A minimal system begin{document}$ (X,T) $end{document} is topologically mildly mixing if for all non-empty open subsets begin{document}$ U,V $end{document}, begin{document}$ {nin {mathbb Z}: Ucap T^{-n}Vneq emptyset} $end{document} is an IPbegin{document}$ ^* $end{document}-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that begin{document}$ (X,T) $end{document} is a topologically mildly mixing minimal system, begin{document}$ din {mathbb N} $end{document}, begin{document}$ p_1(n),ldots, p_d(n) $end{document} are integral polynomials with no begin{document}$ p_i $end{document} and no begin{document}$ p_i-p_j $end{document} constant, begin{document}$ 1le ineq jle d $end{document}. Then for all non-empty open subsets begin{document}$ U , V_1, ldots, V_d $end{document}, begin{document}$ {nin {mathbb Z}: Ucap T^{-p_1(n) }V_1cap T^{-p_2(n)}V_2cap ldots cap T^{-p_d(n) }V_d neq emptyset } $end{document} is an IPbegin{document}$ ^* $end{document}-set. We also give the corresponding theorem for systems under abelian group actions.
A minimal system begin{document}$ (X,T) $end{document} is topologically mildly mixing if for all non-empty open subsets begin{document}$ U,V $end{document}, begin{document}$ {nin {mathbb Z}: Ucap T^{-n}Vneq emptyset} $end{document} is an IPbegin{document}$ ^* $end{document}-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that begin{document}$ (X,T) $end{document} is a topologically mildly mixing minimal system, begin{document}$ din {mathbb N} $end{document}, begin{document}$ p_1(n),ldots, p_d(n) $end{document} are integral polynomials with no begin{document}$ p_i $end{document} and no begin{document}$ p_i-p_j $end{document} constant, begin{document}$ 1le ineq jle d $end{document}. Then for all non-empty open subsets begin{document}$ U , V_1, ldots, V_d $end{document}, begin{document}$ {nin {mathbb Z}: Ucap T^{-p_1(n) }V_1cap T^{-p_2(n)}V_2cap ldots cap T^{-p_d(n) }V_d neq emptyset } $end{document} is an IPbegin{document}$ ^* $end{document}-set. We also give the corresponding theorem for systems under abelian group actions.
{"title":"Topological mild mixing of all orders along polynomials","authors":"Yang Cao, S. Shao","doi":"10.3934/dcds.2021150","DOIUrl":"https://doi.org/10.3934/dcds.2021150","url":null,"abstract":"<p style='text-indent:20px;'>A minimal system <inline-formula><tex-math id=\"M1\">begin{document}$ (X,T) $end{document}</tex-math></inline-formula> is topologically mildly mixing if for all non-empty open subsets <inline-formula><tex-math id=\"M2\">begin{document}$ U,V $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M3\">begin{document}$ {nin {mathbb Z}: Ucap T^{-n}Vneq emptyset} $end{document}</tex-math></inline-formula> is an IP<inline-formula><tex-math id=\"M4\">begin{document}$ ^* $end{document}</tex-math></inline-formula>-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that <inline-formula><tex-math id=\"M5\">begin{document}$ (X,T) $end{document}</tex-math></inline-formula> is a topologically mildly mixing minimal system, <inline-formula><tex-math id=\"M6\">begin{document}$ din {mathbb N} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M7\">begin{document}$ p_1(n),ldots, p_d(n) $end{document}</tex-math></inline-formula> are integral polynomials with no <inline-formula><tex-math id=\"M8\">begin{document}$ p_i $end{document}</tex-math></inline-formula> and no <inline-formula><tex-math id=\"M9\">begin{document}$ p_i-p_j $end{document}</tex-math></inline-formula> constant, <inline-formula><tex-math id=\"M10\">begin{document}$ 1le ineq jle d $end{document}</tex-math></inline-formula>. Then for all non-empty open subsets <inline-formula><tex-math id=\"M11\">begin{document}$ U , V_1, ldots, V_d $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M12\">begin{document}$ {nin {mathbb Z}: Ucap T^{-p_1(n) }V_1cap T^{-p_2(n)}V_2cap ldots cap T^{-p_d(n) }V_d neq emptyset } $end{document}</tex-math></inline-formula> is an IP<inline-formula><tex-math id=\"M13\">begin{document}$ ^* $end{document}</tex-math></inline-formula>-set. We also give the corresponding theorem for systems under abelian group actions.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90235734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Zakharov system in dimension begin{document}$ dleqslant 3 $end{document} is shown to be locally well-posed in Sobolev spaces begin{document}$ H^s times H^l $end{document}, extending the previously known result. We construct new solution spaces by modifying the begin{document}$ X^{s,b} $end{document} spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.
