# All Questions

**1**

vote

**0**answers

45 views

### Physical meaning of the Lebesgue measure

Question (informal)
Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the ...

**0**

votes

**0**answers

44 views

### Functional equation: Can we find a function that satisfies this equation?

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$
Is there a function $f$ from ...

**0**

votes

**0**answers

11 views

### Proof of partial derivative of a distribution

I wanted to proof the formula for the partial derivative of a distribution, but I have an error on my result and I don't understand where I am wrong.
Here is my proof :
I assume that my function f ...

**0**

votes

**0**answers

7 views

### Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps.
In the decomposition
$$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f ...

**6**

votes

**1**answer

50 views

### Union-closed family generated by n 2-sets

I asked this question on Stackexchange, but I got no answer, so I ask it here.
Let us define a $2$-set as a set with exactly $2$ elements. For a natural number $n$, let $l(n)$ denote the least ...

**-1**

votes

**0**answers

37 views

### Connected Lie subgroups of SU(2) and SU(3) [on hold]

Two questions:
Why is every connected subgroup of SU(2) closed?
Can we find a non closed connected subgroup of SU(3)?

**0**

votes

**0**answers

10 views

### One problem about tower stability

Some years ago i asked myself a question that i still can not answer. Here it is. Let be given a tower consisting of finite homogeneous equal to each other cubic blocks staying one on another. What is ...

**1**

vote

**0**answers

28 views

### A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...

**3**

votes

**0**answers

63 views

### What are the sense and reference of the propositions $R$$\notin$$R$,$R$$\in$$R$, where $R$={x|x$\notin$x} in Frege's Grundgesetze?

In their paper, "Frege's New Science" (Notre Dame Journal of Formal Logic, Vol. 41,No. 3, 2000), Antonelli and May give the following quote of Frege, from his paper "Uber die Grundlagen der ...

**0**

votes

**0**answers

20 views

### A question about the definition of complete dg Lie algebras in a paper of Lazarev and Markl

In their paper Disconnected Rational Homotopy Theory, Lazarev and Markl give the following definition (page 23):
Definition: A complete differential graded Lie algebra is an inverse limit of ...

**2**

votes

**0**answers

32 views

### Effective divisor in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-classes

Does anybody knows an effective class in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-coefficients? The standard references; Logan, Farkas or Brill-Noether divisors have all non-negative ...

**2**

votes

**0**answers

36 views

### Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$.
Tilting module theory play an important role in the ...

**0**

votes

**0**answers

18 views

### Experimental Investigations on the Statistics of Infinite, Discrete, Evenly Distributed Pointsets in the Euclidean Plane

I am trying to estimate the distribution of certain planar polygons in the Euclidean plane; to accomplish that, I generate finite set of points, that are evenly distributed in w.l.o.g. the ...

**0**

votes

**0**answers

45 views

### Invariance of Gauss-Bonnet theorem with respect to connection?

I am stuck with a basic understanding of the generalized (and even the ordinary version of) Gauss-Bonnet theorem. For a compact 2-dimensional Riemannian manifold $M$ with boundary $\partial M$, let ...

**-2**

votes

**1**answer

33 views

### Hamiltonian path in countable connected graph such that $\text{deg}(v)=\omega$ for all $v$

Is there a countable connected graph $G=(\omega, E)$ such that $\text{deg}(v)=\omega$ for all $v\in\omega$, but there is no Hamiltonian path in $G$?

**1**

vote

**0**answers

36 views

### Game Theory Cake Cutting

I'm familiar with Game Theory concepts (I took one course at College, but it was rather superficial), but my mathematical skills aren't at the best level though. However, I'd like to hear from more ...

**1**

vote

**0**answers

33 views

### What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?

Let $X$ be a random variable with $|X|\le1$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?
Existing results:
It has been known that ...

**1**

vote

**0**answers

18 views

### On the numerical range of non-self adjoint Gaussian matrix

For a complex $n \times n$ matrix $A$, its numerical range is the set
$$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$
We can further define the ...

**5**

votes

**0**answers

115 views

### Semi-continuity of intersection numbers

I always trusted the following quite vague statement:
If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say ...

**2**

votes

**1**answer

202 views

### Field of definition of an algebraic set

I find this definition in Silverman's book, The Arithmetic of Elliptic Curves:
an algebraic set(in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in ...

**0**

votes

**0**answers

33 views

### Simulated Annealing algorithm for MINLP [on hold]

In the objective function of a mathematical programming model,we have an expression like this:
$$
\biggl(\biggl|X\biggl| \biggl) . Q
$$
in which both X and Q are continuous variables, and $||$ ...

**3**

votes

**0**answers

40 views

### Solving algebraic recurrence relations on a cyclic graph

I have a set of $n$ variables $p_1, \ldots p_n$ with $0 \leq p_i \leq 1$ and a defining equation for each of one of the forms:
$p_i = 0$.
$p_i = 1$
$p_i = p_j p_k$ for some $j, k$ with $i, j, k$ all ...

**0**

votes

**1**answer

54 views

### Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...

