On Common Multiples of Transfinite Numbers

1960 ◽  
Vol 3 (1) ◽  
pp. 31-33 ◽  
Author(s):  
Elias Zakon
Keyword(s):  
Present Note ◽  
Ordinal Numbers

In his well known monograph [l] (p. 81) H. Bachmann indicates that two ordinal numbers > 1 always have a common left multiple > 1, but not always have a right multiple (RM). The monograph does not, however, contain any further analysis of right multiples. The purpose of the present note is to supplement this by formulating the following propositions which, despite their simplicity, seem not yet to be known.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

Evaluation of Second- and Third-Level Variance Proportions in Multilevel Designs With Completely Observed Populations: A Note on a Latent Variable Modeling Procedure

2021 ◽  
pp. 001316442110086
Author(s):  
Tenko Raykov ◽  
Natalja Menold ◽  
Jane Leer
Keyword(s):  
Latent Variable ◽  
Large Scale ◽  
Present Note ◽  
Interval Estimation ◽  
Psychological Research ◽  
Latent Variable Modeling ◽  
Model Choice ◽  
Modeling Methodology ◽  
Level 3 ◽  
Level 2

Two- and three-level designs in educational and psychological research can involve entire populations of Level-3 and possibly Level-2 units, such as schools and educational districts nested within a given state, or neighborhoods and counties in a state. Such a design is of increasing relevance in empirical research owing to the growing popularity of large-scale studies in these and cognate disciplines. The present note discusses a readily applicable procedure for point-and-interval estimation of the proportions of second- and third-level variances in such multilevel settings, which may also be employed in model choice considerations regarding ensuing analyses for response variables of interest. The method is developed within the framework of the latent variable modeling methodology, is readily utilized with widely used software, and is illustrated with an example.

scholarly journals A Note on Units of Algebraic Number Fields

1955 ◽  
Vol 9 ◽  
pp. 115-118 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Existence Theorem ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Algebraic Number Fields

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.

Archilochus and Lycambes

1986 ◽  
Vol 36 (1) ◽  
pp. 60-67 ◽  
Author(s):  
C. Carey
Keyword(s):  
Present Note ◽  
Old Comedy ◽  
English Speaking ◽  
The Voice

A persistent ancient tradition has it that a man named Lycambes promised his daughter Neoboule in marriage to the poet Archilochus of Paros, that he subsequently refused Archilochus, and that the poet attacked Lycambes and his daughters with such ferocity that they all committed suicide. When we reflect that the iambographer Hipponax drove his enemies Bupalus and Athenis and Old Comedy a man named Poliager to suicide, that the ancestress of iambos, Iambe, killed herself, and that all these suicides, like those of Lycambes and his daughters, took the form of hanging, we will not take too seriously the ending of the story of Archilochus' relations with Lycambes and his family.However, it seems now to be generally accepted, at least among English-speaking scholars, that the whole Lycambes tradition is to be rejected. The present note seeks to demonstrate that this extreme scepticism is misguided. I shall begin with a survey of Archilochus' references to Lycambes and his family to ascertain how far the indirect tradition is consistent with the surviving fragments.Lycambes appears to have played a consistent role in Archilochus, as far as the fragments allow us to see. In fr. 38 he appears as the father of two daughters (οἴην Λυκάμβεω παῖδα τ⋯ν ύπερτέρην), in fr. 33 (where the voice of ‘the daughter of Lycambes’ is mentioned) as the father of at least one daughter. In fr. 71 his role cannot be determined. But in fr. 54, if his name is correctly restored in v. 8, he may again figure as the father of a daughter, for a female is mentioned in the fragment, whether for good or ill. If his patronymic is correctly supplied in fr. 57.7, it may be significant that the letters πατρ occur in the same verse.

On a Definition of Ordinal Numbers

1962 ◽  
Vol 69 (5) ◽  
pp. 381-386 ◽  
Author(s):  
Maurice Sion ◽  
Richard Willmott
Keyword(s):  
Ordinal Numbers ◽  
Definition Of

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

Short definitions of the ordinals

1966 ◽  
Vol 31 (3) ◽  
pp. 409-414 ◽  
Author(s):  
Kenneth R. Brown ◽  
Hao Wang
Keyword(s):  
Von Neumann ◽  
Ordinal Numbers

In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms.The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions.

scholarly journals Approximation by Boolean sums of linear operators: Telyakovskiǐ-type estimates

1990 ◽  
Vol 42 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jia-Ding Cao ◽  
Heinz H. Gonska
Keyword(s):  
Upper Bound ◽  
Algebraic Polynomial ◽  
Present Note ◽  
Type Inequality ◽  
Linear Operators ◽  
Polynomial Operators ◽  
Boolean Sums ◽  
General Conditions

In the present note we study the question: “Under which general conditions do certain Boolean sums of linear operators satisfy Telyakovskiǐ-type estimates?” It is shown, in particular, that any sequence of linear algebraic polynomial operators satisfying a Timan-type inequality can be modified appropriately so as to obtain the corresponding upper bound of the Telyakovskiǐ-type. Several examples are included.

General Assembly Resolutions on the Problem of Voting in the Security Council and on Pacific Settlement of Disputes

1949 ◽  
Vol 43 (4) ◽  
pp. 705-715
Author(s):  
Yuen-li Liang
Keyword(s):  
International Cooperation ◽  
Security Council ◽  
Present Note ◽  
General Assembly ◽  
The Political ◽  
Political Field ◽  
Rules Of Procedure ◽  
Settlement Of Disputes

In continuation of the note on “Some Aspects of the Work of the Interim Committee of the General Assembly,” the present note will deal with the five resolutions adopted by the General Assembly during the second part of its third session, held between April 5 and May 19, 1949, on the problem of voting in the Security Council and on the study of methods for the promotion of international coöperation in the political field. These resolutions, which were adopted upon the recommendation of the Interim Committee, concern (1) the problem of voting in the Security Council; (2) restoration to the General Act of September 26, 1928, of its original efficacy; (3) appointment of a rapporteur or conciliator for a situation or dispute brought to the attention of the Security Council; (4) amendments to the rules of procedure of the General Assembly; and (5) creation of a panel for inquiry and conciliation.

scholarly journals A Note on the Variability in Size of Amblyomma hebraeum Koch

Parasitology ◽  
1915 ◽  
Vol 8 (1) ◽  
pp. 11-16 ◽  
Author(s):  
L. E. Robinson
Keyword(s):  
Present Note ◽  
Female Tick ◽  
Amblyomma Hebraeum ◽  
Large Numbers ◽  
Wide Range ◽  
Range Of Variation

Variability in the size and, in a lesser degree, the taxonomic features of male ticks, has arrested the attention of all who have had occasion to examine moderately large numbers of examples of the same species. In the case of the female tick, this variability, though doubtless coextensive with that of the male, is more or less obscured by the wide range of variation in size, depending upon the degree of engorgement; and, also, by the fact that in the female tick the taxonomic characters are, as a rule, less pronounced. The present note is only concerned with variability in the size of the male.

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