Variational Principles and Mean Value Estimates

2022 ◽  
Author(s):  
A. V. Arutyunov ◽  
S. E. Zhukovskiy
Keyword(s):  
Variational Principles ◽  
Mean Value

Fr échet Differentiability of Real-Valued Functions

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principles ◽  
Mean Value ◽  
Lipschitz Functions ◽  
Monotone Functions ◽  
Porous Set ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
The Mean ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the corresponding point of Fréchet differentiability may be found outside any given σ‎-porous set. This new result considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ‎-porous sets is new even in the Lipschitz case. The chapter first discusses the use of variational principles to prove Fréchet differentiability before analyzing a one-dimensional mean value problem in relation to Lipschitz functions. It shows that results on existence of points of Fréchet differentiability may be generalized to derivatives other than the Fréchet derivative.

Introduction

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Variational Principles ◽  
Mean Value ◽  
Lipschitz Functions ◽  
Asplund Space ◽  
Lipschitz Mapping ◽  
Countable Collection ◽  
Fréchet Differentiability ◽  
Nikodym Property ◽  
Frechet Differentiability ◽  
And Variational Principles

This book deals with the existence of Fréchet derivatives of Lipschitz functions from X to Y, where X is an Asplund space and Y has the Radon-Nikodým property (RNP). It considers whether every countable collection of real-valued Lipschitz functions on an Asplund space has a common point of Fréchet differentiability. It also examines the conditions under which all Lipschitz mapping of X to finite dimensional spaces not only possess points of Fréchet differentiability, but possess so many of them that even the multidimensional mean value estimate holds. Other topics include the notion of the Radon-Nikodým property and main results on Gâteaux differentiability of Lipschitz functions and related notions of null sets; separable determination and variational principles; and differentiability of Lipschitz maps on Hilbert spaces.

Long-period modulated superstructures in Pd-12.5 at.%Ce and Pd-15 at.%Ce alloys

1990 ◽  
Vol 48 (4) ◽  
pp. 164-165
Author(s):  
Noriyuki Kuwano ◽  
Masaru Itakura ◽  
Kensuke Oki
Keyword(s):  
Electron Microscopy ◽  
Mean Value ◽  
Heavy Elements ◽  
Arc Furnace ◽  
X Ray ◽  
Long Period ◽  
Ultra High Purity ◽  
Heat Treated ◽  
Atomic Scattering ◽  
The Mean

Pd-Ce alloys exhibit various anomalies in physical properties due to mixed valences of Ce, and the anomalies are thought to be strongly related with the crystal structures. Since Pd and Ce are both heavy elements, relative magnitudes of (fcc-fpd) are so small compared with <f> that superlattice reflections, even if any, sometimes cannot be detected in conventional x-ray powder patterns, where fee and fpd are atomic scattering factors of Ce and Pd, and <f> the mean value in the crystal. However, superlattices in Pd-Ce alloys can be analyzed by electron microscopy, thanks to the high detectability of electron diffraction. In this work, we investigated modulated superstructures in alloys with 12.5 and 15.0 at.%Ce.Ingots of Pd-Ce alloys were prepared in an arc furnace under atmosphere of ultra high purity argon. The disc specimens cut out from the ingots were heat-treated in vacuum and electrothinned to electron transparency by a jet method.

Variational Principles in Mathematical Physics, Geometry, and Economics

2009 ◽  
Author(s):  
Alexandru Kristaly ◽  
Vicentiu D. Radulescu ◽  
Csaba Varga
Keyword(s):  
Variational Principles ◽  
Mathematical Physics

Automated Latex Agglutination and ELISA Testing Yield Equivalent D-Dimer Results in Patients with Recent Myocardial Infarction

1999 ◽  
Vol 82 (11) ◽  
pp. 1412-1416 ◽  
Author(s):  
Wojciech Zareba ◽  
John Horan ◽  
Arthur Moss ◽  
Joel Kanouse ◽  
◽  
...  
Keyword(s):  
Myocardial Infarction ◽  
Elisa Test ◽  
Mean Value ◽  
Latex Agglutination ◽  
Test Results ◽  
Coronary Death ◽  
Elisa Testing ◽  
Strong Linear Correlation ◽  
Highly Correlated ◽  
D Dimer