The Zakharov system in dimension begin{document}$ dleqslant 3 $end{document} is shown to be locally well-posed in Sobolev spaces begin{document}$ H^s times H^l $end{document}, extending the previously known result. We construct new solution spaces by modifying the begin{document}$ X^{s,b} $end{document} spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.
{"title":"Local well-posedness for the Zakharov system in dimension d ≤ 3","authors":"A. Sanwal","doi":"10.3934/dcds.2021147","DOIUrl":"https://doi.org/10.3934/dcds.2021147","url":null,"abstract":"<p style='text-indent:20px;'>The Zakharov system in dimension <inline-formula><tex-math id=\"M1\">begin{document}$ dleqslant 3 $end{document}</tex-math></inline-formula> is shown to be locally well-posed in Sobolev spaces <inline-formula><tex-math id=\"M2\">begin{document}$ H^s times H^l $end{document}</tex-math></inline-formula>, extending the previously known result. We construct new solution spaces by modifying the <inline-formula><tex-math id=\"M3\">begin{document}$ X^{s,b} $end{document}</tex-math></inline-formula> spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"28 10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82720268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger begin{document}$ W^{1,2} $end{document} convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.
We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger begin{document}$ W^{1,2} $end{document} convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.
{"title":"Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction","authors":"G. Colombo, P. Gidoni, Emilio Vilches","doi":"10.3934/dcds.2021135","DOIUrl":"https://doi.org/10.3934/dcds.2021135","url":null,"abstract":"<p style='text-indent:20px;'>We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger <inline-formula><tex-math id=\"M1\">begin{document}$ W^{1,2} $end{document}</tex-math></inline-formula> convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87078211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient begin{document}$ d $end{document} of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling wave. We numerically observed that in the region where the traveling wave exists the disease successfully propagate like traveling wave but in the region of no traveling wave disease stops to invade. We also numerically observed that as begin{document}$ d $end{document} increases the speed of propagation slows down and the parameter region of propagation narrows down.
In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient begin{document}$ d $end{document} of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling wave. We numerically observed that in the region where the traveling wave exists the disease successfully propagate like traveling wave but in the region of no traveling wave disease stops to invade. We also numerically observed that as begin{document}$ d $end{document} increases the speed of propagation slows down and the parameter region of propagation narrows down.
