The symmetric stochastic inverse eigenvalue problem (SSIEP) asks which lists of real numbers occur as the spectra of symmetric stochastic matrices. When the cardinality of a list is 4, Kaddoura and Mourad provided a sufficient condition for SSIEP by a mapping and convexity technique. They also conjectured that the sufficient condition is the necessary condition. This study presents the same sufficient condition for SSIEP, but we do it in terms of the list elements. In this way, we provide a different but more straightforward construction of symmetric stochastic matrices for SSIEP compared to those of Kaddoura and Mourad.
{"title":"Construction of 4 x 4 symmetric stochastic matrices with given spectra","authors":"Jaewon Jung, Donggyun Kim","doi":"10.1515/math-2023-0176","DOIUrl":"https://doi.org/10.1515/math-2023-0176","url":null,"abstract":"The symmetric stochastic inverse eigenvalue problem (SSIEP) asks which lists of real numbers occur as the spectra of symmetric stochastic matrices. When the cardinality of a list is 4, Kaddoura and Mourad provided a sufficient condition for SSIEP by a mapping and convexity technique. They also conjectured that the sufficient condition is the necessary condition. This study presents the same sufficient condition for SSIEP, but we do it in terms of the list elements. In this way, we provide a different but more straightforward construction of symmetric stochastic matrices for SSIEP compared to those of Kaddoura and Mourad.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"72 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147303","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}
Feras Bani-Ahmad, Mohammad Hussein Mohammad Rashid
This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.
{"title":"Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices","authors":"Feras Bani-Ahmad, Mohammad Hussein Mohammad Rashid","doi":"10.1515/math-2023-0185","DOIUrl":"https://doi.org/10.1515/math-2023-0185","url":null,"abstract":"This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"32 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147297","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}
In this study, we solve the system of additive functional equations: h(x+y)=h(x)+h(y),g(x+y)=f(x)+f(y),2fx+y2=g(x)+g(y),left{begin{array}{l}hleft(x+y)=hleft(x)+h(y), gleft(x+y)=fleft(x)+f(y), 2fleft(frac{x+y}{2}right)=gleft(x)+g(y),end{array}right. and we investigate the stability of (homomorphism, derivation)-systems in Banach algebras.
在本研究中，我们求解的是加法函数方程组： h ( x + y ) = h ( x ) + h ( y ) , g ( x + y ) = f ( x ) + f ( y ) , 2 f x + y 2 = g ( x ) + g ( y ) 、 h（x+y）=h（left（x）+h（y）,（g（left（x+y）=f（left（x）+f（y）,（2f（left（frac{x+y}{2}right）=g（left（x）+g（y）,（end{array}right. 并研究了巴拿赫代数中（同态、派生）系统的稳定性。
This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0182_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0182_eq_002.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> <jats:tex-math>nge 3</jats:tex-math> </jats:alternatives> </jats:inline-formula>) with full viscosity in Besov spaces. Under the hypotheses <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0182_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>∞</m:mi> </m:math> <jats:tex-math>1lt plt infty </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0182_eq_004.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:mi>min</m:mi> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>∕</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>−</m:mo> <m:mi>n</m:mi> <m:mo>∕</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo><</m:mo> <m:mi>s</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>∕</m:mo> <m:mi>p</m:mi> </m:math> <jats:tex-math>-min left{n/p,2-n/pright}lt sle n/p</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and the initial condition <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0182_eq_005.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>×</m:mo> <m:msubsup> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>1</m:m
本文主要研究在贝索夫空间中，R n {{mathbb{R}}}^{n} ( n ≥ 3 nge 3 ) 中具有全粘性的布森斯克系统强解的局部和全局存在性与唯一性。在假设 1 < p < ∞ 1lt plt infty 和 - min { n ∕ p , 2 - n ∕ p } <；s ≤ n ∕ p -min left{n/p,2-n/plt sle n/p , 和初始条件 ( θ 0 , u 0 ) ∈ B ˙ p , 1 s - 1 × B ˙ p 、1 n ∕ p - 1 left({theta }_{0},{u}_{0})in {dot{B}}_{p,1}^{s-1}times {dot{B}}_{p,1}^{n/p-1}, 证明布西尼斯克系统有唯一的局部强解。在 n ≤ p < ∞ nle plt infty 和 - n ∕ p < s ≤ n ∕ p -n/plt sle n/p 的假设条件下，或者特别是 n ≤ p < 2 n nle plt 2n 和 - n ∕ p < s <；n ∕ p - 1 -n/plt slt n/p-1 ，初始条件 ( θ 0 , u 0 ) ∈ ( B ˙ p 、1 s - 1 ∩ L n ∕ 3 ) × ( B ˙ p , 1 n ∕ p - 1 ∩ L n ) left({theta }_{0},{u}_{0})in left({dot{B}}_{p、1}^{s-1}cap {L}^{n/3})times left({dot{B}}_{p、1}^{n/p-1}cap {L}^{n}) 具有足够小的规范‖ θ 0 ‖ L n ∕ 3 {Vert {theta }_{0}Vert }_{{L}^{n/3}} 和‖ u 0 ‖ L n {Vert {u}_{0}Vert }_{{L}^{n}} 。 证明布辛斯方程组有唯一的全局强解。
