Sir Charles A. Vector analysis and multiple algebra. Planetary Circulations Project. Scient: Webster's Timeline History - Scientia - 26 pages. Scientia atmospherica Sinica. Scientia Chimica by Franz Runge. Scientia Magna book series, Vol. Vasantha Kadasamy. Scientia Magna international book series , Vol. Zhang Wenpeng. Scientia Magna, international book series, Vol. Scientia Magna, Vol.
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Scientific American: Webster's Timeline History - Scientific and educational books and papers by James George Needham - 32 pages. I, II, , , and in the American exile - diary which is still going on. Literary experiments he realized in his dramas: Country of the Animals, where there is no dialogue! He stated:. Only their ideas and publications counted. We couldn't publish almost anything.
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Then, I said: Let's do literature Let's write Simply: literature-object! Therefore, a mute protest we did! Later, I based it on contradictions. Because we lived in that society a double life: an official one - propagated by the political system, and another one real. In mass-media it was promulgated that 'our life is wonderful', but in reality 'our life was miserable'.
The paradox flourishing! And then we took the creation in derision, in inverse sense, in a syncretism way. Thus the paradoxism was born. The folk jokes, at great fashion in Ceausescu's 'Epoch', as an intellectual breathing, were superb springs. The "No" and "Anti" from my paradoxist manifestos had a creative character, not at all nihilistic.
In he was invited speaker in Brazil Universidad do Blumenau, etc. While "Paradoxist Distichs" introduces new species of poetry with fixed form. Eventually he edited three International Anthologies on Paradoxism with texts from about writers from around the world in many languages. The last drama, that pioneers no dialogue on the stage, was awarded at the International Theatrical Festival of Casablanca His first novel is called "NonNovel" and satirizes the dictatorship in a gloomy way, by various styles and artifice within one same style. His experimental albums "Outer-Art" Vol.
Art was for Dr. Smarandache a hobby. He did:. Levenard, I. Rotaru, A. Twelve books were published that analyze his literary creation, among them: "Paradoxism's Aesthetics" by Titu Popescu , and "Paradoxism and Postmodernism" by Ion Soare In he was nominated for the Nobel Prize in Literature. In mathematics he introduced the degree of negation of an axiom or of a theorem in geometry see the Smarandache geometries which can be partially Euclidean and partially non-Euclidean, , the multi-structure see the Smarandache n-structures, where a weak structure contains an island of a stronger structure , and multi-space a combination of heterogeneous spaces.
He created and studied many sequences and functions in number theory. He generalized the fuzzy, intuitive, paraconsistent, multi-valent, dialetheist logics to the 'neutrosophic logic' also in the Denis Howe's Dictionary of Computing, England and, similarly, he generalized the fuzzy set to the 'neutrosophic set' and its derivatives: 'paraconsistent set', 'intuitionistic set', 'dialethist set', 'paradoxist set', 'tautological set'. Since , together with Dr. In he designed an algorithm for the Unification of Fusion Theories and rules UFT used in bioinformatics, robotics, military.
In physics he found a series of paradoxes see the quantum smarandache paradoxes , and.
Also, considered the possibility of a third form of matter, called unmatter, which is combination of matter and antimatter or quarks and antiquarks :. In philosophy he introduced in the 'neutrosophy', as a generalization of Hegel's dialectic, which is the basement of his researches in mathematics and economics, such as 'neutrosophic logic', 'neutrosophic set', 'neutrosophic probability', 'neutrosophic statistics'. Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra.
This theory considers every notion or idea together with its opposite or negation and the spectrum of "neutralities" i. The and ideas together are referred to as. Other small contributions he had in psychology:. Very prolific, he is the author, co-author, and editor of over books published by thirty five publishing houses such as university and college presses, professional scientific and literary presses, such as Springer Verlag in print , Univ.
Hundreds of articles, books, and reviews have been written about his activity around the world. The books can be downloaded from this. Digital Library of Science:. As a Globe Trekker he visited over 35 countries that he wrote about in his memories. International Conferences:. August , , organized by Dr. Seleacu, University of Craiova, Romania. In this paper, we will consider a structure of principal filter on po-semigroups. By using the relation N which is the smallest complete semilattice congruence on any po-semigroup S, we will observe that N on any po-semigroup S is the equality relation if and only if S is a semilattice and N is the universal relation if and only if S is the only principal filter.
