All Questions
2,027 questions with no upvoted or accepted answers
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Characterization of "PSD-Squared" Matrices
$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...
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Cohomology and higher structures
Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear $G$-action on $M$, there is an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ ...
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Inequalities for trace/eigenvalues of product of multiple 2x2 matrices
Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
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Frobenius algebras of small dimensions
In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension ...
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Prime/irreducible elements in certain (integral) ring extensions
The answer to this question says the following:
Let $R$ be a finitely generated $k$-algebra, where $k$ is a field.
If $p \in R$ is a prime element, then $p$ is a prime element in $\tilde{R}$, the ...
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Generalization of semi-hereditarity
Let $R$ be a ring. A left $R$-module $K$ is called an $N$-th kernel if there are projective left $R$-modules $P_1, \ldots P_N$ and a short exact sequence
$$ 0\rightarrow K \rightarrow P_N \rightarrow \...
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Other kinds of equivalence relations on the set of idempotents of a Banach or $C^*$-algebra or a ring (Can we get a new kind of K-theory?)
The standard equivalent relations on idempotents of a $C^*$ algebra or a Banach algebra are Murray von Neumann, similarity and homotopy equivalent. In this post we consider two other kinds of ...
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Division in the universal enveloping algebra
Let $\mathfrak g$ be a (semisimple) Lie algebra, $\mathfrak b\subset \mathfrak g$ a Borel and $\mathfrak n = [\mathfrak b,\mathfrak b]$. Then I am interested in solving certain division problems in $U(...
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Differential graded Lie algebra over an ordinary Lie algebra
Given a dg (differential graded) Lie algebra $L$ and an ordinary Lie algebra $\mathfrak{g}$,
is there written somewhere a formal definition of $L$ as a dg Lie algebra over $\mathfrak{g}$?
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Matrix series with Hadamard products
Let $A$ and $B$ be hermitian matrices (a special case that would already help would be $A^{-1} = B^T$). I'm looking for a closed form of the series
$$X := \sum_{n=0}^\infty A^n \circ B^n$$
where $\...
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Mysterious identity in cosimplicial $R$-module with Lie brackets
I have a cosimplicial $R$-module $\mathscr{A}:=(A_n)_{n\ge 0}$ and on each $A_n$ a Lie bracket $[-,-]_n:A_n\otimes A_n\to A_n$. Denote the cofaces by $d_i:A_n\to A_{n+1}$ and the codegeneracies by $...
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Cyclic relation algebra
A relation algebra $\mathbf{R}$ is a structure $\langle |\mathbf{R}|, \vee, \neg, \circ, I, (-)^{op} \rangle$ such that:
$\langle |\mathbf{R}|, \vee, \neg \rangle$ is a Boolean algebra,
$\langle |\...
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Do comodules form an exact category?
Let $R$ be a commutative ring, $C$ a coalgebra over $R$. I am asking about the category of $C$-comodules $C$-Comod.
It is clear that if $C$ is a flat $R$-module, then $C$-Comod is abelian. Hence, is ...
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Contractible Banach algebras
A Banach algebra $A$ is contractible if $H^1(A, X)=0$ for all Banach $A$-bimodules $X$. Now to my question
Let $A$ be Banach algebra and $I$ be closed ideal of $A$. If $I$ and $A/I$ are both ...
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How to detect if an element in a completed group algebra is a unit?
Completed group algebras appear in Iwasawa theory and in many situations you want to know if an element is a unit. For example consider $K_n=\mathbb{Q}(\mu_{p^n})$ and $G=\varprojlim \mathrm{Gal}(K_n/\...
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Number of ideals in an algebra
Let $R_{n,m}^q$ be the finite dimensional algebra $K\langle x_1,...,x_n\rangle/J^m$, where the field $K$ has $q$ elements and $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring with ...
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210
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A presentation for a subalgebra
Let $K$ be a field, and let $I=(g_1,\ldots, g_r)$ be an ideal in $A:=K[X_1,\ldots ,X_n]$.
Let $\{f_1,\ldots f_m\}$ be a subset of $A$, and let $B$ be the $K$-subalgebra of $A$ generated by $f_1,\...
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Injective resolution of the ring of entire functions
Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...
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Recovering the bimodule from the trivial extension
Given a ring $S$ with a non-zero $S$-bimodule $M$, the trivial extension of $(S,M)$ is defined as the ring $R:=T_M(S)$ with $R= S \oplus M$ with multiplication $(s,m)(s',m')=(s s', sm' +m s')$.
We ...
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Does linear independence imply algebraic independence for partitioned homogeneous polynomials?
Define a partitioned homogeneous polynomial of degree $d$ to be a polynomial in $$\mathbb Z[x_{11},\dots,x_{1n},\dots,x_{d1},\dots,x_{dn}]$$ with monomials from entries in (polynomials that are $d$-...
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A relation between $Spec((1+I)^{-1}R)$ and $Spec(R/J)$
Let $R$ be a commutative ring with identity and let $I$ and $J$ be two finitely generated ideals of $R$. Clearly $1+I:=\{1+i:i\in I\}$ is a multiplicative closed subset of $R$. We can consider the ...
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129
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How to formulate supercommutativity in a characteristic free way?
