**-1**

votes

**0**answers

14 views

### Algebra working with quadratic and linear equations [on hold]

Hi guys I'm writing my SATthis Saturday, while practicing I came accross this:
Question snapshot
I tried thinking as far as I can but couldn't figure out how to work it out.
I would truly ...

**0**

votes

**1**answer

57 views

### Quotient of a vector space by a linear finite group action

Let the cyclic group $\mathbb{Z}_n$ act on $\mathbb{C}^n$ (or on $\mathbb{R}^n$, I'm interested in both) by permuting coordinates. What does the quotient $Q$ look like? Is there some nice way to ...

**4**

votes

**1**answer

267 views

### Is Lehmer's polynomial solvable?

The degree 10 polynomial
$$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$
given by D.H. Lehmer in 1933 has the property that its largest real root, $\beta = 1.176280 \cdots$ is ...

**2**

votes

**1**answer

14 views

### Differentiability of value function

Suppose $X$ is a process given by -
$dX_t = db_t$
where $b_t$ is a standard Brownian motion with its filtration $(\mathcal{F}_t)$.
Suppose an agent earns a payoff given by
$V(x) = \mathbb{E} ...

**2**

votes

**1**answer

55 views

### Chain divisibility constraints in Frobenius coin problem

Lets say a linear form $ax+by$ represents $n$ if $ax+by=n$ for some $x,y>0$.
Call a pair $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a reasonable coprime pair if linear form $ax+by$ represents ...

**0**

votes

**0**answers

15 views

### Jarník-Besicovitch and outer measure

The set $A_\tau$ of irrational numbers $x$ which are $\tau$-approximable, i.e., that satisfy the estimate
$$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$
for infinitely many rationals $p/q$, ...

**0**

votes

**0**answers

23 views

### Sumsets with distinct numbers, upper bound for maximum element

Let $A$ be a finite set of positive natural numbers with $n$ elements, $|A|=n$, with the property that all sums of two (not necessarily different) elements are distinct, or in the usual notation for ...

**0**

votes

**0**answers

45 views

### Irreducible projective representations of $\mathbb{Z}_2\times\mathbb{Z}_2$

I am currently learning about projective representations of finite groups and their reducibility. According to Schur's theorem (Schur 1904) the degree (dimension) of each irreducible projective ...

**5**

votes

**2**answers

65 views

### Characterizing when matrices are 'dissipative'

An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...

**0**

votes

**0**answers

43 views

### Getting started with quadratic forms [on hold]

I recently studied the Pell-Brahmagupta equation as an addition to my undergraduate project on Number Rings. This exposed me to the idea that binary quadratic forms are determined by their prime ...

**0**

votes

**0**answers

29 views

### Hodge metric on pair (X,D)

I am searching for the definition of log Hodge metric, see definition 2.2 here
But instead of Vector bundle, if we have divisor $D$ with conic singularities then how can we introduce Hodge metric?

**5**

votes

**4**answers

312 views

### What does “higher monodromy” tell us about a principal bundle

Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to ...

**1**

vote

**1**answer

124 views

### cyclotomic polynomials with 7$^{th}$ coefficient greater than 1 in absolute value

It's known that the seventh coefficient of $\Phi_{105}(x)$ is $-2$ and that's the first occurrence of a coefficient with absolute value greater than $1$ for a cyclotomic polynomial. When I did a quick ...

**1**

vote

**0**answers

20 views

### Differential of the adjoint quotient map

My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...

**3**

votes

**1**answer

113 views

### Primitive sequence $a_i$ attaining Pillai's bound on $\sum_{i} 1/a_i$

A primitive sequence $1<a_1<\ldots<a_k\leq n$ is a sequence of integers no one of which divides any other, investigated by Erdos, Behrend and others, over the last 80 years. In fact, $\max ...

**2**

votes

**1**answer

165 views

### Problem related to Frobenius coin problem

Lets say a linear form $ax+by$ represents $n$ if $ax+by=n$ for some $x,y≥0$.
Call a pair $a,b\in\Bbb N$ with $\mathsf{gcd}(a,b)=1$ a good coprime pair if
for some $r,s,t,u>1$ with each of ...

**0**

votes

**0**answers

39 views

### Can we get infinitely often better approximations for algebraic numbers from continued fractions with algebraic partial coefficients?

I am wondering how good approximations to algebraic
numbers one can get via convergents of continued fractions
with algebraic partial coefficients coefficients.
Let $\alpha$ be algebraic integer ...

**0**

votes

**1**answer

80 views

### Regarding a new divergence function of two probability distributions

Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and any constant $0 \leq \alpha \leq 1$, ...

**11**

votes

**1**answer

184 views

### Convergence rate of Fagin's 0-1 law for first-order properties of random graphs

Fagin's 0-1 law for first-order properties of random graphs states that, for every first-order sentence in the logic of graphs, the probability that a uniformly random $n$-vertex graph models the ...