We first recall some basic notions and terminologies from  and . Suppose that S is a po-semigroup and T a subsemigroup of S. For every a S there is a unique smallest filter of S containing a, denoted by N a , which is called the principal filter generated by a. A congruence on a po-semigroup S is a semilattice congruence if for any a, b S, a2 , a and ab, ba. A semilattice congruence on S is called a complete semilattice congruence if for any a, b S, a b implies a, ab. According to , N on a po-semigroup S is the smallest complete semilattice congruence on S.
Then Na is a semiprime ideal of N a. S Lemma 2. Let S be a po-semigroup. Let S be a semilattice. Hence, [a is a subsemigroup of S. To prove that [a is a filter containing a, we suppose that b, c S such that bc [a.
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This shows that b [a , c [a. Since [a [a always holds, [a is a filter containing a, as required. Let T be a filter containing a. By the definition of filters, we have [T T. Since a T , then [a [T T. Suppose that aN b for a, b S. This shows that S is a semilattice as required. Moreover, the partial order on S is the natural order of semilattice.
Theorem 4. Since N is the smallest complete semilattice congruence on S, it is trivial that i iii. Theorem 5. It is clear that S is a semilattice congruence S class of S and so b S. We have proved that Na S. Consequently, Na S. This is exactly the union of some N -classes. This implies that xy S S S and Y. Hence, xy T and T is a subsemigroup of S. Since Y is a semilattice, it is easy to see that Y and Y. Thus, we have x T and y T. This shows that xy S S S. Since is a complete semilattice congruence, we can see that xy, y. From y S , we immediately have xy S.
Hence, x T and [T T as required. We have shown that T is a filter. T is the union of all the S N -classes which are greater than Na. S Conversely, suppose that a, b and a S , then we have b S. The following Corollary is a direct result of Theorem 5.
Corollary 6. Corollary 7. Suppose that is a maximal element in Y. This shows that N a is a subsemigroup of S. From a S we know that S is not empty. The Hasse diagram of S is shown below. From the Cayley table above, we know that S is a semilattice. References  Kehayopulu N. Japonica, 36 , No. Gao and K. Shum, On cyclic commutative po-semigroups, PU. A, 8 , No. Cao and X. Xu, Nil-extensions of simple po-semigroups, Communications in Algebra, 28 , No. Howie, An introduction to semigroup theory, Academic Press, London, M, Yan J.
Z and K. Shum, On Principal Filters of Po-semigroups, to appear. In this paper, we studied the convergent property. In problem 27 of , Professor F. Smarandache ask us to study the properties of b2 n. About this problem, some authors have studied it, for example, Liu Hongyan and Gou Su  used the elementary method to study the mean value properties of b2 n and b2 1 n. Zhang Hongli and Wang Yang  studied the mean value of b2 n , and obtained an asymptotic formula by using the analytic method.
About this problem, many scholars have studied it and obtained some interesting results. For example, Xu Zhefeng  proved the following asymptotic formula: X nx 1 This. That is, we will prove the following: Theorem. Let k 2 be an integer, then for any real number 1, the infinity series. For additive square complements and additive cubic complements, we have the identities.
Proof of the theorem In this section, we will complete the proof of Theorem. Only problems, not Solutions, Xiquan Publ.
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Y and Gou S. On a problem of F. On the mean value of divisor function for square complements, Basic Science Journal of Textile University, 15 , On the additive k-th power complements, Research on Smarandache problems in number theory Collected papers , Hexis, , pp. On the asymptotic property of divisor function for additive complement, Research on Smarandache problems in number theory Collected papers , Hexis, , pp. China Abstract In this paper we give an explicit formula for the n times Smarandache reverse auto correlated sequence of natural numbers.
Recentely, Muthy  proposed the following conjecture: Conjecture. Murthy, Smarandache reverse auto correlated sequences and some Fibonacci derived Smarandache sequences, Smarandache Notions J. Niven, Formal power series, Amer. Monthly, 76 , This study conducted by Computer Algebra System namely, Maple 8.
Using the following procedure, we can verify the number of unrestricted partitions of the Smarandache numbers n is denoted by P s n. With the Maple V. Below the first Smarandache numbers verifying by the above procedure:. By using the above procedure, we can got the first partitions of Smarandache numbers as follows:. Now, the following procedure, we can verify the unrestricted partitions of the Smarandache numbers.
