All Questions
2,026 questions with no upvoted or accepted answers
6
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answers
111
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Factorization to sparse matrices
$\newcommand{\lrank}{\operatorname{lrank}}$
$\newcommand{\rank}{\operatorname{rank}}$
Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it.
Now, given ...
- 3,319
6
votes
0
answers
235
views
A standard name for the algebraic structure on a projective line?
Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...
- 41.8k
6
votes
0
answers
199
views
Zero-one pairings between sets of vectors
Let
$A\subseteq V$ and
$B\subseteq V^\star$
be spanning sets in
a finite-dimensional real vector space $V$ and
its dual $V^\star$.
Suppose that
$$
\langle b,a\rangle\in\lbrace0,1\rbrace
$$
for all
$a\...
- 809
6
votes
0
answers
188
views
Expressing an invertible sparse matrix as a product of few elementary matrices
Let $M$ be an $n \times n$ matrix with integer entries. Suppose that $M$ is invertible (over the integers) and that $M$ has at most $An$ nonzero entries, each of which is less than $B$ in absolute ...
- 18.7k
6
votes
0
answers
179
views
Elementary equivalence for rings
Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...
- 3,318
6
votes
0
answers
270
views
Torsion in the Lie algebra cohomology of gl(n,Z)
What is known about the Lie algebra cohomology $H^*(\mathfrak{gl}_n(\mathbb{Z}),\mathbb{Z})$? After passing to $\mathbb{Q}$-coefficients, the question is classical: $H^*(\mathfrak{gl}_n(\mathbb{Q}),\...
- 4,487
6
votes
0
answers
185
views
How much information can we extract about such a group?
Let $G$ be a group with the property : Given any positive integers $m, n$ and $r$ there exists elements $g, h\in G$ such that $|g|=m, |h|=n$ and $|gh|=r$
Where $|g|$ is the order of the group $\langle ...
- 307
6
votes
0
answers
149
views
Rings with epimorphism from a finitely generated ring
For a commutative ring with unit $R$ let's say it has property $(*)$ if there is an epimorphism in the category of rings ${\mathbb Z}[X_1,\dots,X_n]\to R$, where the former is the polynomial ring in $...
user473423
6
votes
0
answers
340
views
Asymptotically nilpotent Lie sets of matrices
A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$.
Question 1. Assume that $\mathcal{A}$ is the subset of $\textbf{...
- 291
6
votes
0
answers
177
views
Is the monoid of all cancellative finitely generated commutative monoids cancellative?
$\DeclareMathOperator\Mon{Mon}\DeclareMathOperator\Grp{Grp}$Let $\Mon'$ be the set of isomorphism classes of (small) commutative, unital, cancellative ($a + t = b + t$ implies $a = b$) monoids. It is ...
- 1,084
6
votes
0
answers
253
views
Cardinality of a polynomial image $\pmod{p^n}$
Given $P(x) \in \mathbb{Z}[x]$ a polynomial, $n$ a positive integer and $p$ a prime, there is a result that relates $|\text{Im } P \pmod{p^n}|$ with $|\text{Im } P \pmod{p^{n+1}}|$ perhaps in terms of ...
- 61
6
votes
0
answers
106
views
Automorphisms of algebraic Clifford algebra of a Hilbert space
Let $H$ be a real separable, infinite-dimensional Hilbert space and let
$$\mathrm{Cl}(H) = \mathcal{T}(H_{\mathbb{C}}) / \{v\otimes w + w\otimes w - 2\langle v, w\rangle \cdot \mathbf{1} ~|~ v, w \in ...
- 9,853
6
votes
0
answers
167
views
Hölder inequality inside trace
$\DeclareMathOperator\tr{tr}$Suppose we have positive semidefinite matrices $A_1, \dotsc, A_n$ and $B_1, \dotsc, B_n$ of the same dimension. Do we have a Hölder inequality for the trace of the ...
- 565
6
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0
answers
392
views
Divisibility properties of minors of matrices
Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ ...
- 191
6
votes
0
answers
396
views
Typical eigenspectrum of a random projection of a large matrix
Suppose I have a real symmetric $m \times m$ matrix $\Lambda$. This matrix is large ($m \gg 1$) and, for simplicity, we'll assume it's diagonal. I then construct a random $n \times n$ projection
$$ A =...
- 453
6
votes
0
answers
203
views
The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?
I asked this question over a year ago on Math.StackExchange but I didn't get an answer.
In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are ...