{"title":"Traveling wave solution for a diffusive simple epidemic model with a free boundary","authors":"Yoichi Enatsu, E. Ishiwata, T. Ushijima","doi":"10.3934/dcdss.2020387","DOIUrl":"https://doi.org/10.3934/dcdss.2020387","url":null,"abstract":"In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient begin{document}$ d $end{document} of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling wave. We numerically observed that in the region where the traveling wave exists the disease successfully propagate like traveling wave but in the region of no traveling wave disease stops to invade. We also numerically observed that as begin{document}$ d $end{document} increases the speed of propagation slows down and the parameter region of propagation narrows down.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86914322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a blow-up phenomenon for begin{document}$ { partial_t^2 u_ varepsilon} $end{document} begin{document}$ {- varepsilon^2 partial_x^2u_ varepsilon } $end{document} begin{document}$ { = F(partial_t u_ varepsilon)}. $end{document} The derivative of the solution begin{document}$ partial_t u_ varepsilon $end{document} blows-up on a curve begin{document}$ t = T_ varepsilon(x) $end{document} if we impose some conditions on the initial values and the nonlinear term begin{document}$ F $end{document} . We call begin{document}$ T_ varepsilon $end{document} blow-up curve for begin{document}$ { partial_t^2 u_ varepsilon} $end{document} begin{document}$ {- varepsilon^2 partial_x^2u_ varepsilon } $end{document} begin{document}$ { = F(partial_t u_ varepsilon)}. $end{document} In the same way, we consider the blow-up curve begin{document}$ t = tilde{T}(x) $end{document} for begin{document}$ {partial_t^2 u} $end{document} begin{document}$ = $end{document} begin{document}$ {F(partial_t u)}. $end{document} The purpose of this paper is to show that, for each begin{document}$ x $end{document} , begin{document}$ T_ varepsilon(x) $end{document} converges to begin{document}$ tilde{T}(x) $end{document} as begin{document}$ varepsilonrightarrow 0. $end{document}
We consider a blow-up phenomenon for begin{document}$ { partial_t^2 u_ varepsilon} $end{document} begin{document}$ {- varepsilon^2 partial_x^2u_ varepsilon } $end{document} begin{document}$ { = F(partial_t u_ varepsilon)}. $end{document} The derivative of the solution begin{document}$ partial_t u_ varepsilon $end{document} blows-up on a curve begin{document}$ t = T_ varepsilon(x) $end{document} if we impose some conditions on the initial values and the nonlinear term begin{document}$ F $end{document} . We call begin{document}$ T_ varepsilon $end{document} blow-up curve for begin{document}$ { partial_t^2 u_ varepsilon} $end{document} begin{document}$ {- varepsilon^2 partial_x^2u_ varepsilon } $end{document} begin{document}$ { = F(partial_t u_ varepsilon)}. $end{document} In the same way, we consider the blow-up curve begin{document}$ t = tilde{T}(x) $end{document} for begin{document}$ {partial_t^2 u} $end{document} begin{document}$ = $end{document} begin{document}$ {F(partial_t u)}. $end{document} The purpose of this paper is to show that, for each begin{document}$ x $end{document} , begin{document}$ T_ varepsilon(x) $end{document} converges to begin{document}$ tilde{T}(x) $end{document} as begin{document}$ varepsilonrightarrow 0. $end{document}
{"title":"Convergence of a blow-up curve for a semilinear wave equation","authors":"Takiko Sasaki","doi":"10.3934/dcdss.2020388","DOIUrl":"https://doi.org/10.3934/dcdss.2020388","url":null,"abstract":"We consider a blow-up phenomenon for begin{document}$ { partial_t^2 u_ varepsilon} $end{document} begin{document}$ {- varepsilon^2 partial_x^2u_ varepsilon } $end{document} begin{document}$ { = F(partial_t u_ varepsilon)}. $end{document} The derivative of the solution begin{document}$ partial_t u_ varepsilon $end{document} blows-up on a curve begin{document}$ t = T_ varepsilon(x) $end{document} if we impose some conditions on the initial values and the nonlinear term begin{document}$ F $end{document} . We call begin{document}$ T_ varepsilon $end{document} blow-up curve for begin{document}$ { partial_t^2 u_ varepsilon} $end{document} begin{document}$ {- varepsilon^2 partial_x^2u_ varepsilon } $end{document} begin{document}$ { = F(partial_t u_ varepsilon)}. $end{document} In the same way, we consider the blow-up curve begin{document}$ t = tilde{T}(x) $end{document} for begin{document}$ {partial_t^2 u} $end{document} begin{document}$ = $end{document} begin{document}$ {F(partial_t u)}. $end{document} The purpose of this paper is to show that, for each begin{document}$ x $end{document} , begin{document}$ T_ varepsilon(x) $end{document} converges to begin{document}$ tilde{T}(x) $end{document} as begin{document}$ varepsilonrightarrow 0. $end{document}","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"47 4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83189448","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}
Automatic detection of objects in photos and images is beneficial in various scientific and industrial fields. This contribution suggests an algorithm for segmentation of color images by the means of the parametric mean curvature flow equation and CIE94 color distance function. The parametric approach is enriched by the enhanced algorithm for topological changes where the intersection of curves is computed instead of unreliable curve distance. The result is a set of parametric curves enclosing the object. The algorithm is presented on a test image and also on real photos.