References  G. Monagan etc. Combining classical groups with Smarandache multi-spaces, the conception of multi-group spaces is introduced in this paper, which is a generalization of the classical algebraic structures, such as the group, the filed, the body, , etc.. Similar to groups, some characteristics of multi-group spaces are obtained in this paper.
Keywords multi-space; group; multi-group space; Jordan-H older theorem. Introduction The notion of multi-spaces is introduced by Smarandache in  under his idea of hybrid mathematics: combining different fields into a unifying field . Today, this idea is widely accepted by the world of sciences. For mathematics, definite or exact solution under a given condition is not the only object for mathematician. New creation power has emerged. New era for mathematics has come now. A Smarandache multi-space is defined by Definition 1.
The conception of multi-group spaces is a generalization of the classical algebraic structures, such as the group, the filed, the body, , etc. Therefore, G1 is a multi-group subspace of G. For a finite multi-group subspace, we get the following criterion. A subset G f1 , G f1 ; is complete.
Notice that for a multi-group space G, complete. Whence, there exists an Since H T e Gi ; i. Whence, e y H. Then we get the following result by the previous proof. We have the following result. For the operation is Therefore, H e T Gi ; i is a normal subgroup chosen arbitrarily, we know that for any integer i, 1 i n, H of Gi ; i or an empty set. STEP 1: Construct a series e. G e 1l G 1 under the operation 1.
STEP 2: If a series e k1 l. This programming is terminated until the series e n1. G G n n 1 has be constructed under the operation n. The number m is called the length of the series of normal multi-group subspaces. For a series e. For a maximal series of finite then H normal multi-group subspaces, we have the following result. The proof is by the induction on the integer n. Assume the assertion is true for cases of n k. We prove it is also true in the case. Not loss of generality, assume the order of binary operations in O G being 1 2 n and the composition series of the group G1 , 1 being G1.
By Jordan-Holder theorem, we know the length of this composition series is a constant, dependent only on G1 ; 1. According to Theorem 3. Therefore, the length of a subspaces is only dependent on G 1 e maximal series of normal multi-group subspaces is also a constant, only dependent on G. Applying the induction principle, we know that the length of a maximal series of normal e e is a constant under an oriented operations multi-group subspaces of G O G , only dependent e on G itself.
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As a special case, we get the following corollary. Jordan-H older theorem For a finite group G, the length of the composition series is a constant, only dependent on G. Open Problems on Multi-group Spaces Problem 3. In group theory, we know the following decomposition results  and  for a group. Let G be a finite -group. Then G can be uniquely decomposed as a direct product of finite non-decomposition -subgroups.
Each finite Abelian group is a direct product of its Sylow p-subgroups. Then Problem 3. Whether can we establish a decomposition theory for multi-group spaces similar to above two results in group theory, especially, for finite multi-group spaces? Problem 3. For finite multi-group spaces, whether can we find all simple multi-group spaces? For finite groups, we know that there are four simple group classes  : Class 1: the cyclic groups of prime order;.
Class 2: the alternating groups An , n 5; Class 3: the 16 groups of Lie types; Class 4: the 26 sporadic simple groups. Problem 2. Then Problem 2. Birkhoff and S. Vasantha Kandasamy and F. In , Sloane has defined the multiplicative persistence of a number in the following manner. Multiplying together the digits of that number x1 x2 xn , another number N 0 results. If this process is iterated, eventually a single digit number will be produced. The number of steps to reach a single digit number is referred to as the persistence of the original number N. Here is an example: 48 32 6. In this case, the persistence of is 5.
Of course, that concept can be extended to any base b. In , Sloane conjectured that, in base 10, there is a number c such that no number has persistence greater than c. According to a computer search no number smaller than with persistence greater than 11 has been found. In , Hinden defined in a similar way the additive persistence of a number where, instead of multiplication, the addition of the digits of a number is considered.
For example, the additive persistence of is equal to 2. Following the same spirit, in this article we introduce two new concepts: the Smarandache P -persistence and the Smarandache S-persistence of a prime number. If it is a prime then the process will be iterated otherwise not.
The number of steps required to X to collapse in a composite number is called the Smarandache P -persistence of prime X. As an example, lets calculate the Smarandache P -persistence of the primes 43 and 43 55; 23 29 47 75, which is 1 and 3, respectively. Of course, the Smarandache P -persistence minus 1 is equal to the number of primes that we can generate starting with the original prime X.
