3
votes
0answers
22 views

Geometry description of the GSR riffle shuffle model

In 1992 Diaconis and Bayer announced their famous result which is now a well-known folklore: Seven shuffling is enough to randomize a deck of cards. One of the key ingredients in their proof is that ...
1
vote
0answers
16 views

Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ its dual ...
0
votes
0answers
19 views

Get archimedean spiral length by cartesian coordinates

I am currently busy with writing a generator for a Sacks spiral, and the formulas I currently have are these: ...
1
vote
0answers
13 views

Connected representant of a framed cobordism class (reference needed)

Let $N^n\subseteq M^m$ be a submanifold with a framing of the normal bundle, $2n<m$. Then $N^n$ is framed cobordant (in $M^m$) to something connected. I believe it could be proved by directly ...
0
votes
0answers
34 views

Which polynomials define extensions of $k(t)$ unramified at the finite places

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $L$ be the extension of $k(t)$ obtained by attaching a root of an irreducible polynomial $f\in k(t)[x]$. Is there a way to tell ...
2
votes
1answer
34 views

Linear systems on bielliptic surfaces

A bielliptic surface is a surface of type $S=E_1 \times E_2/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group of translations of $E_1$ acting on $E_2$ such that $E_2/G=\mathbf{P}^1$. ...
0
votes
1answer
61 views

Splitting the Hopf map in two

Given the Hopf map $h:S^3\to S^2$ and an inclusion $i:S^2\hookrightarrow S^3$, the map $i\circ h:S^2\to S^2$ has mapping degree zero. Therefore, it is homotopic to the constant map and the image of ...
1
vote
0answers
23 views

Existence of Colimits in the Definition of Locally Presentable Categories

Basically, my question is simple: why does the definition of a locally presentable category require all colimits exist? The motivation for this is that I was learning about algebraic posets, and had ...
1
vote
1answer
44 views

What is the probability of a given induced ordering of a random permutation?

I ran into the following problem in a calculation involving permutations. Let $[n] = \{1,...,n\}$, and assume that $[n]$ is partitioned into equivalency classes. That is, $[n]$ is the disjoint union ...
1
vote
0answers
26 views

Origins of the Jacobi matrix

I have several questions concerning history of Jacobi matrices. Does anybody know why the Jacobi matrix (=symmetric tridiagonal matrix) is named by Carl Gustav Jacob Jacobi? What was his contribution ...
3
votes
1answer
68 views

Diagonalization via the Toda flow

According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...
6
votes
0answers
59 views

Existence of a family of elliptic curves with large torsion subgroup

Andrej Dujella's excellent web-site "High rank elliptic curves with prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/tors.html gives example of the (current) largest known rank of an ...
1
vote
0answers
30 views

Are two $W^*$-algebras $A$ and $B$ that are Morita equivalent (in the sense of Rieffel) also Morita equivalent in the algebraic sense (as rings)?

Let $A$ and $B$ be $C^*$-algebras, I had a question about Morita equivalence for $C^*$-algebras and $W^*$-algebras, I've come across 3 concepts: Morita equivalence for $C^*$-algebras: Equivalence ...
0
votes
0answers
43 views

Does the Fourier transform of a non-strictly positive real kernel $\Phi(t)$ always generate entire function with complex zeros?

Question (1) Does the Fourier transform of a non-strictly positive real kernel $f(t)$ always generate an entire function $g(z)$ with complex zeros? $$g(z)=\int_{-\infty}^{\infty}f(t) ...
3
votes
1answer
52 views

An orthogonal factorization system on 1-Cob?

Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W_{dom}:A\to W$ and $i^W_{cod}:B\to ...
4
votes
0answers
93 views

Is there an infinite J-group?

For a group $G$ let $\operatorname{Sub}(G)$ be the lattice of all its subgroups. A subgroup interval is an interval in the lattice $\operatorname{Sub}(G)$. A group $G$ is called a J-group iff for ...
9
votes
2answers
359 views

Is any particular algebraic number known to have unbounded continued fraction coefficients?