**2**

votes

**1**answer

105 views

### sum over all integer partitions, of the product of the factorials of the terms

I'm looking for something making tractable the sum, over all partitions into k terms of an integer n, of the product of the factorials of all the terms.
Thanks,

**24**

votes

**1**answer

686 views

### Complex manifold with subvarieties but no submanifolds

I previously asked this question on MSE and offered a bounty but received no responses.
There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. ...

**4**

votes

**0**answers

125 views

+100

### Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...

**20**

votes

**1**answer

503 views

### Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...

**5**

votes

**1**answer

120 views

### Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where
\begin{equation}
Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}.
\end{equation}
To ...

**3**

votes

**0**answers

134 views

### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free ...

**1**

vote

**2**answers

862 views

### Encoding vectors of size $n$ in matrices which less than $2n$ rows

I have a set of vectors and each has $n$ nonnegative entries.
Moreover, each entry of a vector has a quality: (1) or (2). It makes $2^n$ different possible patterns.
For example, let's take two ...

**4**

votes

**1**answer

138 views

### Extension of functions from geodesically convex compact sets in a Riemannian manifold

In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...

**5**

votes

**1**answer

170 views

### Spin structures on schemes

This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and ...

**8**

votes

**0**answers

133 views

### Two transfers for ramified or branched covers

Let $\pi: X \rightarrow Y$ be a 2-fold branched cover of complex varieties. I know of (at least) two types of pushforwards associated to this situation:
If I'm not mistaken, there is a pushforward ...

**0**

votes

**1**answer

162 views

### Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$
Of course if $f\in H^s(\mathbb{R}^3)$ ...

**48**

votes

**17**answers

11k views

### Parodies of abstruse mathematical writing

Perhaps under the influence of a recent question
on perverse sheaves,
in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53),
I recalled a long-ago parody of abstruse mathematical language
...

**0**

votes

**0**answers

41 views

### On joint equidistribution of residues

Given $N\in\Bbb N$ we vary $\alpha$ over integers in interval $(N,2N)$ and $\beta,\gamma$ over integers in $(N^{1/c},2N^{1/c})$ where $c>1$ and fix $\delta$ an integer randomly picked from ...

**0**

votes

**0**answers

5 views

### Probability of disjoint cycles

Let $c_1,c_2\in S_n$ be two disjoint cycles of length $|c_1|$ and $|c_2|$ respectively. Let $I(c_i)$ be the coordinates on which permutation $c_i$ acts at $i\in\{1,2\}$. Note by choice we have ...

**3**

votes

**2**answers

496 views

### When a smooth algebra is regular?

Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, ...

**31**

votes

**10**answers

5k views

### de Rham cohomology and flat vector bundles

I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...

**4**

votes

**0**answers

95 views

### Integer sum of distinct reciprocals with no integer subset sum

Question
$\def\nn{\mathbb{N}}$
For any $n \in \nn^+$, is there a finite set $S \subset \nn^+$ such that $\sum_{k \in S} \frac{1}{k} = n$ but $\sum_{k \in T} \frac{1}{k} \notin \nn^+$ for any $T ...

**19**

votes

**7**answers

3k views

### Asymptotic density of k-almost primes

Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is
$$\pi_k(x)\sim\frac{x(\log\log ...

**4**

votes

**1**answer

235 views

### function space in comma category

Let TOP be a category of topological spaces and B be an object of TOP. Is there a notion of function space in the comma category TOP/B.

**4**

votes

**2**answers

540 views

### Is the functor of open subschemes representable?

First some simple observations in order to motivate the question:
The functor $Set^{op} \to Set, X \to \{\text{subsets of }X\}, f \to (U \to f^{-1}(U))$ is representable. The representing object is ...

**0**

votes

**0**answers

7 views

### Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycle (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions).
The ...

**0**

votes

**0**answers

36 views

### How to solve these simultaneous equations? [on hold]

Find the all possible pairs of $x,y,z$ so that
1) $y^3 = x^3 + 9x^2 - 9x +8$
2) $y^2 = z^3 + 17$
Note that, $x,y,z$ are all positive integers.
I've tried many ways to solve this problem but ...

**-4**

votes

**0**answers

18 views

### significance level from 0.10 to 0.05, increase or decrease? [on hold]

A trend test show the trend is downward (alpha<0.10). Then the trend is still downward, but alpha<0.05. Then I describe as follows:
1. The statistical significance is increasing.
2. The ...

**3**

votes

**1**answer

73 views

### How to write the map $ℂ[G/U]↪ℂ[B]$ explicitly?

Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat ...

**4**

votes

**0**answers

32 views

### Points of failure in definition of X- and A-moduli spaces for arbitrary G

In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. ...

**3**

votes

**0**answers

165 views

### Cohomology theory “from” Grothendieck's six operations?

How, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos?
I would like some ...

**0**

votes

**0**answers

13 views

### Densely defined symmetric operators having no self-adjoint extensions

Does there exist a complex Hilbert space H and a densely defined symmetric
operator A on H such that Range(A*+i) is not equal to H?