SummaryOur previous prospective study of post-infarction patients described a strong and significant association of increased plasma D-dimer concentrations in those who experienced a subsequent coronary death or non-fatal myocardial infarction. In the present study, we compare results on stored plasma obtained two months after the index myocardial infarction from 1,038 patients of this trial, using a simple automated latex agglutination (LA) assay in parallel with the standard ELISA test. Results show a somewhat higher mean value for the LA assay (702 ± 1092 vs. 638 ± 986 ng/ml, p = 0.0002), a strong linear correlation of the two assays (r = 0.86) and 88% agreement for values below 500 ng/ml by the ELISA test. D-dimer concentrations determined by each assay were highly correlated in patients with subsequent coronary artery events (p = 0.93) and quartile values for both the LA and ELISA were equally predictive of such events (p = 0.003 and p = 0.001, respectively). This is the first demonstration that a latex agglutination assay for D-dimer can be used to assess the prognostic risk of recurrent coronary thrombotic disease after myocardial infarction

Causes and the way of treatment of patients with chronic leg ulcers

Phlebologie ◽  
2009 ◽  
Vol 38 (05) ◽  
pp. 211-218 ◽  
Author(s):  
C. Wax ◽  
A. Körber ◽  
J. Dissemond ◽  
J. Klode
Keyword(s):  
Treatment Period ◽  
General Practitioners ◽  
Peripheral Arterial Occlusive Disease ◽  
Mean Value ◽  
Arterial Occlusive Disease ◽  
Leg Ulcer ◽  
Leg Ulcers ◽  
Patient Will ◽  
Peripheral Arterial ◽  
Chronic Leg Ulcers

SummaryChronic leg ulcer may have various causes, which are currently not centrally recorded in Germany. It is also unclear who treats patients with chronic leg ulcers in Germany and how the basic implementation of diagnosis and treatment of these patients looks like. Patients, methods: Therefore, we started a survey of 1000 general practitioners and practising specialists in dermatology, surgery and phlebology in five different regions of Germany. We carried out the genesis of a total of 6275 patients from 62 different practising therapists, 33 specialists in dermatology, surgery or phlebology and 29 general practitioners. Results: In 66.1% of all patients we found a venous leg ulcer, in 9.1% a leg ulcer from peripheral arterial occlusive disease, and in 8.5% a mixture of both. Thus there suffered a total of 83.8% of patients on chronic venous insufficiency or peripheral arterial occlusive disease as a major factor in the genesis of the chronic leg ulcer. However, even the rarely diagnosed entities such as exogenous factors, vasculitis, pyoderma gangrenosum or infectious diseases are occur in summation in 16.2% of all patients and should therefore be known and excluded. In addition, the treatment periods and referral routes of patients with chronic leg ulcer should be identified. The analysis showed that the vast majority (86.8%) of patients with chronic ulcers who were investigated by us is treated by specialists. The treatment duration of general practitioners is 6.3 weeks (mean value) before the patient will be referred to a specialist. This treatment period is significantly shorter compared to the treatment period of the specialists, who treat their patients 14.1 weeks (mean value) before the patient will be referred to another specialist or to a clinic. Conclusion: Our results show the current aspects of aetiology and the way of treatment of patients with chronic leg ulcers in Germany.

Partition Coefficient Ratios and Tumour Perfusion Studied with 85mKr and 133Xe

1987 ◽  
Vol 26 (06) ◽  
pp. 253-257
Author(s):  
M. Mäntylä ◽  
J. Perkkiö ◽  
J. Heikkonen
Keyword(s):  
Partition Coefficient ◽  
Partition Coefficients ◽  
Regional Blood Flow ◽  
Blood Perfusion ◽  
Compartmental Analysis ◽  
Malignant Tumour ◽  
Mean Value ◽  
Perfusion Rate ◽  
Biological Variables ◽  
The Mean

The relative partition coefficients of krypton and xenon, and the regional blood flow in 27 superficial malignant tumour nodules in 22 patients with diagnosed tumours were measured using the 85mKr- and 133Xe-clearance method. In order to minimize the effect of biological variables on the measurements the radionuclides were injected simultaneously into the tumour. The distribution of the radiotracers was assumed to be in equilibrium at the beginning of the experiment. The blood perfusion was calculated by fitting a two-exponential function to the measuring points. The mean value of the perfusion rate calculated from the xenon results was 13 ± 10 ml/(100 g-min) [range 3 to 38 ml/(100 g-min)] and from the krypton results 19 ± 11 ml/(100 g-min) [range 5 to 45 ml/(100 g-min)]. These values were obtained, if the partition coefficients are equal to one. The equations obtained by using compartmental analysis were used for the calculation of the relative partition coefficient of krypton and xenon. The partition coefficient of krypton was found to be slightly smaller than that of xenon, which may be due to its smaller molecular weight.

