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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Big Picture: What is the connection of Malliavin calculus with differential geometry?
I know that Paul Malliavin was heavily influenced by ideas from differential geometry while developing his calculus on Wiener space. But what are the concrete analogies between both areas of ...
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Is there a Poincare-Hopf Index theorem for non compact manifolds?
Does Poincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If ...
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2- and 3-body problems when gravity is not inverse-square
Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing
as $1/d^p$ for distance separation $d$ and some power $p$.
Two questions:
Presumably the 2-body ...
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Catenary curve under non-uniform gravitational field
The catenary curve is the shape of a chain hanging between two equal-height poles under the influence of gravity. But the derivation of the (hyperbolic cosine) curve equation from the physics ...
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D-modules over algebraic curves VS differential Galois theory
Disclaimer: I know very little about both of the fields in question.
My question is pretty simple:
What's the relation between differential Galois theory and D-modules
over algebraic curves?
...
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The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game
This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...
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Kontsevich's flow on the space of Poisson structures
In §5.3 of Kontsevich's Formality Conjecture he writes:
This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...
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Vector field built from connection and metric
Consider a smooth finite-dimensional manifold $M$ with metric $g$ and connection $\nabla$. For some local coordinate system, denote by $g^{\alpha \beta}$ the inverse of the metric tensor and by $\...
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does the j-invariant satisfy a rational differential equation?
Let $j$ be the Klein $j$-invariant (from the theory of modular functions).
Does $j$ satisfy a differential equation of the form $j^\prime (z) = f(j(z),z)$ for
any rational function $f$?
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How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?
Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
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Are there alternative proofs for existence/uniqueness of ODE solutions?
Consider the differential equation $\dot x = f(x)$. The standard proofs are
The Picard iteration based proof of existence/uniqueness for Lipschitz $f$.
The Peano existence theorem for continuous $f$...
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Getting a differential equation for a function from a functional equation of its Mellin transform
If $f$ is a locally integrable function then its Mellin transform
$\mathcal{M}[f]$ is defined by
$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$
This integral usually converges in a ...
16
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answer
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What braking strategy is most fuel-efficient?
You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What ...
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The radius of convergence of the p-adic exponential function.
As every number theorist learns, the radius of convergence of $exp(x)$, defined by the usual power series in a neighborhood of zero, is
$$\rho = p^{-1/(p-1)}.$$
This is typically proven by computing ...
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Are there any techniques for solving a differential equation of the form $f ' (x) = f( f( x ) )$?
I am trying to solve the following differential equation
$$f ' (x) = f( f( x ) ),$$
but I have no idea how. I don't think the chain rule is useful for this.
Although I don't think this differential ...
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What justification can you give for the fact that "most ODEs do not have an explicit solution"?
What justification can you give for the fact that "most ODEs do not have an explicit solution"?
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Reference request: the theory of currents
I am a graduate student and want to study the theory of currents. What is a good reference for a beginner?
I should be familiar with the theory of distributions or generalized functions on $\mathbb R^...
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Newton equations, second order equation and (im)possible motions
I am am currently studying Newtonian mechanics from a conceptional and axiomatic point of view. Now, if I am not mistaken, one (but surely not all) statement of Newtons second law about nature is, ...
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An example of a series that is not differentially algebraic?
Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an ...
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Solvability in differential Galois theory
It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative.
The proof I know goes as follows:
Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
15
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Is the heat kernel more spread out with a smaller metric?
Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
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Solution to differential equation $f^2(x) f''(x) = -x$ on [0,1]
I'd like to solve a differential equation $$ f^2(x) f''(x)=-x $$ where $f(x)$ is defined on $[0,1]$ and has a boundary condition $f(0)=f(1)=0$.
I somehow found out that the solution is fairly close ...
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On the non-rigorous calculations of the trajectories in the moon landings
In a paragraph written by a person emphasizing that rigour is not everything in mathematics (I wish I had written down the details), it was stated that the moon landings would have been impossible ...
