# All Questions

**0**

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6 views

### Cut and Fold Polyhedron!

I have two convex polyhedrons with equal side areas. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other polyhedron?
For special case ...

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4 views

### Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...

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7 views

### Double Markovity

Suppose we have a double Markov relation for three random variables $X$, $Y$ and $W$ as follows
$$X\to W\to Y,$$ and $$X\to Y\to W.$$
How to prove that there exist functions $f$ and $g$ such that
...

**1**

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17 views

### Connection between framed cobordisms and zero sets

Let $W\subseteq M\times[0,1]$ be a framed submanifold (a framed cobordism in $M$) and $2w<m-1$ where $w,m=\dim W,M$. Assume that $M$ is compact and that $W\cap M\times \{0\}$ is the zero set of a ...

**-1**

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14 views

### Problem with equation structure and re-arrangement

I am creating a program which used an equation for photo-efficiency in certain types of plants. Because of the limits of the programming (or maybe my abilities with said language). I need to break ...

**-1**

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**1**answer

45 views

### Problem of book Kunen

In $M$: $\left | P \right | \leq \omega_{1}$ and $P$ es c.c.c and $\Diamond$ is hold M. Shows that $\Diamond$ is hold $M[G]$. give one suggestion please

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106 views

### Questions about the Collatz conjecture-also known as the “3x+1 problem” [on hold]

Let "F(k,m)" denote the following recursive function of two positive integer variables. For all k, F(k,1)=k. For all k and all m, if F(k,m) is even, then F(k,m+1)=F(k,m)/2. For all k and all m, if ...

**1**

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**1**answer

27 views

### Contour integral around semi-circle

Can one use contour integration to evaluate $\int^{\pi}_{0} \frac{1}{1-\rho*sin(\theta)}d\theta$ for $0<\rho<1$? This would be trivial if the upper limit were $2\pi$ as we could let ...

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22 views

### If the $L$-series does not vanish

I refer to this paper http://wstein.org/papers/shark/shark.pdf
At the top of page 24, we are dealing with the issue where the $L$-series does not vanish for the case where $p$ is good and ordinary. ...

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21 views

### Hypercovers of sheaves in classical and quasi-categories

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...

**1**

vote

**1**answer

49 views

### Positive definite quadratic forms on Banach spaces

This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it. If $E(x,x)\geq a\|x\|^2$ for some $a>0$ ...

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26 views

### Probability that d-Brownian Motion ,d>3, avoids a set A

In other words, the probability that Brownian motion stays within $A^{c}$. So far I found that it is 1, for random cylinders and thorns.
What about for connected and compact sets (traps)? Any other ...

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66 views

### Could anyone help me with a problem regarding fundamental groups? [on hold]

Let G be a group and x be an element of G. N is the least normal subgroup of G containing x. If there is a normal, path-connected space whose fundamental group is isomorphic to G, then I have to show ...

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38 views

### Counting sets of tuples [on hold]

I am looking to count the size of certain sets created by taking the product of multivariate functions $F,Q,\ldots$, where the input arguments come from finite sets $D_1, D_2$.
For example,
...

**2**

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**0**answers

26 views

### Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...

**2**

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**0**answers

83 views

### Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where ...

**4**

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**0**answers

74 views

### Mukai flops and derived categories

As proved by Kawamata and by Namikawa, if $M$ and $M'$ are complex projective varieties which are related by a Mukai flop (elementary transformation) along projective spaces $W\subset M$ and ...

**7**

votes

**2**answers

249 views

### Maximal ideals are prime (history thereof)

What is the origin of the familiar way of finding prime ideals as maximal ones using Zorn's Lemma? How were prime ideals found before the enactment of the Axiom of Choice? How do constructive ...

**2**

votes

**1**answer

105 views

### Vector Laplace Beltrami operator of the Gauss map

Consider an abstract surface $(M,g)$ embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote by $N:M \to \mathbb{R}^3$ the Gauss map (normal field) of the surface. Write the Laplace Beltrami ...

**1**

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**1**answer

61 views

### Bochner's theorem for measures of positive type

Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying ...

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44 views

### Rank 2^n quadratic form with non trivial invariant e_n

I advise you to have a look this question of mine first
Rank four quadratic Form with non trivial discriminant in I(k)
From quadratic form theory its well known that for a field $k$ and the ...

**1**

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**0**answers

61 views

### Examples of cubic Julia sets

I'm looking for information on explicit cubic filled Julia sets with interesting properties such as:
Having two different finite attractors (such as $f(z)=z^3-1.5z$)
Being disconnected with ...

