Pub Date : 2022-10-17DOI: 10.21136/mb.2022.0029-22
Vandana P. Bhamre, Madhukar. M. Pawar
. The concept of covering energy of a poset is known and its McClelland type bounds are available in the literature. In this paper, we establish formulas for the covering energy of a crown with 2 n elements and a fence with n elements. A lower bound for the largest eigenvalue of a poset is established. Using this lower bound, we improve the McClelland type bounds for the covering energy for some special classes of posets.
{"title":"Covering energy of posets and its bounds","authors":"Vandana P. Bhamre, Madhukar. M. Pawar","doi":"10.21136/mb.2022.0029-22","DOIUrl":"https://doi.org/10.21136/mb.2022.0029-22","url":null,"abstract":". The concept of covering energy of a poset is known and its McClelland type bounds are available in the literature. In this paper, we establish formulas for the covering energy of a crown with 2 n elements and a fence with n elements. A lower bound for the largest eigenvalue of a poset is established. Using this lower bound, we improve the McClelland type bounds for the covering energy for some special classes of posets.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44375693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-11DOI: 10.21136/mb.2022.0155-21
Cyril Gavala, M. Ploščica, Ivana Varga
{"title":"Congruence preserving operations on the ring $mathbb{Z}_{p^3}$","authors":"Cyril Gavala, M. Ploščica, Ivana Varga","doi":"10.21136/mb.2022.0155-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0155-21","url":null,"abstract":"","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42317582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-29DOI: 10.21136/mb.2022.0033-22
Z. Şiar, R. Keskin, Elif Segah Öztas
. Let k > 2 and let ( P ( k ) n ) n > 2 − k be the k -generalized Pell sequence defined by P ( k ) n = 2 P ( k ) n − 1 + P ( k ) n − 2 + . . . + P ( k ) n − k for n > 2 with initial conditions In this study, we handle the equation P ( k ) n = y m in positive integers n , m , y , k such that k, y > 2 , and give an upper bound on n. Also, we will show that the equation P ( k ) n = y m with 2 6 y 6 1000 has only one solution given by P (2)7 = 13 2 .
． 让k n > 2,让(P (k)) k n > 2−be the k -generalized佩尔奈德fi序列n: P (k) = 2 (k) n−1 P + P (k) n−2。。P (k) + n (n−k for > 2与初始条件在这个研究,我们把手the equation P (k) n = y在积极integers n, m、y y这样的那个k, k > 2,和给上束缚在一个n .也会,我们会show that the equation P (k) n = m和y = 2 6 y 1000唯一溶液赐予了:P(2) 7 = 13。
{"title":"On perfect powers in $k$-generalized Pell sequence","authors":"Z. Şiar, R. Keskin, Elif Segah Öztas","doi":"10.21136/mb.2022.0033-22","DOIUrl":"https://doi.org/10.21136/mb.2022.0033-22","url":null,"abstract":". Let k > 2 and let ( P ( k ) n ) n > 2 − k be the k -generalized Pell sequence defined by P ( k ) n = 2 P ( k ) n − 1 + P ( k ) n − 2 + . . . + P ( k ) n − k for n > 2 with initial conditions In this study, we handle the equation P ( k ) n = y m in positive integers n , m , y , k such that k, y > 2 , and give an upper bound on n. Also, we will show that the equation P ( k ) n = y m with 2 6 y 6 1000 has only one solution given by P (2)7 = 13 2 .","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42035013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-08DOI: 10.21136/mb.2022.0187-21
B. Boudine
. A commutative ring R with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length e is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length 2. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length e .
{"title":"Characterization of irreducible polynomials over a special principal ideal ring","authors":"B. Boudine","doi":"10.21136/mb.2022.0187-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0187-21","url":null,"abstract":". A commutative ring R with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length e is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length 2. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length e .","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45469365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-31DOI: 10.21136/mb.2022.0051-22
T. Dube
. Among completely regular locales, we characterize those that have the feature described in the title. They are, of course, localic analogues of what are called cl-isocompact spaces. They have been considered in T. Dube, I. Naidoo, C. N. Ncube (2014), so here we give new characterizations that do not appear in this reference.
