# Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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### Infinitely many planets on a line, with Newtonian gravity

(I previously asked essentially this on physics.stackexchange, but was actually hoping for answers with something closer to a proof than what I got there.) Suppose we have a unit mass planet at each ...
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### What is a Green's function in the language of $\mathcal{D}$-modules?

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...
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As every number theorist learns, the radius of convergence of $exp(x)$, defined by the usual power series in a neighborhood of zero, is $$\rho = p^{-1/(p-1)}.$$ This is typically proven by computing ...
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### Galois groups of classical differential equations

I am currently on the lookout for good motivational examples for differential Galois theory, and I was wondering the following: Is there a book or article devoted (either partially or completely) to ...
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### Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
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### Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
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### Is there a classification of differential equations over the field of fractions of formal power series? (characteristic 0)

Let $k$ be an algebraically closed field of characteristic 0. Consider the field of fractions of formal power series $K:= Frac(k[[T_1,...,T_n]])$. We have the corresponding algebra of differential ...
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### What is the right basis of solutions of the Picard-Fuchs equation of the Legendre family around 0?

I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations. Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the ...
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### Geometric interpretation of energy-momentum tensor and Lagrangian associated to a soliton equation

I have a question for you. BACKGROUND Consider an immersion $F\,:\;\mathscr S\;\longrightarrow\;\mathbb E^3$ of a surface $\mathscr S$ in the $3$-D euclidean plane $\mathbb E^3$ with canonical ...
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### Had this theorem in Tresser's article been proven somewhere?

The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
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### Nonzero solutions to the functional ODE $f'(x)=f(f(x))$

Does $\frac{df}{dx}=f(f(x))$ have nonzero solutions? And if so, what analytic/numerical methods can be used to characterize them?
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### Grothendieck problem

Could you suggest me a book or a link where I can find some information about the Grothendieck problem about differential equations? The Grothendieck problem that I am reffering to is the following: ...
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### Weighted reverse Poincare inequality over a function class of neural networks

We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
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### Conserved positive charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
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### Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle. According to this fact, we search for a related ...
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### Has this functor been studied?

Let $\mathbb{O}$ be the category where the objects are $n$-manifolds, and the morphisms are almost-everywhere smooth cobordisms along with almost-everywhere smooth partial differential equations on ...
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In  (p. 183, paragraph of Eq. 7.8), Hitchin makes an argument which I would summarize as follows. Suppose that $A\in\mathfrak{u}(n)$ (a skew-Hermitian matrix) has distinct eigenvalues and let $S:... 0answers 95 views ### Ricci flow on locally symmetric noncompact manifold As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ... 0answers 126 views ### Orthonormal Basis of Multi-Dimensional Sobolev Space of Different Orders without Reproducing Kernel Let$\Omega$be an open subset of$\mathbb{R}^d$. Under regularity conditions, we know that the$s$-th order Sobolev space$H^s(\Omega)$with$s \geq d/2$is a reproducing kernel Hilbert space. In ... 0answers 307 views ### Spectral Gap of Elliptic Operator Under what conditions on$a(x)$and domain$D$, the spectral gap of the elliptic operator$ \nabla \cdot(a(x)\cdot \nabla)$defined on$D$, can be controlled? The boundary condition is that the ... 0answers 127 views ### An embedding question: Morrey spaces Question. If$u\in L^1$and$Du$is in the dual of the Holder space$C^\alpha$, then is it possible to say$u$belongs to some Morrey space$L^{1, \delta}$? 0answers 108 views ### Origins of the generalized shift operator exp(t*g(z)d/dz) Charles Graves in the 1850s investigated iterated operators of the form$g(x) \frac {d}{dx}\$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
I am trying to reduce the following ODE to Bessel's ODE form and hence solve it: $$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .$$ I tried to solve it via the standard method, i.e.,...