# All Questions

**3**

votes

**2**answers

195 views

### A question on the effective cone

Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$.
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In ...

**1**

vote

**0**answers

15 views

### Compactness of Weyl pseudodifferential operators with integrable symbols

Given a tempered distribution $s \in \mathcal{S}'(\mathbb{R}^{2d})$, define the Weyl pseudodifferential operator of symbol $s$ as the mapping $\mathcal{S}(\mathbb{R}^{d}) \rightarrow ...

**-4**

votes

**0**answers

55 views

### Theory of Numbers. PROOF using division algorithm and well ordering theorem [closed]

I really help with the following two questions.
1) Assume that b>0. Show that there exists k an element of Z, s.t a+kb>0. (Use the division algorithm)
2)Use Theorem 3.8 (well ordering theorem) to ...

**1**

vote

**0**answers

78 views

### Best constant for Maier's theorem?

Maier proved that, for fixed $\lambda>1,$
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1
$$
and in particular
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda ...

**1**

vote

**0**answers

58 views

### Normal bundles of rational equivalent curves

Let $C_1, C_2$ be rationally equivalent curves in a smooth projective variety $P$. Let $$N_i: = \mathcal{H}om(I_{i}/I^2_{i}, \mathcal{O}_{C_i})$$ be the normal bundle of $C_i$, where $I_i$ is the ...

**3**

votes

**1**answer

35 views

### Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension

Call a Borel set $A \subseteq [0,1]$ good if $$0 < \dim(A) \leq \overline{\dim_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim_\text{M}}(A)$ is ...

**-4**

votes

**0**answers

200 views

### Mathematical urban legend - Best second tier mathematician [closed]

A few years ago I heard a story about a talk given at Stanford by a famous probabilist, perhaps Kai-Lai Chung. The speaker got into some sort of argument with a mathematician in attendance, and called ...

**4**

votes

**1**answer

143 views

### liftings of principal bundles

I would like to know what structure has the category of liftings of a principal bundle. Let me be more precise.
Fix $k$ an algebraically closed field and $X$ a smooth projective variety over it (for ...

**-2**

votes

**0**answers

78 views

### Weak Foundation in Math [closed]

I read this article http://www.xamuel.com/five-ways-to-be-better-at-math/ and was wondering if anyone can help me with references or advice to get better at math. I always considered myself weak at ...

**2**

votes

**0**answers

74 views

### Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii

Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape ...

**2**

votes

**0**answers

66 views

### Deformation with fixed ramification

Suppose that $f : X \to Y$ is a finite, surjective morphism of normal varieties. I want to know about the space of first-order deformations of $X$ over $Y$ with fixed ramification, i.e. the ...

**0**

votes

**1**answer

76 views

### Intersection multiplicty and global sections

Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between ...

**1**

vote

**1**answer

63 views

### About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...

**0**

votes

**1**answer

62 views

### Is there an irreducible subfactor with an infinite homogeneous single chain lattice?

We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here).
Is ...

**0**

votes

**0**answers

34 views

### Finding a random variable with a density function [closed]

So I have this homework I'm having a really hard time starting:
For the random variable X with density function
f(x) =
4x , 0 < x ≤ 1/2
4 − 4x , 1/2 < x ≤ 1
0 , otherwise
Determine the ...

**0**

votes

**0**answers

24 views

### Problem implementing Ramez algorithm [closed]

I'm implementing the Ramez algorithm for optimal polynomials, and I'm having problems. That is, I'm getting worse and worse approximations.
Repo: https://github.com/nmiculinic/dismat2[1]
Viewer: ...

**2**

votes

**1**answer

126 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

89 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 [closed]

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

329 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 [closed]

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

188 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

353 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

230 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

136 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

80 views

### Analytic Number Theory [closed]

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

62 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

419 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

113 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

185 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 [closed]

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

85 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

205 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

618 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

156 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

61 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

578 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 $ ...