0
votes
0answers
3 views

Probability of Level Crossing

I am kind of stuck on how to proceed on this. $X_n$ is an IID process with $$f_{X_n}(y)= \frac\lambda2 e^{-\lambda |y|}$$ There is a stationary autoregressive process $Y_n$ defined as $$Y_n=\rho ...
0
votes
0answers
4 views

Get parameters for given point on quadratic bezier triangle

I have a 2 dimensional quadratic bezier triangle described by the position of its corners $v_0$, $v_1$ and $v_2$ and a handle for each side $h_0$, $h_1$ and $h_2$. The parametric equation with the ...
0
votes
0answers
9 views

Prove that $F_1$ and $F_2$ are continuous and that $\int_{\gamma_1}F_1(z) dz = \int_{\gamma_2}F_2(z) dw$

Let $\Omega_1, \Omega_2 \subseteq \mathbb{C}$ and let $\gamma_1: [a,b] \to \Omega_1$, $\gamma_2: [c,d] \to \Omega_2$ be paths. Let $f$ be a continuous function defined on $\gamma_1 \times \gamma_2$ ...
0
votes
0answers
11 views

What is $\int_0^{\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$?

I read this question. The integral has a special property, so it might possibly be evaluable? No one tried evaluating it, so I created this. Not very often I ask question like this, but here it is. ...
1
vote
0answers
6 views

Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic groups ...
0
votes
0answers
11 views

what is the difference between statiscal averagre and average?

I'm reading a book on synthetic aperture radar and it is said that: The term $\sigma^{\circ}$ is the averaged radar cross section per unit area, also called the scattering coefficient or ...
0
votes
0answers
3 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
2
votes
0answers
11 views

Proofs by analysing games

I recently read the following article giving a novel proof of the uncountability of $\mathbb{R}$ by analysing a particular game, amongst other results. ...
2
votes
0answers
19 views

$m(E)=0$ or $m(E^c)=0$

The question comes from former qualifying exam of the graduate school I'm going to attend. Q: Suppose $E$ is measurable and $E=E+\frac{1}{n}$ for every natural number $n\geq 1$. Show that either ...
2
votes
0answers
13 views

Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
3
votes
0answers
27 views

Spaces whose all their metrizations are complete [duplicate]

Which metrizable topological spaces $(X,\tau)$ posses the following property: Every compatible metric (i.e one which induces the same topology $\tau$) is complete. Compact metrizable spaces satisfy ...
0
votes
2answers
28 views

Formula that's only satisfiable in infinite structures

What formula in first order logics can I write that's only satisfiable over infinite structures, over a dictionary without the = sign?
0
votes
0answers
29 views

Does my derivation work?

I've been totally engaged with exponential integrals for a while. I came across to this limit in my work. I started to calculate the limit as below: currently, I am not sure about my handouts. would ...
0
votes
1answer
38 views

Finding the definite integral $\int_{0}^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$

$$\int_0^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$$ My try: $$I=\int_0^\pi \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} dx+\int_\pi^{2\pi} \frac{e^{-\sin x}\cos(x)}{1+e^{\tan x}} dx$$ also ...
3
votes
0answers
35 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
2
votes
0answers
20 views

An easy (or not?) collection of proper sets .

Let $S$ be a finite set. We are given $k$ rows and in each row we have two subsets of $S$ which we call them $A_i$, $B_i$ (for the $i$th row, with $i\leq k$). $A_1$ and $B_1$ $A_2$ and $B_2$ . . ...
1
vote
1answer
21 views

Evaluate $\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$

I need to evaluate the following integral using Green's theorem $$\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$$ $C$: from point $E \to F\to G\to H$ ...
0
votes
3answers
40 views

Continuous but not uniformly continuous example

Let $f(x) = \frac{1}{x}$ for $x > 0$ and take our set at which the function act on $(0,1]$. This function is continuous but not uniformly continuous on $A$. To prove this consider $\epsilon = ...
0
votes
0answers
29 views

Finding sequences in $\pi$ [duplicate]

I recently came across some programs which were able to calculate exactly, when a particular sequence of digits appears the first time in the decimal expansion of $\pi$. This made me wondering, if it ...
1
vote
0answers
7 views

Basic examples of functions in Hörmander class

The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with $$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq ...
0
votes
1answer
17 views

Solving differential equation using Laplace transform

Can this DE be solved using Laplace transform? $\frac{\mathrm{d} y}{\mathrm{d} x}\cos x=y\sin x+\cos ^{2}x$
0
votes
2answers
25 views

Can we obtain the pair $(1,50)$ with these following operations?

It's a problem from some russian competition: We're given a card with two positive integers $(a,b)$ and we have tree machines which generate another card from the one we insert on it(I assume we ...
0
votes
2answers
36 views

Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$

I have problems to demonstrate: $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$. I don't have any clue how can I start to work with it. Any hint will be helpful.
0
votes
1answer
9 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
1
vote
2answers
21 views

Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$

My question is, Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$ for some $a_k$ where $a_k>0$ and $a_k \rightarrow 0$ as $k \rightarrow 0$. ...
0
votes
0answers
14 views

Proof of x = 0 modulo 3 only if the sum of its digits 0 modulo 3 [duplicate]

Okey, lets beggin from a helpfull proposition I've already proved: $$$$ if $a_i\equiv b_i\:\forall 0\le i\le m$ then to any $m$ numbers: $p_1,p_2,...,p_m\in \mathbb{Z}$ $$\sum ...
0
votes
3answers
39 views

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|$

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|.$ This seems quite intuitively correct, but I do not know how to prove this formally, does anyone know how they would go about this?
0
votes
2answers
26 views

Trigonometric Form of Complex Numbers question.

