Index, sub-index and sub-factor of groups with interactions to number theory

2019 ◽  
Vol 19 (06) ◽  
pp. 2050101
Author(s):  
M. H. Hooshmand
Keyword(s):  
Number Theory ◽  
Direct Product ◽  
Prime Numbers ◽  
Additive Number Theory ◽  
Open Problems ◽  
Famous Conjecture ◽  
Future Directions ◽  
Left And Right ◽  
The Difference ◽  
Equivalent Conditions

This paper is the first step of a new topic about groups which has close relations and applications to number theory. Considering the factorization of a group into a direct product of two subsets, and since every subgroup is a left and right factor, we observed that the index conception can be generalized for a class of factors. But, thereafter, we found that every subset [Formula: see text] of a group [Formula: see text] has four related sub-indexes: right, left, upper and lower sub-indexes [Formula: see text], [Formula: see text] which agree with the conception index of subgroups, and all of them are equal if [Formula: see text] is a subgroup or normal sub-semigroup of [Formula: see text]. As a result of the topic, we introduce some equivalent conditions to a famous conjecture for prime numbers (“every even number is the difference of two primes”) that one of them is: the prime numbers set is index stable (i.e. all of its sub-indexes are equal) in integers and [Formula: see text]. Index stable groups (i.e. those whose subsets are all index stable) are a challenging subject of the topic with several results and ideas. Regarding the extension of the theory, we give some methods for evaluation of sub-indexes, by using the left and right differences of subsets. At last, we pose many open problems, questions, a proposal for additive number theory, and show some future directions of researches and projects for the theory.

Complex group algebras of the direct product of non-isomorphic Suzuki groups

2019 ◽  
Vol 19 (02) ◽  
pp. 2050036
Author(s):  
Morteza Baniasad Azad ◽  
Behrooz Khosravi
Keyword(s):  
Direct Product ◽  
Group Algebra ◽  
Irreducible Character ◽  
Prime Numbers ◽  
Product Formula ◽  
Character Degrees ◽  
Group Algebras ◽  
Complex Group ◽  
Complex Group Algebra ◽  
Suzuki Groups

In this paper, we prove that the direct product [Formula: see text], where [Formula: see text] are distinct numbers, is uniquely determined by its complex group algebra. Particularly, we show that the direct product [Formula: see text], where [Formula: see text]’s are distinct odd prime numbers, is uniquely determined by its order and three irreducible character degrees.

scholarly journals A scoping review of chronotype and temporal patterns of eating of adults: Tools used, findings, and future directions

2021 ◽  
pp. 1-51
Author(s):  
Yan Yin Phoi ◽  
Michelle Rogers ◽  
Maxine P. Bonham ◽  
Jillian Dorrian ◽  
Alison M. Coates
Keyword(s):  
Energy Intake ◽  
Scoping Review ◽  
Assessment Tool ◽  
Temporal Patterns ◽  
Time Interval ◽  
Dark Cycle ◽  
Future Directions ◽  
Meal Skipping ◽  
Optimal Health ◽  
The Difference

Abstract Circadian rhythms, metabolic processes, and dietary intake are inextricably linked. Timing of food intake is a modifiable temporal cue for the circadian system and may be influenced by numerous factors, including individual chronotype—an indicator of an individual’s circadian rhythm in relation to the light-dark cycle. This scoping review examines temporal patterns of eating across chronotypes and assesses tools that have been used to collect data on temporal patterns of eating and chronotype. A systematic search identified thirty-six studies in which aspects of temporal patterns of eating including meal timings; meal skipping; energy distribution across the day; meal frequency; time interval between meals, or meals and wake/sleep times; midpoint of food/energy intake; meal regularity; and duration of eating window were presented in relation to chronotype. Findings indicate that compared to morning chronotypes, evening chronotypes tend to skip meals more frequently, have later mealtimes, and distribute greater energy intake towards later times of the day. More studies should explore the difference in meal regularity and duration of eating window amongst chronotypes. Currently, tools used in collecting data on chronotype and temporal patterns of eating are varied, limiting the direct comparison of findings between studies. Development of a standardised assessment tool will allow future studies to confidently compare findings to inform the development and assessment of guidelines that provide recommendations on temporal patterns of eating for optimal health.

