## Classical analysis and Odes – Vasicek model and Sportzins, parameterized by the rate of inversion

By solving an SDE, I want to derive the mean and variance analysis results of the extended Vasicek model process

$$dr = (η-γr) dt + cdX$$

whereas $$γ$$ is the inversion rate and $$s = η / γ$$ the average short rate.

How can I stop $$X (t) = r (t) – s$$ and solve by integration on both sides of the SDE using the integration factor $$e ^ {yt}$$ and in a second step derive the mean and the variance?

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## ca.classical analysis and odes – Under what conditions is this family normal?

To let $$mathcal {S} = {s in mathbb {C} , mid , | Im (s) | <1 }$$ be a strip of the complex plane. To let $$q (s, z)$$ a holomorphic function $$mathcal {S} times mathbb {C}$$, Rent $$mathcal {K}$$ a CD of $$mathbb {C}$$We can accept:

$$sum_ {j = 1} ^ infty || frac {1} {n} q (s- frac {j} {n}, z) || _ { mathcal {K}} < infty$$

For all $$n in mathbb {N}$$With $$n ge 1$$where the border continues as $$n to infty$$ converges evenly into $$s$$ on compact sentences of $$mathcal {S}$$, Note that $$|| h || _ mathcal {K} = sup_ {z in mathcal {K}} | h (z) |$$, Also note that the boundary to the integral converges

$$int _ {- infty} ^ s || q (t, z) || _ { mathcal {K}} , dt$$

Suppose we have a family of holomorphic functions $$F_n (s): mathcal {S} to mathbb {C}$$ in which:

$$F_n (s) to z , , text {as} , , Re (s) to – infty$$

$$F_n (s + frac {1} {n}) – F_n (s) = frac {1} {n} q (s, F_n (s))$$

I want to show that this family approximates the differential equation $$F # (s) = q (s, F (s))$$ with any precision. Or that $$F_n to F$$, To get there, I have to show that this family is normal. My guess is that I lack a condition $$q$$ that provides for normality, or I miss an obvious fact about this family.

Basically, I wonder if there is a limit anyway $$n to infty$$ from $$F_n$$ could escape to infinity, or if the fact that its satisfying equation approaches the differential equation $$y # = q (s, y)$$ Forces the convergence to uniquely solve the differential equation. Where the unique solution is a function $$ell$$ in which $$ell (- infty) = z$$ and $$ell & # 39; = q (s, ell)$$that can be shown to exist in a neighborhood of and to be unique $$– infty$$ by a slight modification of the usual Picard-Lindelöf theorem. Where by the neighborhood of $$– infty$$ I mean $$Re (s) . $$| Im (s) | < tau <1$$ to the $$| R |$$ big enough.

All in all, the family is $${F_n } _ {n = 1} ^ infty$$ normal? If not, what would be a reasonable condition to ensure this?

Any help would be appreciated.

Thanks, Richard.

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## Classical Analysis and Odes – On Exponential Polynomials

Suppose we have the following function $$f: mathbb {R} ^ {+} mapsto mathbb {R}$$
$$f (t) = sum_ {i = 1} ^ k P_i (t) exp ( alpha_i t),$$
from where $$alpha_i$$s are all algebraic numbers and $$P_i (t)$$ are all polynomials with algebraic coefficients and one degree less than $$m$$,

There are several questions that interest me.

on. What is the maximum number $$p$$ so there is one $$t_0$$ and for everyone $$r leq p$$
$$f ^ {(r)} (t_0) = 0.$$
b. Accept $$t_1, cdots, t_q, cdots$$ are the true roots of $$f (t) = 0$$Is it possible to have a convergent sequence of $$t_q$$? In other words, it is possible to have a Cauchy sequence $$t_i$$ so that $$f (t_i) = 0$$? If not, do we have a lower bound on the distance between different roots?

Is there an algorithm for computing all common real roots of $$f (t)$$ and $$f & # 39; (t)$$?

