SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRODINGER PROPAGATOR ON NONCOMPACT MANIFOLDS

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.

A New Browder Fixed Point Theorem in Noncompact Hyperconvex Metric Spaces and Its Applications to Section Questions and Intersection Questions

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.

Isolated right ventricular noncompaction caused recurrent pulmonary embolism

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.

FIXED POINT THEOREMS FOR MEIR-KEELER CONDENSING OPERATORS VIA MEASURE OF NONCOMPACTNESS

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.

NONCOMPACT INFINITE OPTIMIZATION AND EQUILIBRIA OF CONSTRAINED GAMES IN GENERALIZED CONVEX SPACES

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.

STATIONARY DISTRIBUTIONS OF INFINITE PARTICLE SYSTEMS WITH NONCOMPACT STATE SPACE.

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.

Heat flow of harmonic maps from noncompact manifolds

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.

SOLVABILITY OF AN IMPLICIT FRACTIONAL INTEGRAL EQUATION VIA A MEASURE OF NONCOMPACTNESS ARGUMENT

In this paper,we study the existence of solutions to an implicit functional equation involving a fractional integral with respect to a certain function,which generalizes the Riemann-Liouville fractional integral and the Hadaniard fractional integral.We establish an existence result to such kind of equations using a generalized version of Darbo＇s theorem associated to a certain measure of nonconipactness.Some examples are presented.

MEASURES OF WEAK NONCOMPACTNESS AND FIXED POINT THEORY FOR 1-SET WEAKLY CONTRACTIVE OPERATORS ON UNBOUNDED DOMAINS

The main purpose of this paper is to prove a collection of new fixed point theorems and existence theorems for the nonlinear operator equation F（x） = αx （α≥〉 1） for so-called 1-set weakly contractive operators on unbounded domains in Banach spaces. We also introduce the concept of weakly semi-closed operator at the origin and obtain a series of new fixed point theorems and the existence theorems for the nonlinear operator equation F（x） = αx （α≥1） for such class of operators. As consequences, the main results general- ize and improve the relevant results, which are obtained by O＇Regan and A. Ben Amar and M. Mnif in 1998 and 2009 respectively. In addition, we get the famous fixed point theorems of Leray-Schauder, Altman, Petryshyn and Rothe type in the case of weakly sequentially continuous, 1-set weakly contractive （μ-nonexpansive） and weakly semi-closed operators at the origin and their generalizations. The main condition in our results is formulated in terms of axiomatic measures of weak compactness.

Left ventricular noncompaction associated with hypertrophic cardiomyopathy and Wolff-Parkinson-White syndrome

We report a 35-year-old female patient with hypertrophic cardiomyopathy, left ventricular noncompaction, and Wolff-Parkinson-White EKG pattern. Several other family members present the same clinical condition. We speculate that this phenotype is related to the genotypes PRKAG2 and LAMP2 represented by mutations of the genes encoding AMP-activated protein kinase (PRKAG2) and lysosome associated membrane protein 2 (LAMP2).

Metrics on Noncompact Fuzzy Number Space E^n

The theory of metric spaces of fuzzy numbers has been established and found very convenient in many research fields on fuzzy analysis such as fuzzy integrals and differentials, fuzzy differential equations, fuzzy random variables and fuzzy stochastic processes etc.. But, a large part of this theory heavily depends on the condition that fuzzy number has to have compact support set and so fails to analyze and apply noncompact fuzzy numbers. The purpose of this paper is to introduce three classes of metrics on noncompact fuzzy number space and to discuss their basic properties, completeness and separability in detail.

Experience of a Single Center in the Diagnosis and Classification of Cases of Left Ventricular Noncompaction

Objectives: To analyse the clinical profile of consecutive cases of Left Ventricular Non Compaction (LVNC) with particular interest in non-compacted segments valuation. Methods: There were 18,000 patients seen from 2007 to 2010, with a complete evaluation including family history and personal cardiac history, clinical examination and electrocardiography. Diagnosis was based on three published definitions. Results: The diagnosis of LVNC was placed in 1.4% of cases. Clinical and echo-cardiographic data for the 250 cases of LVNC are presented. Trabecular meshwork was observed predominantly at the apex (91.6%), in the lateral and inferior wall (40.4% and 38.0% respectively), and less frequently in the posterior and anterior wall (21.6% and 9.2% respectively). Conclusions: This study suggests that LVNC is a form of cardiomyopathy with higher prevalence and relatively better prognosis than previously reported.

