Background: Left ventricular noncompaction with multiple left ventricular thrombi can be revealed by echocardiography, and early diagnosis seems to be imperative to prevent significant embolic events. Case Report: A 5...Background: Left ventricular noncompaction with multiple left ventricular thrombi can be revealed by echocardiography, and early diagnosis seems to be imperative to prevent significant embolic events. Case Report: A 57-year-old woman presented with symptoms of heart failure. Two-dimensional transthoracic echocardiogram demonstrated a dilated and diffusely hypokinetic left ventricle with severe impaired left ventricular systolic function. Moreover, a markedly thickened endocardium at the left ventricular apex and middle segment with numerous, excessively prominent trabeculations and deep intertrabecular recesses were present. During systole, the ratio of the noncompacted to compacted myocardial layers at the site of the maximal wall thickness was above two, a characteristic finding in left ventricular non-compaction. Multiple mobile, homogeneous, echodense thrombi were identified in the left ventricle, with the largest one in the apical noncompacted segment (dimensions, 32 × 14 mm). Cardiac magnetic resonance imaging confirmed the diagnosis of noncompacted myocardium with the presence of multiple thrombi. After anticoagulant therapy, her symptoms improved and thrombi dissolved. Unexpectedly, she re-admitted to the cardiovascular unit with progressive dyspnea. Transthoracic echocardiogram showed new large right atrial thrombi, with the largest one was 43 × 38 mm compared to the echocardiogram done 11 months ago. The patient was anticoagulated with continuous heparin infusion for several days followed by oral Apixaban. After 4 weeks, the floating thrombi completely disappeared. After a 26-month follow-up, the patient’s condition was stable without embolic complications. Conclusion: Echocardiography was the cornerstone of diagnostic methods for early detecting left ventricular thrombi to eventually prevent embolic events.展开更多
BACKGROUND Left ventricular noncompaction(LVNC)is a genetic cardiomyopathy.It is characterized by intensely developed trabeculae in the ventricles with deep intertrabecular lacunae.LVNC manifests as arrhythmias and he...BACKGROUND Left ventricular noncompaction(LVNC)is a genetic cardiomyopathy.It is characterized by intensely developed trabeculae in the ventricles with deep intertrabecular lacunae.LVNC manifests as arrhythmias and heart failure with a predisposition for thrombus formation.AIM To study predictors of arrhythmic,thromboembolic events and adverse outcomes(death/transplantation)in adult patients with LVNC.METHODS Adult patients with LVNC were included(n=125;mean follow-up:14 months).Electrocardiography,echocardiography,and 24-hour electrocardiography monitoring were performed.Other procedures were conducted for some patients including:Coronary angiography;cardiac magnetic resonance imaging;cardiac computed tomography;genetic testing;myocardial pathological examination;and anti-cardiac antibody level estimation.Primary endpoints were death,heart transplantation,combined endpoint(death+transplantation),and sudden cardiac death.Secondary endpoints were intracardiac thrombosis,embolic events,myocardial infarction,sustained ventricular tachycardia(VT),and implantable cardioverter-defibrillator intervention.RESULTS LVNC manifestations included non-sustained VT,thrombosis/embolism,sustained VT, and sudden cardiac death. Non-sustained VT was associated with the New York Heart Association(NYHA) chronic heart failure (CHF) class, poor R-wave progression, superimposed myocarditis, and highermortality. Thrombosis/embolism was associated with NYHA CHF class ≥ 3, right ventricular end-diastolicdiameter ≥ 3 cm, right atrium volume ≥ 67 mL, left ventricle end-diastolic diameter ≥ 6.3 cm, and velocity timeintegral ≤ 11.2 cm. Sustained VT was associated with premature ventricular contractions (PVCs), low QRS voltage,and atrioventricular block. PVCs > 500/day were predictive of defibrillator intervention. Fatal outcomes wereassociated with E wave/A wave ratio > 1.9, left ventricle ejection fraction < 35%, NYHA CHF class ≥ 3, VT, andmyocarditis.CONCLUSIONFrequent PVCs, non-sustained VT, low QRS voltage, and signs of systolic dysfunction on echocardiogram arepredictors of life-threatening events in patients with LVNC.展开更多
This paper is devoted to the existence results for a class of neutral abstract fractional differential equations involving the composite relaxation process.Based on the Laplace transform,the semigroup theory and the W...This paper is devoted to the existence results for a class of neutral abstract fractional differential equations involving the composite relaxation process.Based on the Laplace transform,the semigroup theory and the Wright functions,we first introduce a definition of mild solutions to the considered problem.By means of the noncompactness of measure and the fixed point technique,we establish existence criteria of solutions.Finally,an example is presented to illustrate our main result.展开更多
This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained b...This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam’s novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.展开更多
In this paper, a new Browder fixed point theorem is established in the noncompact sub-admissible subsets of noncompact hyperconvex metric spaces. As application, a Ky Fan section theorem and an intersection theorem ar...In this paper, a new Browder fixed point theorem is established in the noncompact sub-admissible subsets of noncompact hyperconvex metric spaces. As application, a Ky Fan section theorem and an intersection theorem are obtained.展开更多
Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differenti...Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.展开更多
This paper improves Tychonov ford point theorem and discusses the existence of solutions of nonlinear Fredholm integral equations on [0,+∞] in Banach spaces with Frechet space theory.
