scholarly journals A New Characterization of Boolean Rings with Identity

2015 ◽  
Vol 0076 ◽  
pp. 55-60
Author(s):  
P. Danchev
Keyword(s):  
Boolean Rings

Yap Hian Poh. Postulational study of an axiom system of Boolean algebra. Majallah Tahunan 'Ilmu Pasti—Shu Hsüeh Nien K'an—Bulletin of Mathematical Society of Nanyang University (1960), pp. 94–110. - R. M. Dicker. A set of independent axioms for Boolean algebra. Proceedings of the London Mathematical Society, ser. 3 vol. 13 (1963), pp. 20–30. - P. J. van Albada. A self-dual system of axioms for Boolean algebra. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, series A vol. 67 (1964), pp. 377–381; also Indagationes mathematicae, vol. 26 (1964), pp. 377–381. - Antonio Diego and Alberto Suárez. Two sets of axioms for Boolean algebras. Portugaliae mathematica, vol. 23 nos. 3–4 (for 1964, pub. 1965), pp. 139–145. (Reprinted from Notas de lógica matemática no. 16, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca 1964, 13 pp.) - P. J. van Albada. Axiomatique des algèbres de Boole. Bulletin de la Société Mathématique de Belgique, vol. 18 (1966), pp. 260–272. - Lawrence J. Dickson. A short axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 6 (1967), pp. 253–257. - Leroy J. Dickey. A shorter axiomatic system for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 8 (1968), p. 336. - Chinthayamma . Independent postulate sets for Boolean algebra. Pi Mu Epsilon journal, vol. 4 no. 9 (1968), pp. 378–379. - Kiyoshi Iséki. A simple characterization of Boolean rings. Proceedings of the Japan Academy, vol. 44 (1968), pp. 923–924. - Sakiko Ôhashi. On definitions of Boolean rings and distributive lattices. Proceedings of the Japan Academy, vol. 44 (1968), pp. 1015–1017.

1973 ◽  
Vol 38 (4) ◽  
pp. 658-660
Author(s):  
Donald H. Potts
Keyword(s):  
Boolean Algebra ◽  
London Mathematical Society ◽  
Axiom System ◽  
Dual System ◽  
Boolean Algebras ◽  
Axiomatic System ◽  
Mathematical Society ◽  
Distributive Lattices ◽  
Boolean Rings

A Characterization of Implicative Boolean Rings

1953 ◽  
Vol 5 ◽  
pp. 465-469 ◽  
Author(s):  
A. H. Copeland ◽  
Frank Harary
Keyword(s):  
Boolean Algebra ◽  
Cross Product ◽  
Boolean Ring ◽  
Conditional Probabilities ◽  
Boolean Operations ◽  
Boolean Rings ◽  
The Cross

In the theory of probability, the conditional can be treated by an operation analogous to division. Many properties of the conditional can best be studied by means of the corresponding multiplication (called the cross-product). An implicative Boolean ring is defined [2] in terms of a cross-product and the usual Boolean operations. The cross-product is the only device yet known in which the events corresponding to conditional probabilities are themselves elements of the Boolean ring. The fact that such advice was not introduced by Boole is probably the reason why Boolean algebra has been very little used in the theory of probability, although probability was one of the principal applications which Boole had in mind.

The Categories of Boolean Lattices, Boolean Rings and Boolean Spaces

1964 ◽  
Vol 7 (2) ◽  
pp. 245-252 ◽  
Author(s):  
Hoshang P. Doctor
Keyword(s):  
Axiomatic Characterization ◽  
Boolean Rings ◽  
Equivalence Of Categories ◽  
Boolean Spaces ◽  
Boolean Lattices ◽  
Special Classes

In Theorem 4 of [5] Stone stated that the theory of Boolean rings was "mathematically equivalent" to the theory of Boolean spaces without, however, properly defining the phrase "mathematically equivalent". It is the main purpose of this note to establish a precise reformulation of Theorem 4 in [5]. This is accomplished by introducing special classes of maps between Boolean lattices, Boolean rings and Boolean spaces respectively, and showing the categories arising in conjunction with these maps to be equivalent in the sense of Grothendieck [2]. Thus the notion of equivalence of categories will replace the phrase "mathematically equivalent" in [5]. In addition the well-known axiomatic characterization of meet and complementation of Boolean lattices with unit is discussed in analogous terms.

Morphological Characterization of Mycobacteriophage R1

1974 ◽  
Vol 32 ◽  
pp. 254-255
Author(s):  
B. L. Soloff ◽  
T. A. Rado
Keyword(s):  
Carbon Source ◽  
Stationary Phase ◽  
Growth Phase ◽  
Minimal Medium ◽  
Morphological Characterization ◽  
Logarithmic Growth Phase ◽  
Complete Lysis ◽  
Lysogenic Culture ◽  
Logarithmic Growth

Mycobacteriophage R1 was originally isolated from a lysogenic culture of M. butyricum. The virus was propagated on a leucine-requiring derivative of M. smegmatis, 607 leu−, isolated by nitrosoguanidine mutagenesis of typestrain ATCC 607. Growth was accomplished in a minimal medium containing glycerol and glucose as carbon source and enriched by the addition of 80 μg/ ml L-leucine. Bacteria in early logarithmic growth phase were infected with virus at a multiplicity of 5, and incubated with aeration for 8 hours. The partially lysed suspension was diluted 1:10 in growth medium and incubated for a further 8 hours. This permitted stationary phase cells to re-enter logarithmic growth and resulted in complete lysis of the culture.

