All Questions
49
questions with bounties
1
vote
0
answers
116
views
+100
Simple algebra in rearring terms
I have a very simple mathematical question, and it is just about algebra which seems very tedious. First, let me state my problem from the beginning:
Let $i$ be an index representing countries ($i = {...
- 133
2
votes
2
answers
264
views
+50
Fubini's theorem for transition kernels
In probability theorey there is a different form of Fubini's theorem that includes Markov kernels (regular conditional distributions) that does not need independence. Let $(\Omega, \mathcal{A}, \...
- 343
4
votes
1
answer
132
views
+100
Independence is preserved by joint weak convergence
Suppose a sequence of random vectors $(X_n,Y_n)$ converges jointly to some $(X,Y)$ in the weak topology. Question: If $X_n$ and $Y_n$ are independent for all $n$, are also $X$ and $Y$ independent?
...
- 1,152
2
votes
0
answers
80
views
+100
Inscribing a sphere in an elliptical cone
Suppose you're given the cone whose curved surface is given by
$ (r - V)^T Q (r - V) = 0 $
and its flat surface is $z = 0 $
Question: How would you inscribe a sphere in the cone, such that it is ...
- 19.1k
1
vote
0
answers
154
views
+100
How to show the eigenvectors is preserved under a ODE?
Assume $V$ is a 3-dimsional vector space. And
$$
\varphi(t):V\rightarrow V
$$
is nondegenerate linear map, and smooth respect to $t$. The eigenvalues of $\varphi(t)$ are
$$
\lambda_1(t)\le \...
- 5,735
4
votes
1
answer
94
views
+100
Topology induced by generalised absolute values
In a number of texts (including Cassels' "Local Fields" and Artin's "Algebraic Numbers and Algebraic Functions") I've met the definition of an absolute value on a field that's ...
- 1,333
10
votes
1
answer
312
views
+50
Multiplicative Reversibility = No Primitive Roots
Call a positive integer, $n$, multiplicatively reversible if there exists integers $k$ and $b$, greater than $1$, such that multiplication by $k$ reverses the order of the base-$b$ digits of $n$ (...
2
votes
1
answer
134
views
+100
How do projective representations map elements? Are they mulivalued?
I've seen a definition of projective representation as this: Given a group $G$, a projective representation can be defined with a map
$$\pi:G\rightarrow GL(\mathbb{V})$$
from the group to the linear ...
- 127
1
vote
0
answers
64
views
+100
What are the ingredients of the Sunada's theorem in this example?
I recall what this theorem says:
Let M be a Riemannian manifold upon which a finite group G acts by isometries; let H and K be subgroups of G that act freely. Suppose that
H and K are almost conjugate ...
- 23
1
vote
0
answers
91
views
+50
Justifying steps in Riesz Representation theorem (local compact hausdorff space case)
I am reading the proof of Riesz Representation theorem(1.5.14) on Leon Simon's book:Geometric Measure Theory
And I got stuck at the following higlighted part. I completely understand the note he made ...
21
votes
0
answers
530
views
+250
Where are the other Pythagorean Theorems?
One way to prove the Pythagorean Theorem is by noticing that its altitude divides it into two pieces similar to itself. The theorem immediately follows from the fact that areas scale with the square ...
- 2,239
3
votes
4
answers
221
views
+100
How to show that given two acute angles, the sine ratio of the greater angle is greater than the sine ratio of lesser angle?
How do I show that for angles $\theta ,\psi \in [0^°,90^°$], if $\theta > \psi$, then $\sin \theta > \sin \psi$?
I thought of proving this by creating two right triangles with the same ...
0
votes
1
answer
125
views
+50
What is the formula for the Mode of the Log-Normal distribution followed by the Geometric Brownian Motion?
What is the formula for the Mode of the Log-Normal distribution followed by the Geometric Brownian Motion?
Intro__________________________
I was studying in Youtube this interesting MIT course of math ...
- 1,256
3
votes
1
answer
69
views
+50
Uniform Taylor expansion
$f \colon \mathbb R^n \to \mathbb R$ is differentiable in $x_0$ if there exists a functional $L$
$$f(x_0+h)-f(x_0)-Lh=o(|h|),$$
as $|h|\to 0.$ Here $o(|h|)$ denotes a function going to $0$ faster ...
