A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES

This article is committed to deal with measure of non-compactness of operators in Banach spaces.Firstly,the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication)is a normed semigroup,and the mapping J from C(X)onto F(Ω)is a fully order-preserving positively linear surjective isometry,whereΩis the closed unit ball of X^*and F(Ω)the collection of all continuous and w^*-lower semicontinuous sublinear functions on X^*but restricted toΩ.Furthermore,both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC.The quotient space EC/EK is an abstract M space,hence,order isometric to a sublattice of C(K)for some compact Haudorspace K,and(FQJ)C which is a closed cone is contained in the positive cone of C(K),where Q:EC→EC/EK is the quotient mapping and F:EC/EK→C(K)is a corresponding order isometry.Finally,the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X,thenμ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K),∀T∈B(X).

Hypertrophic cardiomyopathy and left ventricular non-compaction:Distinct diseases or variant phenotypes of a single condition?

Hypertrophic cardiomyopathy(HCM)is a genetically determined myocardial disease characterized by an increased thickness of the left ventricle(LV)wall that cannot be solely attributed to abnormal loading conditions.HCM may present with an intraventricular or LV outflow tract obstruction,diastolic dysfunction,myocardial fibrosis and/or ventricular arrhythmias.Differentiating HCM from other diseases associated with LV hypertrophy,such as hypertension,aortic stenosis,or LV non-compaction(LVNC),can at times be challenging.LVNC is defined by excessive LV trabeculation and deep recesses between trabeculae,often accompanied by increased LV myocardial mass.Previous studies indicate that the LVNC phenotype may be observed in up to 5%of the general population;however,in most cases,it is a benign finding with no impact on clinical outcomes.Nevertheless,LVNC can occasionally lead to LV systolic dysfunction,manifesting as a phenotype of dilated or non-dilated left ventricular cardiomyopathy,with an increased risk of thrombus formation and arterial embolism.In extreme cases,where LVNC is associated with a very thickened LV wall,it can even mimic HCM.There is growing evidence of an overlap between HCM and LVNC,including similar genetic mutations and clinical presentations.This raises the question of whether HCM and LVNC represent different phenotypes of the same disease or are,in fact,two distinct entities.

Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

In this paper, we show that every infinite dimensional Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure. Therefore, it resolves a long-standing question.

Erratum to: Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

[1, Theorem 4.4] states that every infinite dimensional Banach space admits a homogenous measure of noncompactness not equivalent to the Hausdorff measure. Howevere, there is a gap in the proof. In fact,we found that [1, Lemma 4.3] is not true. In this erratum, we give a corrected proof of [1, Theorem 4.4].

On a Measure of Non-compactness for Some Classical Operators

The measure of non-compactness is estimated from below for various operators, including the Hardy-Littlewood maximal operator, the fractional maximal operator and the Hilbert transform, all acting between weighted Lebesgue spaces. The identity operator acting between weighted Lebesgue spaces and also between the counterparts of these spaces with variable exponents is similarly analysed. These results enable the lack of compactness of such operators to be quantified.

Left ventricular non-compaction presenting as cardio-embolic stroke

A 39-year-old male with no known comorbidities presented with sudden onset right-sided weakness.On examination,blood pressure was 128/79 mmHg,National Institutes of Health Stroke Scale score was 4 and there were no signs of heart failure.Emergent computerized tomography demonstrated an ischemic infarct of the left middle cerebral artery distribution and brain magnetic resonance imaging later confirmed it(Figure 1).

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T ： X → X be a uniformly continuous homeomorphism on a non-compact metric space （X, d）. Denote by X＊ = X ∪ {x＊} the one point compactification of X and T ＊ ： X＊ → X＊ the homeomorphism on X＊ satisfying T ＊｜X = T and T ＊x＊ = x＊. We show that their topological entropies satisfy hd（T, X） ≥ h（T ＊, X＊） if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.

Ventricular Non-Compaction: About the First Cases Reported in Lomé(Togo)

Introduction: Ventricular non-compaction, a cardiomyopathy recently described as likely to be rare, belongs to the group of unclassified cardiomyopathy according to European Society of Cardiology. Few studies have been published on the ventricular non-compaction in sub-Saharan Africa. We aim to find out the various aspects, being diagnosis, therapeutic, in Togolese patients carrying the ventricular non-compaction. Methodology: This is a three-year prospective and descriptive study conducted from January 2017 to December 2019 in the two University Hospital of Lomé. Patients having echocardiographic criteria of ventricular non-compaction were included in our study. Results: 10 patients (6 men and 4 women) were diagnosed for ventricular non-compaction during the study period. The mean age of patients was 32.3 years. The most frequent clinical manifestation was heart failure (7 patients). The main electrocardiogram anomaly was left ventricle hypertrophy (9 patients). The preferential segments were: apical (9 cases), apicolateral (8 cases), and septoapical (7 cases). The average ratio of non-compaction/compaction was 3.31. The main complication was thromboembolic event (4 patients). Angiotensin converting enzyme inhibitors and beta-blockers were essentially the medicines used. After a three (3) year follow-up, two (2) of the patients died. Conclusion: Tough ventricular non-compaction has been recently described. It is present in Togo. It displays many clinical manifestations and the prognosis is often guarded.