Clinically, pulmonary embolism （PE） mostly comes from the lower extremity deep vein system, but if such pa- tients have no evidence of deep venous thrombosis （DVT）, other sources of emboli should be considered. It...Clinically, pulmonary embolism （PE） mostly comes from the lower extremity deep vein system, but if such pa- tients have no evidence of deep venous thrombosis （DVT）, other sources of emboli should be considered. It is very rare that embolus comes from the right heart system （in situ thrombosis）, Isolated fight ventricular noncompaction （iRVNC） can cause PE, but it has not been reported so far. Ventricular noncompaction （VNC） is an unusual cause of cardiomyopathy.展开更多
In the past, several authors studied spaces of m-th order difference sequences, among them, H.Polat and F.Basar （[17]） defined the Euler spaces of m-th order difference sequences e r 0 （△ （ m ） ）, e r c （△ （...In the past, several authors studied spaces of m-th order difference sequences, among them, H.Polat and F.Basar （[17]） defined the Euler spaces of m-th order difference sequences e r 0 （△ （ m ） ）, e r c （△ （ m ） ） and e r ∞ （△ （ m ） ） and characterized some classes of matrix transformations on them. In our paper, we add a new supplementary aspect to their research by characterizing classes of compact operators on those spaces. For that purpose, the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0 , c and ∞ . The main tool for our characterizations is the Hausdorff measure of noncompactness.展开更多
This paper deals with the preblem of existence and uniqueness of the stationary distributions (abbr., s. d.'s) for the processes constructed in [4] .The main results are stated in § 1. For the reader's co...This paper deals with the preblem of existence and uniqueness of the stationary distributions (abbr., s. d.'s) for the processes constructed in [4] .The main results are stated in § 1. For the reader's convenience we first restate the existence theorems (Theorem 1 and 2) of the processes given in [4]. Then two existence theorems (Theorem 3 and 4) and a uniqueness theorem (Theorem 5) for the s. d.'s of the processes are presented. The last result (Theorem 6), as an application of the previous ones, is about the Schlgl model which comes from nonequilibrium statisticali physics. The details of the proofs of Theorem 3—6 are given in § 2—4.展开更多
By applying a new existence theorem of quasi-equilibrium problems due to the author, some existence theorems of solutions for noncompact infinite optimization problems and noncompact constrained game problems are prov...By applying a new existence theorem of quasi-equilibrium problems due to the author, some existence theorems of solutions for noncompact infinite optimization problems and noncompact constrained game problems are proved in generalized convex spaces without linear structure. These theorems improve and generalize a number of important results in recent literature.展开更多
In this article, we introduce the notion of Meir-Keleer condensing operator in a Banach space, a characterization using L-functions and provide a few generalization of Darbo fixed point theorem. Also, we introduce the...In this article, we introduce the notion of Meir-Keleer condensing operator in a Banach space, a characterization using L-functions and provide a few generalization of Darbo fixed point theorem. Also, we introduce the concept of a bivariate Meir-Keleer condensing operator and prove some coupled fixed point theorems. As an application, we prove the existence of solutions for a large class of functional integral equations of Volterra type in two variables.展开更多
In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we es...In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter＇s and approximate essential spectrum.展开更多
In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space l（u, v, p; △（m））, which consist of the sequences whose generalized weighted △（m）-di...In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space l（u, v, p; △（m））, which consist of the sequences whose generalized weighted △（m）-difference means are in the linear space l（p） defined by I.J.Maddox. Also, we determine the basis of this space and compute its α-, β- and γ-duals. Further, we give the characterization of the classes of matrix mappings from l（u, v, p, △（m）） to l∞, c, and co. Finally, we apply the Hausdorff measure of noncompacness to characterize some classes of compact operators given by matrices on the space lp（U, v, △（m）） （1 ≤ p 〈 ∞）.展开更多
The global existence of the heat flow for harmonic maps from noncompact manifolds is considered. When L^m norm of the gradient of initial data is small, the existence of a global solution is proved.
