This tag is to be used only when re-tagging highly(!) off-topic questions where none of the actual tags would make sense; all actual tags the questioner has used are removed and something is needed to have some tag, which is enforced by the software, so this tag is used. However note that this tag ...

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10
votes
0answers
448 views

Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups

In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M ...
7
votes
4answers
436 views

Trig functions based on convex curves

Pardon my naivety, but I wonder if much use has been found for trigonometric functions defined in terms of a centrally symmetric convex curve $K$ replacing the circle $C$. For example, here is the ...
0
votes
1answer
95 views

Rate of change of mass of a parameterized region

Let $R_t$ be a family of compact, simply connected regions in the plane defined by $R_t = \{x\in\mathbb{R}^2 : h(x) \leq t\}$ for all $t$, where $h(x)$ is some nicely behaved smooth function. ...
0
votes
2answers
115 views

representation of compact supported distribution

Is this true? Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
2
votes
1answer
147 views

Lipshitz Constant of the convex extension of a submodular function

The title says it all :) Given a submodular function (take the rank function of a matroid, for a concrete example) $f:\{0,1\}^n\rightarrow \mathbb{R}$, we can extend it to a convex function ...
6
votes
0answers
504 views

Hard divisibility problem [closed]

The statement of this problem is elementary... How about the proof ? Let $p$ be an odd prime number. For every integer $a$ , define the number $ ...
4
votes
3answers
2k views

n-dimensional “cross product” reference request

I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to. Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ ...
1
vote
0answers
328 views

elementary exact sequence of normal sheaves

Let $Z \subset Y \subset \mathbb{A^n}$ be a smooth subvarieties of $\mathbb{A^n}$. I'm trying to show that there is an exact sequence of normal bundles. $0 \rightarrow N_{Z/Y} \rightarrow N_{Z} ...
2
votes
0answers
159 views

Computing a double integral [closed]

Hi, I was wondering whether there was an explicit formula for the following integral: $$\int_{B_r}\int_{B_r}|x-y|^{-(d-1)} dx dy$$ Here $dx$ and $dy$ is lebesgue measure on $R^d$ and $B_r$ is ...
6
votes
3answers
527 views

functions with same area

I have two real valued functions $f_1$ and $f_2$ such that $\int_0^Tf_1=\int_0^Tf_2=a_1$ $\int_0^Tf_1^2=\int_0^Tf_2^2=a_2$ $\forall \\ t, f_1(t),f_2(t)\in[0,1]$ Now,I want to construct a family ...
5
votes
1answer
213 views

On the multidimensional generalisation of Gamma function

Gamma function is defined as $$ \Gamma(z) = \int\limits_{0}^{+\infty} x^{z-1} e^{-x} \; dx $$ I'm looking for multidimensional generalisation of this definition. I consider the class $Q$ of ...
5
votes
2answers
520 views

Conservative differential equations “in the wild”

Dear MO world, I'm teaching an undergraduate course on "fun with chaos". As part of a test (on bifurcations in differential equations), I asked students to sketch phase portraits for a family of (2d) ...
2
votes
2answers
405 views

closed form solution to $x\ln (\frac{2}{x})=k(a-x) \ln (\frac{2}{a-x})$

$x\ln (\frac{2}{x})=k(a-x) \ln (\frac{2}{a-x})$ where $a$ and $k$ are positive constants. $a$ is usually small, say, $0< a<0.1$ and $x\in (0,a)$. There are ways to calculate numerical ...
0
votes
1answer
143 views

invariance under dilations

we have that the function (for suitable f) $ F(x)= \sum_{-\infty}^{\infty}f(x+n) $ is INVARIANT under any integer traslation $ y=x+n$ for integer 'n' however my question is can we find a lattice ...
0
votes
0answers
370 views

“closed form” finite sum

If a finite sum has a definite integral representation, for which it can be proved the underlying indefinite integral is not an elementary function, then does this imply the original finite sum can ...
5
votes
0answers
176 views

Does every stack with a connection admit an atlas with a connection?