- 153
6
votes
1
answer
133
views
+100
Alternative q-Analog
Wikipedia gives the following definition of a q-analog: "In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter $q$ that returns the ...
- 65
2
votes
1
answer
337
views
+100
Inequality in an acute-angled triangle
Let $ABC$ be an acute-angled triangle and $M$ a point inside it. We denote by $C_1$, $C_2$, $C_3$ the centers of the Nine-Point circles corresponding to the triangles $BMC$, $CMA$, and $AMB$, ...
- 253
0
votes
0
answers
47
views
+100
What statistical test should I be using for this data?
I have a survey where I've asked people's opinions on a subject, at three points in time - 2021, 2022 and 2023. The same 500 people answered for each year.
The available options were
...
- 29
1
vote
0
answers
156
views
+50
Missing coordinates of a projected rectangle and a mystery circle.
Edit: Judging by the lack of response to the bounty offer, maybe the question is impossible to resolve with the given information. If that is the case then it is allowed to use the additional ...
- 669
1
vote
0
answers
89
views
+50
How did (2+1) KP equation derived from Sato theory?
In section "Connection with Sato theory" of webpage scholarpedia they discussed that KP equation can be derived from taking a pseudo-differential operator $L$ and consider the differential ...
- 738
3
votes
1
answer
204
views
+300
Would We have the preimage of at least one interval taken out from $[0;1]$, inside the any ball taken out from $X$?
This is the theorem:
Let $X$ be separable metric space endowed with non-atomic Borel measure such that $\mu X = 1$.
Using this theorem We can establish isomorphism between $X$ and $[0;1]$.
Denote ...
- 626
1
vote
0
answers
46
views
+50
Can the finite-dimensional projections a Hilbert space valued random variable admit a common density with respect to the Lebesgue measure?
Let $H$ be an infinite-dimensional separable $\mathbb R$-Hilbert space with orthonormal basis $(e_n)_{n\in\mathbb N}$, $$H_d:=\operatorname{span}\{e_1,\ldots,e_d\}\;\;\;\text{for }d\in\mathbb N$$ and $...
- 13.2k
1
vote
1
answer
37
views
+100
Does total unimodularity for this modified assignement problem still preserve?
For a job assignment problem, total unimodularity would guarantee that the solution from a relaxed problem will be the same as the original integer problem. In my case, the data matrix is given and ...
7
votes
0
answers
141
views
+500
A fair 6-sided die is thrown 10 times. What is the probability of rolling a six three times in a row, and the other rolls not being a 6?
In 10 rolls, there are 8 positions where the chain of three can start, so there are 8 permutations since dice rolls are interchangeable (their order doesn't matter).
Therefore, I believe that the ...
2
votes
0
answers
141
views
+50
Different dot product values
I came across the following problem while studying geometry.
Given $v_1, ..., v_7 \in \mathbb{R}^3$ with no three vectors in the same plane. Show that among the dot products $v_i \cdot v_j, i \neq j$ ...
- 71
4
votes
1
answer
68
views
+50
How to show that is unique asymptotic stable
Based on this question:Poincaré-Bendixon show periodic solutions.
Show that the system $x^{'}=x-y-x^{3}$,$y^{'}=x+y-y^{3}$ has a unique periodic orbit on annulus $A:=\{(x,y): 1\le x^2+y^2\le 2\}$...
- 85
0
votes
0
answers
139
views
+50
Bézout theorem from special case
I just realized that one special case of Bézout theorem is very easy.
Bézout theorem: two homogeneous polynomials $P, Q \in \mathbb{C}[X, Y, Z]$ of degree $p, q$ with no common factor define two ...
- 471
3
votes
1
answer
163
views
+100
Prove $\sqrt{a+4b}+\sqrt{b+4c}+\sqrt{c+4a}\le 3\sqrt{abc+4}$ when $ a^2+b^2+c^2=3.$
Let $a,b,c\ge 0: a^2+b^2+c^2=3.$ Prove that$$\color{black}{\sqrt{a+4b}+\sqrt{b+4c}+\sqrt{c+4a}\le 3\sqrt{abc+4}. }$$
Naturally, I use Cauchy-Schwarz as
$$\sqrt{a+4b}+\sqrt{b+4c}+\sqrt{c+4a}\le \sqrt{...
