3
votes
0answers
27 views

Fredholm operators in $K$-theory?

Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ...
3
votes
0answers
14 views

Hausdorff spaces with lattice isomorphism between the topologies

For $i=1,2$ let $(X_i,\tau_i)$ be Hausdorff spaces such that the lattices $\tau_1, \tau_2$ are isomorphic. Does this imply that $(X_1, \tau_1) \cong (X_2,\tau_2)$? (This is a follow-up question to ...
1
vote
0answers
11 views

Generating function of alternating even terms in the Vandermonde Convolution

I have $G(x) = \sum_{i = 0}^{\infty} \sum_{r = 0}^{\infty} (-1)^i \binom{k}{2i} \binom{n-k}{r} x^{2i} x^{r} = \frac{1}{2} \left( (1 + x{\iota})^{k} + (1 - x \iota)^{k} \right)(1 + x)^{n-k}$ where ...
2
votes
1answer
12 views

$T_2$-spaces such that the lattices of open sets can be embedded into each other

Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$. Does this imply that $(X,\tau)\cong (Y,\sigma)$?
1
vote
1answer
32 views

Can the Laplace operator be represented as a sum of second order derivational operators

Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$. Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ ...
2
votes
0answers
19 views

Can hypercompletion be an essential localization?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory. Can it ever happen that this subcategory is ...
0
votes
0answers
19 views

Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here. Let $\Sigma$ be an alphabet and let $y = x_1 ...
0
votes
0answers
31 views

understanding the proof of gauss lemma [on hold]

I was reading the Gauss Lemma from the do carmos Rienmannian geometry book which says that Let $p \in M$ and let $v \in T_pM$ such that $\exp _p v$ is defined Let $w\in T_pM$ is identified with ...
0
votes
0answers
58 views

Is it possible to classify the indecomposable representations wild quiver $\mathbb{F_2}$ using infinite sets of parameters?

I have read several other questions about wild representation type but may I ask..what actually can be done and what have been done (such as partial results) about classifying indecomposable ...
0
votes
0answers
22 views

Average Multivariate Gaussian

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...
1
vote
1answer
41 views

Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials \begin{align} & p_1(x_1,x_2,\ldots,x_n)=0, \\ & p_2(x_1,x_2,\ldots,x_n)=0, \\ & \vdots \\ & p_m(x_1,x_2, ...
5
votes
0answers
53 views

A Collatz-like question about permutations

An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting. Consider all permutations $\pi$ on the natural numbers such that ...
-2
votes
1answer
134 views

Are limits decidable? Should definitions be decidable?

This question is about the Turing computability of the $\epsilon-N$ definition of a limit of an infinite sequence $S$. First, a proposition: There cannot exist a Turing Machine $M$ which, given a ...
4
votes
0answers
44 views

Brownian motion, exists $c < \infty$?

Suppose $B_t$ is a standard Brownian motion. Does there exist $c < \infty$ such that with probability one$$\limsup_{t \to \infty} {{B_t}\over{\sqrt{t \log t}}} \le c?$$I need to know whether or not ...
1
vote
0answers
21 views

Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof): In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...
0
votes
0answers
18 views

Clifford semigroups!

I am trying to prove that if a maximal group image G(S) of a Clifford semigroup S, is abelian-by-finite does the Clifford semigroup S is abelian-by- finte? I am assuming that S is an E-unitary ...
4
votes
1answer
97 views

Number of representations as sums of squares in rings of integers of number fields

Let $K$ be some number field, $\mathcal O_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\alpha \in \mathcal O_K$, and consider the quantity $r_{n,K}(\alpha)$, which denotes ...
1
vote
0answers
20 views

Small open sets around a point intersecting pieces of orbits

Let $T$ be an ergodic rotation on a compact Abelian group. Can one always find a point $x_0$ and a decreasing sequence of open sets $O_n \searrow \{x_0\}$ such that for every $n$ there exists $K \geq ...
3
votes
0answers
73 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

Reposted from math.stackexchange here. The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq ...
1
vote
0answers
27 views

Dense Hopf $*$-subalgebra of compact quantum groups and cancellation laws

Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital ...
-2
votes
0answers
34 views

Arithmetic data in an elliptic curve over a field $\mathbb K$

Note: In this context, $E(K)$ denotes an elliptic curve $E$ over a number field $K$, and $L(E,s)$ denotes the Hasse-Weil L-function (also called zeta function). Is the rank of the abelian group ...
10
votes
0answers
97 views

Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...
0
votes
0answers
37 views

Is there a general theory for the structure of the (semi)group generated by morphisms of an affine space $F_p^3$?

Consider an affine space $\mathbb{F}_p^3$, and assume we have a handful of morphisms $f_i : \mathbb{F}_p^3 \rightarrow \mathbb{F}_p^3$ given by $$f_i(x, y, z) =(P_i(x, y, z), Q_i(x, y, z), R_i(x, y, ...
1
vote
1answer
100 views

Solutions to system of polynomial equations over finite fields

If $P_1$, $P_2$, ..., $P_m$ are $n$-variate homogeneous polynomials of degree d over a finite field $F_q$, where $q$ is much larger than $d$, but much smaller than $n$, then do we know good lower ...
-4
votes
0answers
32 views

Non standerd Algebraic topology3 [on hold]

. A space X is n-connected if π k (X, x 0 ) ∼ = ∗ for all k ≤ n and all x 0 ∈ X.
0
votes
1answer
39 views

Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects. Let $({\cal C},\otimes,*)$ be a semisimple ...
-2
votes
0answers
88 views

Every totally disconnected space is Hausdorff space [on hold]

