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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7
votes
1answer
334 views

What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$: a simplex in $K$ consists of finitely many elements ...
5
votes
0answers
230 views

Azimuthal and polar integration of a 3D Gaussian

Numerical evaluation of the following integral of a 3D gaussian $G$ seems to result in a 1D Gaussian $g$: $$\int_{0}^{2\pi}\int_{0}^{\pi}G(R,\phi,\theta)\sin\theta\ \text{d}\theta \ \text{d}\phi= ...
5
votes
4answers
535 views

Multivariable Calculus Lecture Ideas

I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...
6
votes
11answers
2k views

A function that is defined everywhere but has unknown values [closed]

For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...
3
votes
1answer
397 views

What grows faster? [closed]

What grows faster: Busy Beaver function, TREE function or BEAF largest resursive functions (legions, lugions, lagions...)?
4
votes
0answers
314 views

A question on an intuitive way to look at stacks

I am reading the chapter "Introducing Algebraic Stacks" in The Stacks Projects to get a feeling for them. There is a small point that throws me off. They denote $\mathcal{M}_{1, 1}$ the moduli stack ...
0
votes
2answers
470 views

What's the meaning of pencils in birational geometry?

I see in some books the authors call a one dimensional linear system a pencil, but in other books one call a linear system $|D|$ is not compsited with a pencil if $\dim \phi_{|D|}(X)\geq 2$ and even ...
1
vote
1answer
304 views

Reference for a derivative formula for matrices

I found the identity $$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$ On the matrix cookbook (http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf). It is equation ...
4
votes
2answers
397 views

Can we calculate the inner product of a semicontinous function with the Dirac delta function?

Dear all, It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
4
votes
2answers
192 views

Steinberg Group as a Lattice in a lie group

Given an integral domain $R$, the Steinberg group $St_n(R)$ is the group given by generators $e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda)$, $p\neq q$, $1\leq p,q \leq n$ Subject to ...
2
votes
2answers
300 views

functors unique up to self-equivalence of the source category

Call two functors two functors $H,H':S\longrightarrow T$ weakly equivalent, or equivalent up to a self-equivalence of the source category, iff there exists a self-equivalence of $s:S \longrightarrow ...
2
votes
1answer
292 views

T^i functors are isomorphic for analytically isomorphic isolated singular points

I've been having trouble proving the following: Let $B$ and $B'$ be local rings, essentially of finite type over $k$, having isolated singularities at the closed points. Suppose that they are ...
2
votes
2answers
2k views

Numerical Computation of arcsin and arctan for real numbers [closed]

I'm coding some numerical methods and I do not know what the correct analysis would be for choosing the implementation for $arcsin$ and $arctan$ for real numbers. Here's what I know: Both functions ...
5
votes
1answer
232 views

An extension of Edelstein's Attraction Theorem?

Let $X$ be a metric space. Recall that a function $f: X \rightarrow X$ is contractive if there exists $C \in (0,1)$ such that for all $x,y \in X$, $d(f(x),f(y)) \leq C d(x,y)$; a function $f$ is ...
10
votes
4answers
798 views

compact quotient

Let X be a topological space that is not too bad (let's say "not too bad" = "compactly generated Hausdorff"), and let ∼ be an equivalence relation such that X /∼ is compact Hausdorff. Does there ...
6
votes
1answer
312 views

Evaluating the average distance from a point in the unit disk to the disk

I am interested in finding the average Euclidean distance from a point $(x,y)\in\mathbb{D}_2$, the unit disk $\{(u,v):u^2+v^2\leq 1\}\subseteq\mathbb{R}^2$, to the disk $\mathbb{D}_2$. This amounts to ...
0
votes
0answers
177 views

Matrix Maximization.

Hi everyone, I'm trying to solve/define the following optimization problem: $\max_M f(M)$ s.t. $M \theta = b$ $\sum_j \theta_j = 1$ When: M is an m*n real matrix $\theta$ and $b$ are n*1 column ...
8
votes
2answers
482 views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
0
votes
1answer
230 views

the relationship between integration by parts and surface integrals

Recently, I met an equation about the integration by parts and surface integrals. It says:$$ ...
2
votes
2answers
823 views

An elementary but confusing question in differential geomerty

Let $f:\mathbb{R}\to \mathbb{R}$ be a function, then looking $f$ as a function between manifolds, $df:T\mathbb{R}=\mathbb{R}^2\to \mathbb{R}^2$ and $d^2f:TT\mathbb{R}=\mathbb{R}^4\to \mathbb{R}^4$ are ...
3
votes
0answers
418 views

sine and Archimedes' derivation of the area of the circle

The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...
4
votes
2answers
494 views

Intersection forms of 4-manifolds with boundary

Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...
3
votes
1answer
443 views

The number of simply connected 4-dimension manifold

For a simply connected four-dimension manifold, we know the Freedmen's work. My question is: For every integer N, Is the number of simply connected 4-manifolds which the second betti number is ...
0
votes
0answers
123 views

Hammerstein integral equation with inverse of the solution

In signal processing theory I found this integral equation that I recognized to be of Hammerstein type: $$u(t)-\int_{0}^{1}d\phi cos(\omega t+\phi)\frac{1}{u(\phi)}=0$$ Unfortunately the solution ...
3
votes
2answers
216 views

Minimal right ideals in finite semigroup

Let $E$ be a finite semigroup. According to N. Bourbaki (Algèbre I p. 121 exerc. 14 c), if $M$ and $M'$ are minimal right ideals in $E$, then they are isomorphic. I spent some time browsing through ...
1
vote
1answer
200 views

Order density of smooth functions among continuous functions?

