# All Questions

**65**

votes

**25**answers

24k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**0**

votes

**0**answers

12 views

### canonical metric of toric Kahler manifolds

Let $X$ be non-compact toric Kahler manifold associated to a Delzant polygon $P$
and $g$ be the canonical Kahler metric constructed by Guillemin. Is it true that
the real part of $g$, as a ...

**1**

vote

**0**answers

7 views

### numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if
\begin{align}
\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...

**4**

votes

**2**answers

1k views

### Reed-Muller-Codes

Let $F$ be the field with two elements, $V_m=F^{2^m}$.Let $R(r, m)\subset V_m$ be the
binary Reed-Muller Code. Define $R_m:=R(1, m)$. Then the dimension of $R_m$ is
$1+m$ and its minimal distance is ...

**-1**

votes

**0**answers

62 views

### Disprove this Piece of Jensen's Inquality “Black Magic”

Jensen's inequality states that if a real valued function $f(x)$ is concave, like $f(x)=\ln |x|,$ then $E(f(X))\le f(E(X)).$ A classic application of this is $E(X) \le \ln |E(e^{X})|.$
Now consider ...

**1**

vote

**0**answers

15 views

### LQR solution when there are linear terms in the cost function?

I am trying to solve the following Bellman Equation:
$V(s) =\max_u \left[a'u - (u-s)'Q(u-s) + V(u)\right]$
In the equation above, $s,u,a\in \mathbb R^n$, $Q\in \mathbb R^{n\times n}$ is positive ...

**4**

votes

**0**answers

42 views

+100

### Reduction of self-intersections without reducing the geometric intersection

Let $F$ be a hyperbolic surface. Given a closed curve $a$, let $\bar{a}$ denotes the free homotopy class of $a$. Let $i(\bar{a},\bar{b})$ denotes the geometric intersection number and $i(\bar{a})$ ...

**2**

votes

**0**answers

17 views

### Categories in which an epimorphism applied to a non-monic epimorphism can be monic

Let $\mathcal{C}$ be a category, and let $A$, $B$, and $C$ be objects.
Given $A \xrightarrow{f} B \xrightarrow{g} C$ such that:
$f$ is both epic and monic
$g$ is epic but not monic
$gf$ is epic and ...

**10**

votes

**3**answers

297 views

### “The Two Sheriffs” puzzle -2: threshold for security

I've already asked a question “The Two Sheriffs” puzzle with wrong assumption. Yoav Kallus in his amazing answer using Fano plane showed that the problem has a solution in the case of seven suspects.
...

**-3**

votes

**0**answers

29 views

### Pascals triangle [on hold]

I was out sick for a fair bit and I come back and we are doing this! Can someone explain what I'm supposed to do or show me a video??
"Use Pascal's triangle and the regularity of decreasing powers of ...

**0**

votes

**0**answers

15 views

### Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$

Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:
$K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...

**5**

votes

**0**answers

222 views

### Kernel of Bianchi operator: Is a (smooth tame) Frechet manifold?

Let $M$ be a smooth compact manifold, $\mathcal{S}=\Gamma(\odot^2T^*M)$ the (smooth tame) Frechet space of symmetric $2$-covariant tensors, and $\mathcal{M}=\Gamma(\odot^2_+T^*M)$ the (smooth tame) ...

**3**

votes

**1**answer

60 views

### Real and imaginary part of an holomorphic function

I guess this could be a very elementary question. Anyway I can not find an answer in literature.
Let $f:U\rightarrow\mathbb{C}$ be an holomorphic function on an upen subset $U\subseteq\mathbb{C}$. ...

**4**

votes

**2**answers

741 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**0**

votes

**0**answers

22 views

### Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known a small number (something like $d+1$) points always ...

**1**

vote

**1**answer

118 views

### Deterministic finite-state automaton driven by a Markov chain

I've stumbled on some problem, and I have the feeling that this is closed to something well-studied in dynamical systems. The problem is the following. Consider a finite-state automaton with state ...

**17**

votes

**4**answers

3k views

### Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...

**4**

votes

**2**answers

145 views

### Obtain 4-manifolds by repeating surgeries of submanifolds in $S^4$

In his paper QFT and Jones Polynomials, Witten states: "It is a not too deep result that every 3-manifold can be obtained from or reduced to $S^3$ (or any other desired 3-manifold) by repeated ...

**5**

votes

**3**answers

145 views

### Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$

For $x_i \in \mathbb{Z}$, let $\{x_i\}$ be a fundamental solution to the equations:
$$
\sum_{i= 1}^N x_i = \sum_{i=1}^N x_i^3 = 0
$$
if $x \in \{x_i\} \Rightarrow -x \notin \{x_i\}$.
For instance, a ...

**1**

vote

**0**answers

32 views

### McDuff's classification of symplectic manifolds

According to a theorem of McDuff, if $(X,\omega)$ is a closed symplectic 4-manifold, which contains a symplectic sphere of self-intersection +1, then $X$ is symplectomorphic to $CP^2$ blown up a few ...

**4**

votes

**1**answer

169 views

### Spectrum of the Laplacian on p-forms on the sphere

In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...

**0**

votes

**1**answer

120 views

### Existence of a Lie algebra element orthogonal to the adjoint orbit of another element

Consider a compact, semi-simple, connected Lie group $G$ and its Lie algebra $\mathfrak{g}$. Denote the Killing form by $K$.
Given a single $A \in \mathfrak{g}$ when (i.e. which groups and which $A$) ...

**26**

votes

**7**answers

8k views

### Collecting proofs that finite multiplicative subgroups of fields are cyclic.

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...

**2**

votes

**1**answer

72 views

### Charts needed for an atlas

I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this ...

