scholarly journals Remarks to the Paper “On Montel’s Theorem’ By Kawakami

1956 ◽  
Vol 10 ◽  
pp. 165-169
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Linear Measure ◽  
Measurable Set ◽  
Montel’S Theorem

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined byTheorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

scholarly journals On Montel’s Theorem

1956 ◽  
Vol 10 ◽  
pp. 125-127 ◽  
Author(s):  
Yoshiro Kawakami
Keyword(s):  
Closed Set ◽  
Regular Functions ◽  
Measurable Set ◽  
Montel’S Theorem

In this note we shall prove a theorem which is related to Montel’s theorem [1] on bounded regular functions. Let E be a measurable set on the positive y-axis in the z( = x + iy)-plane, E(a, b) be its part contained in 0 ≦ a ≦ y ≦ b, and ∣E(a, b)∣ be its measure. We define the lower density of E at y = 0 byLEMMA, Let E be a set of positive lower density λ at y = 0. Then E contains a subset E1 of the same lower density at y = 0 such that E1 ∪ {0} is a closed set.

On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

3. On the Great Pyramid of Gizeh, and Professor C. P. Smyth's Views concerning it

1869 ◽  
Vol 6 ◽  
pp. 235-238
Author(s):  
A. D. Wackerbarth
Keyword(s):  
Linear Measure ◽  
Great Pyramid ◽  
Ancient Writer

The author gives a detailed statement of the theories of Professor Smyth, as given in the Transactions of this Society, Vol. XXIII. Part III. He then, after heartily commending the zeal and diligence of the Professor, brings forward objections to some of his views. 1. As to the metron or unit of linear measure. Mr Wackerbarth objects that this measure was utterly unknown to the ancient Egyptians—appearing in no Egyptian document or monument whatever, nor in any ancient writer who describes the condition of the Egyptians.

scholarly journals A question of Gross and the uniqueness of entire functions

1995 ◽  
Vol 138 ◽  
pp. 169-177 ◽  
Author(s):  
Hong-Xun yi
Keyword(s):  
Entire Function ◽  
Nevanlinna Theory ◽  
Entire Functions ◽  
Linear Measure ◽  
Finite Linear

For any set S and any entire function f letwhere each zero of f — a with multiplicity m is repeated m times in Ef(S) (cf. [1]). It is assumed that the reader is familiar with the notations of the Nevanlinna Theory (see, for example, [2]). It will be convenient to let E denote any set of finite linear measure on 0 < r < ∞, not necessarily the same at each occurrence. We denote by S(r, f) any quantity satisfying .

scholarly journals An Item Response Modeling Approach to Cognitive Load Measurement

2021 ◽  
Vol 6 ◽  
Author(s):  
John Fitzgerald Ehrich ◽  
Steven J. Howard ◽  
Sahar Bokosmaty ◽  
Stuart Woodcock
Keyword(s):  
Cognitive Load ◽  
Student Performance ◽  
Item Response ◽  
Item Difficulty ◽  
Linear Measure ◽  
Mental Load ◽  
Relative Difficulty ◽  
Item Response Modeling ◽  
Modeling Approach ◽  
Response Modeling

The accurate measurement of the cognitive load a learner encounters in a given task is critical to the understanding and application of Cognitive Load Theory (CLT). However, as a covert psychological construct, cognitive load represents a challenging measurement issue. To date, this challenge has been met mostly by subjective self-reports of cognitive load experienced in a learning situation. In this paper, we find that a valid and reliable index of cognitive load can be obtained through item response modeling of student performance. Specifically, estimates derived from item response modeling of relative difficulty (i.e., the difference between item difficulty and person ability locations) can function as a linear measure that combines the key components of cognitive load (i.e., mental load, mental effort, and performance). This index of cognitive load (relative difficulty) was tested for criterion (concurrent) validity in Year 2 learners (N = 91) performance on standardized educational numeracy and literacy assessments. Learners’ working memory (WM) capacity significantly predicted our proposed cognitive load (relative difficulty) index across both numeracy and literacy domains. That is, higher levels of WM were related to lower levels of cognitive load (relative difficulty), in line with fundamental predictions of CLT. These results illustrate the validity, utility and potential of this objective item response modeling approach to capturing individual differences in cognitive load across discrete learning tasks.

Sem calibration in the micrometer and submicrometer ranges by means of a periodic linear measure

2000 ◽  
Vol 43 (4) ◽  
pp. 346-352 ◽  
Author(s):  
Ch. P. Volk ◽  
Yu. A. Novikov ◽  
A. V. Rakov
Keyword(s):  
Linear Measure ◽  
Periodic Linear
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