The existence of positive solutions to a boundary value problem of second-order impulsive singular integro-differential equation with integral boundary conditions in a Banach space is obtained by means of fixed point ...The existence of positive solutions to a boundary value problem of second-order impulsive singular integro-differential equation with integral boundary conditions in a Banach space is obtained by means of fixed point theory. Moreover, an application is also given to illustrate the main result.展开更多
In this paper,we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness.Moreover,we apply the obtained results to discuss the incidence of some perturbation re...In this paper,we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness.Moreover,we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces.These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L1 spaces.展开更多
Heart failure is the most common cause of hospitalization in the United States.Just as the prevalence of heart failure has increased,the number of diseases identifi ed that result in the heart failure syndrome has esc...Heart failure is the most common cause of hospitalization in the United States.Just as the prevalence of heart failure has increased,the number of diseases identifi ed that result in the heart failure syndrome has escalated.Certain cardiomyopathies that have previously been regarded as very rare are being recognized with increasing frequency,because of improved imaging techniques and an increased understanding of the pathophysiologic mechanisms that result in these diseases.Improved echocardiographic techniques and methods such as spectral Doppler and 3D image rendering,along with the use of advanced diagnostic tools such as cardiac CT angiography and cardiac magnetic resonance imaging are now common.These advanced imaging methods have led to an increased appreciation of the frequency of diseases such as isolated left ventricular noncompaction and cardiac amyloidosis.Left ventricular noncompaction,once thought to occur in roughly one in one million patients,may actually be seen in fewer than one in 1000 patients.Cardiac amyloidosis,in the senile form,may exist in 80% of 80-year-old patients,although the incidence of clinical symptoms is less.As the genetic alterations that contribute to these diseases are further elucidated,improved diagnosis and a better understanding of the prognosis of these uncommon cardiomyopathies will follow.展开更多
Let L be the Laplace-Beltrami operator.On an n-dimensional(n≥2),complete,noncompact Riemannian manifold M,we prove that if 0<α<1,s>α/2 and f∈Hs(M),then the fractional Schr?dinger propagator e(it|L|α/2)(f...Let L be the Laplace-Beltrami operator.On an n-dimensional(n≥2),complete,noncompact Riemannian manifold M,we prove that if 0<α<1,s>α/2 and f∈Hs(M),then the fractional Schr?dinger propagator e(it|L|α/2)(f)(x)→f(x)a.e.as t→0.In addition,for when M is a Lie group,the rate of the convergence is also studied.These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.展开更多
文摘Background: Left ventricular noncompaction with multiple left ventricular thrombi can be revealed by echocardiography, and early diagnosis seems to be imperative to prevent significant embolic events. Case Report: A 57-year-old woman presented with symptoms of heart failure. Two-dimensional transthoracic echocardiogram demonstrated a dilated and diffusely hypokinetic left ventricle with severe impaired left ventricular systolic function. Moreover, a markedly thickened endocardium at the left ventricular apex and middle segment with numerous, excessively prominent trabeculations and deep intertrabecular recesses were present. During systole, the ratio of the noncompacted to compacted myocardial layers at the site of the maximal wall thickness was above two, a characteristic finding in left ventricular non-compaction. Multiple mobile, homogeneous, echodense thrombi were identified in the left ventricle, with the largest one in the apical noncompacted segment (dimensions, 32 × 14 mm). Cardiac magnetic resonance imaging confirmed the diagnosis of noncompacted myocardium with the presence of multiple thrombi. After anticoagulant therapy, her symptoms improved and thrombi dissolved. Unexpectedly, she re-admitted to the cardiovascular unit with progressive dyspnea. Transthoracic echocardiogram showed new large right atrial thrombi, with the largest one was 43 × 38 mm compared to the echocardiogram done 11 months ago. The patient was anticoagulated with continuous heparin infusion for several days followed by oral Apixaban. After 4 weeks, the floating thrombi completely disappeared. After a 26-month follow-up, the patient’s condition was stable without embolic complications. Conclusion: Echocardiography was the cornerstone of diagnostic methods for early detecting left ventricular thrombi to eventually prevent embolic events.