**21**

votes

**5**answers

1k views

### History of Mathematical Notation

I would like to see a simple example which shows how mathematical notation were evolve in time and space.
Say, consider the formula
$$(x+2)^2=x^2+4{\cdot}x+4.$$
If I understand correctly, Franciscus ...

**4**

votes

**1**answer

96 views

### concentration inequality for $d$-dimensional martingale

Are any concentration inequality available for $d$-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension $d$ in ...

**5**

votes

**0**answers

143 views

### Many-sorted nominal sets as sheaves

The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...

**14**

votes

**1**answer

458 views

### Is the moduli space of graphs simply connected?

The moduli space of graphs $MG_n$ is the quotient of Culler-Vogtmann's outer space $X_n$ by the action of $\mathrm{Out}(F_n)$. It can be thought of as the space of metric graphs homotopy equivalent to ...

**4**

votes

**0**answers

137 views

### Comparison of Constrained Optimization Methods

I am trying to solve a constrained optimization problem using filter methods and came across two papers on the topic that I am having some problems with. The original filter method paper is the ...

**23**

votes

**0**answers

310 views

+50

### Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces

Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?
There are some results of the ...

**14**

votes

**1**answer

260 views

### The density of integers represented by a binary form

Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of ...

**6**

votes

**2**answers

440 views

### Atiyah Singer index theorem and Hodge de Rham operator

When I read about Atiyah Singer index theorem I met the following example: let $M$ is (orientable closed smooth) Riemannian manifold and consider Hodge-de Rham Dirac operator defined by $d+d^*$ ...

**29**

votes

**1**answer

2k views

### “The Two Sheriffs” puzzle

This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler.
Two sheriffs in neighboring towns are on the track of a killer, in a
case involving eight ...

**7**

votes

**4**answers

320 views

### hyperbolic structure on Figure–8 knot complement

I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...

**57**

votes

**38**answers

11k views

### nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...

**117**

votes

**33**answers

66k views

### Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...

**5**

votes

**0**answers

107 views

### Hodge de Rham operator and orientability

Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...

**5**

votes

**1**answer

135 views

### $H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...

**37**

votes

**26**answers

7k views

### Examples of seemingly elementary problems that are hard to solve?

I'm looking for a list of problems such that
a) any undergraduate student who took multivariable calculus and linear algebra can understand the statements, (Edit: the definition of understanding here ...

**77**

votes

**18**answers

7k views

### Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...

**14**

votes

**4**answers

809 views

### Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random ...

**6**

votes

**2**answers

484 views

### Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...

**6**

votes

**2**answers

232 views

### Is an $\mathfrak{sl}_2$-triple determined up to Lie algebra automorphism by the adjoint representation?

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let $\phi_1:\mathfrak{sl}_2(\mathbb{C})\rightarrow\mathfrak{g}$ and ...

**10**

votes

**1**answer

526 views

### Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...

**59**

votes

**8**answers

46k views

### Which are the best mathematics journals, and what are the differences between them?

Suppose you have a draft paper that you think is pretty good, and people tell you that you should submit it to a top journal. How do you work out where to send it to?
Coming up with a shortlist isn't ...

**27**

votes

**14**answers

27k views

### Reading list for basic differential geometry?

I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is ...

**6**

votes

**1**answer

111 views

### Time decay for Hartree equation with Coulomb potential

Are there any time-decay results for the solution of the Hartree equation
\begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in ...

**28**

votes

**2**answers

3k views

### No simple groups of order 720?

I think most students who first learn about (finite) groups, eventually learn about the possibility of classifying certain finite groups, and even showing certain finite groups of a given order can't ...

**11**

votes

**1**answer

3k views

### Are there any non-linear solutions of Cauchy's equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be s.t. $f(x+y) = f(x) + f(y), \ \forall x, y$
It is quite obvious that this implies $f(cx)=cx$ for all $c \in \mathbb{Z}$ and even further: $\forall c \in ...

**13**

votes

**2**answers

677 views

### To what extent can fields be classified?

The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}_p$, or topological completions and ...

**11**

votes

**5**answers

1k views

### reference for Noether's theorem

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?

**15**

votes

**1**answer

3k views

### Evidence for integer factorization is in $P$

Peter Sarnak believes that integer factorization is in $P$. It is a well-known open problem in TCS to identify the real complexity class of integer factorization. Take a look at this link for Peter ...

**3**

votes

**0**answers

220 views

### Global dimension of endomorphism algebra of a coherent sheaf

Let $X$ be a smooth projective variety over algebraically closed field of characteristic zero. In one of the versions of definition of tilting object $\mathcal{F}$ on $X$ there is a requirement that ...

**18**

votes

**6**answers

1k views

### Noncommutative rational homotopy type

Ok, this question is much less ambitious than it might sound, but still:
Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga ...

**5**

votes

**2**answers

1k views

### When does the direct image functor commute with tensor products?

Let $i : U \to X$ be a quasi-compact open immersion of schemes. Under which conditions is the natural map
$i_* M \otimes i_* N \to i_* (M \otimes N)$
for all $M,N \in \text{Qcoh}(U)$ an isomorphism? ...