Blog

A slightly less general version of this question was asked, in a subsequent comment, by the OP of the question at Controlling subsolutions of a second order linear ODE Let $ f:[0,\infty) \to \mathbb{R}$ obey the differential inequality \begin{equation} f” – 2a f’ + (a^2+b^2) f \le 0,\tag{*} \end{equation} where $ a\in\mathbb R$ and $Read more

With the original aim of properly understand the ellipsoidal coordinates $ (\lambda, \mu, \nu)$ , I would like you to help me understanding the concept of “scale factors” in the contest of differential operators. Particularly, I would like you to provide me a relatively fast and intuitive explanation of such concept for someone like meRead more

I would like to fit a set of data (measured value vs time) with a model that contains a differential equation and an integral that has to be evaluated numericaly. The integral also contains Bessel functions (first and second kind) evaluated close to zero which I think does not help to solve the problem becauseRead more

I tried this question twice on math.stackexchange but got no answer so I decided to move it here. Let $ M$ be a smoth manifold. Then $ $ C^\infty(M):=\{f:M\longrightarrow \mathbb R; f\ \textrm{is smooth}\},$ $ is an $ \mathbb R$ -algebra with the pointwise product. If you don’t know anything about smooth manifolds it doesn’tRead more

Let $ X$ be a real $ n\times n$ symmetric matrix of rank $ m\le n$ and let $ Y\in\mathbb{R}^{m\times n}$ be a (=any) representative of the class $ \{Y\in\mathbb{R}^{m\times n} \,:\, X=Y^\top Y\}$ . Let $ A\in\mathbb{R}^{m\times m}$ be a positive definite matrix and $ B\in\mathbb{R}^{n\times m}$ be such that $ YB$ is nonsingular.Read more

Background DiffResolvent.nb, here gives a method that solve quintic equations using differential equations. There’s a progress to transform polynomial equation into differential equation. Then he gave an example: $ $ x^5-5 x^3+5 x-t = 0$ $ The roots of the equation satisfy the differential equation: $ $ 25 \left(4-t^2\right) x”(t)-25 t x'(t)+x(t)=0$ $ Solve thatRead more

Here is a differential equation x”[t] – eps (a^2 – x[t]^2) x'[t] + w^2*x[t] == 0 boundary conditions are x[0]=0, v[0]=0.001 i want to substitute x'[t] to v[t] and then NDSolve the equation to respect of x and v. Right now my code looks like this NDSolve[{ x”[t] – eps (a^2 – x[t]^2) x'[t] +Read more

Let $ M$ be a compact manifold with boundary. If we have two vector bundles $ E, F \to M$ with inner products and a differential operator $ D: C^{\infty}(E) \to C^{\infty}(F)$ then $ D$ admits a formal adjoint $ D^*: C^{\infty}(F) \to C^{\infty}(E)$ . This satisfies, for smooth sections with compact support in theRead more

The Chaplygin sleigh problem is very popular in nonholonomic mechanics (more details can be seen in Bloch, Nonholonomic Mechanics and Control, Springer, New York, 2015, section 1.7, pp 29-30), The Chaplygin sleigh is a rigid body moving on two sliding posts and one knife edge The equations are: $ \qquad v =\dot{x}\cos\theta+\dot{y}sin\theta,$ $ \qquad \ddot{\theta}Read more

For finite-dimensional $ \mathbb{C}$ -algebra $ A$ with basis $ \{e_1,…,e_n\}$ (structure of algebra encoded by structural constants $ \{c_{ij}^k\}_{i,j,k=1}^n$ in a way $ e_ie_j = \sum_{k=1}^n c_{ij}^k e_k$ ) differential of Hohschild complex it is map $ T(A) \to T(A)$ where $ T(A)=\oplus_{i=0}^\infty A^{\otimes i}$ it is tensor algebra, which defined on irreducible tensorsRead more