{"title":"Segmentation of color images using mean curvature flow and parametric curves","authors":"P. Paus, S. Yazaki","doi":"10.3934/dcdss.2020389","DOIUrl":"https://doi.org/10.3934/dcdss.2020389","url":null,"abstract":"Automatic detection of objects in photos and images is beneficial in various scientific and industrial fields. This contribution suggests an algorithm for segmentation of color images by the means of the parametric mean curvature flow equation and CIE94 color distance function. The parametric approach is enriched by the enhanced algorithm for topological changes where the intersection of curves is computed instead of unreliable curve distance. The result is a set of parametric curves enclosing the object. The algorithm is presented on a test image and also on real photos.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83346964","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}
We consider the blow-up problems of the power type of stochastic differential equation, begin{document}$ dX = alpha X^p(t)dt+X^q(t)dW(t) $end{document} . It has been known that there exists a critical exponent such that if begin{document}$ p $end{document} is greater than the critical exponent then the solution begin{document}$ X(t) $end{document} blows up almost surely in the finite time. In our research, focus on this critical exponent, we propose a numerical scheme by adaptive time step and analyze it mathematically. Finally we show the numerical result by using the proposed scheme.
We consider the blow-up problems of the power type of stochastic differential equation, begin{document}$ dX = alpha X^p(t)dt+X^q(t)dW(t) $end{document} . It has been known that there exists a critical exponent such that if begin{document}$ p $end{document} is greater than the critical exponent then the solution begin{document}$ X(t) $end{document} blows up almost surely in the finite time. In our research, focus on this critical exponent, we propose a numerical scheme by adaptive time step and analyze it mathematically. Finally we show the numerical result by using the proposed scheme.
{"title":"Numerical and mathematical analysis of blow-up problems for a stochastic differential equation","authors":"T. Ishiwata, Young Chol Yang","doi":"10.3934/dcdss.2020391","DOIUrl":"https://doi.org/10.3934/dcdss.2020391","url":null,"abstract":"We consider the blow-up problems of the power type of stochastic differential equation, begin{document}$ dX = alpha X^p(t)dt+X^q(t)dW(t) $end{document} . It has been known that there exists a critical exponent such that if begin{document}$ p $end{document} is greater than the critical exponent then the solution begin{document}$ X(t) $end{document} blows up almost surely in the finite time. In our research, focus on this critical exponent, we propose a numerical scheme by adaptive time step and analyze it mathematically. Finally we show the numerical result by using the proposed scheme.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78037633","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}
Kateřina Škardová, T. Oberhuber, J. Tintěra, R. Chabiniok
In this paper we propose a method for locally adjusted optical flow-based registration of multimodal images, which uses the segmentation of the object of interest and its representation by the signed-distance function (OF dist method). We deal with non-rigid registration of the image series acquired by the Modiffied Look-Locker Inversion Recovery (MOLLI) magnetic resonance imaging sequence, which is used for a pixel-wise estimation of T 1 relaxation time. The spatial registration of the images within the series is necessary to compensate the patient's imperfect breath-holding. The evolution of intensities and a large variation of image contrast within the MOLLI image series, together with the myocardium of left ventricle (the object of interest) typically not being the most distinct object in the scene, makes the registration challenging. The paper describes all components of the proposed OF dist method and their implementation. The method is then compared to the performance of a standard mutual information maximization-based registration method, applied either to the original image (MIM) or to the signed-distance function (MIM dist). Several experiments with synthetic and real MOLLI images are carried out. On synthetic image with a single object, MIM performed the best, while OF dist and MIM dist provided better results on synthetic images with more than one object and on real images. When applied to signed-distance function of two objects of interest, MIM dist provided a larger registration error (but more homogeneously distributed) compared to OF dist. For the real MOLLI image series with left ventricle pre-segmented using a level-set method, the proposed OF dist registration performed the best, as is demonstrated visually and by measuring the increase of mutual information in the object of interest and its neighborhood.
