Let I⊂R=K[x1,…,xn] be a monomial ideal, 𝔪=(x1,…,xn), t a positive integer, and y1,…,ys be distinct variables in R such that, for each i=1,…,s, 𝔪∖yi∉Ass(R∕(I∖yi)t), where I∖yi denotes the deletion of I at yi. It is shown in Theorem 3.4 of the article in question that 𝔪∈Ass(R∕It) if and only if 𝔪∈Ass(R∕(It:∏i=1syi)). As an application of Theorem 3.4, it is argued in Theorem 3.6 that under certain conditions, every unmixed König ideal is normally torsion-free. In addition, Theorem 3.7 states that under certain conditions a square-free monomial ideal is normally torsion-free. It turns out that these conditions are not enough to obtain the desired statements in Theorems 3.6 and 3.7. We update these conditions to validate the conclusions of Theorems 3.6 and 3.7. For this purpose, it is enough for us to replace the expression “𝔪∖xi∉Ass(R∕(I∖xi)t)” with the new expression “I∖xi is normally torsion-free”. It should be noted that the previous proofs are still correct.
{"title":"CORRECTION TO THE ARTICLE ON THE EMBEDDED ASSOCIATED PRIMES OF MONOMIAL IDEALS","authors":"Mirsadegh Sayedsadeghi, Mehrdad Nasernejad, Ayesha Asloob Qureshi","doi":"10.1216/rmj.2023.53.1657","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1657","url":null,"abstract":"Let I⊂R=K[x1,…,xn] be a monomial ideal, 𝔪=(x1,…,xn), t a positive integer, and y1,…,ys be distinct variables in R such that, for each i=1,…,s, 𝔪∖yi∉Ass(R∕(I∖yi)t), where I∖yi denotes the deletion of I at yi. It is shown in Theorem 3.4 of the article in question that 𝔪∈Ass(R∕It) if and only if 𝔪∈Ass(R∕(It:∏i=1syi)). As an application of Theorem 3.4, it is argued in Theorem 3.6 that under certain conditions, every unmixed König ideal is normally torsion-free. In addition, Theorem 3.7 states that under certain conditions a square-free monomial ideal is normally torsion-free. It turns out that these conditions are not enough to obtain the desired statements in Theorems 3.6 and 3.7. We update these conditions to validate the conclusions of Theorems 3.6 and 3.7. For this purpose, it is enough for us to replace the expression “𝔪∖xi∉Ass(R∕(I∖xi)t)” with the new expression “I∖xi is normally torsion-free”. It should be noted that the previous proofs are still correct.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1511
Jaspreet Kaur, Meenu Goyal
We give the generalization of α-Baskakov Durrmeyer operators depending on a real parameter ρ>0. We present the approximation results in Korovkin and weighted Korovkin spaces. We also prove the order of approximation and rate of approximation for these operators. In the end, we verify our results with the help of numerical examples by using Mathematica.
{"title":"A NOTE ON α-BASKAKOV–DURRMEYER-TYPE OPERATORS","authors":"Jaspreet Kaur, Meenu Goyal","doi":"10.1216/rmj.2023.53.1511","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1511","url":null,"abstract":"We give the generalization of α-Baskakov Durrmeyer operators depending on a real parameter ρ>0. We present the approximation results in Korovkin and weighted Korovkin spaces. We also prove the order of approximation and rate of approximation for these operators. In the end, we verify our results with the help of numerical examples by using Mathematica.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1553
Ali Rejali, Mitra Amiri
A class of commutative Banach algebras which satisfy a Bochner–Schoenberg–Eberlein-type inequality was introduced by Takahasi and Hatori. We generalize this property for the commutative Fréchet algebra (𝒜,pℓ)ℓ∈ℕ. Moreover, we verify and generalize some of the main results in the class of Banach algebras, for the Fréchet case. We prove that all Fréchet C*-algebras and also uniform Fréchet algebras are BSE algebras. Also, we show that C∞[0,1] is not a Fréchet BSE algebra.
{"title":"BSE PROPERTY OF FRÉCHET ALGEBRA","authors":"Ali Rejali, Mitra Amiri","doi":"10.1216/rmj.2023.53.1553","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1553","url":null,"abstract":"A class of commutative Banach algebras which satisfy a Bochner–Schoenberg–Eberlein-type inequality was introduced by Takahasi and Hatori. We generalize this property for the commutative Fréchet algebra (𝒜,pℓ)ℓ∈ℕ. Moreover, we verify and generalize some of the main results in the class of Banach algebras, for the Fréchet case. We prove that all Fréchet C*-algebras and also uniform Fréchet algebras are BSE algebras. Also, we show that C∞[0,1] is not a Fréchet BSE algebra.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1445
Guowei Dai, Fang Liu, Qingbo Liu
We study the bifurcation structure to Serrin’s overdetermined problem such that −Δu=1 in Ω, u=0, ∂νu= const on ∂Ω. We prove that the bifurcation from the straight cylinder Bλ1×ℝ with λ1>0 is critical at the bifurcation point. Moreover, we obtain the global structure of bifurcation branches. To study the global structure of bifurcation branches, we establish a global bifurcation theorem in finite dimensional space.
