Tagged Questions

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

18k views

What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...
6k views

Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
19k views

why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
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What are reasons to believe that e is not a period?

I hope this rather soft question is suitable for MO, otherwise please migrate it to MSE. In their paper defining periods [1], Kontsevich and Zagier without further comment state that $e$ is ...
13k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...
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I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...
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D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...
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Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
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Gently falling functions

I wonder if it is possible to characterize the class of gently falling functions, which I would like to define as follows. Let $g(x)$ be a $C^2$ function defined on an interval $R \subseteq \mathbb{R}$...
20k views

Is square of Delta function defined somewhere?

Hello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where? In the beginning, this question might look strange. But by restricting ...
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Differentiable functions with discontinuous derivatives

For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...
35k views

Good differential equations text for undergraduates who want to become pure mathematicians

Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate ...
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Why are differential forms called closed and exact?

It seems to me that "exact" relates to exact differential equation. So, why are they called exact?
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Does every ODE comes from something in physics?

Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself. Say I have a nasty ODE, nonlinear, maybe extremely ...
1k views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
8k views

Why have mathematicians used differential equations to model nature instead of difference equations

Ever since Newton invented Calculus, mathematicians have been using differential equations to model natural phenomena. And they have been very successful in doing such. Yet, they could have been just ...
2k views

What kind of Lagrangians can we have?

In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of non-...
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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 ...
2k views

Is this 1974 claim still valid?

In G. F. Simmons' Differential Equations book (p.141), the following claim is made: “... As a matter of fact, there is no known type of second order linear differential equation- apart from those with ...
2k views

Sheaves and Differential Equations

How do sheaves arise in studying solutions to ordinary differential equations? EDIT: Is it possible to construct non-isomorphic sheaves on a domain $D \subset \mathbb{R}^n$ using solution sets to ...
2k views

Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
1k views

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 ...
791 views

Unifying (& “justifying”) the various definitions for differential operators

Reading about differential operators in different sources I've picked up several definitions which are not obviously equivalent (to me). Here they are: Definition 1 ("naive"): Let $X$ be a (real) ...
3k views

PDE on manifolds

I am currently in a PDE course where one of the requirements is to present a paper in PDE. I am wondering if anyone can suggest an early (read foundational, first introductory) paper talking about PDE ...
Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample? Here, Peano's theorem is: Let $E$ be a space with finite dimension. Consider a point $(t_0,x_0) \in \Re \... 4answers 2k views ODE's without a Lipschitz condition When teaching ODE's earlier this semester, one of my students asked the following question for which I didn't know the answer (and none of the textbooks I consulted seem to discuss it). It is ... 6answers 2k views Catenary curve under non-uniform gravitational field The catenary curve is the shape of a chain hanging between two equal-height poles under the influence of gravity. But the derivation of the (hyperbolic cosine) curve equation from the physics ... 1answer 138 views Commuting ODE's implies existence of nonzero vanishing two variable polynomial? Write$\partial := d/dt$, fix$m, n > 0$, and let$$F = \partial^n + f_1(t)\partial^{n-1} + \dots + f_{n-1}\partial + f_0,\text{ }G:= \partial^m + g_1(t)\partial^{m-1} + \dots + g_{m-1}\partial + ... 1answer 584 views The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ... 1answer 869 views Kontsevich's flow on the space of Poisson structures In §5.3 of Kontsevich's Formality Conjecture he writes: This (...) gives a remarkable vector field on the space of bi-vector fields on \mathbf{R}^d. The evolution with respect to the time t is ... 0answers 1k views The radius of convergence of the p-adic exponential function. As every number theorist learns, the radius of convergence of exp(x), defined by the usual power series in a neighborhood of zero, is$$\rho = p^{-1/(p-1)}.$$This is typically proven by computing ... 4answers 2k views does the j-invariant satisfy a rational differential equation? Let j be the Klein j-invariant (from the theory of modular functions). Does j satisfy a differential equation of the form j^\prime (z) = f(j(z),z) for any rational function f? 3answers 941 views What theorem of Liouville's is Gian-Carlo Rota referring to here? I am very curious about this remark in Lesson Four of Rota's talk, Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations: "For second order linear differential ... 2answers 1k views Getting a differential equation for a function from a functional equation of its Mellin transform If f is a locally integrable function then its Mellin transform \mathcal{M}[f] is defined by$$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$This integral usually converges in a ... 2answers 953 views 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"? 4answers 2k views Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc? My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ... 5answers 2k views 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 ... 1answer 1k views 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 ... 5answers 3k views Describing the universal covering map for the twice punctured complex plane As is well known, the universal covering space of the punctured complex plane is the complex plane itself, and the cover is given by the exponential map. In a sense, this shows that the logarithm has ... 1answer 774 views 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 ... 3answers 864 views Are there any nonlinear solutions to f(x+1) - f(x) = f'(x)? Are there any nonlinear solutions to f(x+1) - f(x) = f'(x)? (Asked by bcross at math.iuiui.edu on the Q&A board at JMM.) 9answers 2k views 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, ... 4answers 5k views Can an integral equation always be rewritten as a differential equation? Given an integral equation is there always a differential equation which has the same (say smooth) solutions? It seems like not but can one prove this in some example? Edit: Naively I'm hoping for ... 6answers 882 views 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 ... 1answer 675 views 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 ... 2answers 381 views 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 ... 1answer 488 views 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 ... 1answer 547 views A polynomial recurrence involving partial derivatives Define recursively polynomials f_n(a,b) by$$ f_0(a,b)=1,\ \ f_n(0,b)=0\ \mathrm{for}\ n>0  \frac{\partial}{\partial a}f_n(a,b) = f_{n-1}(b-a,1-a). $$For instance,$$ ... 1answer 682 views 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$\...
Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...