**0**

votes

**1**answer

50 views

### Example s.t. the unbased loop-space is not $\Omega X \times X$

For a connected pointed CW-complex $X$, let us write (as usual) $\Omega X$ for the space of based loops at $X$. I am looking for an example where the space $\Omega' X$ of all (unbased) loops in $X$ is ...

**0**

votes

**2**answers

142 views

### divisor is big iff its birational pullback is big

In Kollár and Mori's Birational Geometry of Algebraic Varieties, the authors say a Cartier divisor is big iff its birational pullback is big (Definition 2.59 below). But I can't understand. Maybe the ...

**5**

votes

**3**answers

380 views

### is there a global obstruction for a diffeomorphism to be an isometry?

Let $V$ be a finite dimensional vector space.
Let us call an automorphism $T:V\rightarrow V$ admissible if there exists an inner product $\langle , \rangle$ on $V$ making $T$ an isometry.
We know ...

**2**

votes

**0**answers

21 views

### How to show the compatibility between Duflo isomorphism and Harish-Chandra isomorphism for semi-simples Lie algebras?

I was told that the Duflo isomorphism is compatible with the Harish-Chandra isomorphism when the Lie algebra $\mathfrak{g}$ is semi-simple. However I cannot see why this is true. All I can show is ...

**1**

vote

**0**answers

14 views

### When is a formula preserved under taking factors in a reduced product or the stalk in a Boolean product?

I want to know if there is a nice characterization of when a formula is preserved under taking reduced factors.
We say that a formula $\phi$ is closed under taking reduced factors if whenever $I$ is ...

**3**

votes

**0**answers

84 views

### Must an algebraic variety with trivial tangent bundle be an abelian variety?

Suppose $X$ is an algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not true due ...

**12**

votes

**4**answers

4k views

### Non-mathematician submitting to top maths journal? [closed]

I am an amateur mathematician, and I had an idea which I worked out a bit and sent to an expert. He urged me to write it up for publication. So I did, and put it on arXiv. There were a couple of ...

**18**

votes

**3**answers

837 views

### Which way for reading the proofs?

I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this ...

**5**

votes

**2**answers

518 views

### Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...

**-2**

votes

**4**answers

300 views

### Studying topology: which first, algebraic or differential? [on hold]

I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...

**6**

votes

**3**answers

984 views

### Does any tensor category correspond to a bialgebra?

I wonder how strong the power of Tannaka philosophy is, and if we accept that a tensor category is a generalized bialgebra, what difficulties we will come up against ?
Edit: Whether most tensor ...

**2**

votes

**1**answer

144 views

### Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...

**3**

votes

**3**answers

351 views

### The free group of a group and the kernel of a canonical morphism

Let $G$ be a group and $F_G$ the free group on the set $G$. Then there exists a canonical surjective morphism ${\rm can}: F_G \to G \to 1$ constructed as follows: let $(e_x)_{x \in G}$ be a copy as a ...

**2**

votes

**1**answer

132 views

### 'Stalk' of vanishing cycles at $k$-point

I have a simple question on notation.
Let $S$ be a Henselian trait with closed point $s$ (with finite residue field $k$) and generic point $\eta$. Let $X/S$ be a variety. Then, we have the functor
...

**3**

votes

**0**answers

111 views

### Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being
$x^2=4+8y^2+13z^2$.
The ideal answer would be a way to parametrize all the integer ...

**-2**

votes

**0**answers

49 views

### Pure algebraic analogy of topological “MEAN”

Please first read the linked paper, then read my question
A $k$- mean on a topological space X is a continuous map $F:X^{k}\to X$ which is invariant under permutations of $k$ elements and its ...

**2**

votes

**0**answers

201 views

### dimension of induced comodule

Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...

**0**

votes

**1**answer

63 views

### Counting the orderings of outward-directed trees where the degree of each vertex is $2$

Let $T$ be a connected directed tree with the following properties:
The degree of each vertex of $T$ is at most $2$ (I am sure there is a name for such a graph but I do not know it).
$T$ has a ...

**0**

votes

**0**answers

11 views

### Equivalence of Graphical model selection algorithms

Suppose, a jointly Gaussian random vector is denoted by $X \in \mathbb{R}^{p}$ and $X$ has a distribution given by $\mathcal{N}(\mu,\Sigma)$. It is known that estimating the graphical model that ...

**0**

votes

**0**answers

66 views

### Volume of a complex manifold

Is there a theorem which states the following?
Let $\mathcal{M}$ be a $k$ -dimensional complex submanifold of $\mathbb{C}^n, \ 1 \le k \le n$. Let $V \subset M$ be open, relatively compact in $M$, so ...

**19**

votes

**0**answers

547 views

### Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?
Related: if such a ...

**1**

vote

**0**answers

62 views

### Why write GRR with the relative tangent sheaf?

The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
...

**1**

vote

**1**answer

85 views

### Given a multivariate polynomial with even degree, can we find its tightest convex polynomial 'envelop'?

To be specific, given a multivariate polynomial function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ with even degree $2d$, can we construct a convex polynomial function $g$, such that:
$\forall ...

**7**

votes

**2**answers

307 views

### Markov processes lacking the Feller property

Let $E$ be a LCH second countable topological space and let $\mathcal{E}$ be its Borel $\sigma$-algebra.
Let $(P_t)_{t \geq 0}$ be a conservative transition function on $(E, \mathcal{E})$.
This ...

**4**

votes

**1**answer

44 views

### Negatively curved metrics minimizing the length of a homotopy class of simple closed curves

Good afternoon everyone !
I have the following question of Riemannian geometry :
Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth ...

**1**

vote

**0**answers

219 views

### Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...