{"title":"Local and global solvability for the Boussinesq system in Besov spaces","authors":"Shuokai Yan, Lu Wang, Qinghua Zhang","doi":"10.1515/math-2023-0182","DOIUrl":"https://doi.org/10.1515/math-2023-0182","url":null,"abstract":"This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0182_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0182_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:math> <jats:tex-math>nge 3</jats:tex-math> </jats:alternatives> </jats:inline-formula>) with full viscosity in Besov spaces. Under the hypotheses <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0182_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi>∞</m:mi> </m:math> <jats:tex-math>1lt plt infty </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0182_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>−</m:mo> <m:mi>min</m:mi> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>∕</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>−</m:mo> <m:mi>n</m:mi> <m:mo>∕</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo><</m:mo> <m:mi>s</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>∕</m:mo> <m:mi>p</m:mi> </m:math> <jats:tex-math>-min left{n/p,2-n/pright}lt sle n/p</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and the initial condition <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0182_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>s</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> <m:mo>×</m:mo> <m:msubsup> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>1</m:m","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"11 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147288","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}
In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0184_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Q</m:mi> </m:math> <jats:tex-math>{mathbb{Q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> cannot be of degree 8. This completes the classification of so-called product-feasible triplets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0184_eq_002.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>left(a,b,c)in {{mathbb{N}}}^{3}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0184_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>a</m:mi> <m:mo>≤</m:mo> <m:mi>b</m:mi> <m:mo>≤</m:mo> <m:mi>c</m:mi> </m:math> <jats:tex-math>ale ble c</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0184_eq_004.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>b</m:mi> <m:mo>≤</m:mo> <m:mn>7</m:mn> </m:math> <jats:tex-math>ble 7</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The triplet <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0184_eq_005.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(a,b,c)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is called product-feasible if there are algebraic numbers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0184_eq_006.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:math> <jats:tex-math>alpha ,beta </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0184_eq_007.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>γ</m:mi> </m:math>
在这篇文章中，我们证明了在 Q {mathbb{Q}} 上，度数为 4 和 6 的两个代数数的乘积不可能是度数为 8 的。这就完成了在{{mathbb{N}}^{3}中 a ≤ b ≤ c ale ble c 和 b ≤ 7 ble 7 的所谓乘积可行三元组 ( a , b , c ) ∈ N 3 left(a,b,c) 的分类。如果在 Q {mathbb{Q} 上存在代数数 α , β alpha , beta , 和 γ gamma 的度数分别为 a , b a,b , 和 c c c ，那么三元组 ( a , b , c ) left(a,b,c)被称为乘积可行。｝ 分别，使得 α β = γ alpha beta = gamma 。在证明过程中，我们使用了一个命题，它描述了 Q [ x ] {mathbb{Q}} 左[x]中所有具有相等模数的四根的一元四次不可还原多项式，并且具有独立的意义。我们还证明了一个更一般的说法，即对于任意整数 n ≥ 2 nge 2 和 k ≥ 1 kge 1，三元组 ( a , b , c ) = ( n , ( n - 1 ) k , n k ) left(a,b,c)=left(n,left(n-1)k,nk) 是乘积可行的，当且仅当 n n 是一个素数。选择 ( n , k ) = ( 4 , 2 ) left(n,k)=left(4,2)也可以恢复 ( a , b , c ) = ( 4 , 6 , 8 ) left(a,b,c)=left(4,6,8)的情况。
{"title":"The product of a quartic and a sextic number cannot be octic","authors":"Artūras Dubickas, Lukas Maciulevičius","doi":"10.1515/math-2023-0184","DOIUrl":"https://doi.org/10.1515/math-2023-0184","url":null,"abstract":"In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0184_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"double-struck\">Q</m:mi> </m:math> <jats:tex-math>{mathbb{Q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> cannot be of degree 8. This completes the classification of so-called product-feasible triplets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0184_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">N</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>left(a,b,c)in {{mathbb{N}}}^{3}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0184_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>a</m:mi> <m:mo>≤</m:mo> <m:mi>b</m:mi> <m:mo>≤</m:mo> <m:mi>c</m:mi> </m:math> <jats:tex-math>ale ble c</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0184_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>b</m:mi> <m:mo>≤</m:mo> <m:mn>7</m:mn> </m:math> <jats:tex-math>ble 7</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The triplet <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0184_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>left(a,b,c)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is called product-feasible if there are algebraic numbers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0184_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:math> <jats:tex-math>alpha ,beta </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0184_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>γ</m:mi> </m:math>","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"21 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140114889","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}
We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.