{"title":"BIFURCATION STRUCTURE TO SERRIN’S OVERDETERMINED PROBLEM","authors":"Guowei Dai, Fang Liu, Qingbo Liu","doi":"10.1216/rmj.2023.53.1445","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1445","url":null,"abstract":"We study the bifurcation structure to Serrin’s overdetermined problem such that −Δu=1 in Ω, u=0, ∂νu= const on ∂Ω. We prove that the bifurcation from the straight cylinder Bλ1×ℝ with λ1>0 is critical at the bifurcation point. Moreover, we obtain the global structure of bifurcation branches. To study the global structure of bifurcation branches, we establish a global bifurcation theorem in finite dimensional space.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1469
Gao Hongya, Zhang Aiping, Huang Miaomiao
We deal with entropy solutions to degenerate elliptic equations of the form { −div 𝒜(x,u(x),∇u(x))=−div(u(x)|u(x)|θ−1E(x))+f(x),x∈Ω,u(x)=0,x∈∂Ω, where the Carathéodory function 𝒜:Ω×ℝ×ℝn→ℝn satisfies degenerate coercivity condition 𝒜(x,s,ξ)⋅ξ≥α|ξ|p(1+|s|)τ and controllable growth condition |𝒜(x,s,ξ)|≤β|ξ|p−1 for almost all x∈Ω and all (s,ξ)∈ℝ×ℝn. We let 1
{"title":"REGULARITY FOR ENTROPY SOLUTIONS TO DEGENERATE ELLIPTIC EQUATIONS WITH A CONVECTION TERM","authors":"Gao Hongya, Zhang Aiping, Huang Miaomiao","doi":"10.1216/rmj.2023.53.1469","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1469","url":null,"abstract":"We deal with entropy solutions to degenerate elliptic equations of the form { −div 𝒜(x,u(x),∇u(x))=−div(u(x)|u(x)|θ−1E(x))+f(x),x∈Ω,u(x)=0,x∈∂Ω, where the Carathéodory function 𝒜:Ω×ℝ×ℝn→ℝn satisfies degenerate coercivity condition 𝒜(x,s,ξ)⋅ξ≥α|ξ|p(1+|s|)τ and controllable growth condition |𝒜(x,s,ξ)|≤β|ξ|p−1 for almost all x∈Ω and all (s,ξ)∈ℝ×ℝn. We let 1<p<n, 0≤τ<p−1, 0≤𝜃<p−1−τ, we let f and E belong to some Marcinkiewicz spaces, and we give some regularity properties for entropy solutions. We derive a generalized version of Stampacchia’s lemma in order to prove the main theorem.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1635
Weidong Wang
Lutwak and Zhang introduced Lp-centroid bodies and established the Lp-centro-affine inequality. Thereafter, Lutwak, Yang and Zhang established the Lp-Busemann–Petty centroid inequality. In this paper, we give the inequalities of dual quermassintegrals, dual affine quermassintegrals and dual harmonic quermassintegrals for polar Lp-centroid bodies.
{"title":"DUAL TYPE QUERMASSINTEGRAL INEQUALITIES FOR POLAR Lp-CENTROID BODIES","authors":"Weidong Wang","doi":"10.1216/rmj.2023.53.1635","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1635","url":null,"abstract":"Lutwak and Zhang introduced Lp-centroid bodies and established the Lp-centro-affine inequality. Thereafter, Lutwak, Yang and Zhang established the Lp-Busemann–Petty centroid inequality. In this paper, we give the inequalities of dual quermassintegrals, dual affine quermassintegrals and dual harmonic quermassintegrals for polar Lp-centroid bodies.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1537
Xinyuan Pan, Xiaofei He, Aimin Hu
We discuss a class of p-Laplacian-type fractional four-point boundary-value problems with a parameter. We use the Green’s function, the Schauder fixed-point theorem and the Guo–Krasnoselskii fixed-point theorem on cones. Some examples are presented to show the validity of the conditions of our main theorem.
