**11**

votes

**0**answers

501 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 ...

**11**

votes

**0**answers

508 views

### Can the similarity between the Riesz representation theorem and the Yoneda embedding lemma be given a formal undergirding?

For example, by viewing Hilbert spaces as enriched categories in some fashion? (I suppose the same idea of considering the inner product of a Hilbert space as a generalized Hom-set has also been ...

**6**

votes

**0**answers

746 views

### Homotopy groups of a Bouquet of n-spheres

Let $$X=\vee_{\alpha\in A} S_{\alpha}^n$$ be a bouquet of $n$-spheres.
Q: How does one compute the homotopy groups $\pi_k(X)$?

**6**

votes

**0**answers

1k views

### Why is mechanical differentiation so hard to get right?

This question is related to this question on differentiation/integration which asks why differentiation is mechanical but integration is an art. The answers given all make a huge assumption: that one ...

**5**

votes

**0**answers

151 views

### Other applications of the 'increment' approach

I would like to hear about other instances of the so-called 'increments' approach, first used by Roth to prove that a subset of $\mathbb{N}$ of positive upper density contained infinitely many ...

**5**

votes

**0**answers

263 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

**0**answers

183 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 ...

**4**

votes

**0**answers

327 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 ...

**4**

votes

**0**answers

439 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

**0**answers

796 views

### integer solutions of $ (n!+1)=m^2$

Consider $4!=24$, if you add one you get $25=5^2$. The same occurs with $5! = 120 = 11^2 - 1$, and $7! = 5040 = 71^2 - 1$. Are there other solutions of the equation $n!+1 = m^2$?
I verified that no ...

**4**

votes

**0**answers

271 views

### Easy to find roots

Is there a smooth function $f:\mathbb{R} \to \mathbb{R}_{\geq 0}$ such that:
1) $\lim_{x \to \infty} = \lim_{x \to -\infty} = 0$
2) $\forall x > 0$, $f'(x) < 0$
3) $\forall x < 0$, $f'(x) ...

**3**

votes

**0**answers

125 views

### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...

**3**

votes

**0**answers

138 views

### Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).
K3 surfaces is obtained by removing a fiber from two copies of $E$ ...

**2**

votes

**0**answers

195 views

### Extension divergence-free, curl-converging vector field

Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...

**2**

votes

**0**answers

861 views

### Derivative of the regularized upper incomplete gamma function

Hello everyone!
I have a question about the derivative of the regularized upper incomplete gamma function. Considering the gamma function and the incomplete gamma function
\begin{eqnarray}
...

**2**

votes

**0**answers

237 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 ...

**2**

votes

**0**answers

581 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 ...

**2**

votes

**0**answers

283 views

### functions on intervals with endpoints

Would most analysts say that $(2/3) x^{3/2}$ is an antiderivative of $x^{1/2}$ on $[0,\infty)$, or
just on $(0,\infty)$?
More generally, is there a standard interpretation of the assertion "$F$ is an ...

**2**

votes

**0**answers

169 views

### open orbits and invariant distributions

Suppose $G$ is a p-adic algebraic group, $P=MN$ a parabolic subgroup of $G$ with its Levi decomposition, $\sigma$ be a irreducible representation of $M$, we use $I(\sigma)$ to denote the unique ...

**2**

votes

**0**answers

349 views

### Statistic of cubic Irreducible polynomial with cyclic Galois group

Van der Waerden proved for monic irreducible polynomials $f$ of degree $n$ with bounded height $X$:
$$
|\{ f(x)\in \mathbb{Z}[x]: Gal(f)\neq S_n \}|=o(X)
$$
Where height of $f$ is defined by maximum ...

**2**

votes

**0**answers

216 views

### volume under induced metric of preimage set of a regular value under a polynomial map

I am interested in the following kind of polynomial map:
$$ f: \mathbb{T}^k \to \mathbb{R}^n$$
where $f$ is a polynomial map of a certain maximum degree $d$, in the sense that if we imbed ...

**2**

votes

**0**answers

657 views

### Vector bundles on some non-projective surfaces

Let $X$ be a smooth projective curve over a field $k$ and let $L$ be a line bundle on $X$.
I will denote by $S$ the total space of $L$ -- this is a smooth surface over $k$ containing
$X$ (as the zero ...

**2**

votes

**0**answers

69 views

### Characterizing local homeomorphisms into an exponent

Let $X$,$Y$, and $Z$ be (compactly generated) spaces. Suppose $f:Z \to Y^X$ is a local homeomorphism. How can we tell this from its adjoint $\tilde f:Z \times X \to Y$? I.e., I want a property $P$ ...

**2**

votes

**0**answers

618 views

### Strong Bezout's Identity?

Let $\{ a_i \}_{i=1}^N $ be a set of elements of the ring of integers, $\mathbb{Z}_D$ and define $g = \text{gcd}(a_1, a_2,\ldots, a_N, D)$. Then Bezout's Identity states that there exists another set ...