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## Differential equations – eigenvalue solution of coupled ODEs

I want an eigenvalue solution from the following paired ODEs: But the code shows errors: BCs are incomplete. ``````                Gamma = 1.67; ep = 0.01; d = 0.05; H = 1.1; Gpara = 0.38;
eqn = {Rho[x]/ Exp[-x/H]* gpara -
1 / gamma / Exp[-x/H]* {D[Rho[Rho[Rho[Rho[x], x]+ D[T[T[T[T[x], x]} +
1,3 * ep * D[V[V[V[V[x], {x, 2}]+ I * lambda * V[x] == 0,
D[Exp[Exp[Exp[Exp[-x/H]* V[x], x]- I * Lambda * Rho[x] == 0,
D[Exp[Exp[Exp[Exp[-x/H]* V[x], x]- V[x]* Rho[x]* gpara -
Gamma * Exp[-x/H]* D[V[V[V[V[x], x]+ gamma * d * D[T[T[T[T[x], {x, 2}]+
I * Lambda * T[x] == 0};
bc = V == 0, V == 0, V & # 39; == 0, V & # 39; == 0, T == 0,
T == 0, T & # 39; == 0, T & # 39; == 0, Rho == 0, Rho == 0;
{vals, funs} =
DEigensystem[{eqn, bc}, {V, Rho, T}, {x, 0, 1}, {lambda}];
vals
plot[Evaluate[funs[V]have fun[Rho]have fun[T]], {x, 0, 1}]
``````

I have to find numerical values ​​of lambda (eigenvalues) and
In the equations above, gpara is considered constant, but it is a variable with 50 rows and 1 column in the gpara.txt file, which is in a different directory. I import gpara by import[“E\ ……\ gpara.txt”] to use in the above equations, but it does not work either. Please help to get the solution.

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## Differential equations – Solution system of ODEs with additional parameters

I would like to solve one $$2 times 2$$ System of the form
$$frac {d} {d theta} T = TA, quad T (0) = Id$$
from where $$theta$$ is real and $$A$$ is of the form
$$A = begin {pmatrix} 0 & frac {e ^ {- i theta}} { lambda} \ frac {1} {36} e ^ {- i theta} left (9 Lambda + 2 ( lambda-1) ^ 2 (6 cos { theta} + cos {2 theta} + 6) right) & 0 end {pmatrix},$$
With $$lambda$$ a free parameter in the unit circle,

I am particularly interested in numerical solutions $$theta = 2 pi$$ depending on the additional parameters $$lambda$$, I'm pretty new with Mathematicaand I've tried it so far:

``````T[θ_] = {{T11[θ]T12[θ]}, {T21[θ]T22[θ]}};
ON[θ_] = {{0, E ^ (- Iθ) / λ}, {1/36 E ^ (- Iθ) (9λ + 2 (-1 + λ) ^ 2 (6 + 6 Cos[θ] + Cos[2 θ])), 0}};
sys = {T & # 39;[θ] == T[θ].ON[θ]};
``````

The previous code sets the system I want to solve, and now I'm trying to solve numerically. I tried it first

``````NSol = NDSolve[{sys, T11 == 1, T12 == 0, T21 == 0, T22 == 1}, {T11[θ]T12[θ]T21[θ]T22[θ]}, {θ}, {θ, 0, 2 Pi}];
``````

that gives me the output

``````NDSolve :: dupv: "Double variable θ found in NDSolve[<[<[<[<<1>>], "
``````

I tried too

``````Nsol2 = ParametricNDSolve[{sys, T11 == 1, T12 == 0, T21 == 0, T22 == 1}, {T11, T12, T21, T22}, {θ, 0, 2 Pi}, {λ}];
``````

what gives me as an issue $$T_ {11}, dots, T_ {22}$$ as Parametric Functions dependent on each other and on $$lambda$$,

I do not know if this is the right approach and if so, how to extract a numeric expression depending on it $$lambda$$ from the last issue – all I've seen in the documentation are examples that are recorded for specific parameter values. Any help is greatly appreciated.

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## Differential Equations – Problem solving the first ODEs with DSolve

The problem I'm facing is the following. I am trying to solve a system of two coupled first order ODEs with DSolve:

``````eqns = {A * f & # 39;[x] == -x * f[x]/ 2 + B * g[x], A * g & # 39;[x] == x * g[x]/ 2 + B * f[x]};
DSolve[EQNS{f[EQNS{f[eqns{f[eqns{f[x]G[x]}, x]
``````

As output this only gives the same command that means from my collection that DSolve can not solve it.

But if I explicitly use the equations, it can suddenly be solved:

``````Fun = f[x] /. To solve[A*g&#39;[A*g'[A*g'[A*g'[x] == x * g[x]/ 2 + B * f[x]f[x]];
dfun = D[fun, x];
DSolve[A*f#39;[A*f'[A*f'[A*f'[x] == -x * f[x]/ 2 + B * g[x] /. {f[x] -> fun, f & # 39;[x] -> dfun},
G[x], x]
``````

Which gives

