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0answers
8 views

Monotone convergence theorem for a generic $f(x,y)$ instead of $f_n(x)$

I was wondering if it is safe to say that the monotone convergence theorem $$\lim_{n\to\infty} \int_X f_n(x) = \int_X \lim_{n\to\infty} f_n(x)$$ (where $f_n(x)$ is a non-decreasing sequence, etc.) is ...
-1
votes
0answers
5 views

Finding integer solutions of K for equation floor(A/K) = B

How to find integer solutions of K for equation floor(A/K) = B, in terms of A and B where A and B are non-negative integers? What I tried: floor(A/K) = B then B <= A/K < B + 1 then BK <= A &...
1
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0answers
7 views

$\forall\delta, \sigma \in F$ where $\delta \land \sigma$ are contrad., $\exists\theta$ so that $\delta\land\neg\theta$ & $\sigma\land\theta$ contrad.

I'm in my first logic class ever and I'm trying to wrap my head around this obscure question... Show that for all pairs $\delta, \sigma \in F$, where $\delta, \sigma$ contradict themselves there ...
-1
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0answers
13 views

Find the minimum value of the expression given that a,b,c are positive and real.

$${(a+3c)\over(a+2b+c)} + {4b\over(a+b+2c)} - {8c\over(a+b+3c)}$$ Here was my attempt: If c tends to infinity and a and b are small we get 1/3. Now , i took $b=0$ and we get $(a-c)^2\over(a+c)(a+3c)$ ...
0
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0answers
13 views

Prove that f is integrable using the Riemann Integral

I am solving the following problem: Define $f$ to $\mathbb{R}$ on $[0,1]$ by $f(x)=\begin{cases}1 ~~~~~~~~\text{if $1-2^{-2k}$$\leq$x$\leq$ $1-2^{-(2k+1) }$ for k=0,1,2,...}\\ 0~~~~~~~~\text{if $1-2^{-...
0
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0answers
3 views

Connecting an interior and boundary point of an open, connected subset of $\mathbb{R}^2$ using a jordan curve.

Suppose $B$ is an open and connected subset of $\mathbb{R}^2$. I want to show that for any interior point $x\in B$ and any boundary point $y\in \partial B$ of $B$, there exists a jordan curve $\Gamma:[...
0
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0answers
3 views

Properties of squared bessel process

How do I verify the following properties of the squared bessel process. I am not getting the right answers and would like to request an overview of the calculation. Please check the attached image ...
0
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3answers
27 views

Solving $e^{2x} + 2e^x = 8$

it's been a while since I've done Analysis and I'm currently trying to solve for $x$. I'll just show what I did to solve to solve : $e^{2x} + 2e^x = 8$ $\leftrightarrow e^x \cdot e^x + 2e^x -8 = 0$ $\...
0
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0answers
9 views

A question on the Hurwitz formula

Let $C$ be a projective curve and $f:C\rightarrow \mathbb{P}^1$ a morphism of degree $d$. By the Hurwitz formula we have that $$2g_C-2 = -2d + deg(R)$$ where $R$ is the ramification divisor of $f$. ...
0
votes
2answers
7 views

write the general term (the (k+1)th term) of the binomial expansion (a+b)^n in terms of n and k, where k < n.

I'm really unsure of what to do here, but I am trying to find the (k+1)th term or (r+1)th term of the binomial expansion (a+b)^n in terms of n and k, where k is less than n (k < n)
-1
votes
0answers
8 views

$C(T_n)$ is dense in $L^p(T_n)$

I am attending a course of functional analysis and I was asked to proof the following statement: $$ C(T_n)\ is\ dense\ in\ L^p(T_n) $$ where $T_n$ is the torus. I don't know what to do. I thought ...
1
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0answers
22 views

$a^4+b^4+c^4+d^4=4abcd$ , prove that $a=b=c=d$.