- 671
5
votes
0
answers
96
views
+50
Question about proof of Kolmogorov inequality for Bernoulli random variables
The question is about one inequality which shows in Kolmogorov's paper (inequality (3.1)) but is not proved. The inequality says that, if we assume $Y_1,Y_2,\ldots$ are i.i.d. Bernoulli random ...
- 81
3
votes
0
answers
90
views
+250
Concerns about the definition of Hawkes process
In the lecture notes I am reading about Point process, when we introduced the Hawkes process several expressions are given and I have some difficulty to understand properly what is the $Z_t$ (defined ...
- 1,116
6
votes
0
answers
196
views
+500
Orbits of f.g. abelian group acting by affine transformations on a free abelian group
Let $A$ be a finitely generated abelian subgroup of $\operatorname{Aff}(\mathbb{Z^n}) = \mathbb{Z}^n \rtimes \operatorname{GL}_n(\mathbb{Z})$. Then $A$ acts naturally on $\mathbb{Z}^n$:
$$A \times \...
- 6,140
2
votes
0
answers
71
views
+100
Well-posedness for ODEs with discontinuous right hand side
Consider the IVP for ODE with discontinuous right hand side,
\begin{eqnarray}
\dot{y}&=&f(y,y) \quad \quad \text{for }t>0,\\
y(0)&=&y_0,
\end{eqnarray}
where, $f\in L^{\infty}(\...
- 496
2
votes
1
answer
72
views
+50
Sheaf on Zariski closed subset $Y$ is well defined
Let $(X,\mathcal{O}_X)$ be a scheme, and $Y$ a Zariski closed subset. For each $U\subset X$ open we defined the ideal of $\mathcal{O}_X(U)$, $I(U)$ to be:
$$I(U)=\{s\in \mathcal{O}_X(U): \forall x\in ...
- 2,136
8
votes
0
answers
216
views
+50
Can this be done? Split Pascal's triangle (without the $1$s) with a straight line into two regions of equal sums.
Consider Pascal's triangle with $n$ rows, without the $1$s, with each number corresponding to a vertex on a pyramid of equilateral triangles, as shown below with example $n=5$.
Can the triangle be ...
- 16.4k
2
votes
3
answers
215
views
+50
How to evaluate $\sum_{i=1}^n i^{2 i}$?
Let,
$$\mathcal{S}(n) = \sum_{i=1}^n i^{2 i}$$
for $n \in \mathbb{N}$
I will be completely honest. When I was returning from my physics tuition center, and suddenly this popped up in my head from ...
- 180
1
vote
0
answers
67
views
+50
On the solution I got from 2D Laplace equation
$f$ is function that $f(0)=1$ and $f:[0,\infty)\to\mathbb{R}$. It's continuous on $[0,\infty)$ and is $C^2$ on $(0,\infty)$. Let $u$ be another function defined on $\mathbb{R}^2$ and $u(x,y)=f(\sqrt{x^...
- 634
0
votes
0
answers
912
views
+50
Continuity of Green's function and its derivatives
We have a differential equation:
$$a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1} y}{dx^{n-1}} + \dots + a_1(x)\frac{dy}{dx} + a_0(x)y = \delta(x-z) $$
And this is satisfied by the Green's ...
- 361
0
votes
0
answers
25
views
+50
Solving logistic regression equation for slope
I've calculated a logistic regression model involving two variables $X_1$ and $X_2$ and their interaction $X_1 \times X_2$ and obtained regression coefficients for each.
The equation takes following ...
- 93
3
votes
1
answer
56
views
+50
Understanding the condition of the bounded variable algorithm in the linear programming
Following is the section 7.3 of Operation Research An Introduction by Hamdy A. Taha,
Define the upper-bounded LP model as, $$\max z=\{CX|(A,I)X=b,0\leq X\leq U\}$$
The bounded algorithm uses only the ...
