All Questions
6,548 questions
1
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121
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Simple algorithm for A107670
Let $T(n, k)$ be A107670 (i.e., matrix square of triangle A107667). Here we define the triangular matrix $P$ by $P(n, k) = \frac{(n+1)^{2(n-k)}}{(n-k)!}$ for $0 \leqslant k \leqslant n$ and the ...
- 4,928
5
votes
1
answer
156
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The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
4
votes
1
answer
230
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$\omega\times\omega$-Hadamard matrices
In the following, we define infinite Hadamard matrices.
Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are approximately orthogonal if $$\...
- 48.8k
19
votes
2
answers
792
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Topology on a module over a topological ring
The questions
Let $R$ be a topological ring, and let $M$ (with no topology) be an $R$-module. Does $M$ somehow "inherit" a topology from the action of $R$?
Here's a proposal for such a ...
- 41.4k
4
votes
1
answer
421
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Visualizing the elements of a finite group and does the Gram matrix determine the finite group?
Let $G$ be a finite group with $n = |G|$ elements. By Cayley's theorem for finite groups, we have an injective homomorphism of groups:
$$
\pi : G \rightarrow S_n, \quad g \mapsto \pi(g)
$$
where ...
- 3,054
10
votes
1
answer
243
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If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 452
0
votes
1
answer
187
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Quotient of a ring by a left ideal
This is a simple algebra question I'm struggling with.
Let $A$ be a ring (with unity) and $I\subset A$ a left ideal and $B\subset A$ a two sided Ideal.
$A/I=B$ and $A/B=I$ (in the category of left $A$...
- 71
7
votes
1
answer
118
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Bound on when sequence of norms of matrix powers starts to decrease
If a matrix $A$ has spectral radius $\rho(A)<1$, it is well-known that $A^n\to0$ as $n\to\infty$, or equivalently $\lVert A^n\rVert\to 0$ for some matrix norm $\lVert\cdot\rVert$; however, it may ...
2
votes
2
answers
227
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Is a probabilistic implementation of unitaries invertible?
Let $\{p_j\}_j$ be a set of probabilities, $\sum_j p_j = 1$, let $\{h_j\}_j$ be a set of $n \times n$ Hermitian matrices, and define $ad_h(A) $ be the adjoint.
Define the following linear mapping
$$ E(...
- 131
15
votes
3
answers
1k
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Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 261
4
votes
1
answer
222
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Recent research on polynomial identities
I work in computational complexity, where I work on the problem of polynomial identity testing over arithmetic circuits. One particular case is when the variables over the polynomial ring don't ...
- 49
2
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1
answer
211
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Direct product of direct sum of a flat module
In the book "Rings and Categories of Modules" by Anderson & Fuller, this problem is given: If $V^A$, i.e. the direct product of the module $V$ by the index set $A$, is flat for all sets $...
- 355
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2
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208
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Real matrix rings and associative hypercomplex numbers
Are there real matrix rings which are not hypercomplex number systems? Is there a canonical form of a real matrix ring?
By a hypercomplex number system I mean a finite-dimensional, unital, associative ...
- 21
4
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0
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158
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Wedderburn-Malcev principal theorem for graded-finite algebras
Let $k$ be a field and $A$ be a noncommutative $k$-algebra with Jacobson radical $J$. If $A$ is finite-dimensional, the Wedderburn-Malcev says that $A$ has a subalgebra $S$ such that $$A = S \oplus J$$...
- 492
1
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0
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48
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How to solve a optimal matching problem where the "quality" of matching is only determined once all nodes are matched
Note: I'm not a mathematician; I'm just a biologist with basic math background trying to solve a scientific problem, so please excuse my ignorance.
The gist of the problem is as follows:
I have two ...
2
votes
0
answers
46
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Are two notions of power-associativity equivalent for loops?
According to Groupprops, a magma $X$ is called power-associative if for every element $x\in X$ there exists a sequence $(x^n)_{n\in\mathbb N}$ of elements of $X$ such that $x^1=x$ and $x^m\cdot x^n=x^{...