AEM analysis of crystallization in Ti-based amorphous alloys

1983 ◽  
Vol 41 ◽  
pp. 270-271
Author(s):  
A.R. Pelton ◽  
A.F. Marshall ◽  
Y.S. Lee
Keyword(s):  
Magnetic Properties ◽  
Amorphous Alloys ◽  
Amorphous Materials ◽  
Isothermal Annealing ◽  
Amorphous Matrix ◽  
Nucleation And Growth ◽  
Current Interest ◽  
Amorphous Films ◽  
Electrical And Magnetic Properties

Amorphous materials are of current interest due to their desirable mechanical, electrical and magnetic properties. Furthermore, crystallizing amorphous alloys provides an avenue for discerning sequential and competitive phases thus allowing access to otherwise inaccessible crystalline structures. Previous studies have shown the benefits of using AEM to determine crystal structures and compositions of partially crystallized alloys. The present paper will discuss the AEM characterization of crystallized Cu-Ti and Ni-Ti amorphous films.Cu60Ti40: The amorphous alloy Cu60Ti40, when continuously heated, forms a simple intermediate, macrocrystalline phase which then transforms to the ordered, equilibrium Cu3Ti2 phase. However, contrary to what one would expect from kinetic considerations, isothermal annealing below the isochronal crystallization temperature results in direct nucleation and growth of Cu3Ti2 from the amorphous matrix.

Characterization of Coherent, Ordered Precipitates in Nickel-Base Alloys

1973 ◽  
Vol 31 ◽  
pp. 144-145
Author(s):  
B. H. Kear ◽  
J. M. Oblak
Keyword(s):  
Stable Phase ◽  
Nickel Base Superalloy ◽  
Alloy 718 ◽  
Commercial Alloy ◽  
Precipitate Morphology ◽  
Continuous Precipitation ◽  
Nickel Base ◽  
Base Superalloy ◽  
Strengthening Precipitate

A nickel-base superalloy is essentially a Ni/Cr solid solution hardened by additions of Al (Ti, Nb, etc.) to precipitate a coherent, ordered phase. In most commercial alloy systems, e.g. B-1900, IN-100 and Mar-M200, the stable precipitate is Ni3 (Al,Ti) γ′, with an LI2structure. In A lloy 901 the normal precipitate is metastable Nis Ti3 γ′ ; the stable phase is a hexagonal Do2 4 structure. In Alloy 718 the strengthening precipitate is metastable γ″, which has a body-centered tetragonal D022 structure.Precipitate MorphologyIn most systems the ordered γ′ phase forms by a continuous precipitation re-action, which gives rise to a uniform intragranular dispersion of precipitate particles. For zero γ/γ′ misfit, the γ′ precipitates assume a spheroidal.

The Use of Advanced Analytical Techniques for Characterization of the Chemical, Physical, and Crystallographic Nature of a Surface

1973 ◽  
Vol 31 ◽  
pp. 132-133 ◽  
Author(s):  
R. E. Herfert
Keyword(s):  
Surface Characterization ◽  
Analytical Techniques ◽  
X Ray Diffraction ◽  
Depth Of Penetration ◽  
X Ray ◽  
The Past ◽  
Transmission Electron ◽  
Scanning Electron

Studies of the nature of a surface, either metallic or nonmetallic, in the past, have been limited to the instrumentation available for these measurements. In the past, optical microscopy, replica transmission electron microscopy, electron or X-ray diffraction and optical or X-ray spectroscopy have provided the means of surface characterization. Actually, some of these techniques are not purely surface; the depth of penetration may be a few thousands of an inch. Within the last five years, instrumentation has been made available which now makes it practical for use to study the outer few 100A of layers and characterize it completely from a chemical, physical, and crystallographic standpoint. The scanning electron microscope (SEM) provides a means of viewing the surface of a material in situ to magnifications as high as 250,000X.

Skeletal elements of crinoid echinoderms: High-resolution structural and microanalytical results

1983 ◽  
Vol 41 ◽  
pp. 194-195
Author(s):  
D. F. Blake ◽  
L. F. Allard ◽  
D. R. Peacor
Keyword(s):  
High Resolution ◽  
Marine Invertebrates ◽  
Sea Urchins ◽  
Structural Features ◽  
Sea Stars ◽  
High Resolution Tem ◽  
Relationship Of ◽  
The Relationship ◽  
Insight Into

Echinodermata is a phylum of marine invertebrates which has been extant since Cambrian time (c.a. 500 m.y. before the present). Modern examples of echinoderms include sea urchins, sea stars, and sea lilies (crinoids). The endoskeletons of echinoderms are composed of plates or ossicles (Fig. 1) which are with few exceptions, porous, single crystals of high-magnesian calcite. Despite their single crystal nature, fracture surfaces do not exhibit the near-perfect {10.4} cleavage characteristic of inorganic calcite. This paradoxical mix of biogenic and inorganic features has prompted much recent work on echinoderm skeletal crystallography. Furthermore, fossil echinoderm hard parts comprise a volumetrically significant portion of some marine limestones sequences. The ultrastructural and microchemical characterization of modern skeletal material should lend insight into: 1). The nature of the biogenic processes involved, for example, the relationship of Mg heterogeneity to morphological and structural features in modern echinoderm material, and 2). The nature of the diagenetic changes undergone by their ancient, fossilized counterparts. In this study, high resolution TEM (HRTEM), high voltage TEM (HVTEM), and STEM microanalysis are used to characterize tha ultrastructural and microchemical composition of skeletal elements of the modern crinoid Neocrinus blakei.

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