A Case Report of Isolated Left Ventricular Non-Compaction Presenting with Congestive Heart Failure and Intramural Thrombus

Isolated left ventricular non-compaction is recently described as a rare form of cardiomyopathy that is associated with a heart failure, life threatening cardiac arrhythmia and thromboembolic complications. The diagnosis is based on echocardiography demonstration of spongy myocardium. Here we report a case of 74 years old female patient diagnosed as an isolated left ventricular non-compaction with congestive heart failure, intramural thrombus and hypertension. There is no specific treatment for LVNC;therapeutic measures are directed towards the patient’s symptom (heart failure, arrhythmia and thrombotic events) and consideration of an implantable cardioverter defibrillator and cardiac transplantation.

Do Antineoplastic Drugs Play an Additional Role in the Progression of Non-Compaction Cardiomyopathy? A Case Report

Non-compaction cardiomyopathy is a rare form of cardiomyopathy;its most common clinical manifestations are heart failure (HF), ventricular arrhythmia, thromboembolism, and sudden cardiac death. We report a rare case of a 63-year-old man with chest tightness, worsening lower leg edema, dyspnea, and decreased exercise tolerance. He had a medical history of gastric cancer treated with subtotal gastrectomy and post-operative chemotherapy with paclitaxel and fluorouracil three years ago. At that time, he was diagnosed with non-compaction cardiomyopathy, and the thickened and reticulated trabecular muscle was exclusively confined to left ventricular apex. Five months ago, he was admitted to our hospital with heart failure and treated for dilated cardiomyopathy, echocardiography revealed severe trabecular noncompact myocardium in both ventricles, which was confirmed by cardiac magnetic resonance imaging (CMR). It is generally accepted that non-compacted myocardium forms in the early embryonic stage, which raises a question in our case whether acquired factors, such as antineoplastic drugs, potentially accelerate the pathological progression of non-compaction cardiomyopathy. Considering there are disparities between current screening tools such as echocardiography and CMR regarding diagnostic criteria, multi-detector CT may be an alternative examination method that could provide a new perspective for diagnosis.

GLOBALLY ATTRACTING SOLUTIONS TO IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS OF SOBOLEV TYPE

We study a generalized Cauchy problem associated with a class of impulsive fractional differential inclusions of Sobolev type in Banach spaces. Our aim is to prove the existence of a compact set of globally attracting solutions to the problem in question. An application to fractional partial differential equations subject to impulsive effects is given to illustrate our results.

Fractional Differential Equations with Initial Conditions at Inner Points in Banach Spaces

This paper is concerned with nonlinear fractional differential equations with the Caputo fractional derivatives in Banach spaces. Local existence results are obtained for initial value problems with initial conditions at inner points for the cases that the nonlinear parts are Lipschitz and non-Lipschitz, respectively. Hausdorff measure of non-compactness and Darbo-Sadovskii fixed point theorem are employed to deal with the non-Lipschitz case. The results obtained in this paper extend the classical Peano’s existence theorem for first order differential equations partly to fractional cases.

An unusual combination of left atrial myxoma and left ventricular non-compaction cardiomyopathy in a male patient:A Casereport

Left atrialmyxoma is a common primary cardiac tumor that is accompanied by organic heart diseases.But left atrial myxoma coexistent with left ventricular non-compaction(LVNC)is extremely rare.A young male patient with left atrial myxoma and LVNC was reported in this study.A 25-year-old manpresented to the emergency department with sudden shortness of breath and syncope,accompanied by fever and cough.He had a history ofacute ischemic strokeone year before hospitalization.Echocardiography revealed that the endocardium of the left ventricle was not smooth with raised muscle trabeculae and deep recesses.There was an oval-shaped strong echo mass with a pedicle in the left atrium attached to the atrial septum.Tumor resection was operated during extracorporeal circulation.Pathological results confirmed left atrium myxoma.In this case report,the patient had heart failure and an ischemic stroke likely of cardiogenic origin.He underwenttumor resection and started on therapeutic anticoagulation.Left atrial myxoma and LVNC are associated with poor outcomes.Early diagnosis and prompt treatment are crucial.

