Questions tagged [vector-spaces]
The vector-spaces tag has no usage guidance.
62
questions with no upvoted or accepted answers
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Dimensions of dual vector spaces
Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
7
votes
0
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293
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A minimal semigroup generating subset of the additive reals
I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...
6
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0
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177
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Growth in a vector space
I am interested in "growth" in finite vector spaces. I have been wondering about the veracity of the following statement:
Statement: For every $\varepsilon>0$ there exists $b\in\mathbb{Z}^+$ ...
5
votes
0
answers
409
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Spectral theory without topology
How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ?
Something along these lines, for example: ...
5
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0
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742
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Is this set empty?
Suppose we have two rank $n-1$ matrices in $\Bbb Z^{(n-1)\times n}$ given by
$$C=\begin{bmatrix}
c_{1}& -d_{1}& 0& 0&\dots 0& 0\\
0& c_{2}& -d_{2}& 0&...
4
votes
0
answers
125
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Hamel basis with all coordinate functionals discontinuous
If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
4
votes
0
answers
1k
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Permutation-invariant matrix representation
The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level ...
3
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0
answers
65
views
Field automorphisms of projective spaces without the axiom of choice
Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...
3
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0
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194
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$2$-vector spaces and algebraic $2$-stacks
I am thinking about higher Artin stacks in the sense of Simpson, concretely I would like to calculate the dimension and compare these two cases:
$\mathfrak{X}_{1}=$ Higher linear stack classifying (...
3
votes
0
answers
223
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Behavior of vector field that is the arg min of a convex function
Say $f(x,y)$ is a real valued function that is strongly convex and twice differentiable in both $x$ and $y$. Here $x$ and $y$ are real vectors of different dimensions. We define $y^\star(x)= \arg \...
3
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0
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193
views
Probability of orthogonal vectors?
Denote $\mathcal I_m=\{-m,-m+1,\dots,0,\dots,m-1,m\}$ where $m\geq0$ is an integer.
Pick a uniformly random vectors $$\hat a=(a_n,a_{n-1},\dots,a_1)\in(\mathcal I_{m_1}+\sqrt{-1}\cdot\mathcal I_{m_2}...
3
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0
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179
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Is there a known generalization of the Schmidt decomposition based on a maximal set of "locally orthogonal" vectors?
I came across the following unusual generalization of the Schmidt decomposition in my work, which I describe below. I would like to know if this structure has been studied before so I can read more ...
2
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0
answers
85
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To show $\{(x,y) \in \mathbb Q^{\geq 0} \times \mathbb Q^{\geq 0}~:~ mn+1 \mid m^x+n^y \}$ is subset of the lattice $\{\vec u+i \vec v+j \vec w\}$?
I am writing two definitions, the $1$st one is a cover in some sense while the $2$nd one is a lattice:
Definition 1: If $m,n$ are integers bigger than $1$, then the set $$A=\{(x,y) \in \mathbb Q^{\geq ...
2
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0
answers
83
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Name for closure property: set of maps closed under taking $(f,g)\mapsto (f-g)/2$
Suppose that $F$ is a collection of functions mapping some set $\Omega$ to $\mathbb{R}$, with the following closure property: whenever $f,g\in F$, we also have $(f-g)/2\in F$. Is there a name for this ...
2
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0
answers
109
views
Right unitor in star-autonomous categories
1.Context
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Here $r$ denotes the right unitor.
Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume ...
2
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0
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127
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Are there any known algebras or vector spaces, where absolute value, modulus or norm is connected to the factors of $\pi$ or $e^{-\gamma}$?
I am currently working on an algebra of divergent integrals and series, and all the elements of that space consist of a regular part (which is a real or complex number) and irregular part (which is ...
2
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0
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125
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Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)
Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...
2
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0
answers
237
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Recover unknown vectors with dot-product queries
Suppose there are $n$ unknown unit vectors in $\mathbb{R}^d$,
$V=\{v_1,\ldots,v_n\}$, no two identical.
