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Showing papers on "Product (mathematics) published in 1971"


Journal ArticleDOI
TL;DR: This article distinguishes determinant attributes from nondeterminant ones and presents a cross-validation method for testing alternative approaches to identifying these key attributes.
Abstract: Not all product attributes are equally important in determining consumer preferences. This article distinguishes determinant attributes from nondeterminant ones and presents a cross-validation meth...

328 citations




Journal ArticleDOI
01 Jun 1971
TL;DR: In this paper it was shown that if X is finite then every element of that is not bijective is expressible as a product of idempotents, and this is indeed the case for all continuous self-maps of a topological space X.
Abstract: In an earlier paper (5) a description was given in set-theoretic terms of the semigroup generated by the idempotents of a full transformation semigroup , one of the results being that if X is finite then every element of that is not bijective is expressible as a product of idempotents. In view of this it was natural to ask whether by analogy every singular square matrix is expressibleas a product of idempotent matrices. This is indeed the case, as was shown by J. A. Erdos (2). Magill (6) has considered products of idempotents in thesemigroup of all continuous self-maps of a topological space X, but a comparable characterization of products of idempotents in this case appears to be extremelydifficult, and no solution is available yet.

122 citations


Journal ArticleDOI

90 citations


Journal ArticleDOI
TL;DR: In this paper, a more conceptual proof was devised by N. P. Dekker, and the one offered below may have some claim to be regarded as "the reason the theorem is true".
Abstract: Paul Halmos expressed [3, p. 110] the general dissatisfaction with the usual proofs of this famous and important theorem. They all make it seem like an accidental product of a computation. A more conceptual proof was devised by N. P. Dekker. In spite of the elegance of his proof, the one offered below may have some claim to be regarded as "the reason the theorem is true".

54 citations


Book
01 Jan 1971
TL;DR: In this paper, the rank of a holomorphic map is defined as a rank of the number of meromorphic lines in a meromorphic line bundle, and a counter example by Kas.
Abstract: Preface.- German letters.- The rank of a holomorphic map.- Product representations.- Meromorphic functions.- Dependence.- Proper, light, holomorphic maps.- The field .- Semi-proper maps.- Quasi-proper maps.- as a finite algebraic extension of .- Quasi-proper maps of codimension k.- Full holomorphic maps.- Globalization.- The schwarz Lemma.- Sections in meromorphic line bundles.- Preparations.- Pseudoconcave maps.- A counter example by Kas.

51 citations


Journal ArticleDOI
TL;DR: A canonical analysis of the relationship of personality traits to product use patterns suggests that the association is significant and complex, involving probable interactions among traits as mentioned in this paper, and suggests that personality traits are correlated with product use.
Abstract: A canonical analysis of the relationship of personality traits to product use patterns suggests that the association is significant and complex, involving probable interactions among traits.

49 citations


Journal ArticleDOI
TL;DR: The cyclic nature of AN codes is defined after a brief summary of previous work in this area is given and new results are shown in the determination of the range for single-error-correcting AN codes when A is the product of two odd primes p_1 and p_2.
Abstract: In this paper, the cyclic nature of AN codes is defined after a brief summary of previous work in this area is given. New results are shown in the determination of the range for single-error-correcting AN codes when A is the product of two odd primes p_1 and p_2 , given the orders of 2 modulo p_1 and modulo p_2 . The second part of the paper treats a more practical class of arithmetic codes known as separate codes. A generalized separate code, called a multiresidue code, is one in which a number N is represented as \begin{equation} [N, \mid N \mid _ {m1}, \mid N \mid _{m2}, \cdots , \mid N \mid _{mk}] \end{equation} where m_i are pairwise relatively prime integers. For each AN code, where A is composite, a multiresidue code can be derived having error-correction properties analogous to those of the AN code. Under certain natural constraints, multiresidue codes of large distance and large range (i.e., large values of N ) can be implemented. This leads to possible realization of practical single and/or multiple-error-correcting arithmetic units.