Isolated non-compaction of ventricular myocardium in a victim of the Wenchuan earthquake with crush syndrome and acute renal failure

Non-compaction of the ventricular myocardium （NVM） is a rare congenital genetic heart defect that was initially reported 17 years ago by means of autopsy; few cases have been published since then. It has been classified as the primary inherited cardiomyopathy by the American Heart Association Scientific Statement in 2006 under contemporary definitions and classification of cardiomyopathies. It is caused by an arrested development of muscle fiber compaction, a process that normally takes place dunng early embryogenesls, which is characterized by numerous sinusoids or trabeculae that are excessive in number and abnormal in prominence and by deep intratrabecular recesses covered by endothelium that exhibits continuity with ventricular endocardium. It usually involves one and/or more segments of the left ventricle, while the clinical manifestations are highly variable,

On the variational principle for the topological pressure for certain non-compact sets

Let (X,d,T) be a dynamical system,where (X,d) is a compact metric space and T:X → X is a continuous map.We assume that the dynamical system satisfies g-almost product property and the uniform separation property.We compute the topological pressure of saturated sets under these two conditions.If the uniform separation property does not hold,we compute the topological pressure of the set of generic points.We give an application of these results to multifractal analysis and finally get a conditional variational principle.

A class of Kazdan-Warner typed equations on non-compact Riemannian manifolds

In this paper,we discuss a Kazdan-Warner typed equation on certain non-compact Rie- mannian manifolds.As an application,we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundles over certain non-compact K(?)hler manifolds.

Whole-exome sequencing reveals twovariants in thegene in two Chinese patients with left ventricular non-compaction cardiomyopathy

Importance:Pathogenic variants in theRBM20 gene are associated with aggressive dilated cardiomyopathy(DCM).Recently,RBM20 was found to be associated with left ventricular non-compaction cardiomyopathy(LVNC).Thus far,only five families with LVNC have been reported to carry variants inRBM20.It remains unknown whether the variants inRBM20 associated with DCM can also cause LVNC.Objective:To elucidate the causativeRBM20 variant in two unrelated patients with both LVNC and DCM,and to identify the clinical characteristics associated with variants inRBM20.Methods:Trio whole-exome sequencing(WES)was performed.Variants were filtered and classified in accordance with the guidelines of the American College of Medical Genetics and Genomics(ACMG).Results:We identified two distinctde novo variants inRBM20(one per patient)in these two patients with LVNC.Both variants have been reported in patients with DCM,without the LVNC phenotype.Patient 1 was an 11-year-old girl who had DCM,LVNC,and heart failure;the ratio of noncompacted-to-compacted myocardium was 2.7:1.Ade novo heterozygous variant c.1907G>A(p.Arg636His)in exon 9 was identified in this patient.Patient 2 was a 13-year-old boy who had clinical phenotypes identical to those of Patient 1;the ratio of noncompacted-to-compacted myocardium was 3.2:1 in this patient.WES revealed ade novo heterozygous variant c.1909A>G(p.Ser637Gly)in exon 9.Both variants were previously characterized as pathogenic,and our study classified them as pathogenic variants based on the ACMG guidelines.Interpretation:We found that two patients with LVNC had variants inRBM20.Our results extended the clinical spectrum of the twoRBM20 variants and illustrated that the same variant inRBM20 can cause DCM,with or without the LVNC phenotype.

Note on families of semistable curves over P^1 with 4 singular fibers whose Jacobian are non-compact

Let f : S → P 1 be a semistable family of curves of genus g 2. We prove that if f admits exactly 5 singular fibers and 4 of them have non-compact Jacobian, then g = 2.

A rare case of isolated right ventricular non-compaction Cardiomyopathy

INTRODUCTION Isolated ventricular non-compaction is a rare congenital cardiomyopathy occurs due to arrest of normal myocardial development during embryogenesis. It is mainly diagnosed by echocar- diography through the appearance of characteristic prominent myocardial trabeculation and deep inter-trabecular spaces. Heart failore,

Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type

We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.Ourfirst existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic.In this context,our result requires only a partial C2 decay of the metric,namely the full decay of the metric in C1 and the decay of the scalar curvature.In particular,no decay of the Ricci curvature is assumed.In our second result we establish that a local volume ratio condition,when combined with negativity of the scalar curvature at infinity,is sufficient for existence of a solution.Our volume ratio condition appears tight.This paper is based on the DPhil thesis of thefirst author.