# All Questions

**0**

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1 views

### If an abelian category $\mathcal{A}$ has enough injectives then so is $\mathrm{Ch}^{\geq 0}(\mathcal{A})$

Well my question is as clear as its title suggests. So here I would like to clarify on the fact that an object $A^\cdot$ in $\mathrm{Ch}^{\geq 0}(\mathcal{A})$ is injective if and only if
...

**0**

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**0**answers

8 views

### An example of optimal control for linear case [on hold]

Does someone have an example of optimal control for the linear case?

**0**

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**0**answers

25 views

### $\delta$-functor and commutativity of pull-back with right derivation

Let $f:X \to Y$ be a faithfully flat projective morphism of noetherial $\mathbb{C}$-schemes. Assume that $Y$ is affine, smooth over $\mathbb{C}$. Let $y \in Y$ be a closed point with residue field, ...

**0**

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62 views

### Is this one of the solutions for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? [on hold]

Let $\ a^3 + b^3 = c^3,\ a, b, c \in \mathbb Z^*,\ $we can assume that all variables are coprime.
Because $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb)= a ^ 3,\ $ so $\ (c - b)\ $ is factor of $a$, let ...

**1**

vote

**1**answer

21 views

### Hypersurfaces with rational self-maps

I'm looking for interesting examples of hypersurfaces $X\subset \mathbb P^n$ with a rational self-map $X\dashrightarrow X$?
Are there such examples for cubic hypersurfaces?

**4**

votes

**1**answer

74 views

### Is there an algorithm to write down the 27 lines of a cubic surface?

Let $S$ be a smooth cubic surface defined by $f\in \mathbb Q[x,y,z,w]$. Is there an algorithm to write down the 27 lines on $S$? Or at least find a field extension of $\mathbb Q$ over which these ...

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28 views

### Permutations on a finite vector space

The question is somewhat related to the theory of permutation polynomials. Let $\mathbb{F}_p$ be a finite field of $p$ elements ($p$ is prime) and $\mathcal{V} = \mathbb{F}_p^2 = \{ (t_1,t_2)\::\: ...

**-2**

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**0**answers

56 views

### Examples of elementary problems that have sophisticated solutions [on hold]

What are some good examples of elementary problems particularly in number theory, algebra and geometry that have insightful interpretations using post-modern (say post 1950) mathematical tools?

**-2**

votes

**0**answers

36 views

### What is a good name for connectives cum set operators?

We have the useful name connective for $\lnot$, $\wedge$, $\vee$, $\rightarrow$ and so on.
What would be a useful and philologically reasonable name for set theoretic operations as $\setminus$, ...

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**0**answers

50 views

### Are there any patterns in simple continued fraction expansions of algebraic real numbers?

As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are there any patterns in simple continued fraction expansions of other algebraic real ...

**2**

votes

**1**answer

84 views

### Whats the prime-to-$p$ Etale fundamental group of the affine line minus two points over $\mathbb{F}_p$?

Background:
I've often heard that the fundamental group of $\mathbb{P}^1/\mathbb{Q}-\{0,1,\infty\}$ is extremely hard to understand. First of all, it has a surjective map to the galois group
...

**4**

votes

**1**answer

49 views

### Whittaker models for $GL_n$ and Fourier coefficients

Let $G$ be a compact abelian group. Then we know, because of the Peter-Weyl theorem, that $L^2(G)$ decomposes as a Hilbert space direct sum of 1 dimensional representations of $G$.
Let $\mathbb{A}$ ...

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55 views

### BSD leading-term coefficient in terms of places without distinction

After reading this blog post, I learned the BSD conjectural formula for the coefficient of the leading term $a_0$ of the L-function of an elliptic curve $E$, namely
$$
a_0 \stackrel{?}{=} ...

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**0**answers

57 views

### does the words normal bundle is nonnegative make any sense? [on hold]

I have a very loose question, does the words normal bundle is nonnegative make any sense. If yes, how do we prove it usually?

**4**

votes

**1**answer

168 views

### About an identity which gives immediate proof of the permanent lemma

Let $A$ be a $n \times n$ matrix over field $F$. Let $a_1, \cdots, a_n$ be the column vectors of $A$. For any subset $S \subseteq [n] = \{1, 2, \cdots, n\}$, let $a_S = \sum_{i \in S} a_i$. Alon's ...