Before proceeding, we must highlight that there will be a class of primes with an infinite Smarandache P -persistence; that is, primes that will never collapse in a composite number. Lets give an. In this case, being the product of the digits of the prime always zero, the prime 61 will never reach a composite number.
In this article, we shall not consider that class of primes since it is not interesting. The following table gives the smallest multidigit primes with Smarandache P -persistence less than or equal to 8: Smarandache P -persistence. By looking in a greater detail at the above table, we can see that, for example, the second term of the sequence 29 is implicitly inside the chain generated by the prime In fact: 29 47 75 23 29 47 75 We can slightly modify the above table in order to avoid any prime that implicitly is inside other terms of the sequence.
Smarandache P -persistence. Now, for example, the prime will generate a chain that isnt already inside any other chain generated by the primes listed in the above table. What about primes with Smarandache P -persistence greater than 8?
Is the above sequence infinite? We will try to give an answer. Lets indicate with L the Smarandache P -persistence of a prime. Thanks to an u-basic code the occurrrencies of L for different values of N have been calculated. The interpolating function for that family of curves is given by: a N eb N L where a n and b n are two function of N. To determine the behaviour of those two functions, the values obtained interpolating the histogram of occurencies for different N have been used:.
In Figure 3, the plot of counting function versus N for 4 different L values is reported. For L 15, the number of primes is very very small less than 1 regardless the value of N and it becomes even smaller as N increases. So the following conjecture can be posed: Conjecture 1. There is an integer M such that no prime has a Smarandache P -persistence greater than M.
In other words the maximum value of Smarandache P -persistence is finite. Following a similar argumentation the Smarandache S-persistence of a prime can be defined. In particular it is the number of steps before a prime number collapse to a composite number considering the sum of the digits instead of the product as done above. For example lets calculate the Smarandache S-persistence of the prime In this case we have a Smarandache S-persistence equal to 4.
The sequence of the smallest multidigit prime with Smarandache S-persistence equal to 1, 2, 3, 4 has been found by Rivera . Moreover by following the same statistical approach used above for the Smarandache P -persistence the author has found a result similar to that obtained for the Smarandache P -persistence see  for details.
Since the statistical approach applied to the Smarandache P and S persistence gives the same result counting function always smaller than 1 for L 15 we can be confident enough to pose the following conjecture: Conjecture 2. The maximum value of the Smarandache P and S persistence is the same. References  N. Sloane, The persistence of a number, Recr. Of course, this function has many arithmetical properties, and they are studied by many people see references ,  and .
In this paper, we shall use the elementary methods to study the solvability of the equation. That is, we shall prove the following main conclusion: Theorem. Proof of the theorem In this section, we shall give the proof of the theorem in two ways, the first proof of the theorem is based on the following: m m m Lemma 1. We use induction to prove this Lemma. This completes the proof of Lemma 1. The second proof of the theorem is based on the Vinogradovs three-primes theorem which we describle as the following: Lemma 2.
Every odd integer bigger than c can be expressed as sum of three odd primes, where c is a constant large enough. Lemma 3. We will prove this lemma by induction. Now we give the second proof of the theorem. This means that the theorem is true for odd integer k 3. This means that the theorem is true for even integer k 4. This completes the second proof of Theorem.
House, Chicago, Introduction In one of my many experiments with numbers I found an interesting property which I will describe in everyday terms before going into a more formal analysis. Imagine a circular putting green on a golf course. A golfer wants to practice putting from the edge of the green. He therefore drops a large number of golf balls on the very edge of the green. He then stands on the edge of the green and is struck by the thought What might be the average distance from here to all these golf balls? We measure with the radius of the green as unit and consider the diameter of a golf ball as negligible.
The amazing result is that: the average distance is A 4. Of course, this result was not obtained by experimentation but through formal treatment of a related problem . But its similarity with the famous Buffons needle experiment  makes it interesting to compare the estimation of obtained by simulating the two experiments. This will be done at the end of the article.
A problem and its solution From a fixed point A a, 0 of a circle C, what is the average distance of all points on the circumference of C? Let dk be the kth arc. As is seen An converges rapidly. Figure 4. As P moves along the periphery of the semi-circle in figure 4 the angle v moves from 0 to.
This expectation value E corresponds to the classical average which deals with a discrete random variable. This result has been obtained through a geometric consideration and a simple integration. It would be interesting to compare this with a proof of this formula by analytical means. Comparing this with the simple way in which the result was found earlier its like using a sledge hammer to kill a mosquito.
