# All Questions

**2**

votes

**1**answer

125 views

### Is this structure a Banach bundle?

Let $X$ be a Banach space. Put $Y=\{ \phi\in X^{*}\mid\;\; \parallel \phi \parallel\leq 1\;\; \&\;\; \phi \neq 0\}$ which is a locally compact Hausdorf space with the weak star topology.
...

**0**

votes

**0**answers

28 views

### normality of truncated arc space

Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field.
For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal?
Same question for ...

**1**

vote

**0**answers

88 views

### A continuous choice of invertible elements

Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map ...

**0**

votes

**0**answers

47 views

### Counting words with pattern and majority constraints [on hold]

Problem: I have an alphabet $X$ with $n$ letters (say $n=8$, $X=\{A, B, C, D, E, F, G, H\}$). I'm looking for words with $m$ letters (say $m=8$), with three constraints:
a given letter (say $A$) is ...

**7**

votes

**1**answer

325 views

### Quest for a human proof of a $q-$binomial identity

Let $$f(n,k) = \sum\limits_{j = - k}^k {{{( - 1)}^{k - j}}}
\binom{n-j}{k-j}\binom{n+j}{k+j}.$$
Then $f(n,k)=\binom{n}{k}$
because it satisfies $f(n,k)=f(n-1,k)+f(n-1,k-1)$ and the obvious ...

**1**

vote

**1**answer

217 views

### Proof of “generic curve of genus at least 2 has no nontrivial maps to a positive genus curve”

I searched for it for a long time, but it seems that everybody is taking this for granted and does not bother to point out a proof. Would it be possible that someone points me to a proof or makes me ...

**-4**

votes

**0**answers

52 views

### closed and exact forms [on hold]

Is the exterior derivative of a 1-form zero? We know that $ω=dψ$; then such an $ω$ is exact and thus $d\omega=0$. Does it mean that here $d\psi$ is a 1 form and the exterior derivative of that is 0?
...

**1**

vote

**0**answers

39 views

### Shift invariance for the distribution of quadratic polynomials

For a probability distribution $X$, supported on integers, define the shift-invariance of $X$, denoted by $shift(X)$ = total variation distance between the random variable $X$ and $X+1$.
Let ...

**3**

votes

**1**answer

187 views

### Equivalence relation defined by the existence of a homeomorphism

Let $(X,\tau)$ be a topological space. We assign to $(X,\tau)$ an equivalence relation $\simeq_{(X,\tau)}$ in the following way:
$x\simeq_{(X,\tau)} y$ if and only if there is a homeomorphism ...

**5**

votes

**1**answer

352 views

### Structure of the automorphism group of a Riemann surface

I was wondering if anything is known about the possible structure of $\mathrm{Aut}(S)$ for a Riemann surface $S$. More precisely, are there known obstructions for a finite group $G$ to be such an ...

**1**

vote

**0**answers

43 views

### Strong solution to parabolic equation without differentiability assumption on coefficient?

Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain
$$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$
$$u|_{\partial\Omega} = 0$$
where $a$ is real-valued and satisfies
$C_1 \leq a(r) \leq C_2$ ...

**3**

votes

**1**answer

210 views

### Existence and uniqueness of a quasi-linear pde system on a surface

I have the following system of first order quasi-linear pde:
$$ -(\Delta+1) a^{\alpha\beta} [b_{\beta\rho} I_{\alpha;\sigma}+b_{\beta\sigma} I_{\alpha;\rho}]
+ a^{\alpha\beta} [(\Delta+1) ...

**3**

votes

**0**answers

100 views

### How to build the smallest regular n-sided polygon that covers an (n-1)-sided polygon? [migrated]

I want to build a figure that contains seven regular polygons, from a triangle up to a nonagon, where each n-sided polygon covers, with the minimal area possible, the n-1 sided one. An added ...

**1**

vote

**1**answer

132 views

### Some general properties of arithmetic groups of simplest type

I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...

**0**

votes

**0**answers

79 views

### Analytic Number Theory [on hold]

Let $\chi_0$ be a principle Dirichlet character modulo q. What is $\lim_{s \rightarrow 0} \Gamma(s) L(s, \chi_0)$?
I know that $L(0, \chi_0) = 0$ and that $\lim_{s \rightarrow 0} \Gamma(s) = ...

**0**

votes

**0**answers

61 views

### Subdivision of a small category

I am reading about subdivision of a category from this paper:
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Delgado.pdf
At the page 4, the author of this paper gives the first example ...