I saw in a websit the following theorem, without any proof, is this theorem true Blockquote Theorem:Every totally disconnected space is Hausdorff space. It was in ...
1
vote
0answers
93 views

An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
2
votes
0answers
17 views

s-arc transitivity and the Moore bound

Given a vertex $x$ of a graph, call the subgraph induced by all vertices of distance $\le n$ from $x$ the $n$-neighbourhood of $x$, denoted $N_n(x)$. Let $G$ be a regular graph of diameter $n+1$. For ...
-1
votes
0answers
33 views

Spectral radius of principal submatrices for the case of hermitian matrix

A principal submatrix of a matrix $\mathbf{A}\in\mathbb{C}^{N\times N}$ is any submatrix from $\mathbf{A}$ for which the same rows and columns have been eliminated. Assume $\mathbf{A}$ is hermitian, ...
0
votes
0answers
10 views

Distance minimization and submodular functions

I have a question about square distance minimization and submodular functions. Suppose I have a random variable $X = (X_1,...,X_n)$ over $\mathbb{R}^n$. Let $x$ be a realization of this random ...
-4
votes
0answers
47 views

What is the algebraic structure of the multiplicative group $\mathbb{F}_q^\times$ after quotienting by the relation $a \sim b$ iff $a/b$ is a square? [on hold]

Define the relation $a \sim b$ by $a/b$ is a square, and consider $S = \mathbb{F}_q^\times / \sim$. What is the structure of $S$?
1
vote
1answer
72 views

Commutator subgroups as normal supplmements

The following question has been asked about a week ago on MathUnderflow (no answers). Let $F$ be a free group and let $N$ be a normal subgroup of $F$ such that \begin{equation*} \tag{*} F = [F,F] ...
-1
votes
0answers
12 views

Formula for computing all possible vertices for a triangle that has the same base length and opposite angle [on hold]

I have a situation where I know the cartesian coordinates of the 2 vertices of a triangle that form its base, hence I know the length of the base and this is fixed. I also know the angle opposite the ...
1
vote
0answers
120 views

A curious limit [on hold]

Let $p$ be an odd prime number and let $a(n,p)$ be the denominator of $p^n/n$. Then it seems that $$\lim_ ...
17
votes
1answer
749 views

What were the main ideas and gaps in Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
-1
votes
0answers
6 views

Mean and variance for unequal samples [migrated]

I have a sampling of variable sized plots. Each plot contains the number of trees present on the plot. Given: $n=$ the number of plots $s_i=$ the size of the $i^{th}$ plot $y_i=$ the number of trees ...
1
vote
0answers
30 views

Degrees of polynomials defining a Jacobian of maximal rank on a variety

Let $f_1,\ldots,f_{n-k} \in \mathbb{R}[x_1,\ldots,x_n]$ be polynomials of degree at most $d$ defining an algebraic set $A \subseteq \mathbb{C}^n$ which contains an irreducible component $V \subseteq ...
0
votes
1answer
44 views

Rational functions on hyperelliptic Riemann surface

Let $\mathcal R$ be an hyperelliptic Riemann surface of genus $g\geq 1$. Is it true that the only possible rational functions on $\mathcal R$ with $\leq g$ poles are the liftings of rational functions ...
1
vote
2answers
39 views

How to determine whether the following sum is nonzero for a given multivariate polynomial?

My research field is combinatorics. I am not very good at Algebra. So I want to ask for a given real multivariate polynomial $f(x_1,x_2,\cdots,x_n)$, is there any algebraic method to compute whether ...
6
votes
2answers
598 views

How is differential geometry used in immediate industrial applications and what are some source to know about it?

Intuitively it might be clear that differential geometry is a very applicable subject in engineering and industry. I'd like to know how some industries/companies use differential geometry. I'd guess ...
1
vote
1answer
100 views

Generalization of a theorem of Steinberg

Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type: ...
5
votes
0answers
82 views

C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$

In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes ...
-1
votes
0answers
40 views

engineering textbooks and punctuating equations [on hold]

( a simplified example}...the following equation that is displayed with a reference number is used to calculate such and such: ...
2
votes
0answers
39 views

Two ODEs, why is one the solution of the other? (Caratheodory ODE)

This question is based on Zeidler II/B, Problem 30.2. Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ ...
-4
votes
0answers
44 views

Please, help the physicist-theorist to calculate the quantum structures (see http://dx.doi.org/10.4236/jmp.2015.67103 ) [on hold]

To calculate, for example, the beginning of nucleosynthesis or the density of electrons on the graphene surface , please, help me to find mathematician who could calculate the average distance ...
1
vote
0answers
89 views

Possible counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves

This might be a counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves. In this paper, the quadratic twist of $f(x)=y^2$ is denoted by $C_d : d y^2=f(x)$ ...
2
votes
1answer
49 views

Conditions for underlying space of an orbifold $\Bbb T^n/\Gamma$ to be a sphere?

Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific ...
2
votes
0answers
45 views

Connected components of a certain real homogeneous space

Let $m>0$ be a natural number. Consider the following semisimple algebraic groups over ${\mathbb{R}}$: $$ G={\mathrm{SU}}(2m,4m),\ \ H={\mathrm{SU}}(2m,2m)\times{\mathrm{SU}}(2m). $$ We embed $H$ ...
0
votes
0answers
29 views

The Fransén-Robinson constant as a limit of integrals

This is the integral of the reciprocal gamma function,$$F=\int_0^\infty dx/\Gamma(x)= e+\int_0^\infty \frac{e^{-t}\ dt}{\log^2 t+\pi^2}$$ $$=e+\int_{-\infty}^{\infty}\frac{\exp(s-e^s)\ ...

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