All Questions
6,548 questions
10
votes
3
answers
2k
views
The ring of finite algebras over Z/p^n
Here is a idea concerning the classification of finite rings (commutative, unital). Related question: Classification of finite commutative rings.
Every finite ring is a direct product of finite ...
CommunityBot
- 1
2
votes
0
answers
495
views
Around the invariant basis property
Hi everybody,
recall that a ring $A$ with unit has the invariant basis property if all bases of a given f.g. free $A$-module have the same number of elements.
It is equivalent to say that every ...
- 91
23
votes
1
answer
1k
views
Codes, lattices, vertex operator algebras
At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention to ...
- 2,150
1
vote
0
answers
138
views
Bases of Ideals With no Monomials
Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
- 345
2
votes
0
answers
282
views
Matrix row selection
Consider an m by n matrix $A$ with entries from some finite field $\mathbb{F}_q$, where $m\geq n$. Let us write matrix $A$ as follows $A=[A_1^T~A_2^T~\ldots A_l^T]^T$, where $A_i$'s are given ...
4
votes
0
answers
261
views
Growth of symmetric positive definite integral matrices.
Given an integer $d$, let $\alpha_d(N)$ denote the number of symmetric integral positive definite matrices of size $d\times d$ with coefficients in $\lbrace -N,-N+1,\dots,N-1,N\rbrace$.
...
- 17.9k
1
vote
2
answers
262
views
How to approx. decompose a sym. p.d. matrix M into X'X?
M: pxp symmetric p.d. matrix with unit diagonals
n: number much smaller than p
Want a nonrandom nxp matrix X such that X'X is
close to M element-wise. If n gets larger, hopefully
difference ...
- 17.9k
3
votes
1
answer
315
views
Group ring and left zero divisor II
Let $K$ be a finite field and $G$ be a discrete group.
Is it true that for every $a=e+a_1+\ldots+a_n,b=e+b_1+\ldots+b_m\in K[G]$ with $b_i\neq e,a_j\neq e$ the condition $ab=0$ implies $ba=0$?
...
CommunityBot
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4
votes
1
answer
1k
views
Cholesky Rank-1 downdate extension
If we have a matrix $K$ we can take do a rank-1 downdate of its Cholesky $L = chol(K)$ to find $L_\star = chol(K - v v^\top)$ in $O(N^2)$ time as opposed to $O(N^3)$ time for doing the Cholesky from ...
- 41
5
votes
0
answers
7k
views
Prime ideals in $\mathbb{Z}[\sqrt{-5}]$ [closed]
First of all, sorry for the noob question, but it's driving me crazy...
I was reading John Stillwell's "Elements of Number Theory" (Springer, ISBN 0-387-95587-9).
In an exercise on page 225, he ...
- 37k
19
votes
1
answer
904
views
Is the norm of a $0-1$ matrix (almost) attained on a $0-1$ vector?
I'd like to state explicitly a problem which was somehow hidden in my three-week-old post:
Does there exist an absolute constant $c>0$ with the property that for any matrix $M\in{\mathcal M}_{m\...
CommunityBot
- 1
1
vote
2
answers
529
views
Dimensionality of a map -- distance
Hello, I am looking for some words to describe what going on here. I'm sure this is not an original thought, so I'd like to read up on more from others who have thought out this topic further.
FORMAT
...
- 940
6
votes
1
answer
2k
views
Matrices that are Hadamard products of $X$ and $X^{-T}$
What are the matrices that you can write in the form $X \odot X^{-T}$, for a complex square matrix $X$, where $X^{-T}$ is the inverse of the complex transpose (not conjugate) and $\odot$ is the ...
CommunityBot
- 1
3
votes
1
answer
303
views
ABA-product of matrices and length of chains of principal inner ideals
Let $k$ be a field, $p,q$ positive integers, and let $R$ be the space of $(p \times q)$-matrices over $k$, and $S$ be the space of $(q \times p)$-matrices over $k$. For every matrix $A \in R$, we ...
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2
votes
2
answers
28k
views
How to write Matlab's dot operators in mathematical expressions?
Matlab has a set of dot operators, such as .*, ./, .^. Each of these operators consists of a dot and a normal algebraic operator. They perform element-wise algebraic operations on a matrix. For ...
CommunityBot
- 1
3
votes
0
answers
474
views
Jacobson-Bourbaki correspondence
The Jacobson-Bourbaki correspondence induces the traditional, finite Galois correspondence by suitable restriction; I've been pondering two things: 1. Are there any (other) interesting applications of ...