- 41.8k
9
votes
2
answers
794
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Why do these finite group Dedekind matrices seem to have integer spectrum when specialized to the order of group elements?
Let $p$ be a prime and let $f_p$ be the permutation on the set $\{1,2,\cdots,p-1\}$ which is given by taking inverses in $\mathbb{Z}/(p)$:
$$x \bmod(p) \mapsto \frac{1}{x} \bmod (p)$$
So for instance, ...
- 3,054
3
votes
1
answer
143
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A problem about matrix inverse and regularization methods
I'm researching the problem of solving the equation $A\mathbf{x}=\mathbf{b}$ with ill-conditioned matrices. We know that if we solve it directly, like $\mathbf{x}=\mathrm{inv}(A)\ast\mathbf{b}$, then ...
- 33
6
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1
answer
206
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What makes the surreals special among other surreal-like fields?
Pre-setup: Let $\kappa = \aleph_\xi$ be an uncountable regular cardinal. Its role in this question is merely to sidestep the technical difficulties surrounding the (imho quite uninteresting) notion ...
- 32.5k
1
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1
answer
87
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An example of a commutative ring which is not SIP
Recall that a module $M_R$ ($R$ is a unital ring) is called an SIP-module if the intersection of any two summands of $M$ is a summand. The ring $R$ is called (left) right SIP-ring if the module (${}...
- 324
3
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36
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Computing the truncations (“ancestors”) of a surreal number from its Hahn series representation (“normal form”)
If $x$ is a surreal number and $\alpha$ an ordinal, let us denote $T_\alpha(x)$ and call $\alpha$-truncation of $x$ the surreal number whose sign sequence is obtained by truncating the sign sequence ...
- 32.5k
6
votes
1
answer
199
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Combinatorial type construction of the free operad
$\DeclareMathOperator\RT{RT}$I am reading the book "Algebraic operads" by J. L. Loday and B. Vallete. The authors have given a combinatorial construction of the free operad over an $\mathbb{...
- 229
4
votes
1
answer
170
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About $CW(512,16^2)$
Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$,
where $I$ is the identity matrix. A circulant ...
- 696
3
votes
0
answers
161
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Generalized dimension property for rings
My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$.
For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
- 223
2
votes
1
answer
153
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What conditions on the rate matrix $Q$ ensure unique convergence in continuous-time Markov chains?
In the study of discrete-time Markov chains, the conditions under which all initial distributions converge to a unique stationary distribution are well-understood. Specifically, if the transition ...
- 807
3
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118
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A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
5
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1
answer
168
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Countably compact Boolean algebras versus distributivity
Let us say that a complete Boolean algebra $B$ is:
countably distributive when for any sequence $(I_n)_{n\in\mathbb{N}}$ of sets and any elements $(u_{n,i})_{n\in\mathbb{N},i\in I_n}$ of $B$ we have
$...
- 32.5k
3
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1
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290
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How do you define the composition of two $\mathbb{S}$-modules?
I am reading the book "Algebraic Operads" by J. L. Loday and B. Vallete. I am stuck with the definition of composition of $\mathbb{S}$-modules in Sec-5.1.6, pg. 99. Below I have written down ...
- 229
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1
answer
66
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Correct conditions for the image of a matrix to intersect a cone?
Given an $m \times n$ real (or rational) matrix $A = (a_{ij})$, what are necessary and sufficient conditions for the image of this matrix to intersect a cone? I am specifically interested in the cone $...
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3
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0
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83
views
Non-commutative Gorenstein Koszul algebras
I was wondering if there exists a characterization of finite-dimensional Gorenstein Koszul algebras in terms of their Koszul dual?
- 359
1
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0
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40
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Bound of entries of inverse of a unimodular matrix whose row sum is bounded
Many questions have been asked about the bound of the entries of the inverse of a matrix subject to certain conditions. Here my condition is slightly different: let $A=(a_{ij})$ be an $n \times n$ ...
- 895
10
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3
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455
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When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 145
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2
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558
views
Whether an isotone bijection from a power set lattice to another sends singletons to singletons
By the work of Paul Cohen (on the continuum hypothesis), one can neither prove nor disprove from the axioms of ZFC that a bijection $f$ from the power set $\mathcal{P}(S)$ of a set $S$ to the power ...
