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-2
votes
0answers
37 views

Integrating a differential form over a box [on hold]

I've only ever seen examples of integrating a differential form over a curve C involving defining a parameterization. I have seen people integrate 1 forms over a box without defining a ...
0
votes
0answers
77 views

Notation for “partial” derivative [on hold]

Say I have a function like: $f(x,c,y) = x+c+y$ say that $c$ is a function of $x$, $c(x)$ (i.e. $x$ and $y$ are the only truly independent variables). How would I write the notation for the ...
0
votes
0answers
34 views

Sufficient Conditions For Monotonic Decreasing of Multivariate Function

I found the following theorem on sufficient conditions for decreasing monotonicity of an absolutely continuous function $\phi : \mathbb{R} \rightarrow \mathbb{R}$, I would like to know if it is ...
2
votes
1answer
64 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{ - ...
0
votes
0answers
20 views

Condition of existence and uniqueness of solution for abel integral equation [migrated]

It is well known that Abel integral equation has a unique continuous solution. For example, $$ f(t)=\int_0^t\frac{g(s)}{(s-t)^{\alpha}}ds , 0<\alpha<1 $$ where f(t) is known. Specifically, ...
3
votes
1answer
150 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
vote
1answer
189 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: ...
3
votes
0answers
141 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 ...
2
votes
0answers
156 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
67 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
197 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 ...
0
votes
0answers
129 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
votes
1answer
882 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
votes
1answer
764 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
votes
0answers
156 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 ...
23
votes
2answers
811 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 ...
2
votes
0answers
2k 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: ...
2
votes
2answers
349 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 ...