**1**

vote

**1**answer

60 views

### A countable tight topological group where every countable subset is metrizable

I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example.
...

**0**

votes

**0**answers

39 views

### Searching for conditions?

I have this operator $$Au(t)=\int_0^1 G(t,s) f(s,u(s)) ds$$defined from $H^1_{0}$ to $H_0^1$ and satisfy the problem: $$\begin{cases} -(Au)''(t)=f(t,u(t)), t\in[0,1]\\Au(0)=Au(1)=0\end{cases}$$
Where ...

**5**

votes

**0**answers

151 views

### Adjunction map for projective surfaces

Before stating my question, let me recall (part of) the classical result on the adjunction map for complex projective surfaces, due in this modern form to Beltrametti and Sommese:
Adjunction ...

**16**

votes

**1**answer

379 views

+50

### Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?

There is a 50 point bounty on this question.
Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define
$$\pi_k(x)=\#\{n\leq x: ...

**7**

votes

**0**answers

79 views

### Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...

**5**

votes

**1**answer

159 views

### Does independence of the sequence $f(A_i, B)$ imply the sequence is independent of $B$?

Suppose $B, \{A_i: i \in \omega\}$ are i.i.d. random variables with uniform distributions on $[0,1]$. If $f$ is a map such that $\{f(A_i, B): i \in \omega\}$ are independent, must $\{f(A_i, B): i \in ...

**3**

votes

**1**answer

111 views

### A question on many-one reducibility

Let $\phi_0,\phi_1,\phi_2,\ldots$ be an acceptable programming system. For each $x\in\mathbb{N}$, let $W_x$ the domain of $\phi_x$, and let $K=\{x\in\mathbb{N}:W_x\neq\emptyset\}$. Is there a ...

**6**

votes

**1**answer

184 views

### A variant to the Hadwiger-Nelson problem

Consider the following graph $G=(V,E)$ where $V=\mathbb{R}^2$ and $E = \{\{x,y\}: x,y \in \mathbb{R}^2 \text{ and } |x-y|\in \mathbb{Q}\}$.
What is $\chi(G)$?
(This is a variant of the ...

**-3**

votes

**0**answers

56 views

### Example of topological vector space [on hold]

Can Someone provide me any link or research paper about the working of definition of topological vector space(open neighborhood definition) on R to become a vector space?

**3**

votes

**0**answers

84 views

### Does the stable category of a nice exact category embed in (the underlying category of) a derivator?

In Derivators, Pointed Derivators, and Stable Derivators, Moritz Groth gives as an example of a non-invertible morphism with trivial cone an inclusion $f:X\to I$. Here $X$ is an object of injective ...

**3**

votes

**1**answer

202 views

### New series for $1/\pi$ based on Ramanujan's ideas

In his classic paper "Modular Equations and Approximations to $\pi$ (1914)", Ramanujan gives a standard technique to obtain a general family of series for $1/\pi$ based on series for $(2K/\pi)^{2}$ in ...

**6**

votes

**0**answers

615 views

### $ n $-Cats-in-a-Bed Problem: Picking $ n $ points in a given planar domain to maximize the sum of their pairwise distances

Let $ C $ be a connected and simply connected compact subset of the plane $ \mathbb{R}^{2} $. How can we pick $ n $ points, denoted $ x_{1},\ldots,x_{n} $, lying in $ C $ such that the total sum $ ...

**-2**

votes

**1**answer

47 views

### Using moment generating functions [closed]

I need to find the mean and variance of a X^2, where X is a gaussian.
By looking up moment generating function of gaussian, I figured out that,
Var(X) = E[X^2] - (E[X])^2 = M''(0) - (M'[0])^2
Using ...

**5**

votes

**1**answer

82 views

### Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?

Is it true that
$$
||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ?
$$
where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...

**-1**

votes

**1**answer

155 views

### Reductive space & Reductive Lie algebra

If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, ...

**0**

votes

**0**answers

41 views

### Show that $(\frac{d}{dt}||S(t)||_{\infty})_{t=0}=0$ where $S(t)$ is the Contraction semigroup for Laplacian [closed]

My Try:
I was able to prove one side of inequality using
$$
||S(t)\phi||_p\leq (4 \pi t)^{-N/2(\frac{1}{q}-\frac{1}{p})}||\phi||_q
$$
take $p=q=\infty$(as inequality is valid as long as $1\leq ...

**1**

vote

**1**answer

60 views

### “Schwarz symmetrization” on annulus

If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we ...