**5**

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**0**answers

43 views

### Ideal classes fixed by the Galois group

Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...

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39 views

### Fourier transform and diagonalization

In light of some apparent discrepancies in this MO-Q, I would like an answer to the following question:
What properties must a linear operator $\hat{O}_x$ have such that
$$\hat{O}_x^n ...

**0**

votes

**1**answer

55 views

### Pushforward of a log canonical pair

This question comes from learning the paper "Existence of minimal models for varieties of log general type", where they define the log terminal model (See Definition 3.6.7). However, the question ...

**3**

votes

**0**answers

59 views

### Is there a geometric meaning of the Major index?

The actual question I want to ask is whether there is a geometric proof of this famous identity
$$\sum_{\sigma \in S_n} q^{\operatorname{inv} \sigma}=\sum_{\sigma\in ...

**2**

votes

**1**answer

87 views

### Spectral multipliers vis-a-vis Differential geometry

Let us mention two papers for examples: this one by Seeger and Sogge and this by Cheeger, Gromov and Taylor. One can also mention papers by Stein, for example, this one. There are also many others of ...

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**0**answers

3 views

### limit of (1+x)^(1/x) when x goes to infinity [migrated]

When it comes to find the limit of
(1+x)^(1/x) when x goes to infinity,
I put 1/x = t and replaced the whole equation with
(1+1/t)^t when t goes to 0.
Hence, I wrote the answer as e, because I ...

**0**

votes

**1**answer

28 views

### Resource on characterizations or properties of traceable graphs

I am looking for some resources that provide information on traceable graphs(paths containing a hamiltonian path). I have found a lot of information on hamiltonian graphs, but none on traceable ...

**2**

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**0**answers

42 views

### Homogenous polynomially convex hull of $[0,1]^n$

I would like to calculate the set of $z\in \mathbb{C}^d$ such that there exists a constant $C >0$ such that for every homogeneous polynomial $p$ in $d$ variables $$|p(z)|\leq C\sup_{x\in [0,1]^d} ...

**2**

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**0**answers

37 views

### Continuity of the solutions of an isogeny in a formal group

Notation for the problem:
Let $E/\mathbb{Q}_P$ be a local field, and $\mu_E$ its maximal ideal. Let $K=E\{\{T\}\}$ be the standard 2-dimensional local field equipped with the Parshin topology and let ...

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**0**answers

81 views

### Tensor product of arbitrary categories [migrated]

I would like to consider a definition of tensor product in the category of (small, finite, whatever is needed) categories, analogous to the tensor product of vector spaces. I will first rewrite ...

**2**

votes

**1**answer

48 views

### laminations and branched surfaces

I am looking for a reference for this question: given a branched surface in a 3-manifold, how we can construct a lamination fully carried by that branched surface.
any comments would be appreciated.

**3**

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**0**answers

37 views

### Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$

Generally, hypergeometric function ${}_2F_1[a,b,c;z]$ is defined using Gauss series ${}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n$ with $|z|<1$, and there seems to be a lot of ...

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27 views

### convergence interval of an infinite series without the general term [on hold]

I am trying to find the convergence of an infinite series of which I do not have the nth term. Instead of applying the ratio test for the nth term, I divided the first two terms, then the next two and ...

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votes

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27 views

### Reference for “Newtonian capacity estimates probability that A is hit by a Brownian motion”

I am looking for the following statement
"In fact, the Newtonian (logarithmic) capacity gives an estimate, up to a constant factor, the probability that A is hit by a Brownian motion started, say, ...

**7**

votes

**1**answer

258 views

### Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform

Suppose that $f$ is in $L^2(\mathbb{R})$ and consider the set of integer translates of this function, $V=\{f(x-k):k\in\mathbb{Z}\}$. This set is linearly independent: taking the Fourier transform of ...

**1**

vote

**0**answers

82 views

### Default Orientation of Vectors [on hold]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...

**2**

votes

**1**answer

116 views

### Reference request for instantons

I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...

**4**

votes

**1**answer

286 views

### Connections between Standard, Hodge and Tate conjectures on algebraic cycles?

What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?

