Convergence of a Monotonic Sequence

2008 ◽  
Keyword(s):  
Monotonic Sequence

scholarly journals Axiomatisations of the average and a further generalisation of monotonic sequences

1974 ◽  
Vol 15 (1) ◽  
pp. 63-65 ◽  
Author(s):  
John Bibby
Keyword(s):  
Bounded Sequence ◽  
Monotonic Sequence

A bounded monotonic sequence is convergent. This paper shows that a bounded sequence which is g-monotonic (to be defined) also converges. The proof generalises one attributed to Professor R. A. Rankin by Copson [1]. The theorem requires two definitions: the first axiomatises the notion of “average“ and the second generalises the concept of monotonicity.

scholarly journals The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 42
Author(s):  
Yuri N. Lovyagin ◽  
Nikita Y. Lovyagin
Keyword(s):  
Line Segment ◽  
Cauchy Sequence ◽  
Research Method ◽  
Axiomatic Theory ◽  
Real Numbers ◽  
Integral Calculus ◽  
Standard Analysis ◽  
Monotonic Sequence ◽  
Basic Concepts ◽  
Non Standard Analysis

This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions.

A Conjectured Monotonic Sequence

SIAM Review ◽  
1983 ◽  
Vol 25 (2) ◽  
pp. 267-268
Author(s):  
R. Cameron ◽  
S. McKee
Keyword(s):  
Monotonic Sequence

scholarly journals Monotonic sequence games

2011 ◽  
pp. 309-328
Author(s):  
M. H. Albert ◽  
R. E. L. Aldred ◽  
M. D. Atkinson ◽  
C. C. Handley ◽  
D. A. Holton ◽  
...  
Keyword(s):  
Monotonic Sequence

On a generalisation of monotonic sequences

1970 ◽  
Vol 17 (2) ◽  
pp. 159-164 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Difference Equation ◽  
Bounded Sequence ◽  
Real Numbers ◽  
Monotonic Sequence ◽  
Image Position

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequalitythen it is convergent. This I was able to prove by considering the corresponding difference equation

scholarly journals Kaplan classes of a certain family of functions

2020 ◽  
Vol 74 (2) ◽  
pp. 31
Author(s):  
Szymon Ignaciuk ◽  
Maciej Parol
Keyword(s):  
Complete Characterization ◽  
Power Functions ◽  
Monotonic Sequence ◽  
Kaplan Classes

We give the complete characterization of members of Kaplan classes of products of power functions with all zeros symmetrically distributed in \(\mathbb{T} := \{z \in\mathbb{C} : |z| = 1\}\) and weakly monotonic sequence of powers. In this way we extend Sheil-Small’s theorem. We apply the obtained result to study univalence of antiderivative of these products of power functions.

scholarly journals Absolute Riesz summability factors for Fourier series

1970 ◽  
Vol 17 (1) ◽  
pp. 65-70
Author(s):  
Prem Chandra
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Summability Factors ◽  
Monotonic Sequence ◽  
Riesz Mean ◽  
Riesz Summability ◽  
Absolute Riesz Summability ◽  
Image Position

Let ∑an be a given infinite series and {λn} a non-negative, strictly increasing, monotonic sequence, tending to infinity with n. We write, for w > λ0,and, for r>0, we write is known as the Riesz sum of “ type ” λn and “ order ” r, andis called the Riesz mean of type λn and order r.

Invariant syllable skeleton, complex segments and word edges

2012 ◽  
Vol 48 (3) ◽  
pp. 685-726
Author(s):  
TOBIAS SCHEER
Keyword(s):  
Special Interest ◽  
Syllable Structure ◽  
General Interest ◽  
Phonological Processes ◽  
Monotonic Sequence ◽  
Distributional Patterns ◽  
Surface Variation ◽  
Paradigm Uniformity ◽  
Empirical Record

San Duanmu'sSyllable Structure: The Limits of Variationraises a number of questions that are of general interest for phonological theory. Of special interest here are: the genesis and management of linearity in complex segments, the place of analogy (or paradigm uniformity) in grammar, the role of morphology in accounting for phonological patterns, the balance of static (distributional patterns) and dynamic (phonological processes) evidence for syllable structure, the role of stress in syllabification, and the import of corpus-based data for phonological analysis. In each case, Duanmu's proposals are evaluated according to their intrinsic consistency, the empirical record and the relevant body of literature. Alternative ways of handling the phenomena are offered, and these are fairly traditional in most cases. Duanmu's book is particularly relevant in the current constitution of the field where the see-saw movement between computation and representations seems to swing back in direction of the latter after having long been immobilised on the computational end. Standing clearly on the representational side, the theory exposed in the book aims to show that all surface strings may be reduced to a fixed and invariant syllable template, C(onsonant)V(owel)X. This enterprise is interesting especially in presence of another representationally-oriented theory, CVCV (Lowenstamm 1996, Scheer 2004), which also aims at reducing surface variation to an invariant syllabic skeleton, made of a monotonic sequence of CV units. However, the CVX and the CVCV templates are quite distinct, and the strategies that are used in order to accommodate the surface string are opposite (shrinking in the former case, expanding in the latter).

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