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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 ...
Pace Nielsen's user avatar
7 votes
0 answers
293 views

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 ...
user107952's user avatar
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6 votes
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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 $b\in\mathbb{Z}^+$ ...
Nick Gill's user avatar
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5 votes
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409 views

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: ...
gdm's user avatar
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742 views

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&...
Turbo's user avatar
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4 votes
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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 $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
Markus's user avatar
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4 votes
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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 ...
Nick's user avatar
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3 votes
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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 ...
THC's user avatar
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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 (...
Martin Hurtado's user avatar
3 votes
0 answers
223 views

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 \...
good bandit's user avatar
3 votes
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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}...
Turbo's user avatar
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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 ...
Jess Riedel's user avatar
2 votes
0 answers
85 views

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 ...
MAS's user avatar
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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 ...
Aryeh Kontorovich's user avatar
2 votes
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 ...
Max Demirdilek's user avatar
2 votes
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127 views

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 ...
Anixx's user avatar
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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 ...
Hvjurthuk's user avatar
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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 ...
Joseph O'Rourke's user avatar
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, ...
Raphael J.F. Berger's user avatar
2 votes
0 answers
85 views

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\...
Turbo's user avatar
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2 votes
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248 views

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 $(m\times n)$...
David Spivak's user avatar
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2 votes
0 answers
95 views

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}, \...
James's user avatar
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2 votes
0 answers
43 views

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 ...
crockeea's user avatar
  • 121
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 ...
haskell looks great's user avatar
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 ...
Paul's user avatar
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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 ...
Nick's user avatar
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2 votes
0 answers
222 views

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, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim V_n$...
Mariano Suárez-Álvarez's user avatar
2 votes
0 answers
393 views

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 ...
Thomas Tam's user avatar
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 ...
Henry Bui's user avatar
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 $2$-...
user488802's user avatar
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. ...
THC's user avatar
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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 ...
HardyHulley's user avatar
1 vote
0 answers
124 views

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 ...
Johnny Cage's user avatar
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1 vote
0 answers
235 views

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 ...
Mamal's user avatar
  • 273
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, $...
Shanks's user avatar
  • 133
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 ...
QC_QAOA's user avatar
  • 121
1 vote
0 answers
24 views

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\...
Jonathan Lee's user avatar
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 ...
Daniel's user avatar
  • 11
1 vote
0 answers
49 views

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 ...
Alpha001's user avatar
  • 143
1 vote
0 answers
143 views

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 ...
Ben Sprott's user avatar
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1 vote
0 answers
103 views

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,\...
Turbo's user avatar
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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 ...
Luka Tinska's user avatar
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 ...
Marko's user avatar
  • 111
1 vote
0 answers
181 views

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 $\mathbb R^n$? ...
MLT's user avatar
  • 213
0 votes
0 answers
84 views

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 ...
JBuck's user avatar
  • 173
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 ...
Ypbor's user avatar
  • 159
0 votes
0 answers
147 views

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 ...
Justin's user avatar
  • 1
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 ...
Mikhail Gaichenkov's user avatar
0 votes
0 answers
50 views

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 ...
xyz's user avatar
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