**3**

votes

**3**answers

56 views

### graded generalization of the Moyal–Weyl product

Has anyone written about the graded generalization of the Moyal–Weyl product/star product, that is, where the original algebra is already graded? Is it just a matter of signs?

**2**

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**1**answer

73 views

### Clifford algebras for quadratic modules over ringed spaces

What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ...

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15 views

### Bivariate skew-normal distribution

I'm using the Gaussian distribution as a weight function for solving pde's.
I'm interested in skewing the function. For one-dimensional problems, it was easy to
derive the resulting skewed Gaussian ...

**4**

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**0**answers

41 views

### First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...

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**0**answers

22 views

### Calculating of the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$? [on hold]

How can i calculate the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$?

**-1**

votes

**0**answers

62 views

### Can a linear map on a finite-dimensional subspace be extended to the whole space “trivially”? [on hold]

I have a question concerning the extension of continuous linear maps.
Let $X$ be a normed vector space and let $U$ be a finite-dimensional subspace of $X$. Furthermore, let $\varphi:U\rightarrow Y$ ...

**7**

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**1**answer

157 views

### Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...

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29 views

### asymptotic notation with graph colouring

This is my first ever post so I hope this is an appropriate question.
Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf
Namely theorem 5.
Now, feel ...

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vote

**2**answers

118 views

### Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there ...

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108 views

### On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces,
$$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...

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43 views

### An integral identity [on hold]

For all $n>1$ and $0<i<n$ we have the following identity?
$$\int\limits_0^{\pi} \frac{\cos(nx)-\cos(i\pi)}{\cos x-\cos(i\pi/n)}dx=0.$$

**4**

votes

**2**answers

115 views

### String diagrams for bimodules over noncommutative algebras?

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...

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13 views

### Gelfand Kirillov dimension of induced modules

How does the Gelfand-Kirillov dimension of induced modules behave?
In patricular, if $S$ is an Ore extension of $R$, how is the GK-dimension of an $R$-module say $N$ related to the GK dimension of $N ...

**3**

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**0**answers

89 views

### Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...

**2**

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**2**answers

145 views

### Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:
"For spectra every cofibration is equivalent to a fibration" (e.g. in the ...

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**0**answers

38 views

### A Density Problem

Let $ \mathscr{D}=\mathscr{D}(\mathbb{R}^n - {0}) $ be the space of all smooth functions with compact support in $ \mathbb{R}^n - {0} $ topologized by the standard Schwartz topology and let $ ...

**-2**

votes

**1**answer

28 views

### Real Analysis : Uniform Convergence of sequences on the real line [on hold]

I want a sequence of continuous functions whose limit function is continuous but the convergence is not uniform. I have an example :
fn(x)=x/n ; The limit function f(x)=0 is continuous but the ...

**5**

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**1**answer

162 views

### Cases where the number field case and the function field (with positive characteristic) are different

In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the ...

**0**

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**1**answer

153 views

### Yang-Mills equations are not elliptic [on hold]

How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic?
Can someone please present a proof of this, or point out where this has been shown in the ...

**3**

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**0**answers

35 views

### Subplanes of Finite Projective Planes

If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a
finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub ...

**3**

votes

**1**answer

117 views

### Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of
$$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$
where $Fr_p$ is a ...

**0**

votes

**1**answer

18 views

### Constructing a transition matrix of a time-homogeneous, finite Markov chain with full support stationary distribution

is there a way to construct a transition matrix of a time-homogeneous, finite Markov chain such that the stationary distribution always has full support (this is equivalent to all states of the chain ...

**1**

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43 views

### Borel class of a set of measures

Let $K$ be a compact Hausdorff space and consider $X=Ball(M(K))$, the unit ball of the space of regular Borel measures on $K$. Endow $X$ with the weak-$*$ topology $\sigma(M(K),C(K))$, regarding ...

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**0**answers

113 views

### Why do I study a lot but still don't understand the material too well? [on hold]

I had a linear algebra test yesterday and I studied a week in advance for it. I did all the assigned homework questions, past exam questions, problem set questions, but I still did poorly on the test. ...

**10**

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**0**answers

144 views

### Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...

**9**

votes

**3**answers

686 views

### When did coordinate plane “as we know it” come into play?

This is a historical question that needs some background to make sense. Let me start with the longer version of the question:
When did negative numbers, algebra and coordinate plane come together?
...

**2**

votes

**1**answer

104 views

### The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$.
One important example of left distributive algebras arises ...

**7**

votes

**1**answer

204 views

### Why polarization of abelian varieties?

Maybe this question is not suitable for here, but I don't think I would receive a satisfactory answer in Math StackExchange.
I could never understand the intuition behind polarization of abelian ...