Dao Huy Cuong, Ngo Nguyen Quoc Bao, Nguyen Duy Khang, Nguyen Nhat Nam
This paper presents a Roe-type numerical scheme for the model of fluid flows in a nozzle with variable cross-section. The proposed scheme is built using the Roe method combined with stationary contact jumps at interfaces. The scheme is proven to capture smooth stationary waves precisely and preserve the positivity of the fluid's density. The numerical tests show that this approach can give considered accuracy to the exact solutions, except where the exact solution crosses a sonic surface.
{"title":"Well-balanced and Positivity-preserving Roe-type Numerical Scheme for the Model of Fluid Flows in a Nozzle with Variable Cross-section","authors":"Dao Huy Cuong, Ngo Nguyen Quoc Bao, Nguyen Duy Khang, Nguyen Nhat Nam","doi":"10.11650/tjm/230804","DOIUrl":"https://doi.org/10.11650/tjm/230804","url":null,"abstract":"This paper presents a Roe-type numerical scheme for the model of fluid flows in a nozzle with variable cross-section. The proposed scheme is built using the Roe method combined with stationary contact jumps at interfaces. The scheme is proven to capture smooth stationary waves precisely and preserve the positivity of the fluid's density. The numerical tests show that this approach can give considered accuracy to the exact solutions, except where the exact solution crosses a sonic surface.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135594690","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}
{"title":"Discretization and Perturbation of Wavelet-like Families","authors":"Michael Wilson","doi":"10.11650/tjm/231001","DOIUrl":"https://doi.org/10.11650/tjm/231001","url":null,"abstract":"","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135310824","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 paper, a new concept of the subdifferential is defined for nondifferentiable (not necessarily) locally Lipschitz functions. Namely, the concept of $E$-subdifferential and the notion of $E$-subconvexity are introduced for $E$-convex functions. Thus, the notion of an $E$-subdifferentiable $E$-convex function is introduced and some properties of this class of nondifferentiable nonconvex functions are studied. The necessary optimality conditions in $E$-subdifferentials terms of the involved functions are established for a new class of nondifferentiable optimization problems. The introduced concept of $E$-subconvexity is used to prove the sufficiency of the aforesaid necessary optimality conditions for nondifferentiable optimization problems in which the involved functions are $E$-subdifferentiable $E$-convex.
{"title":"$E$-subdifferential of $E$-convex Functions and its Applications to Minimization Problem","authors":"Tadeusz Antczak, Najeeb Abdulaleem","doi":"10.11650/tjm/230803","DOIUrl":"https://doi.org/10.11650/tjm/230803","url":null,"abstract":"In this paper, a new concept of the subdifferential is defined for nondifferentiable (not necessarily) locally Lipschitz functions. Namely, the concept of $E$-subdifferential and the notion of $E$-subconvexity are introduced for $E$-convex functions. Thus, the notion of an $E$-subdifferentiable $E$-convex function is introduced and some properties of this class of nondifferentiable nonconvex functions are studied. The necessary optimality conditions in $E$-subdifferentials terms of the involved functions are established for a new class of nondifferentiable optimization problems. The introduced concept of $E$-subconvexity is used to prove the sufficiency of the aforesaid necessary optimality conditions for nondifferentiable optimization problems in which the involved functions are $E$-subdifferentiable $E$-convex.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135557174","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 present Tseng's forward-backward-forward method with extrapolation from the past for pseudo-monotone variational inequalities in Hilbert spaces. In addition, we propose a variable stepsize scheme of the extrapolated Tseng's algorithm governed by the operator which is pseudo-monotone, Lipschitz continuous and sequentially weak-to-weak continuous. We also investigate the algorithm's adaptive stepsize scenario, which arises when it is impossible to calculate the Lipschitz constant of a pseudo-monotone operator correctly. Finally, we prove a weak convergence theorem and conduct a numerical experiment to support it.
{"title":"A Modified Tseng's Algorithm with Extrapolation from the Past for Pseudo-monotone Variational Inequalities","authors":"Buris Tongnoi","doi":"10.11650/tjm/230906","DOIUrl":"https://doi.org/10.11650/tjm/230906","url":null,"abstract":"We present Tseng's forward-backward-forward method with extrapolation from the past for pseudo-monotone variational inequalities in Hilbert spaces. In addition, we propose a variable stepsize scheme of the extrapolated Tseng's algorithm governed by the operator which is pseudo-monotone, Lipschitz continuous and sequentially weak-to-weak continuous. We also investigate the algorithm's adaptive stepsize scenario, which arises when it is impossible to calculate the Lipschitz constant of a pseudo-monotone operator correctly. Finally, we prove a weak convergence theorem and conduct a numerical experiment to support it.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136258065","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}
{"title":"Non Local Weighted Fourth Order Equation in Dimension $4$ with Non-linear Exponential Growth","authors":"Rached Jaidane, A. Ali","doi":"10.11650/tjm/230202","DOIUrl":"https://doi.org/10.11650/tjm/230202","url":null,"abstract":"","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47944714","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}
{"title":"A Class of Viscoelastic Wave Equations with Exponential Source and the Nonlinear Strong Damping","authors":"Menglan Liao","doi":"10.11650/tjm/230705","DOIUrl":"https://doi.org/10.11650/tjm/230705","url":null,"abstract":"","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44276085","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}
Let $k geq 1$ be an integer, and let $(U_{n})$ be the Lucas sequence of the first kind defined by [ U_{0} = 0, quad U_{1} = 1 quad textrm{and} quad U_{n} = kU_{n-1} + U_{n-2} quad textrm{for $n geq 2$}. ] It is well known that $(U_{n})$ is periodic modulo any integer $m geq 2$, and we let $pi(m)$ denote the length of this period. A prime $p$ is called a $k$-Wall–Sun–Sun prime if $pi(p^{2}) = pi(p)$. Let $f(x) in mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $mathbb{Q}$. We say $f(x)$ is monogenic if $Theta = { 1, theta, theta^{2}, ldots, theta^{N-1} }$ is a basis for the ring of integers $mathbb{Z}_{K}$ of $K = mathbb{Q}(theta)$, where $f(theta) = 0$. If $Theta$ is not a basis for $mathbb{Z}_{K}$, we say that $f(x)$ is non-monogenic. Suppose that $k notequiv 0 pmod{4}$ and that $mathcal{D} := (k^{2}+4)/gcd(2,k)^{2}$ is squarefree. We prove that $p$ is a $k$-Wall–Sun–Sun prime if and only if $mathcal{F}_{p}(x) = x^{2p}-kx^{p}-1$ is non-monogenic. Furthermore, if $p$ is a prime divisor of $k^{2}+4$, then $mathcal{F}_{p}(x)$ is monogenic.