{"title":"Signed-distance function based non-rigid registration of image series with varying image intensity","authors":"Kateřina Škardová, T. Oberhuber, J. Tintěra, R. Chabiniok","doi":"10.3934/DCDSS.2020386","DOIUrl":"https://doi.org/10.3934/DCDSS.2020386","url":null,"abstract":"In this paper we propose a method for locally adjusted optical flow-based registration of multimodal images, which uses the segmentation of the object of interest and its representation by the signed-distance function (OF dist method). We deal with non-rigid registration of the image series acquired by the Modiffied Look-Locker Inversion Recovery (MOLLI) magnetic resonance imaging sequence, which is used for a pixel-wise estimation of T 1 relaxation time. The spatial registration of the images within the series is necessary to compensate the patient's imperfect breath-holding. The evolution of intensities and a large variation of image contrast within the MOLLI image series, together with the myocardium of left ventricle (the object of interest) typically not being the most distinct object in the scene, makes the registration challenging. The paper describes all components of the proposed OF dist method and their implementation. The method is then compared to the performance of a standard mutual information maximization-based registration method, applied either to the original image (MIM) or to the signed-distance function (MIM dist). Several experiments with synthetic and real MOLLI images are carried out. On synthetic image with a single object, MIM performed the best, while OF dist and MIM dist provided better results on synthetic images with more than one object and on real images. When applied to signed-distance function of two objects of interest, MIM dist provided a larger registration error (but more homogeneously distributed) compared to OF dist. For the real MOLLI image series with left ventricle pre-segmented using a level-set method, the proposed OF dist registration performed the best, as is demonstrated visually and by measuring the increase of mutual information in the object of interest and its neighborhood.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"102 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76847295","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}
In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference is calculated by the normalized begin{document}$ L^1 $end{document} norm of the difference of step-like functions which are branches of begin{document}$ arg (x) $end{document} whose discontinuities are on the spirals. We find that the differences in the numerical results are small, even though the model equations around the center and the farthest facet are slightly different.
In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference is calculated by the normalized begin{document}$ L^1 $end{document} norm of the difference of step-like functions which are branches of begin{document}$ arg (x) $end{document} whose discontinuities are on the spirals. We find that the differences in the numerical results are small, even though the model equations around the center and the farthest facet are slightly different.
{"title":"Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow","authors":"T. Ishiwata, T. Ohtsuka","doi":"10.3934/dcdss.2020390","DOIUrl":"https://doi.org/10.3934/dcdss.2020390","url":null,"abstract":"In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference is calculated by the normalized begin{document}$ L^1 $end{document} norm of the difference of step-like functions which are branches of begin{document}$ arg (x) $end{document} whose discontinuities are on the spirals. We find that the differences in the numerical results are small, even though the model equations around the center and the farthest facet are slightly different.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86717338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a proof by foliation that the cones over begin{document}$ mathbb{S}^k times mathbb{S}^l $end{document} minimize parametric elliptic functionals for each begin{document}$ k, , l geq 1 $end{document}. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.
We give a proof by foliation that the cones over begin{document}$ mathbb{S}^k times mathbb{S}^l $end{document} minimize parametric elliptic functionals for each begin{document}$ k, , l geq 1 $end{document}. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.
{"title":"A proof by foliation that lawson's cones are $ A_{Phi} $-minimizing","authors":"Connor Mooney, Yang Yang","doi":"10.3934/DCDS.2021077","DOIUrl":"https://doi.org/10.3934/DCDS.2021077","url":null,"abstract":"<p style='text-indent:20px;'>We give a proof by foliation that the cones over <inline-formula><tex-math id=\"M2\">begin{document}$ mathbb{S}^k times mathbb{S}^l $end{document}</tex-math></inline-formula> minimize parametric elliptic functionals for each <inline-formula><tex-math id=\"M3\">begin{document}$ k, , l geq 1 $end{document}</tex-math></inline-formula>. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82976115","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：