What is the following quotient expressed in polar form: $$\frac{10(\cos(35^{\circ})+ isin(35^{\circ}))}{5(\cos(100^{\circ}) +i\sin(100^{\circ}))}?$$ Please enter your answer in cis notation and ...
1
vote
2answers
42 views

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the ...
0
votes
1answer
33 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
0
votes
1answer
14 views

Polynomial with arithmetic values

Can I find a polynomial in a second degree in two variables from the values of which can be found an infinite arithmetic progression? Thank you!
-1
votes
0answers
19 views

What does the Graph of ax0 + bx1+cx2+dx3… look like? [on hold]

What does the Graph of ax0 + bx1+cx2+dx3... look like? How do I plot it in mathematica? I don t have numbers for a,b,c.
2
votes
0answers
14 views

Rigorous definition of an embedded connected sum.

Can someone point me to a rigorous definition of a connected sum of two smooth embeddings? I know about the usual construction, the problem is that I can't find a proof that this construction is well ...
1
vote
2answers
35 views

Does construction of infinite product measure require axiom of choice?

I am learning about infinite (countable) product measure, which the exact statement of the theorem I write below. I was wondering if the theorem requires axiom of choice or not. I would appreciate any ...
1
vote
2answers
18 views

Can we replace the limit of a sequence with that of a function?

Let $f$ be a function defined in $[1,\infty]$. If $\lim_{x\to\infty}f(x) = L$ and $a_n = f(n)$ for integer $n\ge 1$ then $\lim_{n\to\infty}a_n = L$. Found this theorem in many references, but ...
1
vote
2answers
31 views

Show $f: S^1 - {N} \to \mathbb{R} $ $f(x_1,x_2) = \frac{x_1}{1-x_2}$ is Homeomorphism

$S^1$ is a unit circle and $N := \{ (0,1) \in S^1\}$. The question hints that the for any $(x_1,x_2) \in S^1- {N}$, line joining $N$ and $( x_ 1 , x_ 2 )$ meets the $x$ -axis at ($f ( x_ 1 ;x_ 2 ) , 0 ...
1
vote
1answer
29 views

Curl of a vector field cross itself

How we can use the property that $$A×(B×C) = B(A.C)- C(A.B)$$ to prove the relation: $$a×(∇×a) = ∇ (a^2/2) -(a.∇)a.$$ When I use it, the result directly appear to be $$∇(|a|^2 )-(a.∇)a$$ instead of ...
0
votes
0answers
13 views

$ G=(V,E_1 \cup E_2) $ is a triangle free graph, where $ G_1=(V,E_1) $ is planar and $ G_2 = (V, E_2)$ is a tree. Prove that: $ \chi (G) < 7 $

can anyone help with this, any direction could be helpfull? I've tried using the fact that $ G_1 $ satisfies that it's planar and is triangle free because G is. So we should have $|E_1| \leq 2|V|-4 $ ...
2
votes
1answer
37 views

Find the limit of an infinite series

My intuition was to try and see if the series is a Riemann Sum of a function and then see what happens but I can't really see which function fits here. Thanks!
0
votes
1answer
29 views

Prove that if events $A,B$ independent of C then $P(A\cap B\cap C)= P(A\cap B)P(C)$

I am trying to prove why the intersection of two events $A, B$ that are independent of C is also independent of C so that the following equality holds: $$P(A\cap B\cap C)= P(A\cap B)P(C)$$ ...
1
vote
0answers
22 views

confusion about change of variable

If you are integrating $f(x,y)$ over a region and you do a change of variable to $f(u,v)$. The jacobian gives $dx\,dy = du\,dv (\partial x/ \partial u\ \partial y/\partial v - \partial x/\partial v\ ...
2
votes
1answer
21 views

Minimizing long equation with hyperbolic functions

In physics book that I am reading it is said that minimizing the expression $$\phi = - N T k \log (2 \cosh(H \beta)) - \frac{J N}{2} z \tanh^2(H \beta) + H N \tanh(H \beta) $$ with respect to $H$ ...
0
votes
3answers
48 views

What is the limit of this sequence as n->infinity? [on hold]

Find the limit of the following sequence $n^{\ln(n)/n}$ as $n\to\infty$? Please answer without using L'Hopital
7
votes
0answers
48 views

What do we know about the first occurrences of prime gaps?

Are there any conjectures from which we can infer something about the first occurrences of prime gaps length $n$ and their distribution? I've made an interesting graph of these values to make this ...
0
votes
0answers
18 views

Number of ways to get from top to left through a blocked Grid

There is a Grid and a guy , guy can move either down or right in a Grid as shown from starting point (1,1). Let the Grid Dimensions be N x M . (The length of Grid x The Breadth of Grid) There are ...
0
votes
4answers
83 views

Find limit of the following sequence?

Find the limit of $\frac{\log(n+1)}{\log(n)}$ where $n\rightarrow\infty$. Here $n$ is a natural number so I guess we can't use L'Hopital
2
votes
1answer
16 views

Direct Limit of finitely generated groups

Is every group the direct limit of its finitely generated subgroups? This is true for abelian groups, I have not seen this statement for nonabelian groups, so i am wondering if this is true. Seems ...
0
votes
0answers
10 views

Time advance in Adaptive Mesh Refinement method

I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
0
votes
1answer
43 views

Direct Sum Proof

I am reading Axler's Linear Algebra Done Right Book and I am on the part about the direct sum. He gives the following proposition: When he assumes that $a$ and $b$ hold to prove that the proof gives ...
1
vote
0answers
23 views

A question regarding harmonic function.

Can any one provide some hint on the following question? I have being thinking about this for a while but cannot figure out where to start. I have been thinking about Taylor expansion but it seems not ...

15 30 50 per page