**18**

votes

**2**answers

572 views

### Reflection of light from function graph

Let a positive convex decreasing differentiable function $f(x)$ be defined on $\mathbb{R}$ and $\lim_{x \to +\infty}f(x)=0.$ Let the point light source be placed at $ P(x_0,y_0)$ with $ ...

**4**

votes

**1**answer

297 views

### Boardman-Vogt tensor product

Let $\mathbf{sSet}$ be the model category of simplicial sets and $\mathbf{Op}$ the model category of symmetric operads. Equipped with Boardman-Vogt tensor product $ \otimes_{BV}$, the category ...

**2**

votes

**0**answers

114 views

+50

### PRNG and coding theory

Let $k, n \in \mathbb{N}$, $k = (1 - \epsilon)n$ where $1 >\epsilon > 0$.
I want to find $f: \{0,1\}^k \to \{0, 1\}^n$
such that:
1) $f(a) \not= f(b)$ if $a \not=b $
2) for any $x \in ...

**2**

votes

**0**answers

40 views

### probabilistic interpretation of elliptic equation with mixed boundary condition

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions:
Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...

**-2**

votes

**0**answers

65 views

### Can anyone proof legendre transformation? [closed]

Can anyone proof legendre transformation? or introduce a book that i find that in it? the legendre transformation is: //y'=y-zeta1*x1 //y''=y'-zeta2*x2=y-zeta1*x1-zeta2*x2 //. . . ...

**1**

vote

**0**answers

87 views

### Torsion elements in the mapping class group

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...

**-3**

votes

**0**answers

17 views

### Numbers to use in order to calculate the F1 score of precision/recall [closed]

Alright so I want to calculate the F1 score for four pair of values for Precision P and Recall R. These pairs are:
...

**2**

votes

**1**answer

90 views

### Units in a finite semisimple group algebra

Let $G$ be a finite group and $k$ a finite field, with the characteristic of $k$ not dividing the order of $G$. Then $kG$ is a finite semisimple group algebra with the interesting property that an ...

**14**

votes

**1**answer

378 views

### Free Loop-Space Recognition Principle

It is well-known that one can detect based loopspaces using the machinery of operads. Namely, given a group-like space $X$ with an action of $\mathbb{E}_n$-operad, then it is homotopy equivalent as an ...

**1**

vote

**2**answers

306 views

### Examples that the Fermat-Catalan conjecture does not cover

The Fermat-Catalan conjecture states that there are only finitely many sex-tuples $(a, b, c, d, e, f)$ of positive integers such that
(1) $a^d + b^e = c^f$,
(2) $\gcd(a, b, c) =1$,
(3) ...

**0**

votes

**1**answer

146 views

### Hyperelliptic curve of genus 2 over R

I know that the points of an elliptic curve over $\mathbb{Q}$, $\mathbb{R}$ or other field $K$ form a group, particularly the most common example to explain the naive way is with this curve ...

**2**

votes

**1**answer

79 views

### Minimal family of k-sets containing all t-sets

Let $n \ge k \ge t \in \mathbb{N}$, and consider a universe $U$ of size $n$. Let $\mathcal{F}$ be a family of $k$-subsets of $U$, such that every $t$-subset of $U$ is contained in at least one member ...

**0**

votes

**0**answers

51 views

### constant of functional equation of zeta function

Let $C$ be a smooth projective curve, of geometric genus $g$, over a finite field $\mathbb{F}_p$ and consider the zeta function $$
Z(C/\mathbb{F}_p, t)=\exp(\sum_{n=1}^{\infty} |C(\mathbb{F}_{q^n})| ...

**1**

vote

**1**answer

60 views

### Structure of locally compact non discrete topological division algebras without the use of Haar measure

There is a well-known structure theorem for locally compact non discrete topological division algebras, see here
http://math.stackexchange.com/q/1160086/187521
(I repost it here because I think it ...

**0**

votes

**1**answer

92 views

### Maps of balls with fixed value along boundary

Suppose I wish to find the homotopy classes of maps of $B^3 \rightarrow M$ which along the boundary are fixed by a (particular) map $f: S^2 \rightarrow M$. Take $M$ to be a closed orientable ...

**-2**

votes

**0**answers

9 views

### Finding coordinates of pyramid with known base, and known angles for apex [migrated]

I have a regular pyramid, where I know the 3d coordinates of all points on the base, and I know all of the angles associated with the apex. I'm wondering if there's a known method to determine the ...

**0**

votes

**0**answers

71 views

### Explicit formula for Bergman kernel on the unit ball

On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...