All Questions
Tagged with real-analysis or linear-algebra
11,372 questions
2
votes
0
answers
43
views
Distributions and time-kernels
Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
- 1,429
1
vote
1
answer
41
views
Lower spectral radius of matrices with an invariant subspace
Let $\mathcal{A} = \{ A_1, \ldots, A_J, : A_j \in \mathbb{R}^{s\times s}\}$, and define the lower spectral radius (LSR) (aka joint subradius) by
$LSR(\mathcal{A}) = \lim_{n\rightarrow\infty} \inf_{A_{...
- 131
0
votes
0
answers
22
views
Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 698
1
vote
1
answer
90
views
Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
0
votes
1
answer
93
views
A question on finite Fourier series
Let $\mathcal F(N)$ denote the space of finite Fourier series up to frequency $N > 0$, i.e. $f\in \mathcal F(N)$ if and only if it can be written as
$$f(x) = \sum_{k=0}^N a_k\cos(kx+\theta_k)$$
for ...
- 71
1
vote
0
answers
115
views
How can I derive $V$ from the following equation?
$$
[(U-V)^{-1}X^{T}Y]^{T}\;[(U-V)^{-1}X^{T}Y] = I.
$$
Here $U$ and $V$ are symmetric $d \times d$ matrices; $X=[x_1,x_2,...,x_n]$ is an $n \times d$ data matrix ($n$ is the number of samples and $d$ ...
- 11
0
votes
1
answer
71
views
Upper bound on higher order derivatives of $\frac{1}{v(t)}$
Suppose that $ v(t) >l>0$ and
$$
\vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}.
$$
Can we give an upper bound for
$$
(\frac{1}{v(t)})^{(k)}
$$
?
Attempt:
We first compute the first fourth order ...
- 269
1
vote
1
answer
69
views
Exhausting sequences contain a $\pi$ lift of a subset with a $(1-\delta)$ factor
Let $\pi : Y \to X$ be a measurable map between the $\sigma$-finite measure spaces $(Y, \mathcal{B}, \nu)$ and $(X, \mathcal{A}, \mu)$. Suppose there exists $c \in (0, \infty)$ such that for all $A \...
- 105
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
-2
votes
0
answers
64
views
A Problem using Limits of Sequences of Functions
Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
- 1
4
votes
1
answer
172
views
Maximizing trace subject to two equality constraints
I am looking at the following optimization problem
$$\begin{align}
\underset{{\bf X}}{\text{maximize}} \qquad&\mathrm{tr}({\bf AX})\\
\text{subject to} \qquad& \mathrm{tr}({\bf X}) = 1,\\
&...
- 43
0
votes
0
answers
87
views
Is there a name for "applying linear operations to vector sequences from the right"?
Let $v_1,...,v_n\in\Bbb R^d$ be a sequence of vectors. When we say that we "linearly transform" this sequence, we mean that we apply a linear transformation $T\in\Bbb R^{d\times d}$ to each ...
- 13.6k
3
votes
0
answers
95
views
Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 312
2
votes
2
answers
132
views
Invertibility of one matrix constructed by order n subgroup of symmetric group
Let $S_n$ be the symmetric group on $n$ elements $\{ 1,2,\dotsc,n \}$ and $G$ be a subgroup of $S_n$ of order $n$. Denote the elements in $G$ by $\{ \sigma_1,\dotsc,\sigma_n \}$. Let the matrix $A=(\...
- 21
4
votes
0
answers
103
views
Relationship between characteristic polynomials of a matrix and its adjoint representation
Let $A \in \mathrm{M}_n(F)$ be a matrix over a field $F$. Consider its adjoint representation $\mathrm{ad}_A \in \mathrm{End}(\mathrm{M}_n(F))$, defined by
$$
\mathrm{ad}_A(X) = [A, X] = AX - XA.
$$
I ...