{"title":"On locales whose countably compact sublocales have compact closure","authors":"T. Dube","doi":"10.21136/mb.2022.0051-22","DOIUrl":"https://doi.org/10.21136/mb.2022.0051-22","url":null,"abstract":". Among completely regular locales, we characterize those that have the feature described in the title. They are, of course, localic analogues of what are called cl-isocompact spaces. They have been considered in T. Dube, I. Naidoo, C. N. Ncube (2014), so here we give new characterizations that do not appear in this reference.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41594270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-29DOI: 10.21136/mb.2022.0048-21
A. Tripathy
. In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form are established, where m > 0, α > 0, β > 0 are integers and a ( n ), b ( n ), c ( n ), d ( n ), p ( n ) are sequences of real numbers.
{"title":"Oscillation criteria for two dimensional linear neutral delay difference systems","authors":"A. Tripathy","doi":"10.21136/mb.2022.0048-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0048-21","url":null,"abstract":". In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form are established, where m > 0, α > 0, β > 0 are integers and a ( n ), b ( n ), c ( n ), d ( n ), p ( n ) are sequences of real numbers.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48124282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-29DOI: 10.21136/mb.2022.0157-21
J. Alzabut, S. Grace, A. Selvam, R. Janagaraj
. This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form where N 1 − γ = { 1 − γ, 2 − γ, 3 − γ, . . . } , 0 < γ 6 1, ∆ γ is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results.
{"title":"Nonoscillatory solutions of discrete fractional order equations\u0000 \u0000with positive and negative terms","authors":"J. Alzabut, S. Grace, A. Selvam, R. Janagaraj","doi":"10.21136/mb.2022.0157-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0157-21","url":null,"abstract":". This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form where N 1 − γ = { 1 − γ, 2 − γ, 3 − γ, . . . } , 0 < γ 6 1, ∆ γ is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43630341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-29DOI: 10.21136/mb.2022.0010-22
J. Farley
. Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that P is connected and P P ∼ = Q Q imply that Q is connected”, where P and Q are finite nonempty posets. We show that, indeed, under these hypotheses Q is connected and P ∼ = Q .
． Duffus在1978年的博士论文中写道，“P是连通的，P P ~ = Q Q暗示Q是连通的，这是不明显的”，其中P和Q是有限的非空偏集。我们证明，在这些假设下，Q是连通的，P ~ = Q。
{"title":"Does the endomorphism poset $P^P$ determine whether a finite poset $P$ is connected?\u0000 \u0000An issue Duffus raised in 1978","authors":"J. Farley","doi":"10.21136/mb.2022.0010-22","DOIUrl":"https://doi.org/10.21136/mb.2022.0010-22","url":null,"abstract":". Duffus wrote in his 1978 Ph.D. thesis, “It is not obvious that P is connected and P P ∼ = Q Q imply that Q is connected”, where P and Q are finite nonempty posets. We show that, indeed, under these hypotheses Q is connected and P ∼ = Q .","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41537603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The studies considered here are concerend with a linear thermoelastic Bresse system with delay term in the feedback. The heat conduction is also given by Cattaneo’s law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem using the semigroup method. Furthermore, based on the energy method, we establish an exponential stability result depending of a condition on the constants of the system that was first considered by A. Keddi, T. Apalara, S. A. Messaoudi in 2018.
． 本文研究的是反馈中有时滞项的线性热弹性Bresse系统。热传导也由卡塔尼奥定律给出。在时滞权值与阻尼权值之间适当的假设下，用半群方法证明了问题的适定性。此外，基于能量法，我们根据a . Keddi, T. Apalara, S. a . Messaoudi于2018年首次考虑的系统常数条件建立了指数稳定性结果。
{"title":"Stability result for a thermoelastic Bresse system with delay term in the internal feedback","authors":"Bouzettouta Lamine, Baibeche Sabah, Abdelli Manel, Guesmia Amar","doi":"10.21136/mb.2022.0154-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0154-21","url":null,"abstract":". The studies considered here are concerend with a linear thermoelastic Bresse system with delay term in the feedback. The heat conduction is also given by Cattaneo’s law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem using the semigroup method. Furthermore, based on the energy method, we establish an exponential stability result depending of a condition on the constants of the system that was first considered by A. Keddi, T. Apalara, S. A. Messaoudi in 2018.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45786264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-04DOI: 10.21136/mb.2022.0088-21
Y. Echarroudi