Study of the Characteristics of the Pulmonary Vein and Superior Vena Cava of Rabbits

2021 ◽  
Vol 11 (1) ◽  
pp. 112-122
Author(s):  
Pan Wang ◽  
Xin-Chun Yang ◽  
Xiu-Lan Liu ◽  
Rong-Feng Bao ◽  
Huai-Yu Ding ◽  
...  
Keyword(s):  
Action Potential ◽  
Pulmonary Vein ◽  
Refractory Period ◽  
Potassium Current ◽  
Superior Vena ◽  
Vena Cava ◽  
Current Intensity ◽  
Left And Right ◽  
The Difference ◽  
Right Atria

Background: This study aims to (1) investigate the characteristics of the action potential and triggering activity of cardiomyocytes in the pulmonary vein (PV) and superior vena cava (SVC) of rabbits and (2) study the features of cation currents in cardiomyocytes in rabbit PV and SVC-inward rectifier potassium current (IK1), transient outward potassium current (Ito), and non-selective cation currents (INSCC). Methods: The standard glass microelectrode and whole-cell patch-clamp techniques were used to record the action potential and various currents in the above cells. Results: (1) Cardiomyocytes in either PV or SVC had longer action potential durations than in the adjacent atrium, and spontaneous early after depolarization (EAD) could occur in both PV and SVC under normal physiological conditions. (2) The action potential in PV cardiomyocytes had a relative refractory period but did not have an absolute refractory period, and this characteristic enabled a premature beat that triggered a second plateau response, which led to EAD. (3) INSCC was found for the first time in the PV, SVC, and atria. (4) The current intensity of IK1, Ito, and INSCC was significantly lower in the PV and SVC than in the left and right atria, and the difference in the current intensity in INSCC could influence the action potential. Conclusions: PV and SVC can both initiate and maintain AF, but PV is the primary ectopic foci in initiating AF. The present study found that the second plateau response was easily induced in cardiomyocytes in PA shortly after depolarization. This was a specific characteristic of the action potential of PV. In addition, we preliminarily analyzed the differences in the main outward currents and noted a voltage-dependent INSCC in both PV and SVC rabbits’ cardiomyocytes. Furthermore, the current intensities of IK1, Ito, and INSCC were significantly lower in the PV and SVC than in the left and right atria, and the difference in the current intensity of INSCC influenced the action potential. The different permeability of INSCC for cations at different phases may play a role in inducing EAD.

Additive Number Theory

Science ◽  
1936 ◽  
Vol 84 (2176) ◽  
pp. 9-9
Author(s):  
Watson Davis
Keyword(s):  
Number Theory ◽  
Additive Number Theory ◽  
Additive Number

Errors in Estimating the Orientation of Dot Patterns

Perception ◽  
1989 ◽  
Vol 18 (2) ◽  
pp. 237-242 ◽  
Author(s):  
P Lánský ◽  
Naum Yakimoff ◽  
T Radil ◽  
L Mitrani
Keyword(s):  
Pattern Analysis ◽  
Frontoparallel Plane ◽  
The Subject ◽  
Left And Right ◽  
The Mean ◽  
The Difference

The error in estimating the orientation of a dot pattern was measured as the difference between the orientation of the least-squared-distances line (LS-line) of the pattern and the orientation of a line adjusted by the subject to match the perceived orientation of the pattern. Analysis of the mean errors (averaged over ten subjects) obtained for one hundred patterns confirmed that the orientation of the LS-line represents the orientation of elongated dot-patterns. It is shown that estimated orientation was systematically biased towards the nearest 45° oblique meridian. This bias points to the importance of the ±45° directions as natural norms for left- and right-side tilt in the frontoparallel plane.

scholarly journals Rectification Principles in Additive Number Theory

1998 ◽  
Vol 19 (3) ◽  
pp. 343-353 ◽  
Author(s):  
Y. F. Bilu ◽  
V. F. Lev ◽  
I. Z. Ruzsa
Keyword(s):  
Number Theory ◽  
Additive Number Theory ◽  
Additive Number
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