All Questions
2,027 questions with no upvoted or accepted answers
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Information on structure (CI-magma with (non surjective)homorphism) of chemical transformations
Thinking about the mathematical structure of chemical transformations, between all possible components (educts, products) it occurs to me, that this structure is a commutative-idempotent groupoid(=...
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148
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Traces in associative algebras
Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
- 283
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76
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specific sequence of matrices making a strange ratio of matrix norms diverging
For any $t>0$ define $d_t:=\operatorname{diag}j^t=\operatorname{diag}(1^t,2^t,\ldots)$. Now pick up such a $t>0$ and an arbitrary $\theta\in\big(0,\frac12\big)$. For every $k\in\mathbb{N}$ find ...
- 375
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48
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Is simreflexive left-right symmetric?
Call an Artin algebra (we can assume that it is basic) $A$ simreflexive in case every simple A-module is reflexive. Is A simreflexive iff the opposite algebra $A^{op}$ is simreflexive? Or equivalently ...
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67
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When is $R/Soc(R)$ reduced?
Let $R$ be a ring with identity. It is readily checked that when the quotient $R/S_r$ is reduced, the nilpotent elements of $R$ fall into $S_r$, where $S_r$ is the right socle of $R$. Is the converse ...
- 355
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78
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Compute action of the gauge group in deformation theory of an algebra
I am working on Balazs Szendroi's introduction to deformation theory, but I got stuck on the exercise on the bottom of page 6.
Consider a vector space $A$ with a multiplication $m$ that makes it into ...
- 1,509
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138
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When is a Frobenius Algebra a Quasi-Frobenius Ring?
Let $F$ be Frobenius algebra in the monoidal category $\mathcal{C}$ of bimodules over a not-necessarily commutative algebra $A$. When is it true that $F$ is a quasi-Frobenius ring.
For example, this ...
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134
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$\Omega^1_{B/A}=0$ implies $A\subset B$ unramified
Let $(A,\mathfrak{p})\subset(B,\mathfrak{q})$ two local rings, such that $B$ is finitely generated as $A$-module. It's very well known, from Algebraic Geometry, that if $\Omega^1_{B/A}=0$ then the ...
- 1,252
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Threshold rank of matrix summation
Matrices $M_1,M_2,\dots,M_{k-1},M_k\in\{0,1\}^{n\times n}$ with real ranks $r_1,r_2,\dots,r_{k-1},r_k$ respectively.
What is the rank of the matrix $M=M_1\oplus M_2\oplus\dots\oplus M_{k-1}\oplus M_k\...
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$XOR$ function rank of matrices $ $
Matrices $M_1,M_2\in\Bbb Z^{n\times n}$ with ranks $r_1$ and $r_2$ respectively.
What is the rank of the matrix $M=M_1\oplus M_2\in\Bbb Z^{n\times n}$ (not $\Bbb F_2^{n\times n}$) which is defined by:...
- 13.9k
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172
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A vanishing sum of symmetric matrices
Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
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115
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On the complexity of writing down matrices
Consider families of $0/1$ matrices in $\Bbb B$ where $1+1=1$:
$\mathcal M_{1,n,c}$ contains $2^n\times 2^n$ matrices that can be written as Hadamard product of $t=O(2^{(\log n)^c})$ matrices $$(J_n-...
- 13.9k
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Empirical approaches to validate observational bounds on minimum gap between least eigenvalues of $n \times n$ correlation matrix and its submatrices
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an $(n-1) \times (n-1)$ submatrix of $\Sigma$ obtained by eliminating the $i$-th row ...
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312
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Distribution of top singular vector of Bernoulli ensemble
Consider the distribution given by taking the top singular vector of a matrix whose entries are equally likely to be +1 or -1.
This is not well-defined for matrices where the subspace achieving the ...
- 101
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155
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Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem
Is it possible to derive Schur-Zassenhaus theorem from Wedderburn-Malcev theorem? For the existance part I have the following idea: Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $...
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138
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Is every monosemiring an idempotent semiring?
Is every monosemiring an idempotent semiring?
To make my question clear, let me give definitions as follows:
A semiring $(R, +, .)$ is said to be monosemiring if $x.y= x+y$ for all $x, y$ in $R$. And ...
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101
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Operator norm for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$
Suppose $C$ is a $n$ by $n$ real symmetric matrix, and $x\in R^n$. Is there an operator norm of $C$ for $\max\frac{\Vert x \Vert _1}{\sqrt {x'Cx}}$?
If I decompose $C$ into $A'A = C^{-1}$, It seems ...