- 309
0
votes
2
answers
148
views
Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 21
3
votes
2
answers
153
views
On nowhere differentiability of functions that just barely fail to be Lipschitz
By Rademacher’s theorem, Lipschitz functions are differentiable almost everywhere. I am wondering how badly this pointwise differentiability fails for functions that “just barely” fail to be Lipschitz....
- 6,155
7
votes
2
answers
242
views
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
1
vote
2
answers
117
views
If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
4
votes
0
answers
291
views
Lower bound on size of the set of sums and differences of non-orthogonal pairs of vectors over finite field
Consider $\mathbb{Z}_m^n$, an $n$-dimensional vector space over $\mathbb{Z}_m$.
For two sets of vectors $P = \left\{ p^i \right\}$ and $Q = \left\{ q^j \right\}$ and a skew-symmetric matrix $S_{ij}=\...
- 91
1
vote
0
answers
39
views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$:
$$
a^2 u_{xx} - b^2 u_{yy} = f(x, y),
$$
with Dirichlet boundary conditions on $u$.
By using the ...
- 617
0
votes
1
answer
48
views
How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm
If we need to compute a high-degree polynomial expression, such as:
$$
P(y) = \sum_{k=0}^{B} a_k y^k
$$
the Paterson-Stockmeyer algorithm can process the powers in ...
- 3
0
votes
1
answer
157
views
Can the derivative of eigenvectors with respect to its components be taken as zero if all eigenvalues are equal?
I want to ask a couple of follow up questions to the question answered on the thread "Derivative of eigenvectors of a matrix with respect to its components".
I noticed that in the accepted ...
- 23
3
votes
1
answer
111
views
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I found myself needing a generalization of a result of Kostant in his famous paper
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
- 173
-1
votes
0
answers
132
views
Trig conjecture about square roots and Arcsin
Let $r(a,b)$ be a rational number depending on positive integers $a,b$ and $r(a,b)$ being nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$.
Let $C(b)$ be a squarefree positive ...
- 769
1
vote
3
answers
257
views
Eigenvalues of positive matrices in $\mathrm{SL}(d,\mathbb{Z})$
Let $A\in\operatorname{SL}(d,\mathbb{Z})$ be an irreducible positive matrix, i. e. $A=(a_{i,j})_{1\leq i,j\leq d}$ with $a_{i,j}\in\mathbb{Z}_{>0}$. From the Perron-Frobenius theorem, we know that $...
- 11
1
vote
0
answers
67
views
System of linear diophantine equations with many small solutions?
Let $n$ be positive integer, $k$,$B$ fixed positive integers.
Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.
Let $S(f_i,k,B)$ be the set of ...
- 25.4k
-1
votes
0
answers
21
views
Choosing between matrix normal and multivariate normal for Bayesian inference
I’m working on a Bayesian inference problem where I need to estimate a graph structure G with a spike-and-slab prior on each edge of G. My likelihood model is built on observed data R and covariance ...
- 1
3
votes
0
answers
98
views
Square Roots of Non-Negative Even Functions
I'm trying to study properties of maps between quotients of representations of compact Lie groups and I stumbled upon the following problem. Suppose you have a smooth function $f:\mathbb{R}\to\mathbb{...
- 31
1
vote
1
answer
76
views
Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
1
vote
1
answer
188
views
Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?
This question is related to This question.
When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
- 178
2
votes
1
answer
107
views
Linear automorphism preserving a cone
Let $V$ be a finite-dimensional real vector space, and let $C\subset V$ be a closed convex cone, not contained in a hyperplane, and such that $C\cap(-C)=\{0\} $. Let $n$ be a nilpotent endomorphism of ...
- 38k
-1
votes
0
answers
51
views
Existence of continuous crossection
Let $G$ be a second countable locally compact Hausdorff groupoid. Then $G= \sqcup_{u\in G^{0}}G^{u}$ where $G^{0}$ is the unit space of $G$ and $G^{u}=r^{-1}(u)$. Here $r:G \to G^{0}, r(x)=xx^{-1}$.
...
