0
votes
0answers
13 views

What is the inverse of the integrated $\chi^2$ function?

I am implementing some preprocessing of variables in the context of a paper called A Neural Bayesian Estimator for Conditional Probability Densities. It states: 1.) Given a non-linear, a monotonous ...
0
votes
1answer
18 views

Prove that the arithmetic-geometric mean inequality holds for any list of numbers whose length is a power of 2

I am self-studying and currently reading How to Prove it by Velleman. I tried to prove the above by induction (I proved that this holds true for $n=2$), but I think my proof is wrong. I only started ...
2
votes
0answers
10 views

Maple: Symmetries of strange modified heat-equation

I thought about the following PDE: $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(x=0,t).$$ How can i determine the symmetries of this PDE with Maple? And how ...
1
vote
1answer
15 views

What is a primitive element in a Fuchsian group?

I am reading some introductory texts about Selberg's trace formula for hyperbolic surfaces and I have encountered the concept of "primitive element" $\gamma$ in a hyperbolic Fuchsian group $\Gamma \...
2
votes
1answer
22 views

Completion of a normed space using an isometry

I'm practicing some old exams for my functional analysis exam tomorrow, and i'm having trouble with the following: Let $X$ be a reflexive Banach space and let $Y$ be a normed space. Assume there ...
-2
votes
2answers
29 views

Simplify the following question [on hold]

Simplify the following expression: $$(A\cup B)\cap (A\cup C)$$
1
vote
0answers
10 views

Justify $\lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p}$

Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ...
0
votes
1answer
24 views

Show that the closure operation has the following properties

Let $E_1$ and $E_2$ be subsets of $\mathbb{R}$. I need to show that the closure operation has the following properties: a)$(E_1 \subset E_2) \Rightarrow (closure(E_1) \subset closure(E_2))$ b)$...
0
votes
0answers
4 views

Intersection of minimal affine halfspaces

We have a subset S which is the sum of two sets, A and B. (S= A+B) A is the set of a vector: $ x\begin {pmatrix} 0 & 0 & 1\end{pmatrix} \in \mathbb{R}^3$ where $x \geq 0$. B is the set of a ...
1
vote
0answers
38 views

Why does Maple include x in the solution of this definite integral?

I have the following function defined in Maple: $$ f(x) := (2 - a + ax^2) \sqrt{1 + 4a^2x^2} $$ And I want to calculate the definite integral of this from -1 to 1: $$ \int_{-1}^{1}{f(x)dx} $$ I do ...
0
votes
1answer
29 views

Isomorphism between multiplicative group modulo n and that of its factors

I am not entirely sure if this is true, but if it is, I would be done with a very important proof. Let $a$, $b$ and $d$ be pairwise coprime. Prove that: $$|(\mathbb{Z}/ab\mathbb{Z})^*/\langle d\...
0
votes
0answers
18 views

What is the intuition behind Uniform Integrability?

A definition of Uniform Integrability I am currently working with is that: A sequence $X_1, X_2, \ldots$ of random variables is Uniformly Integrable if: $$ \sup_n \mathbb{E}\left(|X_n|\cdot \mathbb{...
1
vote
1answer
23 views

Throwing dice and finding limits

We throw an honest dice $n$ times. Let $S_n$ be the number of throws with even number of dots on the dice. 1.) Calculate the limit $$\lim_{n\rightarrow\infty}P(2S_n \leq n)$$ 2.) Express the value ...
0
votes
0answers
24 views

“A reflection across one line in the plane is, geometrically, just like a reflection across any other line.”How?

How can this statement be represented geometrically?-"A reflection across one line in the plane is, geometrically, just like a reflection across any other line." (i tried it by drawing some ...
0
votes
0answers
9 views

Green's function and strong Markov property for stopped Brownian motion

Let $X(t)$ be a Brownian motion in $\mathbb{R}^n$, stopped at some fixed time $T$. Is there a notion of Green's function for such a Brownian motion? I am guessing that there is, and $G(x, y) : = \...
0
votes
2answers
27 views

Partial summation formula for $\sum_{k=1}^{\infty}\frac{1}{(k+1)\sqrt{k}+k\sqrt{k+1}}$.