The continued fraction $$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be ...
2
votes
0answers
83 views

Are open immersions in analytic geometry transverse?

lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...
5
votes
1answer
205 views

A divisor sum congruence for 8n+6

Letting $d(m)$ be the number of divisors of $m$, is it the case that for $m=8n+6$, $$ d(m) \equiv \sum_{k=1}^{m-1} d(k) d(m-k) \pmod{8}\ ?$$ It's easy to show that both sides are 0 mod 4: the left ...
0
votes
0answers
31 views

Quantile as solution to minimization problem

I posted this on Math Stack Exchange, but since I got no response, I'm trying my luck here. I'm studying basics of quantile regression now and I have trouble proving that $\tau-$th quantile of ...
-3
votes
0answers
43 views

BCL algebra define a partial order. Answer [on hold]

BCL algebra define a partial order. Answer Proof: (i) Reflexivity: If x*x=0, then x⩽x. (ii) Anti-symmety: If x⩽y and y⩽x, then x*y=0 and y*x=0, by axiom (2),we have x=y. (iii) Transitivity: If ...
2
votes
0answers
42 views

A conditional form of Holder's Inequality on Type II-1 von Neumann algebras

Suppose $V$ is a von-Neumann algebra with a trace $\tau$ so that $\tau(I) = 1$. Suppose $W$ is a sub-von Neumann algebra of $V$. Then we can define a conditional expection to be a projection ...
0
votes
0answers
157 views

Why does Neeman avoid t-structures?

I have a simple question: the book "Triangulated Categories" by A. Neeman aims to be an exhaustive reference about the whole (basic) theory of triangulated categories. So why there is only a single ...
-2
votes
0answers
32 views

multiplication of a projection matrix and PSD matrix is a PSD? [on hold]

I have a projection matrix P and X^TAX where A is a diagonal matrix with all strictly positive entries can I tell that PX^TAX is PSD?
9
votes
2answers
512 views

How much of homotopy theory can be done using only finite topological spaces?

Let $X$ be a finite simplicial complex and let $B$ denote the set of barycenters of the simplices of $X$. McCord constructed a $T_0$ topology on $B$ with the property that the inclusion $B \to X$ is ...
10
votes
1answer
209 views

K3 surfaces that correspond to rational points of elliptic curves

In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the ...
2
votes
0answers
102 views

Solving equations with dozens of ceil and floor functions efficiently?

I am tackling a problem which uses lots of equations in the form of: where $q_i(x)$ is the only unknown, $c_i$, $C_j$, $P_j$ are always positive. $C_j < P_j$ for all $j$ 1.How is this type of ...
-6
votes
0answers
48 views

Abstract Algebra connections [on hold]

How can I write a VERY brief statement that explains that you can redefine a vector space as a set V that is an abelian group under + and has scalars and a scalar multiplication that follow five ...
4
votes
0answers
76 views

A question about ordinal definable sets of real numbers revisited [duplicate]

Citing (almost) A question about ordinal definable real numbers If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is ...
4
votes
0answers
63 views

Is every compact monothetic group metrizable?

If $G$ is a compact (Hausdorff) topological group with a dense cyclic subgroup, is it necessarily true that $G$ is first countable? This claim seems to be implicit in a paper that I am reading at the ...
1
vote
0answers
57 views

NP-hard proof of one variant version of exact cover

exact cover is NPC. http://en.wikipedia.org/wiki/Exact_cover#Equivalent_problems Now I am looking into weighted exact cover problem as follows, given a universe $U$, and collection $S=\{s_1, s_2, ...
2
votes
0answers
18 views

Linear control systems

Are there some algorithms to compute the error bound of the difference between the original system and the balance system using the Balance truncation method?
4
votes
1answer
124 views

Automorphisms of SO_n(k,f)

Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let $$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$ and ...
8
votes
2answers
295 views

Fundamental group of $\mathbb{R}^3-F$ where $F\subseteq \mathbb{R}\times \{0\} \times \{0\}$

Maybe not research level. Let $Z\cong \mathbb{R}$ be the $z$-axis of $\mathbb{R}^3$. Clearly $\pi_1(\mathbb{R}^3-Z)\cong \mathbb{Z}$. Now if $F\subset Z$ is a closed non-empty subset, then one easily ...
1
vote
1answer
84 views

When is an exponential functor a bialgebra?

My question is about the notion of exponential functors as they are frequently defined in the literature on (strict) polynomial functors, e.g., the paper "General Linear and Functor Cohomology" by ...
1
vote
2answers
158 views

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

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 ...
1
vote
0answers
79 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
votes
0answers
130 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 ...
3
votes
1answer
152 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?
13
votes
2answers
492 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 ...
4
votes
0answers
116 views

Polynomial dynamical systems

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
votes
0answers
57 views

What is a good name for connectives cum set operators? [on hold]

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 $\setminus$, ...
4
votes
2answers
291 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 ...
4
votes
1answer
170 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 ...
5
votes
2answers
167 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 $\mathbb{A}$ ...
1
vote
2answers
156 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{?}{=} ...
-1
votes
0answers
74 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
1answer
213 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
votes
0answers
59 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 ...
0
votes
0answers
57 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 ...

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