# Tagged Questions

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

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### Lifting a quadratic system to a non vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
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### 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" ...
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### Non-uniqueness of solutions of the heat equation

For the heat equation $(\partial_t-\partial_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the ...
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### Solution of linear ODE

Let $A=A(t)$ be a smooth one parameter family of $n\times n$-matrices, $n\ge 2$. It seems that the solution of linear ODE $$\dot x= Ax$$ can not be written in a closed form using $\int$, $A$, $x(0)$ ...
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### Existence/Uniqueness of solutions to quasi-Lipschitz ODEs

Would the Picard–Lindelöf theorem still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be almost Lipschitz in y? If not, are there any moduli ...
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I'm trying to understand how to establish regularity for elliptic equations on bounded domains with Neumann data. For simplicity, let's presume we are focusing on $-\Delta u = f$ in $\Omega$ and $\... 1answer 2k views ### Skellam distribution: Deep connection between Poisson distributions and Bessel function? The probability mass function for the Skellam distribution for a count difference$k=n_1-n_2$from two Poisson-distributed variables with means$\mu_1$and$\mu_2$is given by: $$f(k;\mu_1,\mu_2)= ... 1answer 655 views ### What is exactly the (singularity) confinement property ? This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata. It seems to be related in certain case to the Painlevé ... 2answers 940 views ### 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... 1answer 835 views ### Solutions of equations characterizing a complex structure Let (S^n,g) denote the unit n-sphere endowed with its induced metric g from its embedding into \mathbb{R}^{n+1}. The Levi-Civita connection of g induces a splitting of the tangent bundle of ... 1answer 589 views ### Proof of the “Neo-classical Inequality” I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in p\geq 1, n: \frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq \frac{(a+... 2answers 4k views ### Looking for the solution of first order non-linear differential equation (y ′+y^{2}=f(x)) without knowing a particular solution. I have been working on Riccati Equation. I have tried many different methods to find a closed form for the solution of first order non-linear differential equation (y'+y^{2}=f(x)) without knowing a ... 3answers 473 views ### Linearization of vector fields Simply, we discuss things on C^{2n} (or R^{2n}) but not on manifolds. Given an anayltic (or real analytic) vector field V on C^{2n} (or R^{2n}), with a zero at the origin . And V_{0} is ... 3answers 637 views ### On discrete version of curve shortening flow One can define an analogous version of the curve shortening flow for polygons in \mathbb R^2, namely defined by the differential equation \dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}, where p_i is the ... 4answers 1k views ### The multiplicity of the max eigenvalue in matrix multiplication Suppose that eigenvalues of two real square matrix A and B are 1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 and 1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \... 1answer 522 views ### A Differential Equation with Nested Functions This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems. I would like to know whether there exists a differentiable function from the (... 2answers 865 views ### duality argument in PDE Can anyone please explain the term 'duality argument', or the difference between this term and the weak formulation in PDE analysis? Or give some references? Occasionally I see this term appears in ... 2answers 456 views ### An algebraic Hamiltonian vector field with a finite number of periodic orbits(1) Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question. Is There a polynomial Hamiltonian H(x,y,z,w)=zP(x,y)+wQ(... 1answer 569 views ### Is there a continuous function f satisfying the following Zygmund condition but not differentiable. Suppose that a continuous function f on the line and satisfies$$ |f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1] $$... 2answers 690 views ### What does it mean for a differential equation “to be integrable”? [duplicate] What does it mean for a differential equation "to be integrable"? Are "integrable" and "solvable" synonyms? The first thing that comes to my mind is to say: it's integrable if we can find the ... 0answers 106 views ### Diffusion equation on mixing of diffusing particles I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles. The picture below shows the idea how permutations and inversion numbers reflect ... 0answers 93 views ### Homogeneous regular manifolds In order to solve the well-known Plateau-Problem on a general (non-compact) Riemannian 3-manifold, Morrey first introduced the condition of homogeneous regularity and defined it in the following way: ... 0answers 87 views ### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2) Is there a polynomial Hamiltonian H:\mathbb{R}^{4}\to \mathbb{R} such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field X_{H} is finite but different ... 1answer 124 views ### Is there an entire solution for the Van der pol equation? Is there a non constant entire function \gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2} which satisfy the following Vander pol dififferential equation?$$\begin{cases}\dot{x}=y-x^{3}\\\dot y=-... 1answer 332 views ### On Harmonic Unit Vector Fields When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold$(M,g)$, then the critical points of this functional are satisfied in$\Delta_g X=...
Edit 1: For a related discussion see this MSE post I apologize in advance, if this question is obvious: 1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...