48 citations


Journal ArticleDOI
TL;DR: Sum and sum-of-product algorithms, some designed to minimize significance error, are compared and recommendations for usage are offered.
Abstract: Sum and sum-of-product algorithms, some designed to minimize significance error, are compared and recommendations for usage are offered. A simplified method of rounding is presented and the benefit of using it demonstrated. Results are presented for the IBM 360 as well as the IBM 7040 computers with obvious application to some other machines.

46 citations




Journal ArticleDOI
TL;DR: In this article, the authors presented a proof of the limiting absorption principle, under suitable hypotheses on E(x) and the A(j) matrices, and showed that the spectrum of Λ is continuous.
Abstract: Many wave propagation phenomena of classical physics are governed by systems of the Schrodinger form-iD t u+Λu=f(x,t) where 1 $$\Lambda = - iE(x)^{ - 1} \sum\limits_{j = 1}^n {(A_j D_j )} $$ , (1) E(x) and the A j are Hermitian matrices, E(x) is positive definite and the Aj are constants. If f(x, t)=e −iλt f(x) then a corresponding steady-state solution has the form u(x, t)=e−i λ tν(x) where ν(x) satisfies (Λ-λ) ν=f(x), xeR n . (2) This equation does not have a unique solution for λeR 1−{0} and it is necessary to add a radiation condition for ¦ x ¦ → ∞ which ensures that ν(X) behaves like an outgoing wave. The limiting absorption principle provides one way to construct the correct outgoing solution of (2). It is based on the fact that Λ defines a self-adjoint operator on the Hilbert space ℋ defined by the energy inner product 2 $$(u,v) = \int\limits_{R^n } {u^* } E{\text{ }}v{\text{ }}d{\text{ }}x$$ . It follows that if ζ=λ+iσ and σ≠0 then (Λ-ζ) ν=f has a unique solution 3 $$v(,\zeta ) = R_\zeta (\Lambda )f \in $$ ℋ where 4 $$R_\zeta (\Lambda ) = (\Lambda - \zeta )^{ - 1} $$ is the resolvent for Λ on ℋ. The limiting absorption principle states that 5 $$v(,\lambda ) = \mathop {\lim }\limits_{\sigma \to 0} v(,\lambda + i\sigma )$$ (3) exists, locally on R n, and defines the outgoing solution of (2). This paper presents a proof of the limiting absorption principle, under suitable hypotheses on E(x) and the A j . The proof is based on a uniqueness theorem for the steady-state problem and a coerciveness theorem for nonelliptic operators Λ of the form (1) which were recently proved by the authors. The coerciveness theorem and limiting absorption principle also provide information about the spectrum of Λ. It is proved in this paper that the point spectrum of Λ is discrete (that is, there are finitely many eigenvalues in any interval) and that the continuous spectrum of Λ is absolutely continuous.


Journal ArticleDOI
TL;DR: In this paper, it was shown that the smooth connected sum of a product of ordinary spheres with an exotic combinatorial sphere is not diffeomorphic to the original product, and this result can be generalized in a nonempty manner.
Abstract: In this paper it is proved that the smooth connected sum of a product of ordinary spheres with an exotic combinatorial sphere is never diffeomorphic to the original product. This result is extended and compared to certain related examples. Before stating the result which provides our point of departure, we recall some standard notation. If Mn is a closed smooth oriented «-manifold, the inertia group I(M) consists of all exotic «-spheres 2 such that M # S is orientation-preservingly diffeomorphic to M. Then in the first section of this paper we shall prove the following result : Theorem A. Ifn^5 and M is a product of ordinary spheres, then I(M)=Q. The special case where M is a product of two ordinary spheres was proved independently by DeSapio [6], Kawakubo [12], and the author [26]. Actually, a weaker version of Theorem A may be derived almost trivially by means of framed cobordism [14], and our proof may be considered an example of the relative difficulty of computing the intersection I{Mn) n 9Pn+1 in general. In the second section we discuss some generalizations of Theorem A, and in 2.3 it is shown that the methods of §1 generalize in a nonempty manner. We are led to conjecture the following result, which is proved in §4 : Theorem C. Let Pk be a product of ordinary spheres, and let 2" be a homotopy sphere («^5). Then the inertia group of£nxPk is equal to the inertia group of 2> x Sk. This result is first proved for Pk = Tk, the fc-dimensional torus, in §3. That section also contains a diffeomorphism classification theorem for all smooth manifolds homeomorphic to Sn x Tk. The result (Theorem B) may be interpreted as an analog of the classification of smooth manifolds homeomorphic to5"x S", particularly as formulated in [17, Proposition 5.7]. Remark. Our methods may be applied to determine whether any two given smoothings of a product of several spheres are diffeomorphic, but any closed Received by the editors March 4, 1970. AMS 1969 subject classifications. Primary 5710, 5720; Secondary 5542, 5731.

