Fundamental units and consecutive squarefull numbers

2016 ◽  
Vol 13 (01) ◽  
pp. 243-252
Author(s):  
Kevser Aktaş ◽  
M. Ram Murty
Keyword(s):  
Abc Conjecture ◽  
Elementary Methods ◽  
Fundamental Units ◽  
Erdös Conjecture

We connect several seemingly unrelated conjectures of Ankeny, Artin, Chowla and Mordell to a conjecture of Erdös on consecutive squarefull numbers. We then study the Erdös conjecture and relate it to the abc conjecture. We also derive by elementary methods several unconditional results pertaining to the Erdös conjecture.

scholarly journals The Mean Values of Character Sums and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 318
Author(s):  
Jiafan Zhang ◽  
Yuanyuan Meng
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Mean Values ◽  
Special Polynomials ◽  
Elementary Methods ◽  
The Mean

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

Research on Biological Materials by Applied Technology with an Algorithm for Identifying and Counting Cells in Microscopic Particle Image

2014 ◽  
Vol 540 ◽  
pp. 352-355
Author(s):  
Sui Yuan Zhang ◽  
Rui Wang ◽  
Xian Qiao Chen ◽  
Ze Wu Jiang ◽  
Xiang Cai
Keyword(s):  
Particle Image ◽  
Dynamic Threshold ◽  
Biological Cells ◽  
Edge Detecting ◽  
Applied Technology ◽  
Channel Information ◽  
High Level ◽  
The Impact ◽  
Fundamental Units ◽  
Rule Out

Cells are fundamental units of life, and the key point in the field of biomaterial. Biological cells are always with high density, small nucleus and much impurities. Based on the technology of image processing, we propose a new method to count cells on the image of microscopic cells with high level of recognition. To precisely count the number, our method includes edge detecting and marking, efficient usage of three channel information of enhanced nucleus, binaryzation of dynamic threshold in separated areas and finally denoising. The experiment shows that the method is precise and quickly-reacted, moreover it can effectively rule out the impact of impurities. With little adjustment, it can apply to some other fields, not only decrease the labor involved, but the budget as well.

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

Preparing Preservice Teachers via Teaching Methods Courses: A Literature Review

2021 ◽  
pp. 1-10
Author(s):  
Fei Wu ◽  
Ashley Phelps ◽  
Michael Hodges ◽  
Yiqiong Zhang ◽  
Xiaofen D. Keating ◽  
...  
Keyword(s):  
Preservice Teachers ◽  
Physical Education ◽  
Teaching Methods ◽  
Elementary Classroom ◽  
National Standards ◽  
Methods Course ◽  
Physical Education Teachers ◽  
Classroom Teachers ◽  
Methods Courses ◽  
Elementary Methods

Purpose: To review past research on teaching methods courses with preservice physical education teachers and preservice elementary classroom teachers. Method: This study was guided by the 2017 National Standards for Initial Physical Education Teacher Education. A thorough literature search was conducted using online databases, and a total of 28 articles were selected for review. Results: About two thirds of the reviewed studies were related to elementary methods courses, and 10.7% of the studies were quantitative. Perceptions and confidence in teaching physical education were the focus of studies for preservice elementary classroom teachers, while pedagogical knowledge development and restructuring was the primary emphasis for preservice physical education teachers via elementary methods course. Research on the secondary methods course yielded no salient themes. Conclusions: Research on the topic has been incongruent with the national standards. More experimental and quantitative studies are needed in the future.

Mechanical Anisotropy of Individual Osteons in Bone Tissue at High Spatial Resolutions

2009 ◽  
Author(s):  
Davide Carnelli ◽  
Haimin Yao ◽  
Ming Dao ◽  
Pasquale Vena ◽  
Roberto Contro ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Cortical Bone ◽  
Mechanical Anisotropy ◽  
Haversian Canal ◽  
Maximum Penetration ◽  
Fundamental Units ◽  
Anisotropy Of Mechanical Properties ◽  
Penetration Depths ◽  
Secondary Osteons ◽  
Mineralized Collagen

Secondary osteons, the fundamental units of cortical bone, consist of cylindrical lamellar composites composed of mineralized collagen fibrils. Due to its lamellar structure, a multiscale knowledge of the mechanical properties of cortical bone is required to understand the biomechanical function of the tissue. In this light, nanoindentation tests were performed along the axial and transverse directions following a radial path from the Haversian canal to the osteonal edges. Different length scales are explored by means of indentations at different maximum penetration depths. Indentation moduli and hardness data were then interpreted in the context of the known microstructure. Results suggest that secondary osteons hierarchical structure is responsible for an observed length scale effect, homogenization phenomena and anisotropy of mechanical properties.

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