scholarly journals Balances in the Set of Arithmetic Progressions

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 350
Author(s):  
Chan-Liang Chung ◽  
Chunmei Zhong ◽  
Kanglun Zhou
Keyword(s):  
Necessary Conditions ◽  
Arithmetic Progressions ◽  
Balancing Numbers ◽  
Sufficient And Necessary Conditions ◽  
Explicit Formulae

This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included.

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

scholarly journals Properties of Certain Subclass of Meromorphic Multivalent Functions Associated with q-Difference Operator

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1035
Author(s):  
Cai-Mei Yan ◽  
Rekha Srivastava ◽  
Jin-Lin Liu
Keyword(s):  
Difference Operator ◽  
Necessary Conditions ◽  
Partial Sums ◽  
Coefficient Estimates ◽  
Radius Of Starlikeness ◽  
Closure Theorems ◽  
Sufficient And Necessary Conditions ◽  
Distortion Theorems ◽  
Growth And Distortion Theorems

A new subclass Σp,q(α,A,B) of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radius of starlikeness and convexity, partial sums and closure theorems, are investigated.

Perfect colorings of patterns with multiple orbits

2019 ◽  
Vol 75 (6) ◽  
pp. 814-826
Author(s):  
Allan Junio ◽  
Ma. Lailani Walo
Keyword(s):  
Necessary Conditions ◽  
Sufficient And Necessary Conditions

This paper studies colorings of patterns with multiple orbits, particularly those colorings where the orbits share colors. The main problem is determining when such colorings become perfect. This problem is attacked by characterizing all perfect colorings of patterns through the construction of sufficient and necessary conditions for a coloring to be perfect. These results are then applied on symmetrical objects to construct both perfect and non-perfect colorings.

scholarly journals On a Periodic Predator-Prey System with Holling III Functional Response and Stage Structure for Prey

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Xiangzeng Kong ◽  
Zhiqin Chen ◽  
Li Xu ◽  
Wensheng Yang
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Iii Functional Response

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

scholarly journals Permanence of Periodic Predator-Prey System with Functional Responses and Stage Structure for Prey

2008 ◽  
Vol 2008 ◽  
pp. 1-15 ◽  
Author(s):  
Can-Yun Huang ◽  
Min Zhao ◽  
Hai-Feng Huo
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Functional Responses ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Ii Functional Response

A stage-structured three-species predator-prey model with Beddington-DeAngelis and Holling II functional response is introduced. Based on the comparison theorem, sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained. An example is also presented to illustrate our main results.

Asymptotic properties of a certain class of Bush–Mosteller learning models

1980 ◽  
Vol 12 (4) ◽  
pp. 922-941
Author(s):  
Peter Findeisen
Keyword(s):  
Necessary Conditions ◽  
Asymptotic Properties ◽  
Response Sequence ◽  
Positive Probability ◽  
Learning Models ◽  
Random Sequences ◽  
Learning Experiment ◽  
Sufficient And Necessary Conditions

One general and three specialized models of the Bush–Mosteller type are presented to describe the kind of learning experiment where the response of the learner is always reinforced. Inhomogeneity is admitted. The random sequences of response probabilities and of responses associated with the different models are considered. Information about the existence and the distribution of asymptotic response probabilities is provided. The stress is on sufficient and necessary conditions for convergence (a.s. or with positive probability) of the response sequence, which is what ‘learning' means.

Optimal allocation of relevations in coherent systems

2021 ◽  
Vol 58 (4) ◽  
pp. 1152-1169
Author(s):  
Rongfang Yan ◽  
Jiandong Zhang ◽  
Yiying Zhang
Keyword(s):  
Optimal Allocation ◽  
Necessary Conditions ◽  
Stochastic Order ◽  
Coherent Systems ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Usual Stochastic Order ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Allocation Strategies

AbstractIn this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

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