On Lower Volume Growth Estimate for f-Minimal Submanifolds in Gradient Shrinking Soliton

2016 ◽  
pp. rnw100 ◽  
Author(s):  
Yong Wei
Keyword(s):  
Volume Growth ◽  
Minimal Submanifolds ◽  
Growth Estimate ◽  
Lower Volume

Growth Estimates on Positive Solutions of the Equation

2001 ◽  
Vol 44 (2) ◽  
pp. 210-222 ◽  
Author(s):  
Man Chun Leung
Keyword(s):  
Positive Solutions ◽  
Volume Growth ◽  
Natural Conditions ◽  
Growth Estimate ◽  
Conformal Metric ◽  
Euclidean Metric ◽  
Slow Decay

AbstractWe construct unbounded positive C2-solutions of the equation in (equipped with Euclidean metric go) such that K is bounded between two positive numbers in , the conformal metric is complete, and the volume growth of g can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on u, we obtain growth estimate on the -norm of the solution and show that it has slow decay.

The relative volume growth of minimal submanifolds

2002 ◽  
Vol 79 (6) ◽  
pp. 507-514 ◽  
Author(s):  
Steen Markvorsen ◽  
Vicente Palmer
Keyword(s):  
Volume Growth ◽  
Minimal Submanifolds ◽  
Relative Volume

On CR volume growth estimate in a complete pseudohermitian 3-manifold

2014 ◽  
Vol 25 (04) ◽  
pp. 1450035 ◽  
Author(s):  
Shu-Cheng Chang ◽  
Ting-Hui Chang
Keyword(s):  
Lower Bound ◽  
Heisenberg Group ◽  
Volume Growth ◽  
Growth Estimate ◽  
Ricatti Equation ◽  
Carathéodory Distance ◽  
Curvature Tensors

In this paper, we first derive the CR Reilly's formula and a CR Ricatti equation for sub-Laplacian of the Carnot–Carathéodory distance in a complete pseudohermitian 3-manifold. As a consequence, we obtain the CR volume growth estimate in a complete pseudohermitian 3-manifold under a lower bound of pseudohermitian curvature tensors. This is a generalization of Nagel, Stein and Wainger's volume growth estimate for the Heisenberg ball in the standard Heisenberg group.

The relative volume growth of minimal submanifolds

2002 ◽  
Vol 79 (6) ◽  
pp. 507-514
Author(s):  
S. Markvorsen ◽  
V. Palmer
Keyword(s):  
Volume Growth ◽  
Minimal Submanifolds ◽  
Relative Volume

A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth

2019 ◽  
Vol 71 (6) ◽  
pp. 1367-1394 ◽  
Author(s):  
Der-Chen Chang ◽  
Shu-Cheng Chang ◽  
Yingbo Han ◽  
Jingzhi Tie
Keyword(s):  
Sobolev Inequality ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Mean Value ◽  
Doubling Property ◽  
Growth Estimate ◽  
Mean Value Inequality ◽  
Volume Doubling ◽  
The Mean

AbstractIn this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.

Relationship between Cross-Border Mergers and Acquisitions and International Trade: The Case of Russian Pharmaceutical Industry

2020 ◽  
pp. 81-94
Author(s):  
Alexandra V. Chugunova ◽  
Olga A. Klochko
Keyword(s):  
International Trade ◽  
Mergers And Acquisitions ◽  
Volume Growth ◽  
Pharmaceutical Products ◽  
Pharmaceutical Companies ◽  
Cross Border ◽  
Export Volume ◽  
Relationship Of ◽  
Geographical Diversification ◽  
The Relationship

This research studies the relationship of cross-border mergers and acquisitions to international trade through the lens of Russian pharmaceutical market. To this aim, the study analyses the woks of foreign economists dedicated to evaluating the link between foreign direct investment and international trade, and the influence of mergers and acquisitions on countries’ export and import flows. The research also presents a correlation analysis between the volume of Russian pharmaceutical exports and imports and cross-border deals performed by foreign pharmaceutical companies in Russia. We characterize these deals and conduct a comparative analysis of the regional structure of Russian pharmaceutical exports and imports as well as of the countries of origin of buyers in cross-border mergers and acquisitions. The results of the analysis indicate a positive relationship between cross-border mergers and acquisitions and Russian pharmaceutical exports, which is reflected in the export volume growth and its geographical diversification. However, it is outlined that particular problems of the industry hinder the amelioration of Russian positions in international exports. Similarly, the relationship between cross-border deals and Russian imports is positive: the major pharmaceutical products supply flow occurs from the countries of origin of buyers in cross-border mergers and acquisitions conducted in the Russian pharmaceutical sector.

Normal cerebral ventricular volume growth in childhood

2020 ◽  
Vol 26 (5) ◽  
pp. 517-524
Author(s):  
Noah S. Cutler ◽  
Sudharsan Srinivasan ◽  
Bryan L. Aaron ◽  
Sharath Kumar Anand ◽  
Michael S. Kang ◽  
...  
Keyword(s):  
Brain Mri ◽  
Volume Growth ◽  
Ventricular Volume ◽  
Growth Patterns ◽  
Diagnostic Process ◽  
Growth Charts ◽  
Normal Brain ◽  
Mr Images ◽  
Ventricular Volumes ◽  
Brain Mr Images

OBJECTIVENormal percentile growth charts for head circumference, length, and weight are well-established tools for clinicians to detect abnormal growth patterns. Currently, no standard exists for evaluating normal size or growth of cerebral ventricular volume. The current standard practice relies on clinical experience for a subjective assessment of cerebral ventricular size to determine whether a patient is outside the normal volume range. An improved definition of normal ventricular volumes would facilitate a more data-driven diagnostic process. The authors sought to develop a growth curve of cerebral ventricular volumes using a large number of normal pediatric brain MR images.METHODSThe authors performed a retrospective analysis of patients aged 0 to 18 years, who were evaluated at their institution between 2009 and 2016 with brain MRI performed for headaches, convulsions, or head injury. Patients were excluded for diagnoses of hydrocephalus, congenital brain malformations, intracranial hemorrhage, meningitis, or intracranial mass lesions established at any time during a 3- to 10-year follow-up. The volume of the cerebral ventricles for each T2-weighted MRI sequence was calculated with a custom semiautomated segmentation program written in MATLAB. Normal percentile curves were calculated using the lambda-mu-sigma smoothing method.RESULTSVentricular volume was calculated for 687 normal brain MR images obtained in 617 different patients. A chart with standardized growth curves was developed from this set of normal ventricular volumes representing the 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles. The charted data were binned by age at scan date by 3-month intervals for ages 0–1 year, 6-month intervals for ages 1–3 years, and 12-month intervals for ages 3–18 years. Additional percentile values were calculated for boys only and girls only.CONCLUSIONSThe authors developed centile estimation growth charts of normal 3D ventricular volumes measured on brain MRI for pediatric patients. These charts may serve as a quantitative clinical reference to help discern normal variance from pathologic ventriculomegaly.

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