Dear all, Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...
11
votes
2answers
4k views

What is “Seetapun Enigma”?

A friend of mine just asked me this very question. While I had some training in combinatorics, I have never heard of the "Seetapun Enigma", which, supposedly, is related to the Ramsey's theorem. A ...
1
vote
1answer
251 views

Simple and general relation between continuant polynomials

Continued fraction $[a_0,a_1,...,a_n]$ may be expressed as quotient of two polynomials of $(a_0,a_1,...,a_n)$, named continuants (see http://en.wikipedia.org/wiki/Continuant_%28mathematics%29 ) ...
3
votes
1answer
227 views

Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \le 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and $\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$ is finite. There are at least two ...
0
votes
1answer
225 views

derivative of a special function in integral form

What is the derivative of $Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e, $$\frac{\partial}{\partial x}Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right), \quad ...
11
votes
2answers
644 views

Interesting result on the Euler-Maschroni constant - what is the background?

Today I entered the following expression in maple: $$a_i = H_{10^i} - ln(10^i) - \gamma$$ Here $H_j$ equals $\sum_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant. When I computed $a_n$ ...
11
votes
1answer
462 views

Is exponent of discrete-analytic function also discrete-analytic?

Lets define a discrete analytic function such a function that is equal to its Newton series: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ Is function $g(x)=e^{f(x)}$ also ...
2
votes
1answer
320 views

An iterated tensor product integral

In "Differential equations driven by rough paths" (Terry Lyons, et al) section 1.4.2 it's claimed that the symmetric part of the tensor: $\int_{0 \le u_1 \le \cdots \le u_j \le t} \mathrm{d}X_{u_1} ...
8
votes
1answer
968 views

Naive definition of surface area doesn't work?

A first stab at a definition of surface area might go like this: Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
2
votes
0answers
230 views

Monge Ampere and Calculus

[ I posted the question on Math StackExchange but didn't get any reply nor comment, so I'm trying here ] I am learning about mass transportation theory and the Monge-Ampere equation, to transport a ...
1
vote
0answers
186 views

Fractional Radon - Nikodym derivative

Given $f$ a function measurable in $[0,\infty]$ defined as $$\frac{d\mu}{d\nu}$$ such that, for every measurable set $A$ we have: $$\mu(A)=\int_A{}fd\nu$$ the function $f$ is called $Radon - ...
0
votes
1answer
384 views

Sequence of smooth functions converging to sgn(x)

I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property: \begin{equation} f_i(x_1+x_2) = g_i(x_1, f_i(x_2)) ...
22
votes
16answers
2k views

functions satisfying “one-one iff onto”

Hello Everybody. I need some more examples for the following really interesting phenomenon: A function from the class ... is one-one iff it is onto. Some ...
1
vote
1answer
175 views

About one series. Are there some related special functions?

Hello, I have the following series: $$ \sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0, $$ where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum ...
18
votes
9answers
4k views

How to motivate and present epsilon-delta proofs to undergraduates?

This would seem to be a common question, but I am surprised not to see it already asked and answered on MO! I am teaching an undergraduate course, and I want to teach them to construct basic ...
7
votes
1answer
323 views

Labeling a Square Array

Suppose that the $n^2$ cells of an $n\times n$ array are labeled with the integers $1, \dots, n^2$. Under the traditional left-to-right and top-to-bottom labeling, the labels of horizontally adjacent ...
1
vote
1answer
445 views

A question on gauge functions

In the second paragraph on Page 71 of the book Matrix Analysis by Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem III 4.4''. How can one get the inequality in Theorem III 4.4 from ...
0
votes
1answer
201 views