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When can Witten-esque moduli spaces be used to define invariants of geometric structures?
I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds.
It is striking ...
14
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1
answer
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Projective-invariant differential operator
This question was originally asked on Math StackExchange.
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align*}
&T(g) = ...
14
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Exactness of 2nd-Order Differential Equations via Differential Forms
This (probably very elementary) question came up the last time I taught differential equations, and I've been toying with it for a while with no success:
A 1st-order differential equation $M(x,y)dx+N(...
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ODE properties true in finite dimension but not in Banach spaces of infinite dimension
Some properties of Ordinary Differential Equations - ODE are true in finite dimension spaces but not in Banach spaces of infinite dimension.
The first one I know is the Peano existence theorem. I ...
14
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2
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Is there a singularity theorem in higher-dimensional Newtonian gravity?
In classical Newtonian gravity with 3 spatial dimensions, it's hard to get two particles to exactly collide, since at short distance the centrifugal force (~1/$r^3$) beats the gravitational attraction ...
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Picard-Fuchs equations for modular functions
Hello, MathOverflow community!
Suppose we have a modular curve of genus $0$, whose rational function field is generated by the modular function $f$. We can view $f$ as the parameter for some pencil ...
14
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conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables
If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D Monge-...
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Modular equations for quasimodular forms
This problem is motivated by this question and by teaching
modular polynomials for the classical modular invariant $j(\tau)$.
The latter implies
that if we consider the fields of modular functions $\...
14
votes
1
answer
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Computing spectra without solving eigenvalue problems
There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
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The perturbation of non-Hamiltonian algebraic vector fields
In this question, we are interested in the number of limit cycles which appears in the following perturbational system:
\begin{equation}\cases{
x'=y -x^{2}+\epsilon P(x,y) \\
y'=-x+\epsilon Q(x,y) }
\...
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Frobenius Theorem for subbundle of low regularity?
Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$...
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Motivation for BMO
At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
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Random N-body problem
Suppose there are $N$ unit-mass particles whose initial positions
are uniformly distributed in a unit-radius disk.
Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...
13
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3
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PDEs, boundary conditions, and unique solvability
I'm interested in a criterion that determines whether a linear scalar PDE (arbitrary order) has a unique solution given vanishing boundary conditions at spatial infinity. I'll try to formulate the ...
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2
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Young-Fibonacci version of Nekrasov-Okounkov
This question addresses a hierarchy of linear recurrences
which arise from an attempt to generalize the Nekrasov-Okounkov
formula to the Young-Fibonacci setting.
A related posting
extensions of the ...
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Poincaré on analytic dependence on parameters of solutions of linear differential equations
There is the following important General Principle: if a parameter enters
in a linear differential equation additively, for example
$$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$
where the parameter is $\...
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Books on the analysis of hyperbolic partial differential equations
Most of the present books on pde analysis deal with the elliptic partial differential equations. Is there some book related to rigorous analysis with hyperbolic pdes, and especially hyperbolic systems ...
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Vector field with holomorphic flow
Let $(M,J)$ be a complex manifold. Suppose that $X$ is a real vector field such that the flow of $X$ is by biholomorphisms.Question Show the flow of $JX$ is by biholomorphisms.
I know one reference ...
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History of ODE and PDE reference request
Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
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Applied mathematics Books (graduate level)
What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems?
There is a lot of books on ...
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Where was/is Compensated Compactness used?
This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
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Reference for a nice proof of "undetermined coefficients"
I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...
12
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2
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curvature flow for loops in S^2
Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, ...
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applications of C$^*$-algebras in the field of PDEs
I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...
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Derivative of the flow for ODEs on manifolds
Let $\mathbf V \colon [0,T] \times \mathbb R^d \to \mathbb R^d$ (for $T>0$) be a given, bounded smooth vector field and let $\mathbf X=\mathbf X(t,x)$ be its flow, i.e. the unique solution to the ...
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Solving ODE via contact geometry
I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential ...
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Differential algebraic geometry vs Diffiety theory
Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations.
...