- 841
6
votes
0
answers
230
views
Gelfand ring in Bourbaki's exercises
In Bourbaki's General Topology, Chapitre III §6 Exercise 11, they define a Gelfand Ring as a topological ring $A$ such that
The set $A^*$ ($=A^{-1}$) of invertibles is open.
The uniform structure ...
- 3,738
6
votes
0
answers
294
views
Independence results on pure algebra
I think that the most celebrated result in this direction is Shelah's famous work on Whitehead's Problem:
Is every abelian group $A$ such that $Ext^1(A, \mathbb{Z})=0$ free?
This is known to be ...
- 3,318
6
votes
0
answers
235
views
How much does Ext tell me about isomorphisms?
So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
- 69
6
votes
0
answers
97
views
Kac-Moody Lie algebra as derivations of associative algebras
The set of derivations of an algebra $\Bbb A$ forms a Lie algebra. This is one aspect of why Lie algebras are interesting. When $\Bbb A$ is polynomial algebra in $n$ variable then $\text{Der } \Bbb A$ ...
- 1,269
6
votes
0
answers
182
views
On properties of an algebra as a bimodule
Let $A$ be a two-sided artinian ring.
Recall that a module $M$ is said to have dominant dimension at least $n$ in case the terms $I_i$ in the minimal injective coresolution of $M$ are projective for $...
- 26.5k
6
votes
0
answers
57
views
A characterisation of weakly symmetric algebras
A finite dimensional algebra $A$ over a field $K$ is called a Frobenius algebra in case there exists a $K$-linear map $f: A \rightarrow K$ such that $ker(f)$ contains no non-zero right ideal of $A$ (...
- 26.5k
6
votes
0
answers
192
views
Bar notation in Bourbaki’s *Lie groups*, Chap. IX
I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they ...
- 31.5k
6
votes
0
answers
141
views
Algorithm to check a conjectural value for the rank of a large matrix
Feel free to suggest a different title, I'm not sure how to phrase this. I'm in the following somewhat specific situation:
I'm checking a conjecture which at the end of the day boils down to the ...
- 8,524
6
votes
0
answers
259
views
Diameter of finite rational matrix groups
Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
- 165
6
votes
0
answers
143
views
Newer versions of Mahler's Lemma
I'm trying to find a way to numerically ensure that two constructible numbers are equal (this would be done by a computer).
The idea is to find a polynomial $p(x)$ that contains both numbers as roots ...
- 563
6
votes
0
answers
96
views
Finding the maximal component of a vector in sublinear time
Given a vector $u \in \Bbb R^n$, finding the value of the largest component of $u$ needs linear time in $n$. However, what if we additionally know that $u$ lies in some linear subspace $U \subset \Bbb ...
- 13.6k
6
votes
0
answers
255
views
Homotopy quotient of groups
Suppose $0\to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 0$ is a short exact sequence of groups.
We have an induced map $k[\iota] : k[A] \to k[B]$ of group algebras over a field $k$.
What ...
- 1,382
6
votes
0
answers
191
views
A curious $q$-identity
Let $[x]_{q}=\frac{1-q^x}{1-q}$ and $\binom{x}{n}_{q}$ denote a $q$-binomial coefficient.
Let $A_n(x,q)$ be the $n\times n $ matrix with entries $$q^\binom{i-j}{2}\binom{i+j+x}{i-j+1}_{q},$$ $0 \le i,...
- 5,612
6
votes
0
answers
695
views
Jacobson radical of a tensor product
Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
- 6,349
6
votes
0
answers
336
views
Legendre's three-square theorem and squared norm of integer matrices
Let $\mathbb{N}$ be the set of non-negative integers. Let $E_n$ be the set of integers which are the sum of $n$ squares. Let $F_n$ be the set of integers of the form $\Vert A \Vert^2$ with $A \in M_n(...
6
votes
0
answers
92
views
Tiling with Horn's polytopes
Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le ...
- 52.3k
6
votes
0
answers
332
views
Independence of characters with respect to polynomials
I came across the following property :
Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors,
$\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\mathcal{...
- 3,738
6
votes
0
answers
128
views
Localizations of group algebras of free groups
$\newcommand{\QQ}{\Bbb Q}$
Let $G$ be a free group on the symbols $x_1, \dots, x_n$, with $\QQ[G]$ its rational group algebra.
Write $\varepsilon: \QQ[G] \to \QQ$ for the augmentation, and for $\...
