All Questions
6,547 questions
3
votes
0
answers
271
views
Algebraic Kneser conjecture?
Recall that Kneser conjecture (now Lovasz theorem) claims that if the family of $k$-subsets (subsets of cardinality $k$) of given $(2k+d)$-set $M$, $d\geq 1$ are colored into $d+1$ colors, then there ...
- 109k
6
votes
7
answers
5k
views
Best way to teach concept of real numbers using a hands-on activity?
I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
- 45.7k
3
votes
1
answer
336
views
Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$.
The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$
has the bracket
$$[xt^r, yt^...
- 138
2
votes
1
answer
757
views
Maximal subfield inside a central division algebra
D is a central division algebra over F. We know that we can always find a maximal subfield K inside D such that K/F is separable. I want to know can we always make it Galois?
- 65.4k
14
votes
2
answers
6k
views
What is the constant of the Coppersmith-Winograd matrix multiplication algorithm
Or at least it's order of magnitude.
I've only ever heard it described as "huge", and a google search turned up nothing.
Also, given that the Strassen algorithm has a significantly greater constant ...
- 216
2
votes
2
answers
535
views
Sequence of constant rank matrices
Let $A_k$ be a sequence of real, rank $r$, $n$ x $m$ matrices such that $A_k$ converges to a rank $r$ matrix $A$. Let $v_k, u_k$ be sequences of vectors such that $u_k\rightarrow u$ and $A_k v_k=u_k$. ...
- 32.4k
15
votes
1
answer
633
views
Introduction to "commutative semialgebra"?
Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on.
However, there are some instances (most obviously tropical geometry) ...
- 85.5k
3
votes
1
answer
538
views
Non-negative matrices with prescribed Perron-Frobenius eigenvectors
In my research I came across the following question.
Let $A$ be an integer non-negative matrix (every entry of $A$ is non-negative) and $x = (x_1,...,x_n)^T$ the probability Perron-Frobenius ...
- 3,929
5
votes
2
answers
752
views
Is there a name for this algebraic structure?
I found myself "naturally" dealing with an object of this form:
X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and ...
- 47.3k
19
votes
2
answers
7k
views
Generalization of the shakehands/condom puzzle?
The classic handshake puzzle goes something like this:
"Given that everyone has a different skin disease, how can you safely shake hands with 3 people when you have only 2 gloves?"
Its common ...
- 27.7k
6
votes
2
answers
2k
views
Examples of one-dimensional non-Cohen Macaulay rings
Can you offer some examples of such rings, other than $\frac{k[x,y]}{(x^{2}, xy)}$. Thanks.
- 30.5k
2
votes
2
answers
718
views
Algebra / unital associative algebra: better terminology?
In Bourbaki an algebra over a commutative ring $k$ is defined to be a $k$-module $A$ together with a $k$-bilinear map $A \times A \rightarrow A$. We then have the obvious notion of morphisms of $k$-...
- 5,233
2
votes
2
answers
423
views
characterization of a submodule
In a vector space $V$ over a field $F$, a nonempty subset $W$ of $V$ is a subspace if it is closed under addition and scalar multiplication. For a module $M$ over a ring $R$ with identity the similar ...
- 1,411
1
vote
1
answer
146
views
equivalence of submodules
I have Z^3/M = Z^3/N = Z_k where M,N are submodules of Z^3 and Z_k is cyclic order k.
I would like to say some SL_3(Z) transformation takes M to N. Is this true? How to show?
- 47.3k
4
votes
1
answer
1k
views
inverse m-matrix
The following is a result by C.R. Johnson appearing every now and then in the literature.
Let $A$ be an $n \times n$ inverse $M$-matrix. Then
All principal minors of $A$ are positive.
Each ...
- 65.4k
0
votes
2
answers
258
views
Existence of an "anti-additive" (or "never linear") map?
(I've edited this question)
I'm searching for a continuously differentiable function $f:\mathbb R^2\to\mathbb R$ such that $f(x)+f(x+u+v)\neq f(x+u)+f(x+v)$ for all $x$ and all linearly independent $...
- 28.9k
9
votes
1
answer
1k
views
First-order UFD (factorial ring) condition / pre-Schreier rings
All rings in this post are commutative and with $1$.
Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...
- 33.8k
1
vote
2
answers
175
views
Freeness of the Canonical $SU(n)$ Action
I have another question about $SU(n)$, again I hope it's not too basic. For $n=2$, the action of $SU(2)$ on $C^2$ is free since it's equal to the group of rotations. In general, the group of rotations ...
- 28k
1
vote
2
answers
252
views
Describing $SU(n,C)$
For $A \in SU(2,C)$, it is clear that $A$ is completly determined by its first row (well any row or column, but let's say first column). In the general $SU(n,C)$-case this is no longer true. In fact, ...
- 25.8k
4
votes
3
answers
388
views
Characterizing nilpotents in a ring by a universal property
This is again a question asked to me by this user. He apparently quit using MO due to a busy time in personal and professional life and resulting difficulties in spending time here with patience. I am ...
CommunityBot
- 1
5
votes
0
answers
539
views
An inverse eigenvalue problem on Jacobi matrices
I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
- 25.8k
0
votes
1
answer
460
views
Sp(2n) intersect Sp(2n,H)? (Please read for explanation of notation)
First let me fix some notation:
Let $O(n)$ be the group of $n \times n$ real matrices $T$ which are "orthogonal", $U(n)$ be the group of $n \times n$ complex matrices $T$ which are "unitary" and $Sp(...
- 54.6k
5
votes
1
answer
551
views
What is an exponential?
Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on
tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential ...
