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4
votes
2answers
154 views

Weight spaces of representations of finite dimensional simple Lie algebras

This question has probably been asked before on this website, but I could not find any solution and neither can I solve this question. So again I am asking the following question: Let $\mathfrak{g}$ ...
1
vote
1answer
41 views

Almost covering every set with few images

Is it possible to choose $k$ fixed point free maps $f_i$ from an arbitrarily large finite set $X$ to itself such that: $$\max_{A\subset X} \vert A \setminus \cup_{i=1..k} f_i(A)\vert = O(\vert X\vert^...
3
votes
0answers
19 views

Disjoint Common Transversals of Two Families of Sets

Let $E$ be a finite set. Let $d,m,n\in\mathbb N$. Let $\mathcal A:=\{A_1,\dots,A_m\}$ and $\mathcal B:=\{B_1,\dots,B_n\}$ be two families of subsets of $E$. A partial transversal of $\mathcal A$ is ...
1
vote
0answers
41 views

Algebraic connectivity of the path $P_n$

Let $G$ be a graph with $n$ vertices. Denote by $L(G)$ the Laplacian matrix of $G$ and $0=\lambda_1\leqslant\lambda_2\leqslant...\leqslant\lambda_n$ its spectrum. The number $\lambda_2$ is called the ...
5
votes
1answer
359 views

Why is an object not defined as identity morphism?

I've seen that there was a single-sorted definition of a category. In some ways, it seems more understandable than the original definition. I don't know much about category theory. But I would like ...
0
votes
0answers
19 views

Compactness lemma: approximate sequence in the space X and the limit not in the same space

Often, when we try to solve some PDE problem, we construct first a sequence of approximate solutions. To construct an exact solution we need to show that a sequence (or at least some subsequence) of ...
2
votes
0answers
32 views

Example of overtwisted contact manifold with finitely many periodic Reeb orbits

Are there examples of overtwisted manifolds with only a finite number of periodic Reeb orbits? An example is given by the irrational ellipsoid in $(\mathbb{R}^4,\omega_\text{st})$, which is not ...
1
vote
0answers
80 views

The Chicken Portioning Problem [migrated]

I thought of this question while portioning chicken in the sandwich shop where I used to work. It may reduce to the bin-packing problem, but I am not certain. It is as follows: You have been given a ...
0
votes
1answer
174 views

What is the symmetry group of this configuration?

This configuration appear as problem 3845 in Crux Mathematicorum. I see it is very beautiful. This configuration are generalization of Pascal theorem and Brianchon theorem: Consider six points $A_1$, ...
2
votes
0answers
30 views

Iterated polyhedron face twisting

Let $Q$ be a polygon in the plane. Modify $Q$ by rotating each edge about its midpoint by $180^\circ$. The result is $Q$ again: No change. This suggests exploring a similar operation in $\mathbb{R}^3$...
-3
votes
0answers
24 views

Calculate probability of multiple events occurring or not occurring using existing data MLB Player [on hold]

I have the data, and the answer, but can't figure out the math driving it. Baseball Hitter has a x% chance of meeting or exceeding 1.5 hits,rbis,and runs in a game (think player proposition bets like ...
1
vote
0answers
69 views

Moment map interpretation of Einstein equation

Einstein's famous equation relates the geometry of a (4-dimensional) manifold to the matter content in that manifold. Is there a way to obtain Einstein's equation as a moment map? More precisely, ...
-4
votes
2answers
246 views

Coordinate free proof of Gauss-Bonnet theorem

Can the theorem be proved invariantly, without any reference to charts basis vectors or coordinates?
3
votes
1answer
113 views

Conceptual explanation for the appearance of entropy in $\frac{d}{dp}\|x\|_p$

For $x\in \mathbb{R}^d$, an elementary computation yields that $$\frac{d}{dp}\log \|x\|_p =\frac{1}{p^2}\sum_{i=1}^d \frac{|x_i|^p}{\|x\|_p^p}\log \frac{|x_i|^p}{\|x\|_p^p}=\frac{1}{p^2}\operatorname{...
5
votes
2answers
78 views

Stationary sets and $\kappa$-complete normal ultrafilters

Let $\kappa$ be a measurable cardinal, and let $u$ be a normal $\kappa$-complete ultrafilter over $\kappa$. It is a standard easy fact that every closed unbounded set must belong to $u$ (notice that ...
3
votes
1answer
101 views

Blowing up vector bundles in the zero section

Assume we are given a scheme $X$ (feel free to add all the needed hypotheses, at this point I’m working with smooth schemes, but the fewer is needed, the better) and a vector bundle $E$ over $X$. I ...
5
votes
0answers
117 views

Is there a classification of finite simple groups of perfect power order?