- 52.9k
3
votes
0
answers
473
views
Infinite Galois correspondence "according to Artin"
Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and ...
26
votes
2
answers
4k
views
Finite subgroups of unitary groups
Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite ...
- 44.4k
5
votes
1
answer
598
views
Associated graded of filtered module-algebra over a Hopf algebra
I ran across the following statement in a paper, and it seems fishy to me:
Lemma: If $A$ is any Hopf algebra, and if $U$ is an $\mathbb{N}_0$-filtered $A$-module algebra, then $U$ and $\mathrm{gr} (...
- 54.6k
6
votes
4
answers
15k
views
how to find one column or one entry of the matrix inversion
Let $A$ be a square $n \times n$ matrix, which is invertible. Now we want to find the $i$-th column of $A^{-1}$ and one $(i,j)-$ entry of $A^{-1}$. Is there any way to compute only a small of portion ...
- 96.4k
61
votes
3
answers
7k
views
Why is there no Cayley's Theorem for rings?
Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...
- 47.1k
9
votes
2
answers
1k
views
Non-Standard Prime
Hello,
My question is about the non-standard models of the integers. If we add to the Peano's axioms $P$ of arithmetic the following axioms for a fixed constant $c$:
$c \neq 0$, $c \neq 1$, $c \neq 1+...
- 47.1k
8
votes
1
answer
3k
views
When a tensor product of two local rings is a local ring?
This is a follow-up to Is tensor product of local algebras local?.
Let $A, B$ and $C$ be local rings (commutative and noetherian). Suppose that we have local ring maps $C \to A$ and $C \to B$.
What ...
CommunityBot
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11
votes
0
answers
286
views
What do Multilinear Forms tell us about Representations?
The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of ...
- 5,191
11
votes
3
answers
7k
views
How to show the matrix exponential is onto? And, how to create a powerseries for log that works outside B(I,1)
Hi,
I've been looking for a clear reference which shows that the matrix exponential is surjective from $M_{n}(C)$ to $Gl_{n}(C)$. Wikipedia claims this is true, but I haven't seen it proven... ...
- 189
10
votes
2
answers
690
views
Examples of Galois-invariant central simple algebras which aren't base change?
Suppose $L/K$ is a Galois extension of number fields, with Galois group $G_{L/K}$. Write $\mathrm{Br}(L)^{G_{L/K}}$ for the subgroup of central simple algebras $A/L$ which are Galois-invariant; ...
- 4,593
2
votes
3
answers
3k
views
Is there any conclusions generalized Singular Value Decomposition into Hilbert Space
Spectrum decomposition can be regarded as the generalizations of the following fact that:
Every Hermitian matrix $A$ can be decomposed into $A=U^{*}\Lambda U$,where $U$ is a unitary matrix
Singular ...
- 1,706
6
votes
1
answer
875
views
When is a ring the ring of adeles of some global field
Given a global field $F$, we can construct the ring of adeles. Given a general locally compact ring $R$, when is it isomorphic to the ring of adeles of some global field $F$ and how can I find $F$ in $...
- 12.6k
12
votes
4
answers
4k
views
Is any $(n-1)\times (n-1)$ submatrix of an $n \times n$ Vandermonde matrix invertible?
Given an $n \times n$ vandermonde matrix $V$ which is invertible, is any $(n-1) \times (n-1)$ submatrix of $V$ invertible also?
I think the answer is yes, but I don't know how to prove.
- 30.1k
2
votes
0
answers
234
views
Group of automorphisms of localization of polynomial ring
Let P_{n} be a polynomial ring P_{n}:=K[x_{1},...,x_{n}] and let us consider
localization of P_{n} by prime ideal I and denote it via B_{n} and also consider
local Weyl algebra: A_{n}:=B_{n}[d_{1},......
- 109
4
votes
1
answer
284
views
When is Out$(SL_n(R))$ a torsion group ?
This question is a follow up question to this question. So my question is:
For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL_n(R))$ a torsion group? A consequence of ...
CommunityBot
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27
votes
1
answer
2k
views
Strong group ring isomorphisms
Background/Motivation
Let $R$ be a commutative ring with unit. If $G$ is a finite (or in general, discrete) group, let $R[G]$ be the group $R$-algebra associated to $G$. The isomorphism problem for ...
- 23k
4
votes
3
answers
422
views
probability that a random element of Z/NZ can be written as a subset sum of others
How could one calculate the probability that any element in $\mathbb{Z}/N\mathbb{Z}$ can be written as a subset sum of $n$ random elements in $\mathbb{Z}/N\mathbb{Z}$?
