The market is a controlled dynamic system. - page 369
You are missing trading opportunities:
- Free trading apps
- Over 8,000 signals for copying
- Economic news for exploring financial markets
Registration
Log in
You agree to website policy and terms of use
If you do not have an account, please register
week_01
.
trac_day_07
.
Very informative.
.
http://mathemlib.ru/mathenc/item/f00/s00/e0000653/index.shtml
VARIATION OF A FUNCTIONAL
FUNCTIONAL VARIATION, the first variation, is a generalization of the notion of a differential of a function of one variable, the main linear part of the increment of the functional along a certain direction; it is used in the theory of extreme problems to obtain necessary and sufficient conditions of an extremum. This is the meaning of the term "V. f.", starting from the work of J. Lagrange [1] (1760). J. Lagrange considered predominantly functionals of classical calculus of the form:
(1)
If we replace the given function x0(t) by x0(t) + αh(t) and substitute it in the expression for J(x), then, supposing continuous differentiability of the integrand L, the following equation holds
J(x0 + αh) = J(x0) + αJ1(x0)(h) + r(α), (2)
where |r(α)| → 0 when α → 0. The function h(t) is often called the variation of the function x0(t) and is sometimes denoted by δx(t). The expression J1(x0) (h), which is a functional with respect to variations of h, is called the first variation of the functional J(x) and is denoted by δJ(x0, h). Applied to the functional (1), the expression for the first variation is
(3)
where
The equality to zero of the first variation for all h is a necessary condition of the extremum of the functional J(x). For the functional (1) the Euler equation follows from this necessary condition and the main lemma of calculus of variations (see Dubois-Reymond lemma):
In a way analogous to (2), variations of higher orders are also determined (see, e.g., in the article Second Variation of a Functional).
The general definition of the first variation in the infinite dimensional analysis was given by B. Gateaux in 1913 (see Gato variation). In essence, Gateaux's definition is identical with the Lagrange definition. The first variation of a functional is a homogeneous, but not necessarily linear functional, V. f. under the additional assumption of linearity and continuity (on h) of the expression δJ(x0, h) is usually called the Gato derivative. The terms "Gato variation", "Gato derivative", "Gato differential" are more widely used than V. f.; the term "V. f." is preserved only for the functionals of the classical calculus of variations (see [3]).
See [1] Lagrange J., Essai d'une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indefinies, Turin, 1762; [2] Gateaux R., 'Bull. Soc. Math. France", 1919, vol. 47, pp. 70-96; [3] Lavrent'ev M.A., Lusternik L.A., A course in calculus of variations, 2nd edition, M.-L., 1950.
В. M. M. Tikhomirov.
Sources:
http://mathemlib.ru/mathenc/item/f00/s00/e0000879/index.shtml
SECOND VARIATION
SECOND VARIATION is a special case of the n-th variation of a functional (see also Gato variation), generalising the notion of the second derivative of a function of several variables; it is used in calculus of variations. According to the general definition of v. in the point x0 of the function f(x) defined in the normalized space X, there is
If the first variation is zero, the non-negativity of V. v. is necessary, and the strict positivity
δ2 f(x0, h) ≥ α ||h||2, α > 0
under some assumptions, it is a sufficient condition for local minimum of f(x) at x0.
In the simplest (vector) problem of classical variational calculus the V. v. of the functional
(considered on vector functions of class C1 with fixed boundary values x(t0) = x0, x(t1) = x1) has the form:
(*)
where 〈⋅, ⋅〉' denotes the standard scalar product in ℝn, and A(t), B(t), C(t) are matrices with coefficients respectively (derivatives are calculated at points of curve x0(t)). It is expedient to consider the functional from h defined by formula (*) not only in the space C1, but also in the broader space W12 of absolutely continuous vector functions with integrable square of the derivative module. In this case non-negativity and strict positivity of V. v. are formulated in terms of non-negativity and strict positivity of the matrix A(t) (Lejandre condition) and absence of conjugate points (Jacobi condition), which gives conditions of weak minimum in calculus of variations.
For the calculus of variations in general, V. v. has been studied for extrema that do not necessarily deliver a minimum (still, however, -when the Lejandre condition is satisfied, see [1]). The most important result is the Morse coincidence of the V. v. index and the number of points conjugate to t0 on the interval (t0, t1) (see [2]).
See [1] Morse M., The calculus of variations in tne large, N. Y., 1934; [2] Milnor J., Morse theory, translated from English, M., 1965.
В. M. Tikhomirov.
Sources:
A competition started today in which I also decided to take part.
start 01.11.2018.
Finish on 30.11.2018.
I set a goal:
To increase the starting deposit by 100 times.
And preferably without losing trades ;)A competition started today in which I also decided to take part.
start 01.11.2018.
Finish on 30.11.2018.
I set a goal:
To increase the starting deposit by 100 times.
And preferably without losing trades ;)or you can bet :-)
is there any monitoring ? at least an impersonal top 10 contest - i will be rooting for you...
or you can bet :-)
Yeah. Okay. Thanks.
I'll have to ask the moderators about the stakes.
Yeah. Okay. Thanks.
You'll have to ask the moderators about the stakes.
yes of course you are welcome...
just ask - don't be like the others - if things don't work out (and they might not work out most of the time), honestly deal with the mistakes right here in public
ps/ or maybe it will work
yes of course you are welcome...
But please don't be like the others - if things don't work out (and they might not work out most of the time), honestly address the mistakes right here in public
ps/ or maybe it will work
it's a deal.
I will give daily progress reports.
Will it work?