If $a,b,c$ and $d$ are positive real numbers satisfying the expression: $$a^4+b^4+c^4+d^4=4abcd$$ then, prove that $a=b=c=d$. Approach: $$a^4+b^4+c^4+d^4=4abcd$$ $$a^4-2a^2b^2+b^4+c^4-2c^2d^2+d^4=...
0
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0answers
10 views

Notation: $\mathbb{P}(N(t) = n)$ unconditional on $t$

This is a question about notation. I find good notation always helps with thinking clearly. Consider a Poisson process with rate $\lambda$, and let $N(t)$ be the number of occurrences at time $t \geq ...
0
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0answers
2 views

Find the Hardy-Littlewood Maximal Function of $\chi_E$ where $E=[0,1]\cup [2,3]$

I am stuck in this problem of trying to find the explicit formula for the Hardy-Littlewood maximal function $f^{*}(x)$ of $\chi_E$ where $E=[0,1]\cup[2,3]$. The Hardy-Littlewood maximal function has ...
0
votes
0answers
7 views

How to solve $|1+x|\frac{|\alpha-\beta|}{|\alpha + \beta|} + \max\{1, |x|\} \leq 1$?

I am trying to solve $$|1+x|\frac{|\alpha-\beta|}{|\alpha + \beta|} + \max\{1, |x|\} \leq 1 \tag{1}, $$ with $\alpha<\beta<0$. Using $$\max(x, y) = \frac{x + y + |x - y|}{2}$$ in $(1)$ we get $$...
0
votes
0answers
3 views

the rank of the Hadamard product

For matrices $D\in C^{d\times p}$ and $E \in C^{d\times p}$ with $d> p$, if $D$ is a full column matrix, for what condition that $D\odot E$ is also a full column matrix where $\odot$ denotes the ...
0
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0answers
15 views

Set-theoretic definition of a matrix? (and related question)

An ordered pair of numbers $(x,y)$ is defined in set theory as the set $\{x,\{x,y\}\}$, and analogously for triplets and so on. What is the "official" set-theoretic definition (or ...
0
votes
1answer
5 views

For $G_1$, $G_2$, $G_3$ simple undirected graphs on the same vertices with disjoint edge sets, if $G_1\cup(G_2\cap G_3)=G_2$, then $G_1=G_2$

If $G_1, G_2$ and $G_3$ are simple undirected graphs on the same set of vertices with disjoint edge sets. If we have a graph equation $$G_1\cup(G_2\cap G_3)=G_2$$ Then we have to show that $G_1=G_2,$ ...
0
votes
0answers
2 views

Discounted occupancy of Markov Chain states

Assuming a Markov Process with states $S$, transition probabilities in the form $p(s_{t+1} | s_t)$ and discount factor $\gamma$, i.e. at any step there is a probability $p(\emptyset | s_t) = 1-\gamma$ ...
0
votes
1answer
21 views

Is it possible to get the area between $y^2 = 3x$ and $y = x^3$?

I am a Grade 12 student taking Calculus. Our lesson for tomorrow is about area between curves. Our teacher gave this question for us to ponder about and do advanced study. Can someone explain this and ...
-2
votes
0answers
15 views

Problem with a limit (no L'Hospital)

I'm stuck with this limit and I don't know how to solve it. $$\lim_{x \to 3} \frac{(x^2-9)^{1/2}-(x-3)}{e^{-1/(x^2-9)}}$$ I'm not allowed to solve it the via Taylor-McLaurin expansion or L'Hospital. ...
0
votes
1answer
6 views

How to find the number of trials for a binomial distribution using a GDC?

I am given a discreet random variable X that satisfies the binomial distribution with p = 0.5 and number of trials 2n. I also know that P(X=n) is 0.273. I have to find the value of n and I am expected ...
0
votes
0answers
2 views

Proving a summation formula using the general Leibniz rule

I am trying to prove the following relations: $$ \partial^{N-2}(f^{N-1}g) =\sum_{n+m=N-2}\frac{(N-2)!}{n!\,(m+1)!}\,\big[\partial^{n}(f^{n}g)\big]\,(\partial^{m}f^{m+1}), \qquad N\geq2, $$ and $$ \...
0
votes
0answers
7 views

Laplace Equation for Brownian Motion

So, I know that there is this theorem (taken from here): For Laplace's equation $\Delta u = 0$ on a domain $D$ and $u=f$ on $\partial D$ (and some regularity conditions on $D$), we have $$ u(x) = \...
0
votes
1answer
10 views

Why is this alternating sequence converge?

the alternating sequence converges when (n+1)th term is less than nth term right? and in this case, when n is 1 the term is 1/3 and when n is 2, which is n+1 the term is 1/2, but this is larger than ...
0
votes
0answers
5 views

Prove energy preservation of implicit midpoint method.