Your task is to determine the vectors in $V$.
The only tool at your disposal is to query a ...
2
votes
0
answers
106
views
Solution of equation on vector field
I have a vector field function $\vec{J}: {\bf R}^3\to {\bf R}^3$ looking like:
$$ \vec{J}(\vec{r}) = (\vec{B} \times \vec{v}(\vec{r}))\rho(\vec{r}) $$
with a (very well behaved) real, positive, ...
2
votes
0
answers
85
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Shortest vectors in tensor product and maximal lattices in tensor product
$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$
$$\|v_1'\|_2\...
2
votes
0
answers
248
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Finite-dimensional vector spaces vs. matrices over a semiring
The category $\mathbf{Vect}_k$ of vector spaces over some field $k$ is weakly equivalent to the category $\mathbf{V}_k$ whose objects are finite sets $n$, and whose morphisms $m\to n$ are $(m\times n)$...
2
votes
0
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95
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Stochastic approximation in two dimensions
I have a question about a stochastic approximation method presented in J. R. Blum, Ann. Math. Stat. 25(4): 737 (1954).
Given a $k$-dimensional vector $\bf x$ and $k$ random variables $Y^1_{\bf x}, \...
2
votes
0
answers
43
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Tensor of Relative Bases
Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of ...
2
votes
0
answers
881
views
A question on vector space over finite field
Let $\mathbb{F}_{2^\sigma}$ be a finite field of $\sigma$-bit elements, and use $\mathbb{F}_{2^\sigma}^{\ell}$ to denote an $\ell$-dimensional vector space over $\mathbb{F}_{2^\sigma}$. Let $V$ be a ...
2
votes
0
answers
88
views
Counting chain maps
I initially asked this question over at math stack exchange, you can find it here. I haven't really gotten any traction and I'm beginning to wonder if maybe its a harder question than I originally ...
2
votes
0
answers
120
views
Terminology for research on distributions of inner products
Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism.
Suppose now that $V$ is ...
2
votes
0
answers
222
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The Euler characteristic of Hilbert series
The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim V_n$...
2
votes
0
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393
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A conjecture about vector space (repost from math.SE)
This post is copied from math.SE in the following link:
https://math.stackexchange.com/questions/456398/a-conjecture-about-vector-space
I have posted the question two days ago, but receive no answer ...
2
votes
0
answers
456
views
Morphisms of Spectral Sequences and alternating products
Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences.
Assume that morphisms $...
1
vote
0
answers
134
views
Reconstructing an object from its shadow
I'm looking into the section "Reconstructing an object from its shadow" in the book Introduction to the Mathematics of Medical Imaging by Charles L. Epstein.
I have two questions
The ...
1
vote
0
answers
153
views
Centraliser of a finite group
Let $G=\operatorname{Sp}(8,K)$ be a symplectic algebraic group over an algebraically closed field of characteristic not $2$.
We have a vector space decomposition $V_8=V_2\otimes V_4$ where the $2$-...
1
vote
0
answers
140
views
General linear group in infinite dimensions
Let $V$ be a vector space over the field $k$. Upon assuming the Axiom of Choice, we know that $V$ has a well-defined dimension $N$, and hence a well-defined basis $B$. Suppose that $N$ is not finite.
...
1
vote
0
answers
601
views
Notation for the space of eventually-zero sequences
An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
1
vote
0
answers
124
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Proving the non-existence of canonical isomorphisms
From time to time, during my undergraduate lectures on linear algebra appears the following question from the most smart students in the class. I asked to my algebra colleagues but I have not received ...
1
vote
0
answers
235
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Vector convolution?
I am working on a research problem which leads to the following optimization problem:
\begin{equation}
\hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
1
vote
0
answers
61
views
Derivative of a function of ordered variables
Can I differentiate
$$(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)^\top(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)$$ with respect to $\pmb{a}$? (I want to minimize the expression with respect to $\pmb{a}$.)