- 52.6k
6
votes
0
answers
145
views
An ideal that there exists a unique ideal maximal with respect to not containing it
Let $I$ be a non-zero ideal of a commutative ring with identity. Is there any equivalent condition to the property that there exists a unique ideal maximal with respect to not containing $I$?
For ...
6
votes
0
answers
79
views
Canonical forms for representations of tame algebras
Recently I have been studying the representation type of algebras (tame-wild dichotomy). A question that I feel is probably well known but that I cannot answer is the following:
Can I define tame ...
- 61
6
votes
0
answers
223
views
Specifying cokernels of all powers of $p$-adic matrix
Given a matrix $A \in M_d(\mathbb{Z}_p)$ (with nonzero determinant), viewed as a map $\mathbb{Z}_p^d \to \mathbb{Z}_p^d$, I am interested in the sequence of abelian $p$-groups $\{coker(A^n)\}_{n \geq ...
- 568
6
votes
0
answers
99
views
What is this matrix decomposition called and does it exist always? - II
Given a rank $2r$ matrix $M\in\Bbb Q_{\geq0}^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank at most $r$ such that $M=M_+-...
- 13.9k
6
votes
0
answers
160
views
m-systems and n-systems in topological rings
Note that throughout rings have a multiplicative identity and are not necessarily commutative
Definition: Let $R$ be a ring and let $M\subseteq R$. Then, $M$ is an m-system iff for every $x,y\in ...
- 1,270
6
votes
0
answers
440
views
Solve this nonlinear matrix equation?
For $M$ and $N$ two invertible square matrices of the same size $n$, consider the equation
$$
\forall i,j, \quad M_{ij}(M^{-1})_{ji} = N_{ij}(N^{-1})_{ji}\ .
$$
Assuming we know $M$, we want to find ...
- 378
6
votes
0
answers
418
views
Alternative Interpretation of the Schwarzian Derivative
The Schwarzian derivative of a holomorphic function $f$ of a complex variable $z$ is defined as $\{f, z\} = \frac{f^{\prime\prime\prime}(z)}{f^\prime(z)} - \frac{3}{2}{\Big(\frac{f^{\prime\prime}(z)}{...
- 192
6
votes
0
answers
138
views
A question on deformation theory of triples of matrices
Let $(x,y,z)$ be a triple of $n \times n$ traceless complex matrices which are simultaneously diagonalizable. We call such a triple regular if $C_x \cap C_y \cap C_z$ is a Cartan subalgebra of $\...
- 5,215
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
6
votes
0
answers
228
views
Lower bound for order of matrix modulo $n$
For a positive integer $n$ and a square matrix $A$ with integral entries, let $\text{ord}(A, n)$ be the smallest positive integer $k$ such that $A^k \equiv \mathbf{1} \bmod n$, if such integer $k$ ...
user40023
6
votes
0
answers
450
views
Can this nonlinear vector equation be solved analytically?
I have the following vector equation:
$$
{\bf Ax} + {\bf b} + {\bf Cx}^{ \circ - 1} = {\bf 0}_n
$$
Where ${\bf x}$ is a vector of unknown variables. ${\bf b},{\bf x}, {\bf x}^{\circ - 1}, {\bf 0}_n ...
- 235
6
votes
0
answers
223
views
Book or survey on Dedekind-finite rings
I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite ...
- 10.5k
6
votes
0
answers
339
views
What is the algebraic equivalent of independent elements?
The definition/notion of independence is always a bit odd in measure theoretic probability theory.
Definition Given a probability space $(\Omega,\mathcal{F},P)$, two sets $A,B\in\mathcal{F}$ are ...
- 8,071
6
votes
0
answers
259
views
Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type?
A fusion ring is a finite dimensional $\mathbb{Z}$-module
$\mathbb{Z}\mathcal{B}$ together with a distinguished basis
$\mathcal{B} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j =
\sum_k n_{ij}^...
6
votes
0
answers
386
views
Is there an explicit construction of Tate-Beilinson residue?
Background: Tate's construction of abstract residues is generalized in Beilinson's constructon as follows:
Let $E$ be a unital, cubically decomposed $k$-algebra, i.e., (a) there are two sided ideals $...
- 493
6
votes
0
answers
136
views
Does a given bound quiver algebra admit an algebra analogous to the preprojective algebra of a path algebra?
Let $Q=(Q_0,Q_1,s,e)$ be a finite quiver.
In the formal construction of the preprojective algebra associated to $Q$, we set $\ Q^*_1=\{\alpha^*:y\rightarrow x| \alpha\in Q_1, \alpha:x\rightarrow y \}...
- 493