Variation of a function

1985 ◽  
pp. 85-114
Author(s):  
Brian S. Thomson
Keyword(s):  
Variation Of A Function

The Fréchet Variation, Sector Limits, and Left Decompositions

1950 ◽  
Vol 2 ◽  
pp. 344-374 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Fourier Series ◽  
Variational Analysis ◽  
Cartesian Product ◽  
Multiple Fourier Series ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Variation Of A Function ◽  
De La Vallée Poussin

1. Introduction. The Fréchet variation of a function g defined over a 2-interval I2 was introduced by Fréchet to enable him to generalize Riesz's theorem on the representation of functionals linear over the space C [7]. Recently the authors have found this variation fundamental in the study of functionals bilinear over the Cartesian product A ⨯ B of two normed linear spaces with certain characteristic properties, and in the further use of this theory in spectral and variational analysis. The recent discovery by the authors of several new properties of the Fréchet variation has made it possible to to give new and natural tests for the convergence of multiple Fourier series generalizing the classical Jordan, de la Vallée Poussin, Dini, Young and Lebesgue tests under considerably less restrictive hypotheses than those now accepted.

Optimal Trajectories and the Accessory Minimum Problem

1965 ◽  
Vol 16 (3) ◽  
pp. 205-220 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Necessary Conditions ◽  
Sounding Rocket ◽  
Necessary Condition ◽  
Minimum Problem ◽  
Multiplier Rule ◽  
Variation Of A Function ◽  
Accessory Minimum Problem ◽  
The First Variation ◽  
The Second Variation

SummaryThe necessary conditions of Clebsch and Weierstrass and of the multiplier rule in the calculus of variations, which arise from the study of the first variation of a function, are summarised. A further necessary condition associated with the second variation is stated. The latter condition is applied to two problems: (i) the determination of the thrust-time programme which maximises the altitude of a sounding rocket, (ii) the determination of the thrust direction programme for a rocket with a known propellant expenditure programme which yields a maximum range. In both problems it is found that the additional necessary condition is satisfied.

The theory of the mean square variation of a function formed by adding known functions with random phases, and applications to the theories of the shot effect and of light

1936 ◽  
Vol 32 (4) ◽  
pp. 580-597 ◽  
Author(s):  
E. N. Rowland
Keyword(s):  
Total Effect ◽  
Mean Square ◽  
Single Event ◽  
Random Events ◽  
Variation Of A Function ◽  
The Mean ◽  
Image Position

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given byWe may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), whereBoth the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.

An integral representation for a generalised variation of a function

1974 ◽  
Vol 11 (2) ◽  
pp. 225-229 ◽  
Author(s):  
A.M. Russell
Keyword(s):  
Integral Representation ◽  
Classical Result ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Variation Of A Function ◽  
Image Position

In this note we present sufficient conditions for the continuity of the total kth variation of a function defined on a closed interval [a, b]. We also give an integral representation for total feth variation, thus obtaining an extension of the classical result

scholarly journals Change of concentration of biochemical markers of dysfunction of endothelium at intake of inhibitors of tyrosinekinase of I and II generations at patients with a chronic myeloid leukemia as risk factor of development of cardiovascular complications

2021 ◽  
Vol 25 ◽  
pp. 4219
Author(s):  
K. V. Naumova ◽  
I. L. Davydkin ◽  
E. G. Lomaia ◽  
T. Yu. Stepanova ◽  
T. P. Kuzmina ◽  
...  
Keyword(s):  
Arterial Hypertension ◽  
Biochemical Analysis ◽  
Chronic Phase ◽  
Cardiovascular Complications ◽  
C Reactive Protein ◽  
Endothelin 1 ◽  
Reactive Protein ◽  
Arterial Blood ◽  
Variation Of A Function ◽  
First Time

CML), the accepting inhibitors tyrosine of kinases (TKI) I and the II generations (TKI1 and TKI2 respectively), and development of arterial hypertension.Material and methods. Examination of 137 patients with CML in the chronic phase (CP) is conducted, the median of age — 47 years. 24 of them were with for the first time the verified diagnosis of CML and earlier did not accept TKI, they have made group of control. Other patients accepted TKI: 39 patients — imatinib 400 mg/day, 36 — dasatinib 100 mg/day, 38 — nilotinib 800 mg/day) more than 6 months. In biochemical analysis of blood indicators of lipidic range were defined. Level detection of ET-1 and VEGF was made by means of enzyme immunoassay. To all patients measurement of the heart rate (HR) and the arterial blood pressure (ABP) on both hands at an interval of 2 minutes from previous was once taken.Results. In group of patients from CML accepting nilotinib authentically significant increase in levels of systolic and diastolic ABP (р<0,001) in comparison with group of control, with group of the patients accepting imatinib and dasatinib is noted. The most serious changes of lipidic range are noted at the patients accepting nilotinib. In all groups statistically significant increase in level of C-reactive protein, fibrinogen, homocysteine, endothelin-1 and VEGF in comparison with group of control is revealed. The most expressed changes are found in group of the patients accepting nilotinib, values of parameters of C-reactive protein, fibrinogen, homocysteine, endothelin-1 and VEGF are changed authentically (р<0,001) and statistically significantly differ in comparison with group for the first time of the revealed patients with CML and groups of reception of imatinib and dazatinib.Conclusion. As a result of the conducted research endothelium variation of a function at patients from CML accepting TKI1 and TKI2 is revealed. The above-stated indicators can be used as additional diagnostic criteria for assessment of risk of development of arterial hypertension in patients with CML at reception of TKI.

scholarly journals Fractional variation of Hölderian functions

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Dimiter Prodanov
Keyword(s):  
Delta Function ◽  
Singular Behavior ◽  
Basic Properties ◽  
Cauchy Sequences ◽  
Variation Of A Function ◽  
Variation Operators ◽  
Local Characterization

AbstractThe paper demonstrates the basic properties of the local fractional variation operators (termed fractal variation operators). The action of the operators is demonstrated for local characterization of Hölderian functions. In particular, it is established that a class of such functions exhibits singular behavior under the action of fractal variation operators in infinitesimal limit. The link between the limit of the fractal variation of a function and its derivative is demonstrated. The paper presents a number of examples, including the calculation of the fractional variation of Cauchy sequences leading to the Dirac’s delta-function.

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