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112
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shifts in Baer*-rings
Let $A$ be a Baer*-ring with the unit $1$. An important point that holds in every Baer*-ring is this: for every element $y\in A$ there is an smallest projection $e_y\in A$, called the left projection ...
- 4,058
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174
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Approximating symmetric matrices by symmetrized low rank matrices
Fix an integer $k$, and suppose $M$ is a real symmetric $n\times n$ matrix admitting a decomposition:
$$
M = A + A^t + B
$$
with $\mathrm{rank}(A)=k$ and:
$$
\|B\|_2 \ll \lambda_{1}(M_{|\mathrm{range}(...
- 121
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125
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Obstruction theory on $A_{\infty}, C_{\infty}$-algebras
Let $\mathcal{P}_{\infty}$ be $A_{\infty}$ or $C_{\infty}$. Let $A=A^{1}\oplus A^{2}$ be a graded vector space concentrated in degree 1 and 2. Let $m_{n}\: : \:{A^{1}}^{\otimes n}\to A^{2}$ be a ...
- 1,424
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124
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Algebra Invariants of Schubert Calculus
For the Grassmannian Gr[N,k] of $k$-planes in $\mathbb{C}^N$, the cohomology ring $H^*(Gr[N,k])$ is a much studied object in an area called Schubert calculus. As a complex algebra, $H^*(Gr[N,k])$ is ...
- 449
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201
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The order of the antipode in a Hopf algebra
As a result of Radford, any finite-dimensional Hopf algebra an antipode of finite order.
My question: How can we classify all finite-dimensional Hopf algebras whose antipode is identity?
Here are ...
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122
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Fullrankness of sum of time shifts
I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now.
Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
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194
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generators for a graded algebra
I have a graded commutative algebra over a field $K$ and I also know that it is not generated by the degree 1 elements. Somehow I could guess a set of generators, is there a criterion to check that ...
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294
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Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
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95
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Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?
Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring.
I want to prove
that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
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69
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A possible conjecture on exponential asymptotics of random recursion relations
I have come to suspect that the following is true (and have confirmed it with some numerical experiments) but I have no idea how to prove it.
Background: Let $f(z) = \sum_{n=0}^N a_n z^n$ be some ...
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67
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References on infinite dimensional graded algebras
I am currently working on some generalisations of known results in 2-category theory, and I have been stymied trying to find references that discuss `graded-finite dimensional' algebras over a field - ...
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Is $L'L_\text{in}+L_\text{in}'L$ positive semi-definite?
Assume that $A$ is the adjacency matrix of a strongly connected directed graph, that is, $A$ is non-negative and irreducible. Let $$L_\text{in}=D_\text{in}-A',\;L=D_\text{in}-A'+D_\text{out}-A$$ where ...
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Cut norm and biclique gap?
Given real $\pm1$ matrix $M\in\Bbb R^{n\times m}$ we have that cut-norm is given by $$\|M\|_C=\max_{\mathcal I\subseteq[n],\mathcal J\subseteq[m]}\Big|\sum_{(i,j)\in\mathcal I\times\mathcal J}M_{ij}\...
- 13.9k
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relationships between $AA^T$ and $[(I-A)(I-A)^T]^{-1}$ with $A$ being strictly lower triangular
I have a matrix $A$ which is strictly lower triangular. Now, I am trying to find some general statements/relationships of following matrices $U,D,V,K$ defined as:
$AA^T=VKV^H$,
$[(I-A)(I-A)^T]^{-1}=...
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650
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Bound of the eigenvalues of a matrix product of two diagonal and one symmetric PSD matrices
Let B be a positive diagonal matrix, and C a PSD symmetric matrix with eigenvalues $\Lambda$, with $\lambda_i = {0,1}$. I'm trying to find a reference for the following bounds or similar
$B^{1\over 2}...
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53
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Generic properties of families of algebras over an infinite dimensional base space
Let $\mathbb{k}$ be an algebraically closed field and let $A$ be a $\mathbb{k}$-algebra which is a free module of rank $r$ over some central subalgebra $Z_0$. If $Z_0$ is affine and $r$ is a finite ...
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59
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A sufficient condition for finite *-rings!
Let $R$ be a unital *-ring. Assume that $R$ has finitely many projections.
Q. Can we conclude that $R$ is finite?!
$\bullet$ We say $R$ is finite if $x^*x=1_R$ implies that $xx^*=1_R$.
$\bullet$ ...
- 4,058
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Hochschild coboundary on the space of alternative forms
Assume that $A$ is a complex algebra. By $C^{n}(A)$ we mean the space of all $n-$linear map $\phi:A^n \to \mathbb{C}$. An alternative $k-$ form is
an element $\phi \in C^{k}(A)$ ...