ly, a functional response can be defined as the relationship between an individual’s rate of consumption (here we talk about a consumption of predator) and food’s density (i.e., prey’s density). This amounts to saying that a functional response reflects the capture ability of the predator to prey or in other words, the functional response is introduced to describe the change in the rate of consumption of prey by predator when the density of prey varies. In the plotting point of view, each type of functional response I, II or III has a special characteristic. In fact, type I, or the linear case of the predator response, is the situation when the plot of the number of prey consumed (per unit of time) as a function of prey density shows a linear relationship between the number of prey consumed and the prey density. The Holling type II, called also concave upward response, is the case when the gradient of the curve decreases monotonically with increasing prey density, probably saturating at a constant value of prey consumption. For information, the Lotka-Volterra model involving this functional response is known as the Rosenzweig-MacArthur model. The type III response is known between the specialists of population dynamics as the sigmoid response having a concave downward part at low food density. Actually, for the Holling III, a sigmoidal behavior occurs when the gradient of the curve first increases and then decreases with increasing prey density. This behavior is due to the “learning behavior” in the predator population. Now, we address some “ecological” interpretations of the three first Holling types functional responses. The type I response is the result of simple assumption that the probability of a given predator (usually the passive one) encountering prey in a fixed time interval [0, Tt] within a fixed spatial region depends linearly on the prey density. This can be expressed under the form Y = aTsX, where Y is the amount of prey consumed by one predator, X is the prey density, Ts is the time available for searching and a is a constant of proportionality, termed as the discovery rate (which is in our case represented by the parameter b). In the absence of need to spend time handling the prey, all the time can be used for searching, i.e., Ts = Tt, and we have the type I response: assuming that the predators (having the density P ) act independently, in time Tt the total amount of prey will be reduced by quantity aTtXP . In addition, if each predator requires a handling time h for each individual prey that is consumed, the time available for searching Ts is reduced: Ts = Tt−hY . Taking into account the expression of Y in response type I, this leads to Y = aTtX − ahXY and this implies Y = aTtX/(1 + ahX) and this is exactly the type II response. Therefore, in the interval [0, Tt] the total amount of prey is reduced by the quantity aTtXP/(1 + ahX). Let us point out that the term “ah” is dimensionless and can be interpreted as the ability of a gen
{"title":"Null controllability of a coupled model in population dynamics","authors":"Y. Echarroudi","doi":"10.21136/mb.2022.0088-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0088-21","url":null,"abstract":"ly, a functional response can be defined as the relationship between an individual’s rate of consumption (here we talk about a consumption of predator) and food’s density (i.e., prey’s density). This amounts to saying that a functional response reflects the capture ability of the predator to prey or in other words, the functional response is introduced to describe the change in the rate of consumption of prey by predator when the density of prey varies. In the plotting point of view, each type of functional response I, II or III has a special characteristic. In fact, type I, or the linear case of the predator response, is the situation when the plot of the number of prey consumed (per unit of time) as a function of prey density shows a linear relationship between the number of prey consumed and the prey density. The Holling type II, called also concave upward response, is the case when the gradient of the curve decreases monotonically with increasing prey density, probably saturating at a constant value of prey consumption. For information, the Lotka-Volterra model involving this functional response is known as the Rosenzweig-MacArthur model. The type III response is known between the specialists of population dynamics as the sigmoid response having a concave downward part at low food density. Actually, for the Holling III, a sigmoidal behavior occurs when the gradient of the curve first increases and then decreases with increasing prey density. This behavior is due to the “learning behavior” in the predator population. Now, we address some “ecological” interpretations of the three first Holling types functional responses. The type I response is the result of simple assumption that the probability of a given predator (usually the passive one) encountering prey in a fixed time interval [0, Tt] within a fixed spatial region depends linearly on the prey density. This can be expressed under the form Y = aTsX, where Y is the amount of prey consumed by one predator, X is the prey density, Ts is the time available for searching and a is a constant of proportionality, termed as the discovery rate (which is in our case represented by the parameter b). In the absence of need to spend time handling the prey, all the time can be used for searching, i.e., Ts = Tt, and we have the type I response: assuming that the predators (having the density P ) act independently, in time Tt the total amount of prey will be reduced by quantity aTtXP . In addition, if each predator requires a handling time h for each individual prey that is consumed, the time available for searching Ts is reduced: Ts = Tt−hY . Taking into account the expression of Y in response type I, this leads to Y = aTtX − ahXY and this implies Y = aTtX/(1 + ahX) and this is exactly the type II response. Therefore, in the interval [0, Tt] the total amount of prey is reduced by the quantity aTtXP/(1 + ahX). Let us point out that the term “ah” is dimensionless and can be interpreted as the ability of a gen","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47907778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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