**5**

votes

**4**answers

774 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 ...

**7**

votes

**11**answers

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, ...

**4**

votes

**0**answers

330 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

**2**answers

658 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

**1**answer

336 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

**2**answers

458 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

**2**answers

202 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

**2**answers

309 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

**1**answer

305 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

**2**answers

3k 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

**1**answer

291 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

**4**answers

948 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

**1**answer

411 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

**0**answers

180 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 ...

**9**

votes

**2**answers

504 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

**1**answer

335 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:$$
...

**3**

votes

**2**answers

873 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 ...

**4**

votes

**0**answers

448 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

**2**answers

590 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

**1**answer

466 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 ...

**3**

votes

**2**answers

262 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

**1**answer

224 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

**1**answer

425 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

**1**answer

474 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

**2**answers

494 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

**1**answer

404 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

**1**answer

288 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

**1**answer

341 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

**4**answers

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

**1**answer

215 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

**4**answers

550 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

**1**answer

423 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 ...

**1**

vote

**0**answers

206 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

**1**answer

307 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
...

**12**

votes

**0**answers

510 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

**1**answer

345 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

**4**answers

525 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

**1**answer

98 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

**2**answers

121 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

**2**answers

873 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

**1**answer

174 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

**0**answers

598 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

**1**answer

369 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

**3**answers

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

**0**answers

636 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

**0**answers

164 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

**3**answers

537 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

**1**answer

294 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 ...

**6**

votes

**2**answers

778 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

**2**answers

420 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 ...