- 356
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Closedness of the range of the distorsion of the multiplicative monoid of a number field
Let $H$ be a multiplicatively written monoid with identity $1_H$. An atom of $H$ is an element $x \in H \setminus H^\times$ such that $a \ne xy$ for all $x, y \in H \setminus H^\times$, where $H^\...
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How to efficiently evaluate the action form of a matrix power?
Binary exponentiation is a well-known method for evaluating positive integer powers of a matrix, $A^p, \; A\in\mathbb C^{n\times n},\,p\in\mathbb Z^+$.
However, I am not seeing an obvious way to ...
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Any link between abelian $R/J(R)$ and 2-primal condition
Let $R$ be noncommutative unital ring such that each element of the quotient $R/Soc(R_R)$ is idempotent. If the nilpotent elements of $R$ form an ideal, is it true that the idempotents of $R/J(R)$ ...
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75
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Does the inequality $x^2\leq x$ hold in Baer*-ring for $0 \leq x \leq 1$?
Let $A$ be a Baer*-ring and $x$ be a positive element with $0\leq x\leq1$ where $1$ is just the unit of $A$. Can we conclude $x^2\leq x$?
- 4,058
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201
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Rank of cross-covariance matrix
Let $\boldsymbol{X}=(X_1,\dots,X_p)^T$ and $\boldsymbol{Y}=(Y_1,\dots,Y_q)^T$ be two random vectors. Denote $r_x=\text{rank}(\text{Cov}(\boldsymbol{X})),r_y=\text{rank}(\text{Cov}(\boldsymbol{Y})), r_{...
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An analogue of Hermitian matrix - does it exist?
Let $k$ be any field and $R\subseteq M_s(k)$ be a subring of $s\times s$ matrices over $k$. Identify $k$ with the scalar matrices, so that $k\subseteq R$. Let $A\in M_n(R)$ be an $n\times n$ matrix.
...
- 3,041
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principal submatrix of a banded matrix with banded inverse
Let $A\in \mathbb{R}^{\mathbb{N}\times \mathbb{N}}$ be a banded matrix ($A_{ij}=0$ for all $|i-j|>K$) with banded inverse (probably different bandwidth $\tilde K$). The question is, if principal ...
- 435
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114
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Intersecting vector spaces defined over different fields
Let $K_1, K_2$ be subfields of $K$, let $k = K_1 \bigcap K_2$, let $V_1$ be a $K_1$-vector space, $V_2$ be a $K_2$-vector space, both of them subsets of a $K$-vector space $V$.
How can I compute a $k$...
- 314
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85
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Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?
There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$,
$k$ is a field of characteristic zero,
the Polarization Theorem, Corollary 5.5 by A. Joseph.
...
- 2,837
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77
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Projective modules over a Hochschild extension of algebras
Let $k$ be a commutative ring and let $R$ be an associative (not necessarily commutative) $k$-algebra with unit. Also, let $M$ be an $R$-bimodule. It is well known that on $S:=R+M$ we can define an ...
- 11
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215
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Non-noetherian coherent local rings
Let $A$ be a non-neotherian UFD such that $A$ satisfies the following condition $(\sharp)$$\colon$
$(\sharp)$ For any prime ideal ${\frak P}$ of $A$ such that ${\mathrm{ht}}({\frak P}) < \infty$, ...
- 781
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129
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Matrix majorization when a diagonal matrix is multiplied from right and left
Let $D_1$ and $D_2$ be two diagonal matrices such that $D_1^2+D_2^2=I$ (identity matrix). Suppose matrix $A$ majorizes matrix $B$. Can we show that matrix $A$ majorizes matrix $D_1 A D_1 + D_2 B D_2$? ...
- 181
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49
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Atomicity of the semidomain (under Dirichlet convolution) of arith. fncs to a subrig of the ring of integers of a totally real NF
Let $S$ be a non-trivial subsemiring of the ring of integers of a totally real number field, and denote by $D_S$ the integral semidomain of all arithmetic functions $\mathbf N^+ \to S$ with the ...
- 10.5k
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187
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A question concerning positive definite matrix functions
Let $C(e^{i\theta})$ be an $m\times n$ ($m\ge n$) matrix-valued continuous function of $\theta\in[-\pi,\pi]$. Let $A_1(e^{i\theta})$ and $A_2(e^{i\theta})$ be two $n\times n$ positive definite matrix-...
- 2,712
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136
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Representations of finite groups over commutative rings-question and reference request
In a textbook of representation theory I have encountered the following statement without proof:
Let $R$ be a commutative ring and $G$ a finite group. If $M$ is a simple $RG$-module then the ...
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