I am stuck in finding the partial summation formula for the following series. I need hint. $$\sum_{k=1}^{\infty}\frac{1}{(k+1)\sqrt{k}+k\sqrt{k+1}}$$
-1
votes
1answer
12 views

Convert currency base from one base to another

Consider a list of currency rates where the base is USD: 1 USD = 0.90390 EUR 0.74771 GBP 1.30101 CAD how can I convert the base from USD to GBP, such that I ...
2
votes
0answers
29 views

Asymptotic vlaue of $ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $

Inspired by this question I tried to find an asymptotic formula for $$ f(n)=\sum_{i=0}^n\lfloor \sqrt{i}\rfloor\binom{n}{i} $$ With the observation: $$ f(n)=\sum_{i=0}^n\frac{\lfloor \sqrt{i}\rfloor+\...
0
votes
0answers
8 views

Homogeneous System of Parameters

Assume that $R$ is a finitely generated $k$-algebra of Krull dimension $n$. Is it true that any set $\{f_{1},f_{2},...,f_{n}\}$ of algebraically independent polynomials is a homogeneous system of ...
2
votes
2answers
31 views

Application Banach-Alaoglu Theorem

When reading about Banach-Alaoglu Theorem on Wikipedia, I read the following assertion: '' Let $f_n$ be a bounded sequence of functions in $L^p$. Then there exists a subsequence $f_{n_k}$ and an $f\...
0
votes
0answers
16 views

integrate equation

I am trying to integrate this equation, however I am not sure which method would be best. $\frac{\dot{a}}{a} = -2 \alpha \frac{\dot{M_1}}{M_1 + M_2}$ All the variables $a, M_1, M_2$ are time ...
0
votes
0answers
5 views

How much larger is the Likelihood Function Under True Model?

Let $X$ be a random variable with probability distribution functions given by $f$, and let $g \neq f$ (on a set of positive measure) be some other distribution. $D=\{x_1, \ldots, x_n\}$ is a set of $n$...
0
votes
2answers
22 views

Lebesgue integral question (double integral)

Let $g,h$ be nonnegative Lebesgue measurable functions on $\mathbb{R}$. Prove that $$\int_{-\infty}^\infty g(x)^2h(x)\,dx=\int_0^\infty\int_{\{t\in\mathbb{R}:g(t)>x\}}2h(t)x\,dtdx.$$ I am lost on ...
1
vote
0answers
10 views

Using Fixed point iterations for solving system of linear equations

Given a system of $n$ linear equations $$ x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find $x_i$. The fixed point iteration define $$ x_i^...
4
votes
2answers
155 views

How to integrate $\int \dfrac{x^{13}\ dx}{x^5 + 1}$

We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method: $$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$ I cant find way to integrate ...
0
votes
0answers
16 views

Rational Solutions to equations like $a^2+3b^2=k^3$

I'm working on the number field $\mathbb{Q}(\sqrt{-3})$, and I want to find elements $\alpha \in \mathbb{Q}(\sqrt{-3})$ such that the polynomial $X^3-\frac{\overline{\alpha}}{\alpha^2}$ be irreducible....
0
votes
1answer
11 views

A condition that the ratio of locations is maximal

Sea $R$ un anillo conmutativo con identidad e $I$ un ideal de $R$ y $m$ un ideal maximal de $R$. Mostrar que $\displaystyle\frac{R_m}{I_m}\neq{0}$ si y solo si $I\subseteq{m}$. Dm: $[\Rightarrow{}]$. ...
0
votes
1answer
22 views

Prove: $\exists (a,b)\in A | \forall (x,y)\in A : \cos (x)+\cos (xy)\leq\cos(a)+\cos(ab)$

$$ A=\left\{(x,y) | -1\leq x\leq 1, |x|\leq y, x^2-2x+y^2\leq0 \right\} $$ Prove: $\exists (a,b)\in A | \forall (x,y)\in A :$ $$ \cos (x)+\cos (xy)\leq\cos(a)+\cos(ab) $$ I don't have any lead so a ...
0
votes
0answers
9 views

Can fourier transform be considered as a sum of a discrete series?

For a non periodic function in x domain, I think the difference between fourier transform and fourier series lies in the coefficients ,if I don't pull T out of the coefficient which makes it dw in ...
1
vote
1answer
12 views

Data point on regression line: Effect on estimates (simple linear regression)

I am concerned with the simple linear regression model, $y_k = a + bx_k + \epsilon_k$, where $(\epsilon_k)$ are iid normal with mean $0$ and $k=1,...,n$; here $n$ is the number of observations. I ...
0
votes
1answer
13 views

Why does $\bar{t_n}-{T_n}=\frac{3}{4^{k_n}}$

We have that $$t=0_4.q_1q_2q_3... \in [0,1]$$ t is in Quaternary form. Let $$T_n=0_4.q_1q_2q_3... q_{k_{n-1}}0$$ and $$\bar{t_n}=0_4.q_1q_2q_3..q_{k_{n-1}}3$$ Why does $\bar{t_n}-{T_n}=\frac{3}{4^{...
0
votes
2answers
24 views

Principale value, how can we consider it?