The finite simple group $\operatorname{PSp}(4,7)$ has order $138297600 = 11760^2$. There also seems to be a description of the $q$ such that $\operatorname{PSp}(4,q)$ has square order, see for ...
2
votes
0answers
53 views

What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?

For integers $n\geq 1$ I denote the Euler's totient function as $\varphi(n)$ and the divisor function $\sum_{1\leq d\mid n}d$ as $\sigma(n)$, that are two well-known mulitplicative functions. We ...
2
votes
1answer
43 views

Regular triangulations of star-convex polyhedra with given boundary

Given an $n$-dimensional star-convex polyhedron $P\subset \mathbb{R}^n$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the ...
7
votes
1answer
101 views

Injectivity of a class of integral operators

Given a probability measure $\mu$ on the interval $[0,1]$, the linear operator $$ T_\mu \! f(y) := \int_0^1 f(yx) \, d\mu(x) $$ takes the space of continuous functions $f: [0, \infty) \rightarrow \...
12
votes
1answer
131 views

Proofs of Young's inequality for convolution

For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|_r\leq \|f\|_p\|g\|_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. ...
4
votes
1answer
166 views

Replacing the Fibre of a Fibration

This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature. Let $p:E\rightarrow ...
3
votes
1answer
39 views

Characterization of Besov space with Lp-modulus of continuity

When reading the characterization of Besov space with $L_p$-modulus of continuity in the 7th chapter “Fractional Order Space” of Sobolev space written by Adams(Page 243), I encounter some small ...
3
votes
1answer
35 views

Existence of topologically mixing (discrete) dynamical system on manifold

If $M$ is a connected $(d\geq 2)$-dimensional smooth closed manifold, then does there exist a class $C^1$-diffeomorphism $\phi$ from $M$ onto itself, such that $(M,\phi)$ is a topologically mixing (...
3
votes
0answers
98 views

Cohomology and base change theorem for non-noetherian schemes

Let $Y$ be a locally noetherian scheme, $f : X \to Y$ proper morphism, $\mathscr{F}$ a coherent module on $X$ which is flat over $Y$. Then we have many theorems about the cohomology of $\mathscr{F}$ ...
1
vote
0answers
69 views

A generalization of $p$-groups

I was wondering if there is a reference studying groups with order $m^k$ where $m,k$ are integers and $m$ is not supposed to be a prime, as a generalization of $p$-groups?
6
votes
3answers
423 views

Real orthogonal and sign [on hold]

I came across the following conjecture, reading a recent paper in the Monthly, an orthogonal matrix of order $n\neq 0 \pmod 4$ has a nonnegative (up to a scalar) row vector. It should be straight in ...
8
votes
1answer
124 views

Petries exotic circle action

In the paper "S^1-actions on homotopy complex projective spaces" by Petrie (Bulletin of the AMS, 1972), Petrie constructs a smooth circle action on $\mathbb{CP}^{3}$ (page 148). The fixed point set ...
-4
votes
0answers
62 views

How to sort differential equation list? [on hold]

Which sorting related with famous sequence for example  sorting differential equation in a list then access the list with famous sequence as index such as using https://oeis.org/ after access with ...
-4
votes
0answers
109 views

Question about the Invariance of Domain Theorem [on hold]

Dear fellow mathematicians, As you know, the Invariance of Domain Theorem states the followiing: "Let f be an injective continuous mapping from Euclidean space R^n to Euclidean space R^n. Let U be ...
-4
votes
0answers
49 views

Topological properties of a subset of $\mathbb R^n$ [on hold]

Consider $\mathbb R^n \quad (n>1)$. A point of $\mathbb R^n$ is a a $n$-uple written $(x_1,\ldots,x_n)$. Consider a set of indices $\{i_1,i_2,\ldots,i_{2p}\}$ of even cardinal $2p$ $(2p\leq n)$ ...
4
votes
2answers
964 views

Prove that this expression is greater than 1/2

Let $0<x < y < 1$ be given. Prove $$4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Big[ \sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}[\pi(y-x)] \Big] \geq \frac{1}{2}$$ I have been working on this ...
-1
votes
0answers
78 views

Probability distributions on algebraic varieties [on hold]

Is there a notion of (algebraic) probability distribution on algebraic varieties? If there is, where can I find it? If not, why not?
1
vote
0answers
78 views

Is there a weak homotopy equivalence between Sp(2n,ℂ)/U(n) and SU(n)?