Let $\mathcal{C}^0([a,b],\mathbb{R})$ be the space of all continuous functions $f:[a,b]\rightarrow\mathbb{R}$ and $\mathcal{C}^\infty([a,b],\mathbb{R})$ the subspace of all smooth functions. Define ...
0
votes
1answer
407 views

Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
4
votes
1answer
377 views

Interchanging min and max for a continuous function of two variables

Let $f:[0,1]\times[0,1]\to\mathbb{R}$ be a continuous function. Define $$ M_x:=\max\limits_{0\leq y\leq 1} f(x,y), \qquad m_y:=\min\limits_{0\leq x\leq 1} f(x,y). $$ Is there a useful set of ...
1
vote
2answers
484 views

How to interpret this class of numbers?

Let's say $ f(p) $ is a number defined as shown below: $ \hspace {10 mm}f(p) = {\sqrt {2} ^ {{\sqrt {2} ^ \sqrt {2}} ^ {... \hspace {1 mm} p\hspace {1 mm} times } }} $ What I understand is: We ...
3
votes
1answer
353 views

On One point Lindeloffication of topological spaces

As you Know when we define a topological space to be the one point compactification of the topological space $X$, we look for a compact space $Y$ such that $X\subset Y$ and $X$ is dense in $Y$ and ...
2
votes
1answer
277 views

Frobenius base change of etale maps

Let $A$ be a characteristic $p>0$ commutative ring. Let $B$ be a finitely presented etale $A$ algebra i.e. $$ B=A[x_1,\ldots,x_n]/(f_1,\ldots,f_n), $$ with $det(\frac{\partial f_i}{\partial ...
1
vote
1answer
319 views

Homology and homotopy of a surface

Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H_1(S, \mathbb Z).$ My question is; does this ...
18
votes
4answers
1k views

The “ds” which appears in an integral with respect to arclength is not a 1-form. What is it?

The only reasonable way to interpret "$ds$" as a functional on tangent vectors has to be that it takes a tangent vector and spits out its length, but this is not linear. So $ds$ is not a 1-form. It ...
1
vote
1answer
209 views

Asymptotic behaviour of a mean

Fix $x>0$ and $c\in\mathbb{N}$. Let $f(x):=\frac{c}{4c-2+2x^2}$ and $$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$ I'm pretty sure $m_N(x)\to\infty$ as $N\to\infty$. I ...
6
votes
4answers
521 views

On Pseudo-finite topological spaces

We recall that a topological space $(X,\tau)$ is Pseudo-finite, if each compact subset of $X$ is finite. One of the classical example of Pseudo-finite topological spaces can be considered as an ...
3
votes
1answer
402 views

asymptotic behaviour of a sum

I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum $$ Z_N(x) := 2^{-N/2} \sum_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$ in order to compute ...
0
votes
0answers
186 views

Evaluating the integral $\int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx$

I'm trying to evaluate or simplify this integral: $$I_{a,b,c} = \int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx $$ with $a,b,c \in \mathbb{R}_+^*$. and $ Ei(x) =\int_{-\infty}^{x} ...
0
votes
1answer
289 views

Bounding a signed sum of complex numbers [closed]

Let $z_i \in \mathbb{C}\:$ for $i=1,\dots, n\;$ be complex numbers, all with absolute value $|z_i|\le 1\;$. Prove (or disprove) that there exists a choice of signs $s_i \in \{\pm 1\}$ such that ...
0
votes
1answer
264 views

Lindelöf subsets of $P$-spaces

A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G_\delta$-subset of $X$ is open .$\($i.e $\tau$ is closed under countable intersection$\)$. Here we recall some ...
11
votes
0answers
478 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 ...
2
votes
1answer
309 views

About subspaces of $F$-spaces

A topological space $X$ is an $F$-space, if Every finitely generated ideal in the ring of all continuous functions on $X$,denoted by $C(X)$, is principal. The text "Rings of continuous functions" ...
7
votes
4answers
455 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
97 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
116 views

representation of compact supported distribution

Is this true? Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
5
votes
2answers
528 views

A question about some special compactifications of $\mathbb{R}$

We Know that the topological space $Y$ is a compactification of the topological space $X$, if the space $Y$ is compact and hausdorff and $X$ is dense in $Y$. If for a positive integer $n$ we have a ...
2
votes
1answer
157 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
521 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 $ ...
3
votes
1answer
344 views

Determining continuous functions on Banach spaces

Let $X$ be a real Banach space. For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
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
382 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} ...