**23**

votes

**3**answers

1k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq 1$ straight ...

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**0**answers

47 views

### Multiplicativity of the Ideal Norm

I asked this question on StackExchange twice, and was never able to get an answer. This is not quite a research-level question, but it is sufficiently difficult and interesting from a pedagogical ...

**1**

vote

**1**answer

104 views

### Why is the Tate local duality pairing compatible with the Cartier duality pairing?

This question is a follow up to Why is the norm map dual to restriction under Tate local duality?
Let $A$ and $B$ be dual abelian schemes over a base scheme $S$. For an integer $n \ge 1$, consider ...

**2**

votes

**1**answer

176 views

### Is the ordinary locus affine?

Let $p$ be a prime number and let $Y$ over $\mathbb F_p$ be a Siegel modular variety, with minimal compactification $X$. It is well known that $X^{\operatorname{ord}}$, the ordinary locus of $X$ is ...

**9**

votes

**1**answer

160 views

### Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?

I asked this question already on math.stackexchange, but maybe it is also useful to ask this here, since it was not answered there.
Suppose we have three directed sequences of $C^*$-algebras, say ...

**2**

votes

**0**answers

40 views

### optimization problem, any solution?

The objective is as follows:
$\min_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - Re\{\mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}\}$
$s.t.\ \ \ Tr(\mathbf{F} \mathbf{F}^H)<p$
where $a$ ...

**0**

votes

**0**answers

32 views

### A solution of a q-difference equation

Is it possible to find a solution of the $q$-difference equation
$$f(q^{-1}x)-f(qx)=x(a-x)f(x),$$
with $f(0)=1$, (perhaps) in terms of basic hypergeometric series? Or in another rather explicit form? ...

**-1**

votes

**0**answers

63 views

### Can first and second fundamental form be considered as differential 1-forms? [on hold]

Let $(M,g)$ be an abstract surface embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote the first and second fundamental form by $I$ and $II$, respectively and the smooth vector fields on ...

**6**

votes

**1**answer

148 views

### Horn's inequalities for n matrices

Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$
i.e. Horn's inequalities for n matrices?
Can such ...

**1**

vote

**1**answer

106 views

### Conjugate surfaces: informations about the orbits

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...

**0**

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**0**answers

129 views

### Basketball shots and stopping rule

Moved over from StackExchange.
You are taken to play a basketball game where you can shoot basketballs at n slots using a machine that is equally likely to shoot the balls into those n slots. You can ...

**5**

votes

**0**answers

129 views

### Are inclusions from spaces of $C^\infty$ sections into spaces of $C^k$ sections homotopy equivalences?

[EDIT: The answer to my original question was obviously no, as user56365 pointed out. Here is what I should have asked.]
For finite-dimensional smooth manifolds $E,M$, let $E\to M$ be a smooth fibre ...

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votes

**0**answers

30 views

### discrete dynamical system [on hold]

sir,
I am doing research work in dynamical systems in the area alpha limit sets.I came across a situation where i can find some properties of alpha limit sets analogous to omega limit sets.I am ...

**3**

votes

**0**answers

195 views

### Multiplicative Identity for all elements in SU(n)

Let $\{P_i\}$ be a subset of $SU(n)$ such that for any $U$ in another subset (or perhaps subgroup) $H$ of $SU(n)$: $$P_1UP_2U\cdots P_mU=I$$ where $I$ is the identity element. Is there a sequence ...

**4**

votes

**1**answer

56 views

### Reference request: Continuity of unique maximizer of linear functional on convex set

Does anyone know reference for a theorem of the following sort:
Proposition: Let $K \subset\mathbb {R}^n$ be a compact convex set, and assume that
$$f(w):=\operatorname{argmax}_{x\in K}w(x) $$ is ...

**0**

votes

**0**answers

43 views

### Involution on $\chi^{\infty}(\mathbb{R}^{n})$

Is there a Lie algebraic involution $\theta$ on $\chi^{\infty}(\mathbb{R}^{n})$ which restriction to linear vector fields is the involution
$\theta(A)=-A^{tr}$ for $A\in M_{n}(\mathbb{R})$?
...

**6**

votes

**0**answers

76 views

### Can $Ded(\kappa)$ be a supremum?

Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$.
It ...