Showing papers on "Prime (order theory) published in 1970"

Statistical methods for research workers

Abstract: The prime object of this book is to put into the hands of research workers, and especially of biologists, the means of applying statistical tests accurately to numerical data accumulated in their own laboratories or available in the literature.

Fitting continuous probability density functions over [0, infty ) using information theory ideas (Corresp.)

Abstract: The problem of estimating a probability density function p(x) over [0, \infty) when several low-order moments are known is considered. A computational procedure, based on information theory, is used to obtain the maximum entropy estimates of p(x) in a number of cases. The situation when the first two moments only are known is considered in some detail. A table is included for estimating p(x) when given \mu_1 ^{\prime} , \mu_2 ^{\prime} with \mu_2 ^{\prime} \leq 2(\mu_1 ^{\prime}) ^ 2 .

A table of the first factor for prime cyclotomic fields

Abstract: The first factor of the prime cyclotomic fields for all primes < 200 is computed by means of a determinantal formula, correcting some errors in tables of Kummer. Let p = 2m + 1 be an odd prime. Let f be a primitive pth root of unity, and let R be the field of rationals. It is well known that if h is the class number of the cyclotomic field Ä(f), and h0 the class number of the totally real subfield Ä(f + 1/f), then h is divisible by h0. The quotient h/ho is denoted by h*, and is known as the first factor of h. A complete discussion of these matters is to be found in the beautiful book [1] by Borevic and Safarevic, where a table (uncredited) of h* is given for all odd primes p < 100. Presumably, this table is due to Kummer, who computed h* for all odd primes ^ 163 (see [2] and [3]). The numbers h* are quite difficult to compute, and it is of some interest to verify and to extend the above-mentioned tables. The importance of h* stems from the fact that the prime p is irregular if and only if p divides h*. It turns out that Rummer's tables are not error-free: the values of h* corresponding to p = 103, 139, and 163 are incorrect. The fact that p = 103 is irregular, and was correctly identified to be so by Kummer, is explicable by his method of computation. Let g be a primitive root modulo p. Define 9n = gn p[gn/p] , n = 0, 1, 2, • • • . Let 8 be a primitive (p — l)st root of unity. Then the first factor h* is given by the formula 1 7747 /ys\\ 7-7 //.3\\ T4/Qp—2\\ where h* = ——; F(0)F(tf) • • • F(0 ) , (2p)m-1 V—2 Fix) = X) 9\"Xn ■

Individual differences in "definition of the situation":

Abstract: State action is the action taken by those acting in the name of the state. Hence, the state is its decisionmakers. State X as actor is translated into its decision-makers as actors. It is also one of our basic choices to take as our prime analytical objective the re-creation of the "world" of the decisionmakers as they view it. The manner in which they define situations becomes another way of saying how the state oriented to action and why [Snyder et al., 1962, p. 65; italics in the original].

A search for simple groups of order less than one million

Abstract: Publisher Summary This chapter describes the search for simple groups of order less than one million. In 1900, L. E. Dickson listed 53 known simple groups of composite order less than one million. Three more groups have been added to this list since that time. A group of order 29,120 was discovered by M. Suzuki in 1960, the first of an infinite class, and one of order 175,560 was discovered by Z. Janko in 1965, which appears to be isolated. Z. Janko announced that a simple group with certain properties would have order 604,800 and have a specific character table. The chapter describes the construction of a simple group of order 604,800. The notation for the classical simple groups presented in this chapter is essentially that used in Artin. GF(q) denotes the finite field with q elements where q = pr, p being a prime. The chapter also discusses the role of Brauer theory of modular characters in this search, particularly the theory for groups whose order is divisible by exactly the first power of a prime.

A convergent splitting of matrices

Abstract: LetA, M, N ben ×n real matrices, letA = M? N, letA andM be nonsingular, letM? y ? 0 implyN? y ? 0, and letA? y ? 0 implyN? y ? 0 (where the prime denotes the transpose). Then the spectral radius?(M ?1 N) ofM ?1 N is less than one, and the iterative processx i+1 =M ?1 N x i +M ?1 b converges to the solution ofA x = b starting from anyx 0.

Note on the distribution of Ramanujan’s tau function

Abstract: According to a conjecture of Sato and Tate, the angle 0 whose cosine is ITi(p)p-11/2, where r is Ramanujan's function and p a prime, is distributed over (0, 7r) accord- ing to a sin20 law. The paper reports on a test of this conjecture for the 1229 primes under 10000. Extreme values of 0 are also given.

Simultaneous Solution of Line-Out and Open-End Line-to-Ground Short Circuits

Abstract: One prime objective in developing a computer program is to minimize the running time requirements. For short-circuit programs, one major obstacle is the time absorbed in processing mutually coupled elements for line-out and open-end conditions.

