the Smarandache Function Chapter 4: A functional recurre... ... 11 Chapter 3: A Congruence with the Smarandache function Smarandache’s function is def... ... the factor ) ( 1 + k p P = 0. Therefore we have the following recurrence relation: ∑ ∏ + = + = + + + = k p k p m m k p ... ... i j si j i j i | 0 | 1 1 1 , , 2 , 1 ≥ = i i j L We deduce of this relation: ∑ =       − −     ... ...r people who love numbers: Smarandache Function applied to perfect numbers, congruences. Also, the Smarandache Prime and Coprime functions in connec...

... in the Zentralblatt für Mathematik, Germany. Chapter VII. A k-k additive relationship involves the Smarandache function S(n) which is defined as... ...ence of function values S(n), S(n+1)+ … +S(n+2k-1) satisfies a k-k additive relationship if S(n)+S(n+1)+ …+S(n+k-1)=S(n+k)+S(n+k+1)+ 5 ... ...+k)+S(n+k+1)+ 5 …+S(n+2k-1). The analysis of these types of relations leads to the conclusion that there are infinitely many 2-2 ad... ...ations leads to the conclusion that there are infinitely many 2-2 additive relations and that k-k relations exist for large values of k. Only the f... ...require 2 k+1 (ax+1), which means that we are looking for solutions to the congruence (1) ax ≡ -1 (mod 2 k+1 ) In case 4 we write m+1=... ...n case 4 we write m+1=ax and require 2 k+1 (ax-1). This corresponds to the congruence (2) ax ≡ 1 (mod 2 k+1 ) If x=x 1 is a solution to... ...in the interval 2 k <x< 2 k+1 then 2 k+1 -x 1 is a solution to the other congruence which lies in the interval 0 <x< 2 k . So we have m=ax or m=a... ... b a G b a ) 1 k x ( x i 0 0 , j j i j m 1 k 1 k ≡ = − + ∑ + − − l l l The congruence (5) ) G (mod b a kx i 0 0 k ≡ is soluble iff (k, G i )|a ... ...rimes p=2, 3, 5, 7, … to p≤k max . One of three things will happen: 1. All congruences are soluble modulus G i for k≤k max for all p i ≤k max . ...

...× µ ) (x, y) = min {λ(x), µ (y)} for every (x, y) ∈ X × Y. A fuzzy binary relation R λ on a set X is defined as a fuzzy subset of X × X. The co... ...X is defined as a fuzzy subset of X × X. The composition of two fuzzy relations R λ and R µ is defined by (R λ o R µ )(x, y) = sup X t∈ {min R... ... y)}, for every x, y ∈ X. DEFINITION 1.1.12: Let R λ be a fuzzy binary relation on a set X. A fuzzy subset µ of the set X is said to be a pre cla... ... {µ (x), µ (y) } ≤ R λ (x, y) for every x, y ∈ X. 11 A fuzzy binary relation R λ on a set X is said to be a similarity relation on the set X i... ...of µ . For the fuzzy normal subgroup µ of G and for t ∈ [0, 1], µ t is a congruence relation on the group G. In view of all these the reader is e... ...[70]: Let µ be a fuzzy normal subgroup of a group G and µ t be a t-level congruence relation of µ on G. Let A be a non-empty subset of the group G. ... ...uence relation of µ on G. Let A be a non-empty subset of the group G. The congruence class of µ t containing the element x of the group G is denoted... ... THEOREM (EXISTENCE THEOREM): Let µ be a fuzzy subgroup of a group G. The congruence class [x] µ of µ t containing the element x of the group G exis... ...zzy normal subgroup of a group G then the t-level relation µ t of µ is a congruence relation on the group G and hence the congruence class [x] µ of...

...f a 1 , a 2 , . . . , a n-1 are n-1 distinct natural numbers given by the relation a 1 = 2 a 2 = a 1 + 1 13 a 3 = a 1 a 2 + 1 . . . ... ...o prove that the elements of a Principle Reciprocal Partition satisfy the congruences a 2t ≡ 3 mod(10) and a 2t+1 ≡ 7 mod(10), for t ≥ 1. Co... ..., Hardy, Ramanujan and others. Other properties of the function involving congruences are also known. In the previous sections, the concept of th... ...s. Definition: A number is said to be a Balu number if it satisfies the relation d(n) = F’(n) = r and is the smallest such number. Examples: ... ... 18 2 *( T n-1 - T n-2 ) , (n > 2). Note: There seems to be a direct relation between d and the coefficient of ( T n-1 - T n-2 ) (or the com... ...ma Divisor Prime Sequence: The sequence of primes p n , which satisfy the congruence n-1 Σ p r ≡ 0(mod p n ). r=1 The first few terms of the... ...ngle can be found in the book by Krishnamurthy. This provides us with a relationship between the Stirling numbers of the first kind and those of t... ... which are a valid digits sum. It can also proved using the properties of congruence that the digits sum of the product of two numbers is the produc... ...problem: Find a formula for the sum of the rows. Open problem: Search for congruence properties when n is composite. The Smarandache function S(n) ...