``````{{G[x] ->
C ParabolicCylinderD[B^2/A, (I x)/Sqrt[A]]+
C ParabolicCylinderD[(-A - B^2)/A, x/Sqrt[A]]}}
``````

That makes me think that I'm just doing something wrong on the first line, which is probably a stupid notation error. If someone could help me, I would be very grateful.

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## Differential equations – How do you solve these ODEs with NDSolve?

The calculation changes `t = 0` there `sin[ψ`

``` eq1 = ω1 eq2 = ω2 and construct the linear combinations, eq1n = Simplify[eq1Sin[eq1Sin[eq1Sin[eq1Sin[ψ (* Cos[ψ (If this were not possible, the equations themselves could not be solved in principle.) Now replace eq1, eq2 by eq1n, eq2n, I1 = 2; I2 = 3; I3 = 4; s = NDSolveValue[{I1*ω1&#39;[{I1*ω1'[{I1*ω1'[{I1*ω1' I2 * ω2 & # 39; I3 * ω3 & # 39; eq1n == 0, eq2n == 0, ω3 ω1 == 2, ω2 == 3, ω3 == 4, ψ == 0, φ == 0, θ == Pi / 6}, {ω1 plot[Evaluate@s[[1 ;; 3]], {t, 0, 120}, ImageSize -> Large]plot[Evaluate@s[[4 ;; 6]], {t, 0, 120}, ImageSize -> Large]  Incidentally, the original equations can be solved by slightly changing the initial state ψ == 0 to ψ == 10 ^ -6, Another approach is to use the option. Method -> {"EquationSimplification" -> "Residual"} Everyone gives the same answer. ```
``` ```
``` Author AdminPosted on March 17, 2019Categories ArticlesTags differential, equations, NDSolve, ODEs, solve ```
``` How do I solve these Odes with NDSolve? I have six odes and can not use DSolve. So I tried NDSolve. But it does mean that there may be mistakes. The code looks like this: I1 = 2; I2 = 3; I3 = 4; NDSolve[{I1*ω1' I2 * ω2 & # 39; I3 * ω3 & # 39; 0, ω1[ t] == φ & # 39; sin[ψ[ψ[ψ[ψ t] == φ & # 39; cos[ψ[ψ[ψ[ψ t] == φ & # 39; 0] == 2, ω2 == 3, ω3 == 4, ψ == 0, φ == 0, θ == Pi / 6}, {ω1, ω2, ω3, ψ, φ, θ}, {t, 0, 120}] I want to know how to avoid this mistake. Author AdminPosted on March 17, 2019Categories ArticlesTags NDSolve, ODEs, solve Classical analysis and Odes – Algebraic Riccati and WKB It is a one-liner to show that the algebraic Riccati equation (ARE) and the lowest-order form of WKB are the same for a linear ode. But I've looked everywhere on the Internet and it seems that this connection, despite its triviality is not recognized. It seems that there is a total separation between physicists who make many WKB and control people who do a lot of Riccati. Another trivial point is that a linear ode in psi has the Lie symmetry psi -> lambda psi, so the Riccati variable is the Lie invariant, and this leads to a reduction in order (or more precisely, a reduced one Order) followed by a quadrature.) Is this so obvious that it can never be mentioned? This finding means that not only linear equations are used, but every equation that is homogeneous in degree 1 and whose order is reduced by the Riccati transformation. Author AdminPosted on March 16, 2019Categories ArticlesTags algebraic, analysis, classical, ODEs, Riccati, WKB Classical analysis and odes – Failure of a Falconer distance problem in one dimension I am told that the Falconer distance guess in one dimension is trivial, but I really can not find any clue. I ask exactly the following question: For a compact set $$E subseteq mathbb R ^ n$$we define the set distance $$Delta (E) subseteq[0infty)[0infty)[0infty)[0infty)$$ his: $$Delta (E) = {| x-y |: x, y in E }.$$ Then, when $$n = 1$$do we ask the following questions? Can we find a compact set? $$E subseteq [0,1]$$ so that $$mathrm {dim} _H (E)> 1/2$$ but $$mathcal L ^ 1 ( Delta (E)) = 0$$? Can we find a compact set? $$E subseteq [0,1]$$ so that $$mathrm {dim} _H (E) = 1$$ but $$mathcal L ^ 1 ( Delta (E)) = 0$$? Author AdminPosted on March 7, 2019Categories ArticlesTags analysis, classical, dimension, distance, failure, Falconer, ODEs, problem Posts navigation Page 1 Page 2 Next page ```