## Stochastic differential equations: expected value, variance and ode

I wonder if it is possible to compute statistics from a stochastic differential equation. I start with a simple question about the stochastic linear differential equation. Leave $$dot {w} = z$$Y $$z$$ a normal distribution function, how can I calculate the expected value and standard deviation? I suspect that its standard deviation is 0, but I'm not sure about its stddev.

## Differential Geometry – Commutation Covariant Derivatives of a Variation

I am trying to understand the proof of the first and second variation of the Arclength formulas for Riemannian manifests. I want a check that the following covariant derivatives switch. I find it intuitive but I also want to have a formal test.

Some notation: Let $$gamma (t, s): overbrace {(a, b)} ^ {t} times overbrace {(- epsilon, epsilon)} ^ {s} rightarrow M$$ be a variation of $$gamma_0 (t)$$ with $$| dot { gamma_ {0} |} = lambda forall t in (a, b)$$ Y $$V = gamma _ {*} left ( frac { partial} { partial s} right)$$ The variational vector field.

So I would like to demonstrate that $$frac {D} {ds} ( dot { gamma_t}) = frac {D} {dt} V$$ or in other words that $$frac {D} {ds}, frac {D} {dt}$$ commute. Note that $$frac {D} {ds}$$ is the covariant derivative throughout the map $$gamma$$ therefore we will use the property for connections across maps that $$nabla ^ { gamma} _X (Z circ gamma (p)) = nabla _ { gamma _ * (X)} Z | _p$$.

Indeed $$frac {D} {ds} left ( dot { gamma_t} right) = frac {D} {ds} ( gamma _ * ( frac {d} {dt}) circ ( gamma (s, t))) = D _ { gamma _ * left ( frac {d} {dt} right)} gamma _ * ( frac {d} {ds}) = D _ { gamma _ * left ( frac {d} {ds} right)} gamma _ * ( frac {d} {dt})$$.

The last equality follows from $$D _ { gamma _ * left ( frac {d} {dt} right)} gamma _ * ( frac {d} {ds}) – D _ { gamma _ * left ( frac {d} {ds} right)} gamma _ * ( frac {d} {dt}) = left ( gamma _ * ( frac {d} {dt}), gamma _ * ( frac {d} {ds}) right) = gamma _ * (( frac {d} {ds}, frac {d} {dt})) = 0$$ as $$frac {d} {ds}, frac {d} {dt}$$ they are vector coordinate fields.

## smooth collectors: when does the retracement of differential forms preserve the cohomology classes: $F ^ *[alpha]=[F^* alpha]$

Leave $$M ^ n$$ be a smooth, compact collector without limit. Consider a dipheomorphism $$F: M rightarrow M.$$

The retrocess $$F ^ *$$ defines a linear isomorphism in the de Rham cohomogy spaces $$F ^ *: H ^ k_ {dR} (M) rightarrow H ^ k_ {dR} (M)$$ where $$0 leq k leq n.$$

My question is: under what conditions is this the identity map in $$H ^ k_ {dR} (M)$$? Also if $$M$$ it is a complex closed complex, under which it conditions a holomorphic dipheomorphism $$F$$ would give the identity in $$H ^ r, r} (M)$$?

Any help is really appreciated!

## differential equations: how to model diffusion across a membrane?

This is a follow-up to How to handle the discontinuity in the diffusion coefficient?

Consider dissemination of $$u (t, x)$$ in the domain $$x in (0.2)$$ with some simple boundary conditions like $$u (0) = 2, u (1) = 1$$.

Our domain is divided into two parts: $$(0.1)$$ to the left and $$(1,2)$$ on the right, with different diffusion coefficients, p. $$D ^ text {left} = 1, D ^ text {right} = 3$$.

The diffusion equation is:
$$partial_t u = partial_x (D partial_x u)$$

So far, this is the summary of the linked question.

This time we also have a membrane in $$x = 1$$, imposing the following condition on the flows in $$x = 1$$:
$$D ^ text {left} partial_x u ^ text {left} = D ^ text {right} partial_x u ^ text {right} = d ^ text {membrane} (u ^ text {right} – u ^ text {left})$$

What is the cleanest way to model this with NDSolve? Is there a way to preserve severe conditions in $$x = 1$$? Perhaps an approach that could be used is to consider a membrane of finite thickness, which has a very high diffusion coefficient. However, this is really a trick. Is it possible to solve the equation in the two half domains "separately" and couple the boundary conditions in $$x = 1$$?

## general solution of the first-order differential equation dy / dx = (x + y – 1) ^ 2, where x, y are real?

Could someone help me solve this.