In other words, say I pick $n$...
- 30.1k
3
votes
1
answer
218
views
decompositions of matrices over $\mathbb{Q}$
Given a matrix $A\in GL_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL_n(\mathbb{Z}[1/p])$ and $C\in GL_{n}(\mathbb{Z}_{(p)})$, where $ \mathbb{Z_p}$ denotes the ...
- 20.8k
41
votes
7
answers
6k
views
Why don't ideals and quotients work well for categories?
Ideals are intimately related to quotients and congruence relations. They clearly play a very important role in ring theory and order theory. So do normal subgroups in group theory. (Enriched) ...
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8
votes
2
answers
1k
views
Quasi-Lie algebras in nature?
A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand,...
CommunityBot
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0
votes
2
answers
2k
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How to accelerate/avoid multiplication for large matrices in Matlab? [closed]
The setting is here.
X: 6000x8000 non-sparse matrix
B: 8000x1 sparse vector with only tens of non-zeros
d: positive number
M: is sparsified X'X, i.e. thresholding the elements smaller than d ...
- 96.4k
4
votes
1
answer
497
views
Is the functor of divided powers a weakly monoidal functor?
Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\...
- 22.6k
2
votes
3
answers
997
views
About solvable groups
Is it possible for a group (non-simple and non-abelian) that solvability of all of its proper subgroups leads the whole group to be solvable?
- 7,105
9
votes
2
answers
1k
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Question on eigenvalue square root subadditivity
ORIGINAL QUESTION
Let $\lambda_{1}\left(\cdot\right)$ be the larger eigenvalue of a
$2\times2$ matrix and $\lambda_{2}\left(\cdot\right)$ the smaller
eigenvalue of a $2\times2$ matrix. Is it true ...
- 9,486
4
votes
1
answer
643
views
An application of Zorn's lemma.
Suppose that $R$ is a commutative ring, an $R$-module $M$ is said to be finitely embedded if $M$ has a finitely generated essential socle. Now Let $M$ be finitely embedded and not
artinian, let $S$ ...
- 21
2
votes
0
answers
520
views
Eigenvector of infinite matrix
I consider the system of reaction-diffusion PDEs in a ball
with Robin boundary condition.
It is a Steklov eigenvalue problem
(see G Auchmuty (2004) "Steklov eigenproblems and the representation
of ...
- 31
4
votes
1
answer
688
views
Subgroups of U(M_n)
can any subgroup of the unitary group of full matrix alg $M_d(\mathbb{C})$ be approximated on finite
sets by a finite subgroup?
i.e. is the following True or false?
Let $n, d$ be positive integers ...
CommunityBot
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0
votes
0
answers
148
views
Bogus(?) proof for equivalence of normal and dual distributivity conditions in lattices
I am attempting to prove the equivalence of the following two definitions of distributive lattices:
$(a \lor b) \land c = (a \land c) \lor (b \land c)$
$(a \land b) \lor c = (a \lor c) \land (b \lor ...
- 101
27
votes
1
answer
1k
views
Understanding "infinite" relations in groups
Consider the matrices $A = \frac{1}{5}\begin{pmatrix}5&0&0\\\ 2&2&1\\\ 2&1&2\end{pmatrix}$, $B = \frac{1}{5}\begin{pmatrix}2&2&1\\\ 0&5&0\\\ 1&2&2\end{...
CommunityBot
- 1
3
votes
2
answers
344
views
Pseudo-idempotent matrix generating a free module
Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\...
- 6,919
6
votes
1
answer
1k
views
Arnoldi method to compute the dominant eigenvector
Hi, everyone!
I have a problem of computing the dominant eigenvector. When I want to approximate the dominant eigenvector of a large sparse matrix via the famous Arnoldi method, I am wondering how to ...
CommunityBot
- 1
4
votes
2
answers
691
views
Induced p-norm of a Random matrix
This question is related to my earlier question
here .
Given an $n\times n$ random matrix $A$, is determining the properties (mean, variance,moments,etc.) of its induced $p$-norm ($p\neq 0,1,2,\...
- 11.4k
1
vote
0
answers
688
views
The sum of a nilpotent left ideal and a nil left ideal
In class, we recently saw that the sum of 2 two-sided nil ideals is a nil ideal. We were asked to show that the sum of a niplotent left ideal and a nil left ideal is a nil left ideal.
I am having ...
- 33.8k
8
votes
0
answers
916
views
duality between universal enveloping and function algebra for GL(n)
Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular ...
CommunityBot
- 1