I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, ...
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Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...
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Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
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Rank of binary matrix related to the number of positive squarefree integers less than $n$
I posted this question at the Mathematics SE, but received no response there so I am posting it here.
The following fact is stated in the comments-section of sequence A013928 in the OEIS.
Let $C$ ...
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How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
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Generalizing Autonne-Takagi factorization
Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that:
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
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4
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108
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Does this fact about the minimal polynomial give an efficient diagonalizability criterion?
I am ready to agree beforehand that this looks more like a math.SE question. I posted it there a week ago without any feedback (except for 27 views and 2 upvotes).
Besides, I really need an answer. ...
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467
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Quaternion algebras in characteristic 2
Let $k$ be a field and let $Q$ be a quaternion algebra over $k$.
It is well known that, if $\mathrm{char}\,k\neq 2$, one can define $Q$ as the $k$-algebra of dimension $4$ generated by elements $x,y$ ...
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Totally Unimodular matrix edited from ordinary matrix
Given a matrix $M\in\{0,1\}^{m\times n}$ is there an algorithm to tell if we can convert some of $1$s to $-1$s and make $M$ Totally Unimodular and output such a Totally Unimodular in polynomial in $mn$...
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An upper bound on the Jordan condition number of a matrix
The Jordan condition number of a matrix $A$ is defined to be $\min_{V}\kappa(V)$, where $V$ ranges over complex matrices that satisfy $A = VJV^{-1}$ for $J$ being the unique Jordan normal form matrix ...
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Maximizing a certain eigenvalue ratio
Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
- 2,712
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Closing Subsets Under Operations
My question is about closing sets under operations. First, I need a definition:
Definition: Let $A$ be a set and take a function $f : A^n \rightarrow A$ for $n \in \mathbb{N}_{\geq 0}$. For a set $S$,...
user30211
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A Toeplitz variant of the Hilbert matrix
It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries
$$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$
determines a bounded operator on $\ell^{2}(\mathbb{N}...
- 2,188
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tensor products of noetherian domains
Under what conditions is the tensor product of two non-commutative Noetherian domains also a Noetherian domain? To be more precise about the problem, I am looking at rings of fractions on such tensor ...
- 1,143
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Distribution of min/max row sum of matrix with i.i.d. uniform random variables
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $1$.
all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables ...
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Question on $n$-torsionless modules
Let $A$ be a finite dimensional algebra. Recall that a module $M$ is called $n$-torsionfree in case $Ext_A^i(D(A),\tau(M))=0$ for all $i=1,...,n$ when $\tau$ denotes the Auslander-Reiten translate. ...
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Serre duality graded singularity category
Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that ...
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Non-singularity of a series of matrices
Let $A_1$, $A_2$ be $n\times n$ real matrices. Suppose that $A_1$ and $A_2$ are Schur stable (i.e., their eigenvalues are strictly inside the unit circle in the complex plane). Let $B_1$, $B_2$ be two ...
- 2,712
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"Singularly convex" cones of matrices
The ambient space if ${\bf M}_n({\mathbb R})$.
Let us begin with facts.
1- The cone of positive semi-definite symmetric matrices is convex. 2- It is a little subtler that the cone $K^+$ of matrices ...
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On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix
(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows:
$$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$
Let $...
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Extending a Theorem of Brualdi to Matrices with Infinitely Many Rows
This question is about extending a result on transportation polytopes from Brualdi regarding $m\times n$ matrices to the case when $m=\infty$.
Notation: Denote an $m\times n$ matrix by $A=[a_{i,j}]$, ...
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Sextic resolvent rings of quintic rings
In Higher composition laws IV: The parametrization of quintic rings M. Bhargava gave an explicit parametrization of quintic rings by quadruples of $5\times5$ skew-symmetric matrices. His proof hinges ...
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149
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Zero diagonal nonsymmetric block checkerboard matrix: orbits and numerical ranges
Let $A \in \mathbb{R}^{m \times m}$ be a nonsymmetric zero diagonal matrix with a zero/non-zero pattern which is symmetric and persymmetric (i.e. symmetric in the northeast-to-southwest diagonal).
If ...
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Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
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127
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Injective dimension is infinite?
Let $A$ be a non-selfinjective finite dimensional algebra and $M$ a nonprojective module with $Ext^{i}(M,A)=0$ for all $i \geq 1$. It is easy to see that $M$ has infinite projective dimension. Does $M$...
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Is there a nice way to express a matrix exponential when rows are proportionally scaled?
Assume I am given an $n \times n$ matrix $A$ with real or complex coefficients. Its matrix exponential is denoted by $\exp(A)$ and is calculated as usual. Assume further that I want to rescale the ...
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How can one prove that the algebra of smooth functions is semisimple?
I have read in some differential geometry works that the ring of smooth functions $C^{\infty}(U)$ is a semi-simple ring, for $U\subseteq\mathbb{R}^n$ an open set; right now I can cite a remark ...
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What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
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Perturbation of a rank-restricted product of matrices
I formulated a statement, which is hopefully true (at least I'm not knowledgeable enough to see a reason for it not to be). However, I'm struggling to come up with a proof.
Let $W_i \in \mathbb{R}^{...
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