- 2,756
2
votes
2
answers
3k
views
Statement of Lagrange's theorem on determinants(elementary question).
Apologies for this elementary question; but I was unable to find a reference otherwise.
Let $A, B, C$ be square matrices of the same dimension. Then,
$$\begin{vmatrix} A & C \\\ 0 & B \end{...
- 33.8k
3
votes
2
answers
611
views
Computing Integral Closures
I'm wondering if there's an algorithm, or a program I can use, to compute integral closures. Specifically, what I have in mind are variants of questions of the sort: what is the integral closure of &#...
- 156k
4
votes
2
answers
322
views
A hands-on description of a "completion" of the free commutative monoid on countably many generators
This is basically an I'm-weak-at-algebraic-geometry question. I asked it as a warm-up question here, but Ilya N asked me to break that post up into several questions.
Consider the free commutative ...
CommunityBot
- 1
9
votes
3
answers
987
views
Octonionic Unitary Group?
Hi all.
I was wondering if anyone has any references on work related to the Octonionic Unitary group. I would imagine that such a group would be generated by Octonionic skew-Hermitian matrices (at ...
- 33.3k
5
votes
1
answer
2k
views
Self-similar matrices? [closed]
Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?
- 7,800
11
votes
2
answers
1k
views
What are important examples of filtered/graded rings in physics?
Hi,
what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics?
Of ...
- 33.3k
1
vote
2
answers
922
views
Extremum under variations of a traceless matrix
Sorry for my precedent tentative, I was a little hasty:
Ok, I think I'd better put the original problem:
I have an action of three fields: $A$ which is the spin-connection, $B$ an skew-symmetric 2-...
- 33.3k
5
votes
1
answer
572
views
Non-smooth algebra with smooth representation variety
A not necessarily commutative algebra A (over C, say) is called formally smooth (or quasi-free) if, given any map $f:A \to B/I$, where $I \subset B$ is a nilpotent ideal, there is a lifting $F:A \to B$...
- 4,267
10
votes
1
answer
835
views
what was Hilbert's geometric construction in his 17th problem?
Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
- 3,396
2
votes
1
answer
160
views
Lower bound for characteristic variety
Let K be an algebraically closed field of char. 0, let A_n(K) be the Weyl algebra. Let I in A_n(K) be a left ideal generated by p elements. Set M := A_n / I.
Does the following then hold?
dim Ch(M) \...
- 3,131
3
votes
2
answers
889
views
computation, algebra, logic
So a really simple way of describing a digital computer is to say that it is a device for performing boolean operations. You feed it a bunch of bit strings, which is a description of the problem and ...
- 236k
11
votes
1
answer
410
views
An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
- 2,600
6
votes
1
answer
875
views
What should Spec Z[\sqrt{D}] x_{F_1} Spec \bar{F_1} be?
What should be $\text{Spec } \mathbb{Z}[\sqrt{D}] \times_{\mathbb{F}_1} \text{Spec } \overline{\mathbb{F}}_{1}$?
Sure, there's more than one definition.
I'm looking for any answer that uses at least ...
- 2,941
2
votes
1
answer
651
views
Splitting matrix of rank one
Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc
Let A=(a i j ) a matrix with elements in R and dimension n x m.
Suppose
rank A=1 ↔ all 2 x ...
- 156k
0
votes
2
answers
119
views
Properties of adjacent submatrixes [closed]
Hi!
I've encountered a matrix problem when designing an algorithm, which I cannot seem to figure out. I have a (square) matrix with the following properties:
j<k → aij<aik, aji<aki
aij&...
- 5,992
17
votes
3
answers
1k
views
R2 and S3 for rings.
For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition ...
- 15.1k
13
votes
1
answer
5k
views
What are tame and wild hereditary algebras?
What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...
- 47.3k
5
votes
5
answers
5k
views
Notions of Matrix Differentiation
There are a few standard notions of matrix derivatives, e.g.
If f is a function defined on the entries of a matrix A, then one can talk about the matrix of partial derivatives of f.
If the entries of ...
- 2,284
4
votes
1
answer
135
views
(in-)compatible gradings of an associative algebra tell us...?
If an associative algebra A is $\mathbb{Z}$-graded, then it is automatically $\mathbb{Z}\_2$ (aka $\mathbb{Z}/2\mathbb{Z}$) graded by defining $A\_{\bar{0}}$ to be ...
- 156k
1
vote
4
answers
385
views
Is it that only with normal matrices, the transition matrix to its [del: inherent] [ins: own] basis is unitary?
Does this even make sense what I translated into english?
PS. I am probably gonna delete this question eventually
- 13
1
vote
2
answers
5k
views
Elliptical rotation matrix [closed]
We can rotate a point 'circularly' about an arbitrary axis:
the equation is here, but this site doesn't trust me enough yet to post an image.,
But as we walk theta 0 -> 2PI this takes the point ...
- 9,366
2
votes
1
answer
978
views
What is the comultiplication of a matrix frobenius algebra?
One of the easiest examples I can think of for frobenius algebras is a plain ol' matrix algebra with tr : V → k as the co-unit (or equivalently, tr(a⋅b) as the frobenius form). This is ...
- 54.6k
6
votes
3
answers
4k
views
Intuitive Example of a Jacobson Radical
Can anyone explain what a Jacobson radical is using an intuitive example? I can't quite understand Wikipedia's explanation.
- 8,681
9
votes
1
answer
643
views
Determinant of a pullback diagram
Suppose that X and Y are finite sets and that f : X → Y is an arbitrary map. Let PB denote the pullback of f with itself (in the category of sets) as displayed by the commutative diagram
PB &...
- 25.2k