- 10.5k
0
votes
1
answer
113
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Constructing an adjunction between algebras and differential graded algebras
Fix a ring R. I am looking for a construction of the adjunction between R-algebras and differential graded R-algebras. I am looking for a reference which constructs the left adjoint to the functor ...
user534056
6
votes
1
answer
127
views
Intersection of integral points with a unipotent and its opposite
This is a follow-up to Does the bruhat decomposition induces decomposition on integral points (on an open cell)?
Given a split connected reductive group $G$ over a $p$-adic local field $F$ with ring ...
- 503
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0
answers
80
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Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
- 193
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1
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75
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Lemma of Harada and Sai on sums of modules with a "chain" of monomorphisms between them
I am trying to get a contradiction from the following set of hypotheses:
Let $R$ be a ring. Let $M$ be a direct sum of non-zero $R$-modules $M_1$, $M_2$, $\dotsc$. For each $i\ge1$, let $f_i:M_i\to M_{...
- 1,644
1
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1
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51
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Exceptional Lenz-Barlotti classes IVa.3 and IVb.3
On this web-site, devoted to the Lenz-Barlotti classification of projective planes, it is written that the class IVa.3 (and its dual IVb.3) is somewhat exceptional, because it contains exactly one ...
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13
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Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?
Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
- 3,772
2
votes
0
answers
71
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Lexicographically largest incidence matrix
I have simple algorithmic question, but I can't find any source where this algorithm is explained in details.
Let's assume that we have incidence (with 0 and 1 values) matrix of size $m\times n$. Let ...
- 501
2
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0
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104
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$\mathrm{Tor}$'s for submodules of division rings
Let $R$ be a ring, $D$ a division $R$-ring in which $R$ embeds, and $M$ a finitely generated $R$-submodule of $D$. What, if anything, can be said about the finiteness properties of $M$? $\mathrm{Tor}^...
- 163
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434
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How to define $\mathbb{R}^\frac{1}{2}$?
The Cayley-Dickson construction generates higher-dimensional hyper complex numbers from lower-dimensional ones, producing algebras of dimension $2^n$.
I want to generate an algebra of dimension $2^{-1}...
- 75
4
votes
0
answers
79
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Closed character formula for the module $L(a\omega_i)$
Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\omega_1, \omega_2, \dots, \omega_n\in\mathfrak{h}^{*}$ is the ...
- 41
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43
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Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_\text{min}$, of $A$?
What I want is something like: $\sigma_\...
1
vote
0
answers
134
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Number of ways to place 4 kings on nxn chessboard
I have a $n\times n$ chessboard and 4 kings inside it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example:
In the case where the $4$...
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1
vote
1
answer
391
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How one can show that this matrix is full rank?
Fix $d\in\mathbb{N}$ and consider $e_{i,j}\in\mathbb{C}$ for $i=1,\dots,d+3$ and $j=1,\dots,d-1$. Suppose to have the following matrices
$$N_{i,1}=\begin{pmatrix}
1 & 0 \\
e_{i,1} & 1
\end{...
- 11
1
vote
0
answers
34
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An algebraic characterization of dual translation projective planes
It is well-known that translation projective planes are coordinatized by quasifields. More precisely, a projective plane is translation if and only if it has a ternary-ring $R$ which is linear, the ...
- 41.8k
1
vote
0
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73
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Relationship between units and primitive characters 2
This is a follow up to this question.
Let $(R,+,\cdot)$ be a finite ring.
Definition Given the dual group $\widehat{R}$ of $(R,+)$, a character $\chi\in\widehat{R}$ is said to be primitive with ...
1
vote
1
answer
215
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Reference about cancellation property for semigroups
Have the semigroups with the following cancellation property been studied?
Property: Let $S$ be a semigroup and $x,y\in S$ such that $xz=yz,$ for all $z\in S,$ then $x=y$.
- 313
7
votes
0
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131
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Approximation of a continuous curve on commuting matrices
I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that
$[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
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