The hyperquasicenter of a finite group. II

Abstract: The role of Theorem B of Hall and Higman has been explained in detail to complete the proof of the fact that the hyperquasicenter is the largest supersolvably immersed subgroup. Other results included contain some sufficient conditions for supersolvability of a group and for the nontriviality of its center. I. The proof of Theorem 1.2 in The hyperquasicenter of afinite group. I (Proc. Amer. Math. Soc. 26 (1970)) needs to be elaborated. One needs to consider only the case when r is positive and is a prime and induction may be applied to show that x7 is a p-group. Once this is done the proof may be completed by breaking it up into two cases: (i) r=p. In this case the proof may be completed by induction if lxl is divisible by a prime different from p, or else one needs to consider the case when G=(x)(y) is a p-group. In this case the proof may be completed as given in the paper by taking Z to be a subgroup of order p. (ii) r$p. In this case one needs to use Ore's theorem [1, p. 438, Theorem 16] namely that every quasinormal subgroup of a group is subnormal. In this connection, it may be said that it was brought to the author's notice that the proofs of Theorems 1.2 and 1.8 needed to be modified. These imperfections are attributable to complete breakdown of communications between the author and the referee. The proof of Theorem 1.8 follows from Theorem 6.1 of Baer in [4], and that requires that G induces in every chief factor contained in P, a Sylowp-subgroup of the quasicenter, an abelian group of automorphisms. Th& verification of the fact that G does satisfy this requirement is crucially dependent on the well-known Theorem B of Hall and Higman in [2]. Several other results have been included without proof. They are not difficult to verify.

Structures determined by prime ideals of rings of functions

Abstract: Introduction. Let 3 and Y31 respectively denote the category of commutative rings with unity and the category of completely regular Hausdorff spaces; also, ' denotes the full-subcategory of -31 whose objects are compact, and (TD the full-subcategory of ' whose objects are totally-disconnected. The collection of prime ideals of A E 3 is the underlying set for an object KA E ,TD and K is contravariantly ftinctorial. If C denotes the contravariant functor which assigns to each X E S31 the ring C(X) of real-valued continuous functions on X, then the resulting functor KC is the domain of a natural transformation 0: KC -> /3, where / denotes the Stone-Cech reflection of -31 into W. The prime z-ideals of C(X) also furnish such a space (X, functor 4 and natural transformation 0: 4 --> /. In the appropriate category, 4 fills in a diagram which exhibits / as a push-out. Topological properties of KC(X) and (X are studied and (X is characterized as a certain compactification of Xp, the P-topology on X, which helps establish the place of (X between gXp and gX. The above results are applied in an investigation of the continuous and orderpreserved image of 4 Y in both of (X and gX arising from f: Y -> X. As one consequence, the prime z-ideal structure and the minimal prime ideal structure associated with X is illuminated by the corresponding structures associated with certain subspaces of X; as another, a convenient simplification and unification is provided for approaching several types of problems found in the literature on prime ideals of C(X).

Generalized Combining of Elements From Finite Fields

Abstract: In a recent paper Raktoe [1969] has presented a new approach and also a generalized technique to combining elements from distinct finite fields. The results however were related only to distinct prime fields i.e. each of the Galois fields in question consisted of residue classes modulo a prime. This paper solves the problem of combining elements of the most general case, i.e. the fields are not necessarily based on distinct primes and they can be prime powered.

Two examples in varieties of monoids

Abstract: This note exhibits an infinite independent set of laws for monoids (viz. ( x p y p ) 2 = ( y p x p ) 2 , p prime) and a variety of groups defined by one law (which is x 2 y 2 = y 2 x 2 ) such that the smallest containing variety of monoids is not finitely definable.

The levitzki radical in jordan rings

Abstract: The main purpose of this paper is to give an external characterization of the Levitzki radical of a Jordan ring 2f as the intersection of a family of prime ideals W. This characterization coincides with that of associative rings which was given by Babic in [1I]. Applying this characterization, it is easy to see that the Levitzki radical of a Jordan ring contains the prime radical of the same ring. For associative rings the same statement is well known, since the prime radical in associative rings is called the Baer radical. If the minimal condition on ideals holds on Jordan ring 2, then the Levitzki radical, L(2f), and the prime radical, R(2f) of 2f coincide. Throughout this paper, any Jordan ring 2f, that is a (nonassociative) ring satisfying (1) ab = ba, and (2) a2(ab) =a(a2b) for all a, b in 2t, and any of its subrings satisfy the conditions, (3) 2a = 0 implies a = 0 and (4) if a is in a subring C of 2 then there exists a unique element x in C such that 2x = a. In a Jordan ring, the following identity (*) is well known. One can find the proof in [3 ].

A Remark about Noncommutative Integral Extensions

Abstract: Let B be a ring with unity, A a imitai subring of the centre Cof B. Suppose further that B is A-integral. (That is, every element of B satisfies a monic polynomial with coefficients in A.) Under these assumptions, Hoechsmann [2] showed that "contraction to A" is a mapping from: (1) The prime ideals of B onto the prime ideals of A, (2) The maximal ideals of B onto the maximal ideals of A. In this note we show that, under additional assumptions, a noncommutative version of the rest of the Cohen-Seidenberg "going up theorem" can be established.