Maximum value of a function with iterated logarithms

Consider this function $f(x)=-x\log\log(bx)+x\log\log\log(bx)+ax$, where $a$ and $b$ are positive constants, and the logarithms are 2-based. Is it possible to find the maximum value (or even with ...
2
votes
0answers
469 views

Tamagawa number for functional fields

Let $G$ be a split semi-simple simply connected group over a global field $F$ and let $\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well known that ...
1
vote
1answer
143 views

integral of a rational function (1+a_i s)^-1/prod((1+a_j s)^k)

Is there any closed-form expression for the following integral: $\int_0^\infty \frac{1}{(1+a_i s) \prod_{j=1}^n (1+a_j s)^k} ds $ where the ai are >0 and k is a positive integer. And, if k is not an ...
1
vote
0answers
207 views

Homotopy-Fibre Sequence of Classifying Spaces

Let $G$ be a topological group and $H$ be a normal subgroup of $G$ (I think $H$ is required to be admissible in the sense that the quotient map $G\to G/H$ is a principal $H$-bundle, am I right?). Then ...
2
votes
1answer
407 views

A real algebraic curve in $\mathbb{R}^3$ is the intersection of zero sets of polynomials in $\mathbb{R}[x,y,z]$. Can we choose the polynomials in a way, that, seen in $\mathbb{C}[x,y,z]$, the intersection of their zero sets is a complex alg. curve?

By "algebraic curve" here (both in $\mathbb{R}^3$ and in $\mathbb{C}^3$), I do not only mean that the dimension of the set as an algebraic variety is 1, but also that its dimension is 1 in a ...
1
vote
2answers
565 views

expectation of log(1+x) if x is a gamma random variable

I would like to know if there is a closed form expression for the expectation of log(1+x) when x is a gamma random variable. Thank you.
0
votes
1answer
406 views

Maximum sum of 3 consecutive numbers in a permutation [closed]

Given that $X = \{0, 1, 2, ..., 7, 8, 9\}$, and $P$ is a permutation on $X$. Let $M(P)$ be the maximum sum of 3 consecutive elements. For example, if $P = (0, 2, 4, 1, 5, 7, 9, 3, 8, 6)$, then $M(P)$ ...
0
votes
1answer
286 views

Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
6
votes
2answers
468 views

Computational complexity of integration in two dimensions

I have an algorithm for solving a certain problem that requires that I compute two-dimensional integrals as a subroutine, and I'd like to make some kind of statement about its running time. Suppose ...
3
votes
1answer
160 views

bounding the probability that a polynomial is near 0

Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...
8
votes
3answers
549 views

Variable-centric logical foundation of calculus

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...
3
votes
1answer
384 views

Definition of Pontrjagin Classes

I am studying characteristic classes recently and find some quesions about Pontrjagin classes. Firstly the definition of Pontrjagin classes is not that natural. When we talk about Pontrjagin classes ...
15
votes
8answers
5k views

Interesting Applications of the Classical Stokes Theorem?

When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
4
votes
3answers
237 views

Operation on Isospectral graphs

Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that obtain by binary operation between $G$ and $H$? For example,in special case, is ...
2
votes
2answers
359 views

branch points of modular parametrization of an elliptic curve

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to ...
1
vote
2answers
421 views

Uncountability of the “Peculiar” sets:

I call a set PECULIAR, if its elements are uncountable, pairwise disjoint subsets of R (the real number system). As for example, the set {(0,1),(3,5),[8,9]\Q},where Q denotes the set of rationals, is ...
0
votes
0answers
431 views

A product sum inequality question

For any $x_{1},x_{2},\cdots x_{6}$ with $\sum_{i=1}^{6}x_{i}^{2}=1$ and $y_{1},y_{2},\cdots y_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}y_{i}^{2}=1$, do there always exist $z_{1},z_{2},\cdots z_{6}$ in ...
1
vote
1answer
225 views

spectrum and degree sequence

We have the spectrum and the degree sequence of one graph. Can we uniquely determine the graph with these given information?