**2**

votes

**0**answers

262 views

### Drawing a combinatorial 3-configuration of points and lines with pseudolines

This question is related to the question of drawing a combinatorial 3-configuration of points and lines with straight lines. We only relax the condition and admit drawings with pseudolines. Let us ...

**1**

vote

**0**answers

497 views

### Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...

**1**

vote

**0**answers

103 views

### Existence of open dense subset in a Lie group

Let $G=S\cdot R$ be a Levi-Malcev decomposition of a connected complex Lie group
and $\Gamma$ a discrete subgroup of $G$ such that the subgroups
$\Gamma_g := \Gamma\cap (gSg^{-1})$ are finite for all ...

**1**

vote

**0**answers

610 views

### Area Under Generalized Parabolas and Hyperbolas without Calculus

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...

**1**

vote

**0**answers

203 views

### Average weighted value of a linear functional over increasing bounded subsets of Z^n

Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...

**1**

vote

**0**answers

343 views

### The used symbols for equality and equivalence

Background: I am currently developing a general purpose programming language which allows formal verification (i.e. correctness proofs) of programs. During the development it came out that a lot of ...

**1**

vote

**0**answers

112 views

### A question about smoothness

$f$ is a smooth function on $[0,+\infty)$ and $f(x)>0$ for all $x>0$. Then does the following equivalence hold :
$\phi(x,y)=f(\sqrt{x^2+y^2})$ is smooth if and only if $f^{(k)}(0)=0$ for all ...

**1**

vote

**0**answers

545 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} ...

**1**

vote

**0**answers

199 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 - ...

**1**

vote

**0**answers

219 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 ...

**1**

vote

**0**answers

139 views

### Morphisms, inducing isomorphisms on global functions

Let $f:X\to Y$ be a surjective morphism of complex varieties with $Y$ affine. Assume that every fiber of this morphism has the property that all the global functions are constant.
What else do I need ...

**1**

vote

**0**answers

307 views

### Some help with differential operators

Hello
I am trying to follow the derivation of the Generalized Cornish Fisher Expansion from this paper.
I dont see how the author goes from equation (21) to (22).
For one to be able to do this ...

**1**

vote

**0**answers

390 views

### Recursive algorithm for integration of complex functions

Integrals of the form $\int_{0}^{\pi} d\theta \sin\theta f(r(\theta)) j_{\ell_{1}} (a r(\theta))y_{\ell_{2}}(br(\theta))P_{\ell_{3}} ^{m} (\cos(\theta))P_{\ell_{4}} ^{m}(\cos(\theta))$, where $f$ is a ...

**1**

vote

**0**answers

142 views

### Do higher dimensional maxima of a real valued multivariable function form a cell complex?

Suppose $f: R^n \rightarrow R$ is a positive real valued function. Let $\lambda_1, \ldots , \lambda_i$ be the first $i$ ordered eigenvalues of the Hessian $Hess(f)$. Let $v_1, \ldots, v_i$ be the ...

**1**

vote

**0**answers

408 views

### A transformation of infinite series

Suppose I have a convergent infinite series $\sum_{n=0}^\infty (-1)^n a_n = S_0$ and $0 < S_0 < 1$. Write $s_n$ for the $n$-th partial sum. ($s_n = \sum_{k=0}^n (-1)^k a_k$) Now consider the ...

**0**

votes

**0**answers

84 views

### name for a class of functions

I would like to ask whether is used some name for functions $g:A\to\mathbb{R}$, $A\subset \mathbb{R}$, for which $$\exists \lambda>1:\;\; \lim_{x\to 0^+}\frac{\lambda g(x)}{g(\lambda x)}>1.$$

**0**

votes

**0**answers

143 views

### Foliation over characteristic positive

Ekedahl wrote about foliation in characteristic positive, over the field $/frac{Z}{pZ}$ as a subsheaft of the tangent sheaft, that is closed over involution and $p$-power, My question is if there ...

**0**

votes

**0**answers

239 views

### Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...

**0**

votes

**0**answers

179 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 ...

**0**

votes

**0**answers

198 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

**0**answers

521 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 ...

**0**

votes

**0**answers

220 views

### Properties of morphisms induced by divisors on curves

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim ...

**0**

votes

**0**answers

314 views

### Systems of linear octonionic equations

Is there theory of determinants, rank of matrices and systems of linear equations with octonionic coefficients? Does anybody could indicate references? I want to know mainly does there exist a ...

**0**

votes

**0**answers

259 views

### Is there an intuition for the exterior calculus identity dd = 0

Considering an exterior derivative $d$, which may be discretized as the transpose of the boundary operator on a simplicial mesh, I have read and seen that $dd = 0$, but I have not seen an intuitive ...

**0**

votes

**0**answers

414 views

### covariant derivative complex manifold

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...

**0**

votes

**0**answers

486 views

### Integrating the product of two functions one of which has a positive non-integer power

I'm looking to integrate several functions having the form
$\int_0^T \frac{ sin(\omega \tau) }{\omega} \tau^{2H} d\tau$
where $2H \ge 0$ but may not be an integer. I'd like to know if the machinery ...