{"title":"Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function","authors":"Mohamed Jleli, Bessem Samet","doi":"10.1515/math-2023-0178","DOIUrl":"https://doi.org/10.1515/math-2023-0178","url":null,"abstract":"We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"46 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140114987","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}
在本文中，我们将证明以下结果。设 n ≥ 3 nge 3 是某个固定整数，设 R R 是质环，char ( R ) ≠ ( n + 1 ) ！ 2 n - 2 {rm{char}}left(R)ne left(n+1)!{2}^{n-2} 。假设存在一个加法映射 D : R → R D:Rto R 满足关系式 2 n - 2 D ( x n ) = ∑ i = 0 n - 2 n - 2 i x i D ( x 2 ) x n - 2 - i + ( 2 n - 2 - 1 ) ( D ( x ) x n - 1 + x n - 1 D ( x ) ) + ∑ i = 1 n - 2 ∑ k = 2 i ( 2 k - 1 - 1 ) n - k - 2 i - k + ∑ k = 2 n - 1 - i ( 2 k - 1 - 1 ) n - k - 2 n - i - k - 1 x i D ( x ) x n - 1 - i begin{array}{rcl}{2}^{n-2}Dleft({x}^{n})&；=& left(mathop{displaystyle sum }limits_{i=0}^{n-2}left(genfrac{}{}{0.0pt}{}{n-2}{i}right){x}^{i}Dleft({x}^{2}){x}^{n-2-i}right)+left({2}^{n-2}-1)left(Dleft(x){x}^{n-1}+{x}^{n-1}Dleft(x)) & &+mathop{displaystyle sum }limits_{i=1}^{n-2}left(mathop{displaystyle sum }limits_{k=2}^{i}left({2}^{k-1}-1)left(genfrac{}{}{0.0pt}{}{n-k-2}{i-k}right)+mathop{displaystyle sum }limits_{k=2}^{n-1-i}left({2}^{k-1}-1)left(genfrac{}{}{0.0pt}{}{n-k-2}{n-i-k-1}right){x}^{i}Dleft(x){x}^{n-1-i}end{array} for all x∈ R. xin R. In this case, D D is a derivation.这个结果与赫斯坦的一个经典结果有关，它指出在质环上任何char ( R ) ≠ 2 {rm{char}}left(R)ne 2 的乔丹导数都是一个导数。
具有边 e 1 , ... , e m {e}_{1},ldots ,{e}_{m} 的图 G G 的原子-键总和-连通性（ABS）指数是 , e m {e}_{1},ldots ,{e}_{m} 是 1 - 2 ( d e i + 2 ) - 1 sqrt{1-2{left({d}_{e}_{i}+2)}^{-1}}}在 1 ≤ i ≤ m 1le ile m 上的数字之和，其中 d e i {d}_{e}_{i}} 是与 e i {e}_{i} 相邻的边的数目。在本文中，我们将研究具有给定参数的图中 ABS 指数的最大值。更具体地说，我们确定了具有固定(i)最小度、(ii)最大度、(iii)色度数、(iv)独立性数或(v)下垂顶点数的给定阶数连通图的最大 ABS 指数。我们还描述了在所有这些类别中达到最大 ABS 值的图的特征。
{"title":"On the maximum atom-bond sum-connectivity index of graphs","authors":"Tariq Alraqad, Hicham Saber, Akbar Ali, Abeer M. Albalahi","doi":"10.1515/math-2023-0179","DOIUrl":"https://doi.org/10.1515/math-2023-0179","url":null,"abstract":"The atom-bond sum-connectivity (ABS) index of a graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0179_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> with edges <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0179_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo form=\"prefix\">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{e}_{1},ldots ,{e}_{m}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the sum of the numbers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0179_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msqrt> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msqrt> </m:math> <jats:tex-math>sqrt{1-2{left({d}_{{e}_{i}}+2)}^{-1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0179_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≤</m:mo> <m:mi>m</m:mi> </m:math> <jats:tex-math>1le ile m</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0179_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:math> <jats:tex-math>{d}_{{e}_{i}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the number of edges adjacent to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0179_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{e}_{i}</jats:tex-math> </jats:alter","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"80 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139758563","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}
Ahmed Salem Heilat, Ahmad Qazza, Raed Hatamleh, Rania Saadeh, Mohammad W. Alomari
This study systematically develops error estimates tailored to a specific set of general quadrature rules that exclusively incorporate first derivatives. Moreover, it introduces refined versions of select generalized Ostrowski’s type inequalities, enhancing their applicability. Through the skillful application of Hayashi’s celebrated inequality to specific functions, the provided proofs underpin these advancements. Notably, this approach extends its utility to approximate integrals of real functions with bounded first derivatives. Remarkably, it employs Newton-Cotes and Gauss-Legendre quadrature rules, bypassing the need for stringent requirements on higher-order derivatives.