Which of the following is the general solution of the first-order differential equation dy / dx = (x + y – 1) ^ 2, where x, y are real?

(A) Y = 1 + x + tan ^ -1 (x + c), where c is a constant.

(B) Y = 1 + x + tan (x + c), where c is a constant.

(C) Y = 1-x + tan ^ -1 (x + c), where c is a constant

(D) Y = 1-x + tan (x + c), where c is a constant.

## dg. differential geometry: how sectional curvature is affected by multiplying the Riemann metric by constant

I have the following short question: The section curvature of a plain is defined in terms of the Riemann curvature tensor, which is defined in terms of the Levi-Chivita connection, which is defined in terms of the Riemann matrix.

My question is, how does multiplying the Riemann metric by a constant affect the sectional curvature?
Thank you.

## differential equations: how to define the boundary condition in 1D heat transfer

I am trying to calculate the head transfer between a 1-D rod, with an insulated end while the right end is submerged on a surface of constant temperature T = 0. Suppose the initial temperature of the rod is T = 1. The bar length is 5. I configured the equation like this:

$$frac { partial ^ 2 u (x, t)} { partial x ^ 2} – frac { partial u (x, t)} { partial t} = 0 \ u (x, 0) = 1 \ frac { partial u (5, t)} { partial x} = 0 \ u (0, t) = 0$$

sol = NDSolve({
eqn = D(u(t, x), t) - D(u(t, x), {x, 2}) == 0,
u(0, x) == 1,
u(t, 0) == 0,
(D(u(t, x), x) /. x -> 5) == 0
}, u, {t, 0, 50}, {x, 0, 5})

Plot3D(Evaluate(u(t, x) /. %), {t, 0, 50}, {x, 0, 5},
PlotRange -> All)


Unfortunately, I got something like this:

NDSolve :: ibcinc: Warning: Initial limits and conditions are inconsistent.

Can anyone help me with the limit value problem?

## differential equations: using WhenEvent to bind variables

I would like to use the WhenEvent (link here) to set a limit on a dynamic variable in an EDO set. That is, when the variables k21(t) Y k12(t) reach above the specific value aI would like to put them back at that value. Seems like a simplified version of the problem

w1 = 6/24.5;
w2 = 6/23.5;
a = 0.1;

Eqs = {
x1'(t) == w1 + (k21(t)/2)*Sin(x2(t) - x1(t)),
x2'(t) == w2 + (k12(t)/2)*Sin(x1(t) - x2(t)),
k21'(t) == a*(Cos(x2(t) - x1(t) + (Pi)) + 1),
k12'(t) == a*(Cos(x1(t) - x2(t) + (Pi)) + 1)};

ICs = {x1(0) == 3/2, x2(0) == 3/4, k21(0) == 0.0001, k12(0) == 0.0001};

events = {WhenEvent(Abs(k21(t)) > a, k21(t) -> a), WhenEvent(Abs(k12(t)) > a, k12(t) -> a)};

EqsICs = Join(Eqs, ICs, events);

SolutionValue(t_) = NDSolveValue(EqsICs, {x1(t), x2(t), k21(t), k12(t)}, {t, 0, 10^6});


And I trace the solutions as:

Show(
Plot(SolutionValue(t)((3)), {t, 0, tmax}, PlotRange -> {{0, tmax}, {-0.1, 0.1}}, AxesOrigin -> {0, 0}),
Plot(SolutionValue(t)((4)), {t, 0, tmax}, PlotRange -> {{0, tmax}, {-0.1, 0.1}}, AxesOrigin -> {0, 0})
)


However, my WhenEventdoes not work The graph shows that the values ​​of kij continues to grow beyond a. Is my syntax incorrect? Thank you 🙂

## differential equations: analytical resolution of nonlinear ODE

Thanks for contributing with a response to Mathematica Stack Exchange!

But avoid

Leave $$(M ^ 3, g)$$ be a Riemannian variety (closed) and leave $$u: M a S$$ be a harmonic function where $$S$$ It is a closed steerable surface. Yes $$omega$$ is a $$2$$-form in $$S$$What are the sufficient conditions for $$omega$$ in order $$u ^ * omega$$ be a harmonic $$2$$-form in $$M$$?
The specific case I am analyzing is the following: we have $$u_1, u_2: M to mathbb {S} ^ 1$$ harmonic functions, $$u = (u_1, u_2): M to mathbb {S} ^ 1 times mathbb {S} ^ 1$$ Y $$omega = d theta_1 wedge d theta_2$$, where $$d theta_i$$ denotes the shape of the volume in each $$mathbb {S} ^ 1$$.