... n . (D 2n is called the dihedral group of order 2n given by the following relation, D 2n = {a, b/ a 2 = b n = 1; bab = a}. 13. Give an examp... ...ces used in this book. DEFINITION 1.2.1: Let A and B be non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to B are c... ...lation R from A to B is a subset of A × B. Relations from A to B are called relations on A, for short, if (a, b) ∈ R then we write aRb and say that ... ...ions on A, for short, if (a, b) ∈ R then we write aRb and say that 'a is in relation R to b’. Also if a is not in relation R to b, we write b R a / ... ...ness the definition of several types of special semirings like ∗-semirings, congruence simple semirings and so on. We do not intend to give all defin... ...efinitions and results quoted in this book. DEFINITION (MONICO, CHRIS): A congruence relation on a semiring S is an equivalence relation ~ that al... ... x 2 1 2 1 2 1 2 1 2 1 for all x 1 , x 2 ∈ S. A semiring that admits no congruence relation other than the trivial ones, identity S and S × S is... ...ce relation other than the trivial ones, identity S and S × S is said to be congruence simple or c-simple. He has proved if S is a finite field t... ...n a polynomial semiring. PAMS 56 45-50, 1976. 3. Monico, Chris On finite congruence simple semiring. http://arxiv.org/PS_cache/math/pdf/0205/020...

... Other Objects 24 Chapter 2. Hilbert’s Axioms 27 Incidence 27 Betweenness 35 Congruence 41 Parallels 45 Chapter 3. Smarandache Geometries 53 Pa... ...s-manifolds are a very restricted subclass of the 7 polyhedral surfaces. The relationship between polyhedral surfaces and Riemannian manifolds is... ... The non-Euclidean vertices introduce a sort of curvature, and this affects the relationships between s-lines. The topology of an s-manifold can all... ...lines being parallel or not parallel at different points along them, since this relationship between s-lines changes as 21 we move from point to ... ...y, to change from region to region, this does not necessarily carry over to the relationships between lines, or 26 geodesics. This is, in fact, o... ...ll make it impossible to S-deny all of Hilbert’s axioms, and our choice to have congruence to be an equivalence relation further reduces our ability ... ...7. Here the s- lines a and AEB pass on either side of an elliptic vertex. 41 Congruence Hilbert’s axioms of congruence are as follows [14]. II... ...uent if they have the same length, so this definition agrees with the notion of congruence of segments in Euclidean geometry. Figure 19. The s-s... ... s-segment AB (the longer one) is not s-congruent to the s-segment BA. With s-congruence defined this way, an s-segment AB is always s-congruent to...

...ncircle) we obtain p 0 = r R p: (4) 9 Now, Euler’s inequality 2r R gives relation (3). For the proof of (2) we shall apply the standard algebraic in... ...Geometric inequalities (Hungarian), Ed. Dacia, Cluj, 1988. 4. J. S andor, Relations between the elements of a triangle and its podaire triangle, Mat.... ...ctively. Then AB AC = MB MC sinv sinu : (1) We wish to mention here that relation (1) appears also in my book [3] on page 112, where it is used for ... ... of a triangle, we get BD CD = A(ABD) A(ACD) = AB sin AC sin(A ) (i.e. relation (1) with u = , v = ). Similarly one has BE CE = AB sin(A ... ... the title is y 2 =x p + 1 (p> 3; prime) (2) In 1964 Chao Ko [1], by using congruence-theory has shown that this equation hasn’t solutions in positive... ...rs, Smarandache No- tions J., 10(1999), no.1-2-3, 114-115. 2. S.M. Ruiz, A congruence with Smarandache’s function, Smarandache Notions J., 10(1999), n... ...ality (31). Remark. We note that there exist in the literature a number of congruence properties of the function ’. E.g. it is known that nj’(a n 1)... ...u)j’(v) implies (known property of ’) what we have stated. The most famous congruence property of ’ is the following Conjecture. (D.H. Lehmer (see [4]... ...onjecture. (D.H. Lehmer (see [4])) If ’(n)j(n 1), then n = prime. Another congruence property of ’ is contained in Euler’s theorem: mj(a ’(m) 1) fo...