Simulation experiments A frame work in which we can compare simulation experiments for Buffons needle experiment, which will be described below, and our golf ball experiments will be set up. I will henceforth refer to the two cases as Needles and Golf Balls respectively. The same random number generator is used and randomized in the same way in the two experiments.
Ten experiments were carried out in each case. In each experiment needles respectively golf balls were dropped. Refer to figure 5. Our random variables are r and v. The average value 3. Calculation is made for a random distribution of balls. The difference from the true value to 5 decimals is only 0. Needles and Golf Balls - Comparison The simulation programs used earlier were adapted for multiple runs, in each case. Each run consists of tossing the needle and the golf ball times each.
The average and. In addition the frequency of simulation values of for each interval of length 0. Figure 6 shows the result for needles: Average simulation result for 3. The difference between the simulation result and to four decimals is 0. Figure 7 shows the result for golf balls: Average simulation result for 3. At first sight it is surprising that d in 12 can be chosen arbitrarily as long as it is not smaller that L. The explanation is that the larger we choose d the larger will be the standard deviation, i.
The golf ball experiment is much better behaved. For the same number of tosses the average is much closer to with a trend curve that is closer to the average. Combining Smarandache multi-spaces with classical metric spaces, the conception of multi-metric spaces is introduced. Some characteristics of multimetric spaces are obtained and the Banachs fixed-point theorem is generalized in this paper. Introduction The notion of multi-spaces is introduced by Smarandache in  under his idea of hybrid mathematics: combining different fields into a unifying field  , which is defined as follows.
Definition 1. By combining Smarandache multi-spaces with classical metric spaces, a new kind of spaces called multi-metric spaces is found, which is defined in the following. For terminology and notations not defined here can be seen in  ,  for terminologies in the metric space and in ,   for multi-spaces and logics. Characteristics of multi-metric spaces For metrics on spaces, we have the following result. We only need to prove that F 1 , 2 , , m satisfies the metric conditions for x, y, z M. Now by i and iii , we get that. Therefore, F 1 , 2 , , m is a metric on M.
According to Theorem 2. Whether a convergent sequence can has more than one limit point? The following result answers this question. Any convergent sequence in a multi-metric space is a bounded points set. The necessity of these conditions is by Theorem 2. Now we prove the sufficiency. According to Theorem 4. For a completed multi-metric space, we obtain two important results similar to the metric space theory in classical mathematics. For a -disk sequence Theorem 2. By the Proof. N such that if m N , then xm Ml by Theorem 2. If f is connected at every point of M f f said a continuous mapping from M1 to M2.
Denoted by T the x M f. Then on M. By definition, we know that for any integer n, n 1, there exists an integer i, 1 i m such that xn , yn Mi. Whence, we inductively get that 0 i xn , yn n 1 x0 , y0. Therefore, there exists an integer N1 such that xn , yn Mi0 if n N1. Now if n N1 , we have that. Whence, we know that. Since n. Similar consider the points in Mi , 2 i m, we get that 1 T m.
Banach Let M be a metric space and T a contraction on M. Then T has just one fixed point.
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Open problems for a multi-metric space On a classical notion, only one metric maybe considered in a space to ensure the same on all the times and on all the situations. Essentially, this notion is based on an assumption that spaces are homogeneous. In fact, it is not true in general. Multi-Metric spaces can be used to simplify or beautify geometrical figures and algebraic equations.
One example is shown in Fig. Generally, in a multi-metric space we can simplify a polynomial similar to the approach used in projective geometry. Whether this approach can be contributed to mathematics with metrics? Whether can it be used for classifying 3-dimensional manifolds? Simplify equations or problems to linear problems. Abraham, J.
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Marsden and T. Chern and W. On a equation for the square complements Li Zhanhu1,2 1. China 2. China Abstract For any positive integer n, let a n denotes the square complements of n. That is, a n is the smallest positive integer such that na n is a perfect square number. In this paper, we use the analytic method to study the number of the solutions of the equation involving the square complements, and obtain its all solutions of this equation. Introduction and Result For any positive integer n, the square complements a n is defined as the smallest positive integer such that na n is a perfect square.
Smarandache asked us to study the properties of a n. About this problem, some authors had studied it before. In this paper, we use the analytic method to study the number of the solution of the equation involving square complements, and give all solutions of the equation. The equation. Some lemmas To complete the proof of the theorem, we need the following lemmas.