Unanswered Questions
49,209 questions with no upvoted or accepted answers
0
votes
1
answer
199
views
Intersection between a line and an n-dimensional parallelotope
Suppose that I have a line in an $n$-dimensional space described by
$$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$
here $A$ is known and I want to find all the possible vectors $B$ ...
- 1
0
votes
1
answer
246
views
question regarding double summations
I'm looking for a reference and/or table for double summations. The sum I'm trying to compute is
$$\sum_{k=1}^\infty \sum_{m=1}^\infty \frac{1}{km(ak^2+bm^2)}$$ for real numbers $a$, $b$.
0
votes
1
answer
257
views
How to find $K,W,S$ in the Mostow decomposition theorem?
The Mostow decomposition theorem states:
Let $Z$ a nonsingular complex matrix, then $Z$ can be factored as:
$$Z=We^{iK}e^S$$ where: $W$ is unitary, $K$ is real and skew symmetric and $S$ is real and ...
- 399
0
votes
1
answer
226
views
Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra
Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of $L^{1}(\...
- 175
0
votes
1
answer
268
views
Nonnegative Matrix
Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ${\...
- 303
0
votes
1
answer
431
views
Efficient isomorphic subgraph matching with similarity scores
I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...
- 223
0
votes
1
answer
440
views
Variation on Fatou's lemma for Sobolev norms
Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions
$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$
If I am not ...
- 1
0
votes
1
answer
180
views
(probably simple) optimization question
Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
- 6,962
0
votes
1
answer
221
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Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
- 1
0
votes
1
answer
1k
views
Alternative proofs of Euclid-Euler theorem
What are some alternative methods of proof for the necessity direction of the above theorem, ie $n$ an even perfect number $\Rightarrow n$ is of form $2^{a-1} (2^a - 1)$ where $2^a - 1$ is a Mersenne ...
0
votes
2
answers
747
views
Number of Dyck paths with k returns and b peaks
The number of Dyck paths from the origin to $(2n,0)$ which touch the $x$-axis $k+2$ times ($k$ internal touches) is given by
$$\frac{k}{2n-k}{2n-k \choose n}.$$
The number of Dyck paths from the ...
- 21
0
votes
1
answer
568
views
Non-cohomological proof that a noetherian scheme $X$ is affine if its reduction $X_{red}$ is affine
Can we prove that a noetherian scheme $X$ is affine if its reduction $X_{red}$ is affine without using cohomology?
Remark
Here is a similar question.
- 1,169
0
votes
1
answer
198
views
Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$
In March 2018, I formulated the following somewhat curious question.
Question 1. Whether for any integer $n>1$ there is a nonnegative integer $k$ such that $n^2-2\times 4^k$ or $n^2-5\times 4^k$ ...
- 15.6k
-1
votes
0
answers
33
views
Tightest decreasing majorant
I had asked this question here but received no answer.
Let $O$ be an operator that maps sequences to sequences such that the elements of the sequence $O(a)$ are given by
$$\bigl(O(a)\bigr)_n ~{}={}~ \...
- 349
-1
votes
0
answers
18
views
What is the expected value of the set when N elements are chosen from the same probability distribution?
Suppose we have a parameter that follows some probability distribution $f(x)$. When simulating an $N$-body with that parameter as an attribute, how should values of the parameter be chosen?
Let each ...
- 1
-1
votes
0
answers
44
views
How to prove the following theorem by distribution function and series
Let $g$ be nonnegative and measurable function in $\Omega$ and $\mu_{g}$ be its distribution function, i.e.,
$$
\mu_{g}(t)=\left|\left\{ x\in \Omega:g(x)>t\right\}\right|,\; t>0.
$$
Let $\eta>...
- 1
-1
votes
0
answers
42
views
Homomorphism from field of hyperreals to field of reals?
I am curious if it is possible to construct a homomorphism from a field of hyperreal numbers to the field of real numbers? (Similarly, a homomorphism from the surreals to the reals?)
Assuming that ...
- 55
-1
votes
0
answers
58
views
Solving special multivariable limits by Euclidean geometry
General Problem: Inspect function $L(a)$ for $a\in \mathbb{R}^{n}$, given that:
$$L(a)=\lim_{x\to 0_{+}}\frac{x^{a}}{F(x)}$$
Notation legends:
$x=(x_1,\ldots,x_n),\ a=(a_1,...,a_n),\ x^a=x_{1}^{a_1}\...
- 231
-1
votes
0
answers
45
views
Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
-1
votes
0
answers
23
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A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
- 49
-1
votes
0
answers
26
views
Estimate the value of the PDF $P(f)$ at the minimal $f_0$ of the random-variable function $f(\mathbf{x})$
Let $f(\mathbf{x})=f(x_1,x_2,\dotsc,x_N)$ with $N>2$ be a real and continuous function and $f(\mathbf{x})\ge f_0$ for any $\mathbf{x}\in\mathbb{R}^N$. Now let $x_1,x_2,\dotsc,x_N$ be the i.i.d. ...
- 375
-1
votes
0
answers
115
views
Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
-1
votes
0
answers
35
views
Different definition of Feller semi-group
(This is a crosspost of a question on MathStackExchange which did not receive any answer.)
Let $E$ be a locally compact metric space, let $C_0(E)$ be the set of real-valued continuous functions of $E$ ...
-1
votes
0
answers
81
views
Can the higher order corrections to energy be calculated the same way in degenerate perturbation theory as with non-degenerate perturbation theory?
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Now from non-degenerate perturbation ...
- 23
-1
votes
0
answers
95
views
Relation between properties of functions/sets and Grzegorczyk's hierarchy
I know for example that the first level of the Grzegorczyk hierarchy contains the functions which enumerate the c.e sets and that it has an interesting relation to the provably total functions in ...