{"title":"EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A CLASS OF p-LAPLACIAN-TYPE FRACTIONAL FOUR-POINT BOUNDARY-VALUE PROBLEMS WITH A PARAMETER","authors":"Xinyuan Pan, Xiaofei He, Aimin Hu","doi":"10.1216/rmj.2023.53.1537","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1537","url":null,"abstract":"We discuss a class of p-Laplacian-type fractional four-point boundary-value problems with a parameter. We use the Green’s function, the Schauder fixed-point theorem and the Guo–Krasnoselskii fixed-point theorem on cones. Some examples are presented to show the validity of the conditions of our main theorem.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1571
Wanting Sun, Shuchao Li, Xuechao Li
A graph is split if its vertex set can be partitioned into a clique and an independent set. A split graph is (x,y)-bidegreed if each of its vertex degrees is equal to either x or y. Each connected split graph is of diameter at most 3. In 2017, Nikiforov proposed the Aα-matrix, which is the convex combination of the adjacency matrix and the diagonal matrix of vertex degrees of the graph G. It is well-known that a connected graph of diameter l contains at least l+1 distinct Aα-eigenvalues. A graph is said to be lα-extremal with respect to its Aα-matrix if the graph is of diameter l having exactly l+1 distinct Aα-eigenvalues. In this paper, using the association of split graphs with combinatorial designs, the connected 2α-extremal (resp. 3α-extremal) bidegreed split graphs are classified. Furthermore, all connected bidegreed split graphs of diameter 2 having just 4 distinct Aα-eigenvalues are identified.
{"title":"COMPLETE CHARACTERIZATION OF THE BIDEGREED SPLIT GRAPHS WITH THREE OR FOUR DISTINCT Aα-EIGENVALUES","authors":"Wanting Sun, Shuchao Li, Xuechao Li","doi":"10.1216/rmj.2023.53.1571","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1571","url":null,"abstract":"A graph is split if its vertex set can be partitioned into a clique and an independent set. A split graph is (x,y)-bidegreed if each of its vertex degrees is equal to either x or y. Each connected split graph is of diameter at most 3. In 2017, Nikiforov proposed the Aα-matrix, which is the convex combination of the adjacency matrix and the diagonal matrix of vertex degrees of the graph G. It is well-known that a connected graph of diameter l contains at least l+1 distinct Aα-eigenvalues. A graph is said to be lα-extremal with respect to its Aα-matrix if the graph is of diameter l having exactly l+1 distinct Aα-eigenvalues. In this paper, using the association of split graphs with combinatorial designs, the connected 2α-extremal (resp. 3α-extremal) bidegreed split graphs are classified. Furthermore, all connected bidegreed split graphs of diameter 2 having just 4 distinct Aα-eigenvalues are identified.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1337
Maria N. F. Barreto, Gastão Frederico, José Vanterler da Costa Sousa, Juan E. Napoles Valdes
Using the recently defined generalized derivative, we present a generalized formulation of variation of calculus, which includes the classical and conformable formulation as particular cases. In the first part of the article, through the properties of this generalized derivative, we discuss the generalized versions of the Bois–Reymond lemma, a Tonelli-type existence theorem, Euler–Lagrange equation, d’Alembert principle, du Bois–Reymond optimality condition and Noether’s theorem. In the second part, we discuss the Picard–Lindelöf theorem, Grönwall’s inequality, Pontryagin’s maximum principle and Noether’s principle for optimal control. We end with an application involving the time fractional Schrödinger equation.
{"title":"CALCULUS OF VARIATIONS AND OPTIMAL CONTROL WITH GENERALIZED DERIVATIVE","authors":"Maria N. F. Barreto, Gastão Frederico, José Vanterler da Costa Sousa, Juan E. Napoles Valdes","doi":"10.1216/rmj.2023.53.1337","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1337","url":null,"abstract":"Using the recently defined generalized derivative, we present a generalized formulation of variation of calculus, which includes the classical and conformable formulation as particular cases. In the first part of the article, through the properties of this generalized derivative, we discuss the generalized versions of the Bois–Reymond lemma, a Tonelli-type existence theorem, Euler–Lagrange equation, d’Alembert principle, du Bois–Reymond optimality condition and Noether’s theorem. In the second part, we discuss the Picard–Lindelöf theorem, Grönwall’s inequality, Pontryagin’s maximum principle and Noether’s principle for optimal control. We end with an application involving the time fractional Schrödinger equation.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1371
Belkis Bordj, Abdelouaheb Ardjouni
Our purpose is to extend the work of Lois-Prados and Precup (2020) and use the Krasnoselskii-type homotopy fixed-point theorem to prove the existence of positive periodic solutions of nonautonomous Lotka–Volterra dynamic systems with a general attack rate on time scales.
{"title":"POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS LOTKA–VOLTERRA DYNAMIC SYSTEMS WITH A GENERAL ATTACK RATE ON TIME SCALES","authors":"Belkis Bordj, Abdelouaheb Ardjouni","doi":"10.1216/rmj.2023.53.1371","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1371","url":null,"abstract":"Our purpose is to extend the work of Lois-Prados and Precup (2020) and use the Krasnoselskii-type homotopy fixed-point theorem to prove the existence of positive periodic solutions of nonautonomous Lotka–Volterra dynamic systems with a general attack rate on time scales.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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