{"title":"A New Condition for $k$-Wall–Sun–Sun Primes","authors":"Lenny Jones","doi":"10.11650/tjm/231003","DOIUrl":"https://doi.org/10.11650/tjm/231003","url":null,"abstract":"Let $k geq 1$ be an integer, and let $(U_{n})$ be the Lucas sequence of the first kind defined by [ U_{0} = 0, quad U_{1} = 1 quad textrm{and} quad U_{n} = kU_{n-1} + U_{n-2} quad textrm{for $n geq 2$}. ] It is well known that $(U_{n})$ is periodic modulo any integer $m geq 2$, and we let $pi(m)$ denote the length of this period. A prime $p$ is called a $k$-Wall–Sun–Sun prime if $pi(p^{2}) = pi(p)$. Let $f(x) in mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $mathbb{Q}$. We say $f(x)$ is monogenic if $Theta = { 1, theta, theta^{2}, ldots, theta^{N-1} }$ is a basis for the ring of integers $mathbb{Z}_{K}$ of $K = mathbb{Q}(theta)$, where $f(theta) = 0$. If $Theta$ is not a basis for $mathbb{Z}_{K}$, we say that $f(x)$ is non-monogenic. Suppose that $k notequiv 0 pmod{4}$ and that $mathcal{D} := (k^{2}+4)/gcd(2,k)^{2}$ is squarefree. We prove that $p$ is a $k$-Wall–Sun–Sun prime if and only if $mathcal{F}_{p}(x) = x^{2p}-kx^{p}-1$ is non-monogenic. Furthermore, if $p$ is a prime divisor of $k^{2}+4$, then $mathcal{F}_{p}(x)$ is monogenic.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135211250","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}
Ideal submanifolds have been studied from various aspects since Chen invented $delta$-invariants in early 1990s (see [12] for a survey). In centroaffine differential geometry, Chen's invariants denoted by $delta^{sharp}$ are used to determine an optimal bound for the squared norm of the Tchebychev vector field of a hypersurface. We point out that a hypersurface attaining this bound is said to be an ideal centroaffine hypersurface. In this paper, we deal with $delta^{sharp}(2,2)$-ideal centroaffine hypersurfaces in $mathbb{R}^{5}$ and in particularly, we focus on $4$-dimensional $delta^{sharp}(2,2)$-ideal centroaffine hypersurfaces of type $1$.
{"title":"$delta^{sharp}(2,2)$-Ideal Centroaffine Hypersurfaces of Dimension 4","authors":"Handan Yıldırım, Luc Vrancken","doi":"10.11650/tjm/230706","DOIUrl":"https://doi.org/10.11650/tjm/230706","url":null,"abstract":"Ideal submanifolds have been studied from various aspects since Chen invented $delta$-invariants in early 1990s (see [12] for a survey). In centroaffine differential geometry, Chen's invariants denoted by $delta^{sharp}$ are used to determine an optimal bound for the squared norm of the Tchebychev vector field of a hypersurface. We point out that a hypersurface attaining this bound is said to be an ideal centroaffine hypersurface. In this paper, we deal with $delta^{sharp}(2,2)$-ideal centroaffine hypersurfaces in $mathbb{R}^{5}$ and in particularly, we focus on $4$-dimensional $delta^{sharp}(2,2)$-ideal centroaffine hypersurfaces of type $1$.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135262144","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 paper, a modified alternately linear implicit (MALI) iteration method is derived for solving the non-symmetric coupled algebraic Riccati equation (NCARE). In the MALI iteration algorithm, the coefficient matrices of the linear matrix equations are fixed at each iteration step. In addition, the MALI iteration method utilizes a weighted average of the estimates in both the last step and current step to update the estimates in the next iteration step. Further, we give the convergence theory of the modified algorithm. Last, numerical examples demonstrate the effectiveness and feasibility of the derived algorithm.
{"title":"A Modified Iterative Method for Solving the Non-symmetric Coupled Algebraic Riccati Equation","authors":"Li Wang, Yibo Wang","doi":"10.11650/tjm/231101","DOIUrl":"https://doi.org/10.11650/tjm/231101","url":null,"abstract":"In this paper, a modified alternately linear implicit (MALI) iteration method is derived for solving the non-symmetric coupled algebraic Riccati equation (NCARE). In the MALI iteration algorithm, the coefficient matrices of the linear matrix equations are fixed at each iteration step. In addition, the MALI iteration method utilizes a weighted average of the estimates in both the last step and current step to update the estimates in the next iteration step. Further, we give the convergence theory of the modified algorithm. Last, numerical examples demonstrate the effectiveness and feasibility of the derived algorithm.","PeriodicalId":22176,"journal":{"name":"Taiwanese Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135660055","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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