- 379
0
votes
0
answers
120
views
+100
Optimize $\sqrt{a^2+5bc}+\sqrt{b^2+5ca}+\sqrt{c^2+5ab}\ge \sqrt{a^2+b^2+c^2+17(ab+bc+ca)}.$
Question. Let $a,b,c\ge 0.$ Prove that $$\sqrt{a^2+5bc}+\sqrt{b^2+5ca}+\sqrt{c^2+5ab}\ge \sqrt{a^2+b^2+c^2+17(ab+bc+ca)},$$when $c=\min\{a,b,c\}$ and $(a-b)^2\ge (c-b)(c-a).$
Here's what I done so ...
- 758
1
vote
0
answers
38
views
+100
Monte carlo estimation, but function value can only be estimated indirectly
I have the following setup.
I want to Monte Carlo estimate a big sum $$F=\sum_x p(x) f(x)$$ by drawing $\\{ x_1,\dots, x_M \\}$ from the distribution $p(x)$ and averaging $f(x_i)$.
However, in my case ...
- 291
5
votes
2
answers
247
views
+50
Conditional Probability and Borel-Cantelli theorems
I am given a sequence of events $A_1, A_2, \dots$ that satisfy
$$
\sum_{n \geq k} \mathbb{P}\left(A_n \ \middle|\ \bigcap_{i=k}^{n-1} A_i^c \right) = \infty, \forall k \in \mathbb{N}
$$
and asked to ...
- 3,755
15
votes
0
answers
293
views
+150
Can nonisomorphic groups have near-identical Cayley tables?
Nonisomorphic groups can have very similar multiplication (Cayley) tables. For example, the two groups
\begin{align*}
\mathbb{Z}/9\mathbb{Z}&=\{\overset{a}{0},\overset{b}{1},\overset{c}{2},\...
- 111
6
votes
0
answers
100
views
+50
Exercise 5.18 Isaacs’ Character Theory of Finite Groups
The exercise is as follows:
Prove that the following polynomial takes integer values when evaluated at integer points: $$f(x) = \frac{1}{|G|} \sum_m a(m) x^{\frac{|G|}{m}}$$ where $a(m) = |\{g \in G \...
- 2,309
7
votes
0
answers
455
views
+50
Estimate true probabilities from weighted sampling without replacement
Suppose we have a random event with $n$ outcomes, with (unknown) true probabilities $p_1, p_2, \dots, p_n$. We have performed a study of this event, sampling it in batches of $k < n$. From this we ...
- 10.3k
2
votes
0
answers
97
views
+50
A question about divisor function and primes
$\sigma_1(n)$ denotes the divisor function which sums the divisors of a natural number $n > 1$.
We calculate the remainder of the division of $\sigma(n+\sigma(n+1+\sigma(n+2)))$ by $n+1$. $r$ is ...
- 47
1
vote
0
answers
39
views
+150
Are finitist systems the ones with a proof-theoretic ordinal of at most $\omega^\omega$?
The proof-theoretic ordinal of $\mathsf{EFA}$ and $\mathsf{RCA}_0^*$ are $\omega^3$ and the one of $\mathsf{PRA}$, $\mathsf{I\Sigma1}$, $\mathsf{RCA}_0$, etc. is $\omega^\omega$.
See https://ncatlab....
- 39
1
vote
0
answers
39
views
+50
Show the convergence of a sequence of random variables
Consider some discrete random variables $X_1,X_2,...$ each with support $A$. Take another sequence of random variables $Y_1,Y_2,\dots$
Given $x\in A$, assume:
A1: $\Pr(X_t=x| X_{1},\dots, X_{t-1}, Y_1,...
- 152
4
votes
2
answers
2k
views
+50
Why is only the first (highest) term of the divisor in polynomial long division used to divide?
There is one small matter that has always stumped me with polynomial long division. In the example from the Wikipedia on Polynomial long division, why is the equation only divided by the first/highest ...
- 41
2
votes
0
answers
65
views
+500
Kruskal's weak tree theorem and the size of tree(4)
For background please see this question: lower bound for Kruskal's weak tree function
I didn't have difficulty finding the same lower bound that user Deedlit did for tree(3), which is still in the ...
- 51.3k