Journal ArticleDOI
TL;DR: In this article, the variety of all abelian groups is defined as the product of a product of (finitely many) varieties each of which is either soluble or cross.
Abstract: Let denote the variety of all abelian groups and, for each prime p, let p be the variety of all elementary abelian p-groups. Let be a subvariety of a product of (finitely many) varieties each of which is either soluble or Cross.

Journal ArticleDOI
Evan O. Kane1
TL;DR: In this paper, an attempt was made to fit cyclotron masses and principal energy gaps for silicon using a Heine-Abarenkov-type determination of the core-valence interaction fitted to the atomic spectra of ${\mathrm{Si}}^{3+}$.
Abstract: An attempt is made to fit cyclotron masses and principal energy gaps for silicon using a Heine-Abarenkov-type determination of the core-valence interaction fitted to the atomic spectra of ${\mathrm{Si}}^{3+}$. The valence-valence exchange and correlation potential is approximated by a local potential. The masses and gaps are found to obey a $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}''$ product relation under variations of the local potential. The theoretical product is 10-25% smaller in absolute value than the experimental product. We conclude that a local approximation to exchange and correlation is inadequate for silicon. If the masses are fitted the gaps are in error by 0.5-0.7 eV. We suggest that screened Hartree-Fock exchange may provide the nonlocality required to overcome these fitting difficulties.

Journal ArticleDOI
TL;DR: In this article, it was shown that Strassen's algorithm is a representation of a product of (2, 2) matrices by a Hadamard product in a space of 7 dimensions.
Abstract: The purpose of this paper is to prove that Strassen's algorithm is a representation of a product of (2,2) matrices by a Hadamard product in a space of 7 dimensions. For matrices (n, n) it is possible to obtain this representation in spaces withn 3?n+1 dimensions.

Journal ArticleDOI
01 Mar 1971
TL;DR: Theorem 5.1 as mentioned in this paper shows that A is a dual algebra if and only if A' = A* and for each xEA, the mapping T:.f--f *x is a weakly completely continuous operator on A*.
Abstract: Let A be a commutative B*-algebra, 'D its carrier space and A* the conjugate space of A. Let A' be the closed subspace of A * spanned by (. We show that A is a dual algebra if and only if A'=A* and for each xEA, the mapping T:.f--f *x is a weakly completely continuous operator on A*. This improves an early result by B. J. Tomiuk and the author. A similar result holds for general B *-algebras. 1. Notation and preliminaries. Notation and definitions not explicitly given are taken from [7]. Let A be a Banach algebra and A* its conjugate space. For each xEA and fCA*, define (f * x)(y) =-f(xy) (y E A). Thenf * xEA*. Let T. be the operator from A* into itself given by TX(f)= f*x (fE A*). The mapping Tx is called weakly completely continuous on A* if for every bounded net {ft } CA *, there exist a subnet { fk } C { ft } and an elementfEA* such that T.(fk) ->T(f) weakly; i.e., F(f * x) = lim F(fk * x) k for all FCA**, where A** denotes the second conjugate space of A. In this paper, all algebras and spaces under consideration are over the complex field C. 2. Lemmas. Let A be a B*-algebra. It is well known that A is Arens regular and A** is a B*-algebra under the Arens product * (see [1, p. 869, Theorem 7.1]). LEMMA 2.1. Let A be a B*-algebra and r the canonical mapping of A into A**. Then A is a dual algebra if and only if 7r(A) is a closed two-sided ideal of A**. PROOF. This is Theorem 5.1 in [7]. Received by the editors March 6, 1970. AMS 1969 subject classifications. Primary 4650; Secondary 4655.