文摘BACKGROUND Left ventricular noncompaction(LVNC)is a genetic cardiomyopathy.It is characterized by intensely developed trabeculae in the ventricles with deep intertrabecular lacunae.LVNC manifests as arrhythmias and heart failure with a predisposition for thrombus formation.AIM To study predictors of arrhythmic,thromboembolic events and adverse outcomes(death/transplantation)in adult patients with LVNC.METHODS Adult patients with LVNC were included(n=125;mean follow-up:14 months).Electrocardiography,echocardiography,and 24-hour electrocardiography monitoring were performed.Other procedures were conducted for some patients including:Coronary angiography;cardiac magnetic resonance imaging;cardiac computed tomography;genetic testing;myocardial pathological examination;and anti-cardiac antibody level estimation.Primary endpoints were death,heart transplantation,combined endpoint(death+transplantation),and sudden cardiac death.Secondary endpoints were intracardiac thrombosis,embolic events,myocardial infarction,sustained ventricular tachycardia(VT),and implantable cardioverter-defibrillator intervention.RESULTS LVNC manifestations included non-sustained VT,thrombosis/embolism,sustained VT, and sudden cardiac death. Non-sustained VT was associated with the New York Heart Association(NYHA) chronic heart failure (CHF) class, poor R-wave progression, superimposed myocarditis, and highermortality. Thrombosis/embolism was associated with NYHA CHF class ≥ 3, right ventricular end-diastolicdiameter ≥ 3 cm, right atrium volume ≥ 67 mL, left ventricle end-diastolic diameter ≥ 6.3 cm, and velocity timeintegral ≤ 11.2 cm. Sustained VT was associated with premature ventricular contractions (PVCs), low QRS voltage,and atrioventricular block. PVCs > 500/day were predictive of defibrillator intervention. Fatal outcomes wereassociated with E wave/A wave ratio > 1.9, left ventricle ejection fraction < 35%, NYHA CHF class ≥ 3, VT, andmyocarditis.CONCLUSIONFrequent PVCs, non-sustained VT, low QRS voltage, and signs of systolic dysfunction on echocardiogram arepredictors of life-threatening events in patients with LVNC.
基金Supported by the Guangxi Science and Technology base and Talent Project(AD23026249,AD22080047).
文摘This paper is devoted to the existence results for a class of neutral abstract fractional differential equations involving the composite relaxation process.Based on the Laplace transform,the semigroup theory and the Wright functions,we first introduce a definition of mild solutions to the considered problem.By means of the noncompactness of measure and the fixed point technique,we establish existence criteria of solutions.Finally,an example is presented to illustrate our main result.
基金supported by Ministry of Human Resource and Development(MHR-02-23-200-429/304)
文摘This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam’s novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.
基金Supported by the Scientific Research Foundation of Bijie University(20072001)
文摘In this paper, a new Browder fixed point theorem is established in the noncompact sub-admissible subsets of noncompact hyperconvex metric spaces. As application, a Ky Fan section theorem and an intersection theorem are obtained.
文摘Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.
文摘This paper improves Tychonov ford point theorem and discusses the existence of solutions of nonlinear Fredholm integral equations on [0,+∞] in Banach spaces with Frechet space theory.
文摘Clinically, pulmonary embolism （PE） mostly comes from the lower extremity deep vein system, but if such pa- tients have no evidence of deep venous thrombosis （DVT）, other sources of emboli should be considered. It is very rare that embolus comes from the right heart system （in situ thrombosis）, Isolated fight ventricular noncompaction （iRVNC） can cause PE, but it has not been reported so far. Ventricular noncompaction （VNC） is an unusual cause of cardiomyopathy.
基金supported by the research project#144003 of the Serbian Ministry of Science, Technology and Development
文摘In the past, several authors studied spaces of m-th order difference sequences, among them, H.Polat and F.Basar （[17]） defined the Euler spaces of m-th order difference sequences e r 0 （△ （ m ） ）, e r c （△ （ m ） ） and e r ∞ （△ （ m ） ） and characterized some classes of matrix transformations on them. In our paper, we add a new supplementary aspect to their research by characterizing classes of compact operators on those spaces. For that purpose, the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0 , c and ∞ . The main tool for our characterizations is the Hausdorff measure of noncompactness.