I am using the Runge Kutta implicit midpoint method $$m_{n+1}=m_n + \frac{h}{2}(\frac{m_n + m_{n+1}}{2} \times (T^{-1}\frac{m_n + m_{n+1}}{2}).$$ To solve a free rigid body problem, where $T$ is the ...
1
vote
0answers
11 views

Repeated eigenvalue = 0

I have this quite straightforward system: $$ \displaystyle \begin{array}{l} \frac{d x}{d t}=\left[\begin{array}{cc} 3 & 9 \\ -1 & -3 \end{array}\right] x \\ \text { with } x(0)=\left[\begin{...
0
votes
0answers
7 views

Ramification question in compositum of cyclotomic and degree 5 extension.

I am reviewing some past exam questions and I have a problem solving the following example: Let $F = \mathbb{Q}(\zeta_5, \sqrt[5]{75})$, a field of degree 20 over $\mathbb{Q}$. Determine the ...
0
votes
0answers
9 views

induced fundamental group homomorphism by continuous map from torus $T^2$ to $\mathbb{RP}^1$

What is possible description of fundamental group homomorphism induced by continuous map from torus $T^2$ to $\mathbb{RP}^1$?
-1
votes
1answer
8 views

Exponential probability distribution formula

Why is it that for exponential distribution, when P(X<x) the probability is 1-exp(-λx) while for P(X>x), the probability is exp(-λx)? Normally, if P(X<x), wouldn’t the probability just be the ...
0
votes
0answers
7 views

Chernoff bounds - delta too large

I am trying to solve some bounds with Chernoff bounds. My problem is that the values i get for $\delta$ are usually so large that it is not possible to compute what is the result. As an example here ...
0
votes
0answers
11 views

why is the residue defined on the circle in complex analysis?

I'm confused by the definition of the residue,which is as follows(from the book complex bariables and applications by James Ward Brown) I don't know why the author restricted the z in $0<|z-z_0|&...
1
vote
1answer
10 views

$Tor(A,B) = 0$ if $A,B$ is torsion free

I don't understand the proof given in Hatcher p.265 of $Tor(A,B) = 0$ if $A,B$ is torsion free. The proof is the following : The line I don't get is "This means [...] can be reduced to $0$ by a ...
0
votes
1answer
23 views

What is $DE.DA$ here?

In $\triangle ABC$, let $D$ be the midpoint of $BC$ and $E$ be a point on $AD$ so that $\angle BEC +\angle BAC = 180^\circ$. If $BC = x$, then find $DE.DA$. How to approach this? Please do not work ...
0
votes
0answers
8 views

Entropy for partition with respect to sum of Dirac measures

Let $(X,\mathcal{A},m)$ be some probability space where $m=\frac{1}{p}\sum_{j=0}^{p-1}\delta_{f^jx}$ for some fixed $x\in X$ that is $p$-periodic with respect to the measure-preserving transformation $...
1
vote
1answer
27 views

Solving the system $ab=9-12i$, $ac=-16-12i$, $db=36$, $dc=-48i$ for complex $a$, $b$, $c$, $d$

I need to find four Unknown Variables, $a$, $b$, $c$, $d$, and I have four equations: $$\begin{align} ab &= \phantom{-1}9-12i \tag1\\ ac &=-16-12i \tag2\\ db &=\phantom{-}36 \tag3\\ dc &...
-1
votes
0answers
19 views

Does this group G have a subgroup isomorphic to G/Z(G)?