3
votes
0
answers
57
views
Maximizing a Gaussian quadratic form
Let $u$ denote a fixed unit vector in $\mathbb{R}^n$ and $g$ a standard Gaussian vector (in $\mathbb{R}^n$).
Consider the map
$$
f_n(X) = \mathbb{E} \langle (X^{-1} + gg^T)^{-1} u, u\rangle,
$$
...
- 380
5
votes
0
answers
231
views
Avoiding Cartan subalgebra in a Lie algebra
Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.
What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ ...
- 309
2
votes
0
answers
104
views
Find an $a \times m$ submatrix of an $n \times m$ matrix with smallest rank
Given a matrix $n \times m$, I want to find the submatrices $a \times m$ by selecting $a$ columns such that their rank is minimal. Can this problem be solved efficiently?
- 1,207
2
votes
0
answers
90
views
Representation of Dirac-delta distribution in subspace of functions
Suppose I have a subspace $V\subset L^2(\Omega)$ where $\Omega\subset \mathbb{R}^d$ is a bounded and closed set. $V$ is defined by
\begin{align}
V=\text{span}(\{\varphi_i(x): i=1,2,\dots,n\})
\end{...
- 93
0
votes
0
answers
44
views
Sufficient conditions for a homogeneous polynomial to have a continuous right inverse
this is a question that continues a series of questions I'm coming up with on homogeneous polynomials, like for example this one.
For now I can prove that a homogeneous polynomial $f:\mathbb R^n\to \...
- 311
3
votes
1
answer
212
views
$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$
I want to solve the equation:
$$
\begin{cases}
\nabla \times (F\times\mathbf v)=g, \\
\operatorname{div}(\mathbf v)=0,
\end{cases}\label{1}\tag{1}
$$ where $F$ and $g$ are given vector fields. The ...
- 617
0
votes
1
answer
98
views
Only special permutations result in a constant expression when permuting coefficients in a sum involving binomials?
Fix $n\geq 1$ and let $p_k(x) := x^k(x-1)^{n-k}$.
Suppose $\pi$ is a permutation on $\{0,1,\dotsc,n\}$, such that
$$
\sum_{k=0}^n (-1)^k \binom{n}{k} p_{\pi(k)}(x) \text{ is a constant}.
$$
Must it be ...
- 15.8k
11
votes
2
answers
386
views
Bounds for the difference in the number of ones in $M$ and $M^{-1}$
If $M$ is a full rank $n$ by $n$ binary matrix over $\mathbb{F}_2$, how much larger or smaller can the number of $1$s in $M^{-1}$ be, compared to the number of $1$s in $M$?
Clearly the identity matrix ...
- 3,377
0
votes
2
answers
364
views
Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
- 178
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
0
votes
2
answers
97
views
Optimization algorithms for Kronecker approximation of high-dimensional covariance matrices
I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable.
Here's the setup:
I have a graph $G$ represented by a $D\...
- 1
6
votes
1
answer
568
views
Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 178
4
votes
1
answer
256
views
Approximating an $L^1$ function with Riemann sums
Note: Here all functions are genuine functions, i.e. pointwise defined measurable functions instead of defined only a.e.
Let $f: [0, 1] \to \mathbb R$ be an arbitrary $L^1$ function. Of course, $f$ is ...
- 6,155
1
vote
1
answer
112
views
Bounding a Riemann sum by its integral limit?
Let $M_{n}(\mathbb{C})$ denote the space of complex $n \times n$ matrices and, for $a>0$, $a \in \mathbb{R}$ fixed, let $A: [0,a) \to M_{n}(\mathbb{C})$ be a given function. I will write $A(t) = (...
6
votes
2
answers
772
views
Finiteness of an integral
In a paper I am reading, the following seems to be claimed:
Let $f:[0,\infty)\to [2,\infty)$ be a continuous, monotonically increasing function with $\lim_{x\to\infty}f(x)=\infty$ and let $\alpha>3/...
- 460