Journal ArticleDOI
TL;DR: The general uncertainty relation for real time functions in communication theory is derived, where the product of pulse duration and spectral width referred to the positive frequency spectrum only, is not less than 1.1802….
Abstract: The general uncertainty relation for real time functions in communication theory is derived. The product of pulse duration and spectral width referred to the positive frequency spectrum only, is not less than 1.1802…, as compared with 2 in the Heisenberg and Gabor cases. This minimum is reached with a pulse whose time and spectral functions are numerically evaluated.

Journal ArticleDOI
TL;DR: In this article, a recursive construction for quasigroups orthogonal to their transposes is given, which is based on the singular direct product (SDP) construction of A. Sade.
Abstract: In [2], A. Sade gives a construction for quasigroups which he calls the singular direct product. In this paper we generalize Sades' construction. As an application we obtain a recursive construction for quasigroups orthogonal to their transposes. All quasigroups considered in this paper will be finite.

Journal ArticleDOI
TL;DR: In this paper, the ground state of the permanganate ion was studied by an ab initio SCF MO calculation using a minimal basis set of contracted gaussian functions of near Hartree-Fock accuracy.

Journal ArticleDOI
TL;DR: In this article, the product from the reaction of 2,2-diphenylchromen with a 1,1-diarylethylene has been shown by spectroscopic methods and by unambiguous synthesis to be a 2 2 -diaryl-4-(2,2 -dimethylvinyl)chroman.
Abstract: The product from the reaction of 2,2-diphenylchromen with a 1,1-diarylethylene has been shown by spectroscopic methods and by unambiguous synthesis to be a 2,2-diaryl-4-(2,2-diphenylvinyl)chroman. Similar reactions with related compounds have been carried out.

Journal ArticleDOI
01 Jan 1971
TL;DR: For n > 5, every cell in En contains a tame arc and every k-dimensional polyhedron P CBm-k X Ik is tame in En for product cells Bm-KXIkCEn-kXEkC = En as discussed by the authors.
Abstract: We prove here that, for n >5, every cell in En contains a tame arc and that, for product cells Bm-kXIkCEn-kXEk = En, every k-dimensional polyhedron P CBm-k X Ik is tame in En.



Journal ArticleDOI
TL;DR: In this paper, the kernel and the nucleolus of a product of two simple games are given in terms of the kernels and nucleoluses of the components of the component games.
Abstract: The kernel and the nucleolus of a product of two simple games are given in terms of the kernels and the nucleoluses of the component games.



Journal ArticleDOI
TL;DR: In this paper, it was shown that a ring satisfying (x, y, y) 0 and (y, y and x) 0 is an alternative for a ring R* if and only if it is satisfied by R*.
Abstract: Preliminaries. A ring satisfying (x, y, x) = 0 is called flexible. Trivially in flexible rings i) is satisfied and ii) reduces to (x, [y, z], w) = ([y, z], w, x). It is clear that i) and ii) hold in commutative rings. A ring satisfying (x, y, y) 0 and (y, y, x) 0 is called alternative. One easily sees that an alternative ring is flexible and satisfies ii). More generally every subring of the direct product of a family of rings satisfying i) and ii) also satisfies i) and ii). If one changes the product xy of x, y E R into yx one obtains the ring R* anti-isomorphic to R. Any one of the conditions i), ii), iii) is satisfied by a ring R if and only if it is satisfied by R*.