Here, $...
1
vote
0
answers
45
views
Characterization of dimension over $\mathbb{Q}$ of infinite sums of rational functions
Let $P(n)=(n+r_1)(n+r_2)...(n+r_k)$ be a polynomial with simple, rational, negative roots (i.e. $r_i>0$) and degree $k\geq 2$ (I stick with negative roots as I don't have to worry about dividing by ...
1
vote
0
answers
24
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Weird subspace/equality-constrained LP problem/variant of change-making problem
Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve
$$\sum_{i=1}^n c_i\leq\delta$$
$$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$
where
$0\...
1
vote
0
answers
64
views
Function for unique volume element
This is an issue that I'm am trying to solve for a fine-tuning measure in particle physics, but it is purely mathematical. Consider three vectors $\{v_1, v_2, v_3\}$ in $\mathbb{R}^3$. I would like a ...
1
vote
0
answers
49
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Existence of subspace which is totally non-invariant under unitary transformation
Given a complex Hilbert space $\mathcal{H}$ of dimension $d$ - interpreted as vectorspace over $\mathbb{R}$ with dimension $2d$. And the space $L(\mathcal{H},\mathbb{C}^4)$ of all linear operators ...
1
vote
0
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143
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Does the 2 category of Groupoids Admit the Vector Space Monad?
We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad.
3.2. Vector spaces. For a semiring S one can define the ...
1
vote
0
answers
103
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On dimension of Segre embedding of lattice translations
Consider three lattices $L_1$, $L_2$ in $\Bbb Z^{n+1}$ and $L$ in $\Bbb Z^{2n+1}$.
Let $L_1+v_1$, $L_2+v_2$ and $L+v$ be their respective translationsfor some $v_1,v_2\in\Bbb Z^{n+1}\backslash\{(0,\...
1
vote
0
answers
453
views
Orthonormal basis of matrices
I am asking if somebody knows how to do or is aware of the following construction:
Let $n \in \mathbb{N}$ be given, then take an arbitrary matrix $A \in \mathbb{C}^{n \times n}$. Then, there are maps ...
1
vote
0
answers
72
views
Dedekind complete quotients of Riesz spaces
Suppose that $E$ is a Dedekind complete Riesz space and let $J$ be an order ideal of $E$. Then one can form a quotient Riesz space $E/J$. I am interested which properties $J$ needs to possess so that ...
1
vote
0
answers
181
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Grassmannian frames in the Grassmannian
I am new to the Grassmannian. I have read about Grassmannian frames in $\mathbb R^n$. My question is can we define Grassmannian frames in a Grassmannian space $Gr(k,n)$ just like in $\mathbb R^n$? ...
0
votes
0
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84
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Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
0
votes
0
answers
86
views
Construct a vector space whose elements are sets
I would like to construct a vector space whose elements are convex and closed subsets of $\mathbb{R}^n$.
A natural idea is as follows.
For any two sets $S_1, S_2 \subseteq \mathbb{R}^n$, define the ...
0
votes
0
answers
147
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Given optimality of L1 norm, prove that absolute value of sum of a vector with proper sign is less than 1?
Problem:
Given a domain $\mathcal{D}\subset\mathbb{R}^{l}$, we can find $l$
points $\boldsymbol{v}_{i}\in\mathcal{D}$, $i=1,\cdots,l$. Each
point is a column vector with dimension $l\times1$. They ...
0
votes
0
answers
83
views
Arithmetic triangles and unimodality of its rows
Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence.
How to prove that the coefficients form an ...
0
votes
0
answers
50
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A query regarding complex vector decomposition
Given a complex vector $V$ of length $n^2$. Each complex entry in the vector is of size (number of digits or bits required to express the complex number) $c$ for some constant $c$.
Is it always ...