Effect of trabeculated myocardial mass on left ventricle global and regional functions in noncompaction cardiomyopathy

BACKGROUND Left ventricular(LV)noncompaction cardiomyopathy is a rare cardiomyopathic subtype that has been recognized in recent years and is being diagnosed at an increased rate.There is no consensus regarding the diagnosis of the disease,and increased trabeculation rates that meet the existing diagnostic criteria may even be present in healthy asymptomatic people.This indicates that differentiating criteria for diagnosis are needed.AIM To examine the increase in myocardial trabeculation and the change in left ventricular global and regional functions.METHODS This retrospective study included 65 patients(28 females,37 males)diagnosed with LV noncompaction cardiomyopathy who underwent cardiac magnetic resonance imaging between January 2011 and August 2016 and had a noncompacted/compacted myocardial thickness ratio of over 2.3 in more than one segment in the left ventricle.The distribution and ratios of trabeculations in apical,midventricular,and basal regions were examined in short-axis images obtained from cardiac magnetic resonance.In addition,by using short-axis cine images,regional ejection fraction(EF)and global EF were calculated using the Simpson method in the left ventricle at apical,basal,and midventricular levels.RESULTS While the number of trabeculated segments were similar at the apical(3.2±1.0)and midventricular levels,a statistically significant level of involvement was not observed at the basal level(0.4±0.9)(P>0.05).The highest noncompacted/compacted(trabeculation)ratio was observed at the apical level(3.9±1.4),while this ratio was higher at the anterior(59%-89.4%)and lateral(62%-84.8%)segments(P>0.05).Global EF was positively correlated with apical,midventricular,and basal regional EF(P<0.05).However,there was no significant correlation between regional EF and the number of trabeculated segments or trabeculation ratio in all three regions;nor was there a significant correlation between regional EF and the number of trabeculated segments or trabeculation ratio in the entire LV(P>0.05).CONCLUSION No global or regional relationship was observed between LV dysfunction and trabeculation rate or the number of trabeculated segments.This limits the usefulness of change in LV functions in the differentiation between normal and pathological trabeculation.

A Case of Recurrent Multiple Left Ventricular Thrombi without Thromboembolism in Noncompacted Myocardium

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.

Representation of measures of noncompactness and its applications related to an initial value problem in Banach spaces

This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).

Existence of Solutions of a Convolution Integral Equation

In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball Br belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set M∈ B r the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.

Incremental value of contrast echocardiography in the diagnosis of left ventricular noncompaction

Contrast echocardiography with left ventricular opacification （LVO） improves the definition of endocardium in two-dimensional echocardiography （2DE）. This study was aimed to determine whether LVO offered added diagnostic value in noncompaetion of left ventricular myocardium （NCVM）. A total of 85 patients （40± 20 years, 54 males） with suspected NCVM were subjected to transthoracic 2DE and LVO, and 40 healthy volunteers were examined with 2DE and assigned as control subjects. The location of NCVM, the thickness ratio of noncompacted to compacted myocardium （NCR）, and the cavity size and ejection fraction of LV were quantified. Results revealed that NCVM was mainly located in the LV medium （53.2%）, apical （46.2%） segments, and lateral wall （39.8%）. The NCR obtained through LVO was greater than that detected through 2DE （4.2 ±1.3 vs. 3.3 ±1.2, P 〈 0.001）, and higher inter-correlations and less intra- and inter-observer variabilities were determined in the former than in the latter. The NCVM detection rates were also increased from 63.5% via 2DE to 83.5% via LVO and 89.4% via 2DE combined with LVO （2DE ＋ LVO） （P = 0.0004）. The LV cavity size was greater and the LV ejection fraction （LVEF） was lower in the NCVM patients than in the control group （P 〈 0.01）. In the NCVM group, the LV cavity size was higher and the LVEF was lower in LVO than in 2DE （P 〈 0.01）. In conclusion, contrast echocardiography contributes significant sensitivity and reproducibility to routine transthoraeic echoeardiography in NCVM diagnosis. Therefore, this technique should be clinically performed to diagnose suspected NCVM.

一类中立型随机积微分方程mild解的存在性

该文在Hilbert空间中研究了一类具有非瞬时脉冲的中立型随机积微分方程非局部问题mild解的存在性.利用预解算子理论、非紧性测度理论和Darbo不动点定理,建立了所研究问题mild解的存在性定理,丰富和发展了已有的相关结果.

非紧致度量空间上的持久性

考虑非紧致度量空间上同胚的持久性问题,利用同胚的持久性、等度连续性、强拓扑稳定性、持久跟踪性等定义,证明:等度连续且拓扑稳定的同胚是持久的;同胚有持久跟踪性当且仅当该同胚是持久的且有伪轨跟踪性;有持久跟踪性的可扩同胚是强拓扑稳定的.

A GLKKM Type Theorem for Noncompact Complete L-Convex Metric Spaces with Applications to Variational Inequalities and Fixed Points

In this paper,a new GLKKM type theorem is established for noncompact complete L-convex metric spaces.As applications,the properties of the solution set of variational in-equalities,intersection point sets,Ky Fan sections and maximal element sets are shown,and a Fan-Browder fixed point theorem is obtained.