文摘This paper deals with the preblem of existence and uniqueness of the stationary distributions (abbr., s. d.'s) for the processes constructed in [4] .The main results are stated in § 1. For the reader's convenience we first restate the existence theorems (Theorem 1 and 2) of the processes given in [4]. Then two existence theorems (Theorem 3 and 4) and a uniqueness theorem (Theorem 5) for the s. d.'s of the processes are presented. The last result (Theorem 6), as an application of the previous ones, is about the Schlgl model which comes from nonequilibrium statisticali physics. The details of the proofs of Theorem 3—6 are given in § 2—4.
文摘By applying a new existence theorem of quasi-equilibrium problems due to the author, some existence theorems of solutions for noncompact infinite optimization problems and noncompact constrained game problems are proved in generalized convex spaces without linear structure. These theorems improve and generalize a number of important results in recent literature.
文摘In this article, we introduce the notion of Meir-Keleer condensing operator in a Banach space, a characterization using L-functions and provide a few generalization of Darbo fixed point theorem. Also, we introduce the concept of a bivariate Meir-Keleer condensing operator and prove some coupled fixed point theorems. As an application, we prove the existence of solutions for a large class of functional integral equations of Volterra type in two variables.
文摘In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter＇s and approximate essential spectrum.
文摘In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space l（u, v, p; △（m））, which consist of the sequences whose generalized weighted △（m）-difference means are in the linear space l（p） defined by I.J.Maddox. Also, we determine the basis of this space and compute its α-, β- and γ-duals. Further, we give the characterization of the classes of matrix mappings from l（u, v, p, △（m）） to l∞, c, and co. Finally, we apply the Hausdorff measure of noncompacness to characterize some classes of compact operators given by matrices on the space lp（U, v, △（m）） （1 ≤ p 〈 ∞）.
基金Supported by the National Natural Science Foundation of China (1057115610671079+1 种基金10701064)the Zijin Project of Zhejiang University
文摘The global existence of the heat flow for harmonic maps from noncompact manifolds is considered. When L^m norm of the gradient of initial data is small, the existence of a global solution is proved.
基金Supported by the Foundation for Outstanding Middle-Aged and Young Scientists of Shandong Province (GrantNo. BS2010SF004)a Project of Shandong Province Higher Educational Science and Technology Program (GrantNo. J10LA53)+1 种基金the China Postdoctoral Science Foundation (Grant No. 20110491154)the National NaturalScience Foundation of China (Grant No. 10971179)
文摘The existence of positive solutions to a boundary value problem of second-order impulsive singular integro-differential equation with integral boundary conditions in a Banach space is obtained by means of fixed point theory. Moreover, an application is also given to illustrate the main result.
文摘In this paper,we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness.Moreover,we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces.These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L1 spaces.
文摘Heart failure is the most common cause of hospitalization in the United States.Just as the prevalence of heart failure has increased,the number of diseases identifi ed that result in the heart failure syndrome has escalated.Certain cardiomyopathies that have previously been regarded as very rare are being recognized with increasing frequency,because of improved imaging techniques and an increased understanding of the pathophysiologic mechanisms that result in these diseases.Improved echocardiographic techniques and methods such as spectral Doppler and 3D image rendering,along with the use of advanced diagnostic tools such as cardiac CT angiography and cardiac magnetic resonance imaging are now common.These advanced imaging methods have led to an increased appreciation of the frequency of diseases such as isolated left ventricular noncompaction and cardiac amyloidosis.Left ventricular noncompaction,once thought to occur in roughly one in one million patients,may actually be seen in fewer than one in 1000 patients.Cardiac amyloidosis,in the senile form,may exist in 80% of 80-year-old patients,although the incidence of clinical symptoms is less.As the genetic alterations that contribute to these diseases are further elucidated,improved diagnosis and a better understanding of the prognosis of these uncommon cardiomyopathies will follow.
基金supported partly by the NationalNatural Science Foundation of China(12071437)the Natural Science Foundation from the Education Department of Anhui Province(KJ2020A0044)+2 种基金the Research Fund Project of Chaohu University(KYQD-2023016)the High Level Scientific Research Achievement Award Cultivation Project of Chaohu University(kj20zkjp04)the Key Construction Discipline of Chaohu University(kj22zdjsxk01)。
文摘Let L be the Laplace-Beltrami operator.On an n-dimensional(n≥2),complete,noncompact Riemannian manifold M,we prove that if 0<α<1,s>α/2 and f∈Hs(M),then the fractional Schr?dinger propagator e(it|L|α/2)(f)(x)→f(x)a.e.as t→0.In addition,for when M is a Lie group,the rate of the convergence is also studied.These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.