{"title":"An application of Hayashi's inequality in numerical integration","authors":"Ahmed Salem Heilat, Ahmad Qazza, Raed Hatamleh, Rania Saadeh, Mohammad W. Alomari","doi":"10.1515/math-2023-0162","DOIUrl":"https://doi.org/10.1515/math-2023-0162","url":null,"abstract":"This study systematically develops error estimates tailored to a specific set of general quadrature rules that exclusively incorporate first derivatives. Moreover, it introduces refined versions of select generalized Ostrowski’s type inequalities, enhancing their applicability. Through the skillful application of Hayashi’s celebrated inequality to specific functions, the provided proofs underpin these advancements. Notably, this approach extends its utility to approximate integrals of real functions with bounded first derivatives. Remarkably, it employs Newton-Cotes and Gauss-Legendre quadrature rules, bypassing the need for stringent requirements on higher-order derivatives.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"32 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462653","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}
本文研究指数多项式的唯一性，并主要证明：设 n n 为正整数，设 p i ( z ) ( i = 1 , 2 , ... , n ) {p}_{i}left(z)hspace{0.33em}left(i=1,2,ldots ,n) 是非零多项式，并且让 c i ≠ 0 ( i = 1 , 2 , ... , n ) {c}_{i}ne 0hspace{0.33em}left(i=1,2,ldots ,n) 是不同的有限复数。假设 f ( z ) fleft(z) 是一个全函数、 g ( z ) = p 1 ( z ) e c 1 z + p 2 ( z ) e c 2 z + ⋯ + p n ( z ) e c n z gleft(z)={p}_{1}left(z){e}^{c}_{1}z}+{p}_{2}left(z){e}^{c}_{2}z}+cdots +{p}_{n}left(z){e}^{c}_{n}z} 。如果 f ( z ) fleft(z) 和 g ( z ) gleft(z) 共享 a a 和 b b CM（计算乘数），其中 a a 和 b b 是两个不同的有限复数，那么必须出现以下情况之一： (i) n = 1 n=1 。 如果 a ≠ 0 ane 0 , b = 0 b=0 , 那么要么 f ( z ) ≡ g ( z ) fleft(z)equiv gleft(z) 或者 f ( z ) g ( z ) ≡ a 2 fleft(z)gleft(z)equiv {a}^{2} ； 如果 a = 0 a=0 , b ≠ 0 bne 0 , 那么要么 f ( z ) ≡ g ( z ) fleft(z)equiv gleft(z) 或者 f ( z ) g ( z ) ≡ b 2 fleft(z)gleft(z)equiv {b}^{2} ； 如果 a ≠ 0 ane 0 ， b ≠ 0 bne 0 ，那么要么 f ( z ) ≡ g ( z ) fleft(z)equiv gleft(z) 或者 f ( z ) g ( z ) ≡ ( a + b ) g ( z ) - a b fleft(z)gleft(z)equiv left(a+b)gleft(z)-ab 。 (ii) n ≥ 2 nge 2 , f ( z ) ≡ g ( z ) fleft(z)equiv gleft(z) . 这是对 1974 年早先关于微变函数的研究中得到的结果的扩展。
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