... Preface 5 1. Preliminary notions 1.1 Binary Relation 7 1.2 Mappings 9 1.3 Semigroup and Smarandache Semigroup... ...marandache semigroups. In Chapter one, we introduce some basic notation, Binary relations, mappings and the concept of semigroup and Smarandache sem... ...appings and the concept of semigroup and Smarandache semigroup. 1.1 Binary Relation Let A be any non-empty set. We consider the Cartesian pr... ... 8 diagonal of A × A. A subset S of A × A is said to define an equivalence relation on A if (a, a) ∈ S for all a ∈ A (a, b) ∈ S implies (b, ... ... Instead of speaking about subsets of A × A we can now speak about a binary relation (one between two elements of A) on A itself, by defining b t... ... = a (a -1 h -1 ) = h -1 ∈ H since H is a subgroup of G. By the definition of congruence mod H this implies that ha ∈ [a] for every h ∈ H, and so H...

...love to give pleasure back to the host. Another example of a giving-receiving relationship is be- tween parents and children. Actually, the child is ... ... 62 T H E FA M I LY U N I T Dr. Satinover: What is the Kabbalistic approach to relation- ships between men and women at the start of the 21 st centur... ...an be together, marching together on the path of self-correction and reaching congruence with the Up- per Force. By doing that, they will complement ... ... corrections, they will come to the right connection in such a way that their relationship will resemble the Upper Force. The difference between what... .... As the ego runs amuck, people cannot stand to be near one an- other. Family relationships in general, and spousal relationships in particular, are... ...t we feel in reality adheres to the principle of “equivalence of Form,” the “congruence principle.” In each of our five senses, we perceive a certai... ...e fantasy. The purpose of the wisdom of Kabbalah is one: to bring humanity to congruence with the Creator through gradual correction. Kabbalah is su... ...induced in the Light itself. All that we perceive is our measure of internal congruence with the Light, nothing more. From all we have said thus far... ..., and their systems. But for a researcher to discover all that, there must be congruence of form between the researcher and the level of these Force...

...p (S-strongly commutative loop). The author is requested to obtain the relations between these concepts. THEOREM [72]: Let L be a power associ... ...easily obtained. Hence left for the reader to prove. We now give some relation between the quasi regular elements and idempotents, how the exist... ...dempotent element cannot be right quasi regular. 25 Now we give the relation between the J(FL) and W(FL) in a loop ring. THEOREM 2.2.10: Le... ...rongly ideal ring (S-strongly ideal ring). We do not have any form of relations existing between them but we have several problems in this direc... ...ould be if L is a loop having P (L) to be the pseudo commutator; find the relation between P (RL) and R (P (L)). DEFINITION 2.4.20: Let RL be a l... ...f H is an ideal of L, then the quotient L/H which consists of all cosets (congruence classes mod H) is defined. For X ∈ L denote the congruence class...

...hereA,B,C are analytic functions. I emphatically remark that the geometric relations between the components of the metric tensor of (2a) are precisely... ...lthough the absolute value is suppressed in (1) and (7a)). The geometrical relations between the components of the metric tensor are inviolable. There... ...ue arbitrary nature of the origin r 0 . Therefore, the correct geometrical relations have gone unrecognized by the orthodox analysis. I note, for inst... ...located at somer =r 0 in parameter space, wherer, r 0 ∈ . The geometrical relationships between the components of the metric tensor of (1) must be pr... ... q p C(D)+ q p C(D)−α K , (20) D=|r−r 0 |, K = const. The relationship betweenr andR p is, asr→r ± 0 , R p (r)→0 + , or equivalently,... ...follows from this. A smooth feld on time-like vectors of each frame defnes congruence of lines that are tangent to this feld. We say that each line is...

...R be a ring. We say R is a Smarandache filial ring (S-filial ring) if the relation S-ideal in R is transitive, that is if a S-subring J is an S-ideal... ... a ring, a pair x, y ∈ R is said to have a Smarandache subring right link relation (S-subring right link relation) if there exists a S-subring P in ... ... \ {x, y} such that x ∈ Py and y ∈ P x. Similarly Smarandache subring link relation (S-subring link relation) if x ∈ y P and y ∈ x P. If it has both ... ... if x ∈ y P and y ∈ x P. If it has both a Smarandache left and right link relation for the same S-subring P, then we say x and y have a Smarandache ... ... using groupoids over rings. These non-associative rings enjoy a varied relation only when they satisfy the special types of identities like Moufa... ...ngs, Contrib. Gen. Alg., 8, 189-204 (1992). 38. MONICO, Chris, On finite congruence simple semiring. http://arxiv.org/PS_cache/math/pdf/0205/020508...