How can we consider principal value where as, it's something that "doesn't exist". For example, $$\int_{-\infty }^\infty \frac{1}{x}\mathrm d x$$ doesn't exist, but the principal value is nulle. What ...
3
votes
1answer
55 views

Why do we need axiom of choice for this?

The answerers on this question say that we need AoC (or some variant thereof) to prove that every infinite set has a countably infinite subset. In my view, choice is not needed but the answerers are ...
0
votes
0answers
4 views

$p \vee q \leq p+q$ for $p,q$ projection

I am wondering if $p \vee q \leq p+q$ for $p,q$ projection acting on some Hilbert space $H$. In particular, I wonder if the set of finite trace projections is upwards directed with the usual ordering ...
-2
votes
1answer
51 views

A proposition about a stochastically continuous process with independent increments.

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with filtration $F=\left\{ \mathcal{F_t} \right\}_{t\geqslant0}$ , and $X=\left\{ X_t \right\}_{t\geqslant0}$ is an ...
0
votes
1answer
15 views

Proving that the following is a convex subset

It is given that $I(S)=\{z|\exists x,y \in \mathbb{R}$ such that $(x,y,z)^t \in S$}$\subseteq \mathbb{R}$. How do I show that I(S) is a convex subset? We know that $S \subseteq \mathbb{R}^3$ is ...
1
vote
1answer
32 views

Local sections of $\mathcal{O}(1)$

Let $\mathcal{O}(-1)$ be the tautological bundle over $\mathbb{P}^n$ and $\mathcal{O}(1)$ its dual bundle, also known as the hyperplane bundle. I know that there is a bijection between $\Gamma(\mathbb{...
0
votes
0answers
20 views

Probability of Elements belonging together

Imagine a camera scene, where an algorithm labels the person which are inside from 1 to n. Now, imagine there is not just one perspective, but multiple. That means multiple cameras looking at the same ...
1
vote
6answers
50 views

How to prove $\gcd(a-bc, b) =\gcd(a,b)$ for $a,b,c \in \mathbb{Z}$? [duplicate]

I'm not really sure how to approach the problem, since I'm not really sure I understand the mechanisms why it is true aside from putting some numbers in and see that it works. Qualitatively, I'd try ...
0
votes
0answers
8 views

How do great circles project on the mercator projection?

Given a great circle connecting two points on a sphere, what is the function describing it's Mercator projection? In other words, given two longitudes and latitudes $(\phi_1, \theta_1)$ and $(\phi_2, \...
0
votes
0answers
6 views

image of composition of upper triangular integral matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $im(AB)$ in terms of $im(A)$, $im(B)$, unions, intersections, determinants, etc?
-1
votes
0answers
14 views

Combinatorial commutative algebra

Let G is simple graph and Δ(G) is clique complex, IΔ(G) has a 2-linear resolution if and only if, for any subset W ⊂ [n], one has H˜i(Δ(G)W ; K) = 0 unless i=0
2
votes
1answer
54 views

How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
0
votes
1answer
15 views

Calculating the normalization constant in spherical harmonics?

Anyone know a presentation of the calculation of the normalization constant in spherical harmonics. Specifically, how has $$\sqrt{\frac{2l+1}{4 \pi} \frac{(l-m)!}{(l+m)!}}$$ been found in $$Y_l^m(\...
0
votes
0answers
15 views

Explaination of this particular approach to form differential equation.

Consider we need to form differential equation of the following: $Ax^2 + By^2 = 1$ where A & B are constants. Now one approach I know of is two differentiate this equation 2 times (since 2 ...
2
votes
3answers
50 views

For how many 3-digit prime numbers $\overline{abc}$ do we have: $b^2-4ac=9$?

For how many 3-digit prime numbers $\overline{abc}$ do we have: $b^2-4ac=9$? The only analysis I did is: $(b-3)(b+3)=4ac \implies\ b\geq3 $ $b=3\implies\ c=0\implies impossible!!$ So I deduced that $...
0
votes
1answer
15 views

Can we define flat connection on any given smooth manifold?

For example, a sphere S^2 in R^3 is apparently not flat with respect to the Euclidean connection, but can we define a flat connection and thus with affine charts on S^2?
-1
votes
2answers
30 views

If $\{a_n\}$ is a sequence then show that $\liminf (-a_n ) = -\limsup(a_n)$

Could any one please help me figure out this question: If $\{a_n\}$ is a sequence then show that $\liminf (-a_n ) = -\limsup(a_n)$. Please, I am having trouble understanding it.

15 30 50 per page