Reliability of a Single β-Thromboglobulin Measurement in a Diabetic Population : Importance of PGE1 in Anticoagulant Mixture

1987 ◽  
Vol 57 (02) ◽  
pp. 201-204 ◽  
Author(s):  
P Y Scarabin ◽  
L Strain ◽  
C A Ludlam ◽  
J Jones ◽  
E M Kohner
Keyword(s):  
In Vitro Release ◽  
Mean Value ◽  
Reliable Estimate ◽  
Diabetic Patients ◽  
Single Measurement ◽  
Diabetic Population ◽  
The Difference ◽  
Vitro Release

SummaryDuring the collection of samples for plasma β-thromboglobulin (β-TG) determination, it is well established that artificially high values can be observed due to in-vitro release. To estimate the reliability of a single β-TG measurement, blood samples were collected simultaneously from both arms on two separate occasions in 56 diabetic patients selected for a clinical trial. From each arm, blood was taken into two tubes containing an anticoagulant mixture with (tube A) and without (tube B) PGE!. The overall mean value of B-TG in tube B was 1.14 times higher than in tube A (p <0.01). The markedly large between-arms variation accounted for the most part of within-subject variation in both tubes and was significantly greater in tube B than in tube A. Based on the difference between B-TG values from both arms, the number of subjects with artifically high B-TG values was significantly higher in tube B than in tube A on each occasion (overall rate: 28% and 14% respectively). Estimate of between-occasions variation showed that B-TG levels were relatively stable for each subject between two occasions in each tube. It is concluded that the use of PGEi decreases falsely high B-TG levels, but a single measurement of B-TG does not provide a reliable estimate of the true B-TG value in vivo.

Hereditary Heparin Cofactor II Deficiency and the Risk of Development of Thrombosis

1987 ◽  
Vol 57 (02) ◽  
pp. 196-200 ◽  
Author(s):  
R M Bertina ◽  
I K van der Linden ◽  
L Engesser ◽  
H P Muller ◽  
E J P Brommer
Keyword(s):  
Liver Disease ◽  
Venous Thrombosis ◽  
Healthy Volunteers ◽  
Mean Value ◽  
Age Group ◽  
Normal Range ◽  
Heparin Cofactor Ii ◽  
History Of ◽  
Group 2 ◽  
Hereditary Nature

SummaryHeparin cofactor II (HC II) levels were measured by electroimmunoassay in healthy volunteers, and patients with liver disease, DIC, proteinuria or a history of venous thrombosis. Analysis of the data in 107 healthy volunteers revealed that plasma HC II increases with age (at least between 20 and 50 years). HC II was found to be decreased in most patients with liver disease (mean value: 43%) and only in some patients with DIC. Elevated levels were found in patients with proteinuria (mean value 145%). In 277 patients with a history of unexplained venous thrombosis three patients were identified with a HC II below the lower limit of the normal range (60%). Family studies demonstrated hereditary HC II deficiency in two cases. Among the 9 heterozygotes for HC II deficiency only one patient had a well documented history of unexplained thrombosis. Therefore the question was raised whether heterozygotes for HC II deficiency can also be found among healthy volunteers. When defining a group of individuals suspected of HC II deficiency as those who have a 90% probability that their plasma HC II is below the 95% tolerance limits of the normal distribution in the relevant age group, 2 suspected HC II deficiencies were identified among the healthy volunteers. In one case the hereditary nature of the defect could be established.It is concluded that hereditary HC II deficiency is as prevalent among healthy volunteers as in patients with thrombotic disease. Further it is unlikely that heterozygosity for HC II deficiency in itself is a risk factor for the development of venous thrombosis.

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