Single-Error-Correcting Nonbinary Arithmetic Codes

Abstract: Except for some elementary definitions and fundamentals, the theory of AN code is by and large the theory of binary (radix = 2) arithmetic codes. It is often believed (erroneously) that this theory can be readily generalized to any nonbinary radix. The very fundamental theorems of Brown and Peterson on single-error-correcting codes have been derived for the binary case only. Whereas a generalized version of Brown's theorem can be stated and proved relatively easily (as shown here), the one for Peterson's theorem is not forthcoming. However, we have succeeded in deriving a theorem for the ternary case (radix = 3) somewhat along the lines of the Peterson's theorem as follows. Let M_3 (A, d) denote the smallest positive integer such that the arithmetic weight of A M_3 (A, d) in ternary representation is less than d . Also ley A = 2p for some odd prime p . Then 3 is a primitive element of GF(p) if and only if \begin{equation} M_3 (A, 3)=(3^{(p-1)/2} + 1)/A. \end{equation}

On Cut Elimination in Intuitionistic Systems of Analysis

Abstract: Publisher Summary This chapter focusses on cut elimination in intuitionistic systems of analysis. Certain intuitionistic systems of analysis, formulated in terms of the sequential calculus are considered in the chapter. The proofs of certain results, in part on cut elimination, about these systems are outlined. The chapter reviews the Gentzen's second consistency proof. A prime sequent is called verifiable if it is true whenever the free variables are replaced by constant terms. A prime sequent is called saturated if it contains only prime formulas t = q with t, q saturated. A saturated prime sequent is called numerically true if it is true under the usual interpretation.

Primary factorization in a weak bezout domain

Abstract: It is well knlown that in a weak Bezout domain each prime factorization of an element is uiniquie up to similarity. In this paper, a corresponding extension to primary factorizaLions is ob- tained. An integral domain with unity is called a weak Bezout domain' by P. M. Cohn in (2 ) if the sum and intersection of any twTo principal right ideals that have nonzero intersection are again principal. There it is shown that in such a ring any prime factorization of an element is unique up to order of factors and similarity. This generalizes the familiar result for commutative wTeak Bezout domains (called Bezout rings) that any prinme factorization of an element is unique up to order of factors and associates. Just as in the commutative case one also con- siders primary decompositions, and when they exist the question of uniqueness arises. In this note we study primary factorizations in a weak Bezout domain, and we show that any primary factorization of an element is unique up to order of factors and similarity (associates for the commutative case). In what follows R denotes a weak Bezout domain, R* R\ 0}, and U denotes the group of units of R. We assume the reader is famil- iar with the gcld (a, b)1, lcrm (a, bl,r, etc., and similarity a-b of a, bER*. In particular, if alb'=ba' in R* then (a, b)i= (a', b')r=l iff (a, b)-= (a', b')l=ab'= ba', in which case a-a' and b-b'. We call c an S-factor of b iff c-c' and c' is a factor of b. Left and right S-factors are defined in the obvious way. If pER*\U, p is called prime iff in every factorization P=PIP2 either p'i U or P2E U. An element b ER*\ U is called p-prinVary for some prime p iff every prinme factor of b is similar to p and every nonunit r- or i-factor of b has at least one prime factor. We call c = cIc2 cn, a primary factorization of c iff

On a result of Cassels

Abstract: Let α be an irrational algebraic number of degree k over the rationals. Let K denote the field generated by α over the rationals and let a denote the ideal denominator of α. Then Cassels [3] has shown that for sufficiently large integral N > 0 distinctly more than half the integers n , are such that ( n +α)a is divisible by a prime ideal p n which does not divide ( m +α)a for any integer m ≠ n satisfying . The purpose of this note is to point out that minor modification of Cassel's proof enables the extension of the interval for n from to , and to derive results on the proportion of values n , for which the values f(n) of a given integral polynomial in n are divisible by a prime p > N .

A characterization of the nil radical of a ring.

Abstract: It is shown that all primitive ideals are of the form p = P(Γ, S, ψ) and that a ring R is nil if and only if it has no prime ideals of the form P = P(Γ9 S, ψ). It is shown that the nil radical of any ring is the intersection of ail prime ideals P = P(Γ, S, φ). It is shown that if P = P(Γ, S, φ) for all prime ideals P Q R then the nil and Baer radicals coincide for all homomorphic images of R. If the nil and Baer radicals coincide for all homomorphic images of R, it is shown that any prime ideal P of R is contained in a prime ideal P' — Pr(Γ, S, φ). Finally, by consideration of prime ideals P = P(Γ, S, φ), two theorems are proved giving information about rings satisfying very special conditions.