This question, Is there a weak homotopy equivalence between Sp(2n,ℂ)/U(n) and SU(n)?, is at the end of a long string of my comments in https://math.stackexchange.com/questions/3296373/is-sp2n-mathbbc-...
1
vote
0answers
134 views

What arithmetic information is determined by the $j$-invariant of an elliptic curve?

It is known that the complex isomorphism class of an elliptic curve $E/\mathbb{C}$ is uniquely determined by its $j$-invariant. One way to define it algebraically for a curve $$\displaystyle E : y^2 =...
25
votes
0answers
341 views

A curious relation between angles and lengths of edges of a tetrahedron

Consider a Euclidean tetrahedron with lengths of edges $$ l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34} $$ and dihedral angles $$ \alpha_{12}, \alpha_{13}, \alpha_{14}, \alpha_{23}, \alpha_{24}, \...
1
vote
0answers
24 views

Equivalence between Gibbs measures and conformal measures

I was reading an article about Gibbs measures, but the author defines Gibbs measures in a different way than the usual (which is done by using conditional expectations). The way that he defines I have ...
1
vote
0answers
15 views

Non existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$

Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question: Is there a complete vector field $X$ on $\mathbb{R}^2\setminus\{p,q\}...
4
votes
2answers
199 views

Existence of algebraic integers with certain properties

Is the following statement true? ($\star$) Given integers $n > k > 0$, there exists a monic polynomial of degree $n$ with integer coefficients and constant term $\pm 1$, irreducible over $\...
6
votes
1answer
139 views

Zariski closure of set of units in a number ring

Let $\mathcal{O}$ be a number ring. Letting $r$ and $2s$ be the number of real and complex embeddings of $\mathcal{O}$, the number ring $\mathcal{O}$ is a lattice in $\mathcal{O} \otimes \mathbb{R} \...
0
votes
0answers
99 views

Weak k-Tuple conjecture form and what we should prove

Let $d \in \mathbb{N}, d \geq 2$ and consider the tuple $G(d,c) = (c_1,c_2,\cdots,c_d)$, with $c_1<c_2<\cdots<c_d$ are evens positive integers. I am trying to found an asymptotic formula for ...
0
votes
0answers
25 views

Distance between colored rooted graphs

My question is what is the rough meaning of $\alpha_{1,2}$? I think of it in this way: Consider two balls one is around $o_1$ $B_{G_1}(o_1,[r])$ in the first graph and the other $B_{G_2}(o_2,[r])$ is ...
-1
votes
0answers
33 views

Find an equation for a random set of points [on hold]

Let's say there is a random set of points, say 100, scattered over a 2d Cartesian plane, where no x-coordinate have more than one y-coordinate. Is there a way to find an(only 1) equation that will ...
0
votes
1answer
41 views

Non negative solutions of 3 linear equations and n unknowns [on hold]

This might seem a bit elementary but it applies to a real world problem in nutrition that I am trying to solve. I simply need a numerical method for solving a system of three linear equations with an ...
5
votes
3answers
251 views

Definition of $E_n$-modules for an $E_n$-algebra

The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more ...
5
votes
2answers
101 views

Finite-dimensional Hilbert $C^*$-modules

Does there exist a classification, or characterization, of finite-dimensional Hilbert $C^*$-modules? More generally, does there exist a characterization of countable direct sums of finite-dimensional ...
1
vote
0answers
52 views

Numerical and rational equivalences on intersection of divisors

Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...
1
vote
0answers
11 views

Alcove address characterization of weak order reference request

Let $\Phi$ be the root system of type $A$. Let $\mathcal{A}$ be an alcove of the corresponding affine arrangement. The address (or Shi coordinates) of $\mathcal{A}$ is a function $k:\Phi^+ \rightarrow ...
-3
votes
1answer
87 views

Is there a function that calculates an average with a higher weight on the 'newer' values? [on hold]

I have a set of values (blood pressure values) taken over a period of time. I want to calculate an 'average' value from that sequence that put more weight on the later ('newer' or more recent) values ...
2
votes
0answers
53 views

The rank of a special matrix

Suppose that $P$ is a polynomial of degree $d:=\deg P$ over a field $\mathbb F$ of zero characteristic, splitting completely into pairwise distinct linear factors, and $B,C\subset\mathbb F$ are sets ...

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