- 893
-1
votes
0
answers
80
views
Internalization and enrichment
I am starting with the ideas of higher categories. I have encountered the notions of internalization and enrichment. I haven't quite understood how do they lead to (n+1,r+1) categories starting from (...
- 331
-1
votes
0
answers
27
views
Number variance of random points (and deviations for empirical processes)
Let $X_1, X_2, \dots$ be i.i.d. random variables having uniform distribution on $[0,1]$. Write $I_{t,x}$ for the indicator function of an interval of length $x$ with center $t$. Consider
$$
V(N,x) = \...
- 2,217
-1
votes
0
answers
53
views
convergence of convolution in Bochner space
I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$
let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
-1
votes
0
answers
73
views
Analog of ceil and floor of $\sqrt{a(a+1)}$ in modular arithmetic
If we take ceil and floor of $\sqrt{x(x+1)}$ (when it exists) we get $x$ and $x+1$ respectively. Is there an analog of this assuming roots exist in modular arithmetic (at least modulo primes)?
...
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-1
votes
1
answer
87
views
how take weak derivative of norms in hilbert spaces?
Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$.
Let $u∈L^2 ([0,T];V); ...
-1
votes
1
answer
147
views
Is it possible to have square-free order(s) in $\mathbb{Z}^\times_N$?
Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.
Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random ...
-1
votes
1
answer
215
views
Best approximation of the modulus function
While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...
- 149
-1
votes
2
answers
87
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
-1
votes
1
answer
129
views
(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes
Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$,
\begin{equation}
\int_0^te^{-\lambda ...
- 411
-1
votes
1
answer
199
views
Isomorphism between subgroups by preserving index
Let $C$ be a subgroup of a group $A$ such that $C\cong A/\{\pm1\}$ and $D$ be a subgroup of $B$ such that $D\cong B/\{\pm1\}$. Let $\pi:A\to B$ be a group homomorphism, and let $\pi$ induce a ...
- 309
-1
votes
1
answer
77
views
Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
- 287
-1
votes
1
answer
68
views
Bound for an expectation of random matrix with quantized random variable
Let random matrix $\mathbf{X} \in \mathbb{C^{\mathrm{m} \times \mathrm{n}}}$ and random vector $\mathbf{y} \in \mathbb{C^{\mathrm{m} \times 1}}$ are unknown distributed, but their covariance and ...
- 25
-1
votes
1
answer
825
views
How to calculate determinants of such types?
Consider next determinant that we want to expand around $h=1$
\begin{eqnarray}
Z_q \ = \ h^{N N_f} \ \ \left ( \prod_{n=1}^{N} \ \sum_{l_n=0}^{N_f -q} \ h^{2l_n+q} \ \binom{N_f}{l_n} \right ) \ \...
-1
votes
1
answer
104
views
Singularities of Painlevé II
It is known that Painlevé II has only simple poles $t_n$ as singularities and these poles have non-movable property, i.e., they depend only on the equation rather than initial conditions. One can try ...
- 593
-1
votes
1
answer
297
views
The distribution of the sum of values from a normal and a truncated normal distribution
Using R to extract truncated normal distribution samples and normal distribution samples separately, when they are combined, the image drawn by the hist function is very similar to a normal ...
-1
votes
1
answer
230
views
Determine whether the center of a $C^*$-algebra is 0
Let $G$ be a locally compact group and $A$ be a non-unital $C^*$-algebra. )$(A,G,\alpha)$ is a $C^*$-dynamical sysytem. The space of all continous functions from $G$ to $A$ with compact support is ...
- 451
-1
votes
1
answer
170
views
Empirical distribution of vector
Currently, I am reading about the empirical distribution and doubting a statement, which is my own intuition. Let's say we have a vector $\mathbf{x} \in \mathbb{R}^n$ converges in distribution to the ...
- 87
-1
votes
1
answer
551
views
Lower bound of an expectation
Suppose a random variable $X$ has unit variance i.e. $\sigma^{2} = 1$. Is there a positive constant $c > 0$ such that
$$\mathbb{E}[\ | X - \mathbb{E}[X] | \ ] \ge c $$
My attempt of a solution is ...
- 643
-1
votes
1
answer
555
views
Noetherianity assumptions in Hartshorne's book
It seems that noetherian assumptions are not necessary in many results by Hartshorne, in his book "Algebraic Geometry". How much is this true? Could you please give examples?
- 1,422
-1
votes
1
answer
74
views
Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail
Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of ...
- 1
-1
votes
1
answer
437
views
Harmonic function in infinite domain in $\mathbb{R}^3$, constant on the boundary and decaying as $1/r^2$
EDIT: Let $\Omega\subset \mathbb{R}^3$ be a bounded domain with smooth connected boundary. Let $f\colon \mathbb{R}^3\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\...
- 21.8k
-1
votes
1
answer
280
views
A question on assigning finite values to divergent sums involving expression of primes
We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
- 149
-1
votes
1
answer
65
views
A follow-up question in a proof in a paper on complete multipartite graphs
A follow-up question from the following article/paper:
"Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion"
by Shaowei Sun and Kinkar Chandra ...
- 199
-1
votes
1
answer
205
views
How to combine estimator with different variances?
Consider independent random variables $X_1,X_2,\ldots,$ that have the same expectation $\mathbb x=\mathbb E[X_1]=\mathbb E[X_2]=\ldots$
Further, assume that we know that $Var[X_i]=\sigma_i^2$.
In the ...
- 127
-1
votes
1
answer
60
views
Linear operator over a simplex space in a multinomial distribution parameter estimation problem
This is actually a variant of a well-known problem of how the parameters of a multinomial distribution can be estimated by maximum likelihood, and this arises from a final year project I undertook ...
- 1