Assume that 1° $G$ is a group of order 180; 2° $Z(G)$ has order 3; 3° $G/Z(G)$ is isomorphic to $A_{5}$ (fifth alternating group); 4° every nontrivial characteristic subgroup of $G$ has order ...
0
votes
1answer
19 views

Prove by induction for $n\geq1, n \in \mathbb{N}$ $, 2^{2n+1}\equiv 9n^2-3n + 2(\mod54)$.

Question taken from the book by Andre Weil, titled Introductory number theory, chapter 5, question #V.3. Prove by induction that, if $n$ is a positive integer, then $2^{2n+1}\equiv 9n^2-3n + 2(\mod54)$...
-2
votes
0answers
12 views

What is the first and second order derivative of the following function?

What is the first and second order derivative of the following function? $$\sum_{l=1}^{k}(\sum_{j\in B_l} Y_j^{\frac{1}{\Theta_l}})^{\Theta_l}$$
0
votes
0answers
10 views

Finding analytic continuatiuon of a branch

I'm warning you that this post relates of a topic I'm not comfortable with, so in order to solve the problem I will write below, I'm very interested in understanding other cases/general cases. The ...
0
votes
0answers
9 views

Probable existence of an almost integer contained in a limit

I found this almost integer in studying the limit : $$\lim_{x\to \infty}\Gamma\left(\sin^2\left(\frac{1}{x}\right)\right)\Gamma\left(\sin\left(\frac{1}{x}\right)\right)-x^3=-\infty$$ Well my goal was ...
0
votes
0answers
6 views

How to construct a bump function from a given diffeomorphism

I encountered a problem from, An introduction to chaotic dynamical systems 2nd edition by Robert. L. Devaney(pdf version available online), which reads: Using a bump function, show that the ...
0
votes
0answers
11 views

Positive function not vanishing in a neighbourhood

If a positive function in $C[-1,1]$ does not vanish in any neighbourhood of $-1$, then it has to be strictly positive in some neighbourhood of $1$. It seems obvious to me when I try draw pictures, ...
0
votes
0answers
4 views

Sign Error: solved differential equation $y'' = -b\,y'-g$

I got solved a differential equation here, but it still seems to differ from the textbook solution by a wrong sign: again it's about $y'' = -b\, y'-g$. The solution process goes like: \begin{align*} &...
0
votes
1answer
18 views

Showing that $F(x) = x + f(x)$ defines a homeomorphism when $f : E \to E$, and where $E$ is a Banach space.

Let $E$ be a Banach space and $f : E \to E$ a contraction. Show that the equation $F(x)=x+f(x)$ defines a homeomorphism $F:E \to E$ that is Bilipschitz. Since $f$ is a contraction the following to ...
0
votes
0answers
13 views

$\lim_{n \to \infty}\sup_{k \geqslant n} (\frac{1+a_{k+1}}{a_k})^k \geqslant e$ [duplicate]

Prove: $$\lim_{n \to \infty}\sup_{k \geqslant n} (\frac{1+a_{k+1}}{a_k})^k \geqslant e$$ for any sequence $\{a_n\}$ where $a_n > 0$. I made a likely proof myself, but it's so complicated that ...
0
votes
0answers
9 views

Survival probability of a random walk

I was looking for the probability that a discrete random walk stays below a certain level for $n$ steps. $x_0$ denotes the initial position and the position at the $i$-th step is $x_i=x_{i-1} + \eta_i$...
0
votes
0answers
6 views

Supporting hyperplane for convex set in Hilbert space

So, I have a closed and convex set $C$ in a Hilbert space $H$. Say, we have a point $x_0\in\partial_{\rm rel}C$ (relative boundary of $C$). Does there exists a vector $y\in H\backslash\{0\}$ such ...
-2
votes
0answers
30 views

Mathematical research institutes similar to Banff and Oberwolfach

What other institutes such as these two exist for a visit by a scientist for an undisturbed period of short research? Ideally with a good landscape. Dagstuhl is another one I found.
0
votes
1answer
32 views

Tangents to circles: What do I do now?

everyone! I am confused about what to do here and how to do it. May I have some help? So, I know that $\overline{FI}$ and $\overline{IE}$ are both radii of the small circle and $\overline{JK}$ and $\...

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