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0
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0answers
12 views

Setting bound on particular integral when proving properties of Bogovskii operator

I am reading the proof of the properties of Bogovskii operator in the book Introduction to the Mathematical Theory of Compressible Flow. Let $B^\epsilon(y) = \{ x : |x-y| > \epsilon \}$, $f$ and $...
1
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1answer
203 views

An answer to this system of PDE's

Planning of the question: Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle The isotropic almost complex structures $J_{\delta , \sigma}$ were introduced by Aguilar on the ...
1
vote
1answer
42 views

Approximate largest eigenvalue of Monodromy matrix

Does anyone know the procedure (or have pseudo code) to approximating the largest eigenvalue of a monodromy matrix? Or even to approximate the monodromy matrix itself? There is no explicit solution ...
1
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0answers
49 views

Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$: $f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $ where $P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq j}\{\Phi[\sqrt{w}(v-u_k)]\}...
0
votes
0answers
33 views

Reparametrisation of a PDE with arclength

Suppose I have the following PDE: $\frac{\partial y}{\partial t} = \frac{\partial}{\partial x}\left[(1-y)^3y^3\left(\sin \theta + \frac{\partial y}{\partial x}\cos \theta\right) \right]$ I notice ...
3
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0answers
34 views

how to study the size of basins of attraction on a graph

I have a certain finite (but huge and without an apparent pattern, so that only numerical studies seem feasible) graph $G = (V,E)$, and a function $f: V \rightarrow \mathbb{R}$. On each edge $e = (u,v)...
0
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0answers
51 views

Monotone functions in ordered Banach spaces

Let $(X,\preceq)$ be a real ordered infinite-dimensional Banach space and $f:X\to \mathbb{R}$ is a Fréchet differentiable function. It is said to be a monotone non decreasing map if $$ x\preceq y \...
1
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0answers
22 views

Computing skewness derivative in terms of variance

In the Portilla Simoncelli paper (page 18): http://www.cns.nyu.edu/pub/lcv/portilla99-reprint.pdf They go about calculating the derivative of the skewness $\eta(x)$ of a distribution (2D matrix in ...
0
votes
1answer
166 views

$P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable. $Z$ is eigenvalue of $P(Z )$ if $\...
1
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0answers
30 views

Divergence of a second order tensor [closed]

I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. I am not sure which is correct. Suppose you would like to compute the components of a ...
-1
votes
1answer
49 views

A question on decreasing function [closed]

Let $t\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} + f(t) = 0$ $f(t) $ is continuous decreasing function of $t$. $a_i\ge0$ for all $i$. $y(t)$ is positive real zero of the first equition. Can ...
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votes
1answer
209 views

Does differentiation widen, or narrow, the class of functions?

Let $\cal F^k$ be a set of functions, each of class $C^k$, i.e., both, for every function in $\cal F^k$: $k^{\textrm{th}}$ derivatives exist, and are continuous. Let $D(\cal F^k)$ be the set of ...
6
votes
1answer
281 views

Law of unconsious statistician: application in characteristic function

Let $g(x)=(x-a)\mathbf 1_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought ...
-1
votes
2answers
739 views

What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$? [closed]

I happened to ponder about the differentiation of the following function: $$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$ Now, while I do know how to manipulate power towers to a certain extent,...
5
votes
3answers
663 views

A definition of the fractional derivative

I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it? $$\frac{d^n}{dx^n}f(x) = \lim_{h \...
2
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2answers
297 views

Minimum of an apparently harmless function of two variables

DISCLAIMER: I already posted this question on Mathematics a month ago, here. However, since it has not been solved yet on that platform, I decided to ask it also here on mathoverflow. At a first ...
-1
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1answer
59 views

growth of an entire solution of a differential equation

Let $d$ be an integer and $\sum_{k=0}^dP_k(z)y^{(k)}(z)=0$ be a differential equation over $\mathbb C$, where the $P_k$ are polynomials of degree $\le d$. Consider (if it exists) an entire solution $f$...
2
votes
1answer
130 views

Monotonicity of the integral

Let $R(x)$ be the residual function associated to the normal probability density, i.e. $$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$ Define $$\phi(...
0
votes
1answer
90 views

Why does optimization of a sum of two terms result in “neat” answers? [closed]

This is a somewhat vague and philosophical question. Consider the following three problems: Problem 1: Minimize over all real-valued $x,$ the function $f(x) = bx-ax^2$ where $a,b>0.$ Ans:...
2
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4answers
656 views

Intrinsic definition of arc length [closed]

Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?
2
votes
1answer
73 views

Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over $\mathbb{R}^3$...
5
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2answers
308 views

Common roots of polynomial and its derivative

Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots then,...
2
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2answers
148 views

About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? If ...
2
votes
1answer
195 views

Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$

Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...
-1
votes
1answer
142 views

How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable? In this context PSD means symmetric PSD.
2
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0answers
121 views

Slice a compact C1 surface in R3 by a moving transverse plane. Does the length of the slice depend C1 on the plane?

To be more precise I am interested in questions similar to the one below (I asked the question below on math.stackexchange last week but got not answer.) I have a $C^1$ function $f:[0,1]^2 \to \...
2
votes
1answer
101 views

Is first term of my cost function convex?

I have an optimization problem in the form of [\begin{array}{l} \mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...
3
votes
1answer
177 views

A functional equality

I don't know if this is known, but I was fiddling around with this equality : $$f:(-1,1)\to (-1,1)\quad \text{satisfies}\quad(f(z)+1)^s=\sum_{j=0}^{\infty}\dbinom{s}{j}f(z^k) \quad \forall z\in (-1,1),...
1
vote
1answer
236 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: $$\det[\...
4
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0answers
253 views

Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
3
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0answers
168 views

Looking for author of calculus quote

When I was a lowly calculus student many many years ago, my calculus teacher quoted some famous mathemtician: "Calculus is the last course in arithmetic and the first course in mathematics that one ...
3
votes
1answer
94 views

Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$. I would like to know how many extrema $p$ has on the standard simplex ...
1
vote
1answer
224 views

Request for help with two integrals

It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...
1
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0answers
180 views

Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...
2
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1answer
2k views

The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
8
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1answer
1k views

Differential Calculus and the De Rham Homotopy Operator

Suppose $M$ is a smooth manifold, $\Omega^k(M)$ the vector space (or $C^\infty(M)$-module in case this is better suited) of $k$-differentialforms on $M$ and $$I: \Omega^k(M\times \mathbb{R}) \to \...
4
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0answers
188 views

Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them

Hello! I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I ...
22
votes
2answers
886 views

Is there a convenient differential calculus for cojets?

I understand the basics of exterior differential geometry and how to do calculus with exterior differential forms. I know how to use this to justify the notation dy/dx as a literal ratio of the ...
17
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4answers
7k views

How did Bernoulli prove L'Hôpital's rule?

To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...
1
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0answers
3k views

What is the geometric meaning of the third derivative of a function at a point? [closed]

What is the geometric meaning of the third derivative of a function at a point? This question is now asked on the sister site: http://math.stackexchange.com/questions/14841/what-is-the-meaning-of-the-...
2
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2answers
357 views

Finding the Universal Ideal of a (Covariant) Differential Calculus

Let $(\Omega,d)$ be a differential calculus over an algebra $A$. It is easy to show that $\Omega$ is always equal to a quotient of $\Omega_u(A)$, the universal calculus over $A$, by some ideal $N$ of $...