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I have a function $$f(\lambda)=n\ln(1-p+pe^{\frac{\lambda}{n}})-\lambda p$$
I need to prove that $$f(\lambda)\leq \frac{\lambda^2}{8n}$$ using Taylor expansion.
I have used the taylor expansion for a function of the form $\ln(1-x)=-x-\frac{x^2}{2}-\frac{x^3}{3}-\dots$
Therefore: $f(\lambda)=n\ln[1-p+pe^{\frac{\lambda}{... |
Let $$f(x) = \frac{\sin x}x+\frac{\sin(x-1)}{2(x-1)}+\frac{\sin(x+1)}{2(x+1)}.$$
Find the common denominator and use common trigonometric identities to establish that
$$|f(x)| \le \frac C{|x^3|}$$
for large values of $x$. Note that $C$ is a positive constant.
Following these instructions, I find that
$$|f(x)| =... |
Assume that $f$ is in $L^1 (\mathbb{R})$ and $g(x)= e^{2iπx}$. Compute $f * g$
I just need a hint and not the entire answer. How can I compute the convolution when I don't know what $f$ is?
|
Let us define the following recurrence relations as so. $$a_1=6, a_{n+1}=a_n!$$
$$b_1=6, b_{n+1}=6^{b_n}$$
So, which of the following is larger? $a_{b_2}$ or $b_{a_2}$?
To clarify, I am trying to compare $6^{6^{6^{6^6\dots}}}$ ($720$ times) and $(((6!)!)!)!\dots$($46656$ times)
While I tried to use a calculator to de... |
Find the equation to the pair of angle bisectors of the pair of lines $(ax+by)^2=3(bx-ay)^2$.
Efforts:
$$(ax+by)^2=3(bx-ay)^2$$
After simplifying, I got:
$$x^2(a^2-3b^2)+8abxy+y^2(b^2-3a^2)=0$$
Now, what should I do next?
|
Does the following function can be simplified or solved?
$$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$
where S is a compact subset of a two-dimensional surface which has positive Lebesgue measure. $w(y)$ and $h(x)$ are weights, $w(y):S\righta... |
Can anyone please explain why the following is true? And what is the intuition behind it?
$$\chi_A(x) = \begin{cases}1 &, x \in A\\ 0 &, x \notin A.\end{cases}$$
Then we have
$$\chi_{\liminf A_n}(x) = \liminf \chi_{A_n}(x); \quad \chi_{\limsup A_n}(x) = \limsup \chi_{A_n}(x)$$
for all $x\in X$.
Thank you.
|
a) The $(x>0, y< -1)$ region of the curve $y= -\frac{1}{x}$ rotated about the $y$-axis.
The instructions say that one should use the formula:
$V = \int 2πxf(x) dx$
I used another method and got an answer of $π \text{ units}^3$. The formula that I used was $\int π* 1/y² dy$ (and I used an upper bound of $-1$, lower ... |
everybody.
I got this exercise from Jacobson.
Let $M$ be a monoid generated by a set $S$ and suppose every element of $S$ is invertible. Show that $M$ is a group.
Proof:
every element of $M$ has the form $s_1,s_2,...,s_n$ which is invertible with inverse $s^{-1}_1,s^{-1}_2,...,s^{-1}_n \in M$. Hence, $M$ is a group.
|
I want to ask what does it mean to say a group $A$ acts on $N$ via automorphisms. It is a notion used in M.Isaacs book and I am not familiar with. I tried to find how it is defined but a scanned e book is so hard to go through.
Suppose $*: A\times N \to N$ is a group action of $A$ on $N$, thus we have homomorphism $\ph... |
Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ and $\alpha \in BV([a,b])$
$$
\beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]}
$$
Why is the additional assumption $f$ continuous and what theorem if any is needed to show the we can substitute $d\beta$ in the following?
$$
\int_a^b... |
Let $F$ be a field of order $32$. Then find the number of non-zero solutions $(a,b)\in F\times F$ of the equation $x^2+xy+y^2=0$.
As , $|F|=32$ , so $(F\setminus\{0\},.)$ forms a group of order $31$, which is prime. So , $F\setminus \{0\}\simeq\mathbb Z_{31}$.
Then how I proceed ?
|
We say a morphism is a monomorphism if $fg=fh$ implies $g=h$.
So if $f$ is a monomorphism, is it necessarily an injection? i.e. $f(x)=f(y)$ implies $x=y$.
My approach is to consider a specific morphism $g_x$ who map every elements to $x$, thus $f(x)=f(y) \Rightarrow fg_x=fg_y \Rightarrow g_x=g_y \Rightarrow x=y$. But n... |
What is the widely used notation to denote the class of Lipschitz continuous functions?
i.e. suppose $f$ is continuously differentiable, then $f \in C^1$
|
Prove that for nonnegative $x,y,z$ that $\frac{1}{3}(x+y+z)^2 \geq xy + yz + xz.$
I saw this result in a problem but didn't know how to prove it. I tried expanding and collecting to get the trivial inequality but it didn't work.
The reason I am asking this is because it comes from the solution below where it claims i... |
In an exercise from Munkres-Topology Article 30 the author writes that there is a very familiar space which is NOT first countable but every point is a $G_\delta $ set. What is it?
Though there are answers posted on this site to the above question, I don't find the spaces familiar to what has been taught in the book u... |
To prove the first one, can I just say that $\lim_{n\to\infty} {a^n/n^p}$= $(\lim_{n\to\infty} a^n)(\lim_{n\to\infty} 1/n^p)$ and when the absolute value of $a$ is less than or equal to one, $(\lim_{n\to\infty} a^n)=0$ so the limit of the whole sequence is $0$?
|
I have seen in a paper to claim the following:
If $\sum\limits_n \sqrt{\mathrm{Var}(X_n)} < \infty$, then $\sum\limits_n (X_n - E[X_n])$ converges in $L^2$, for any sequence of random variables $\{X_n\}$.
As $\{X_n\}$ need not be independent, there will be covariance term, not understanding how is it possible ?
Wha... |
In one book on differential equations and dynamical systems I read that if (1) $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then (2) $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$. But I don't see how (2) follows from (1). Can someone please explain?
|
I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions
$$ \begin{bmatrix}
(1\times1) & (n\times1)\\
(n\times1) & (n\times n)
\end{bmatrix} .$$
I need to compute the determinant.
I've tried to understand what is shown ... |
This is an A level question. For better understanding, I will attach a screenshot of the question and the mark scheme.
Question:
Here's what I have done:
$$A(OBA) = \frac 12r^2α$$ [basic formula for area of sector]
$$A(ONB) = \frac 12r \sin(\frac {BN}{r})$$ [basic formula for area of triangle]
$$2\frac 12r \sin(\fra... |
I know the answer is $-10$ but I don't know where the negative sign is coming from.
This is what I ended up with. $$\frac{(x-25)(\sqrt{x}+5)}{x-25} = (1)\sqrt x+5 = 10
$$
Like I said I'm not sure where the negative sign comes from.
|
Can someone reconcile between the channel capacity in bits/symbol and in bits/second? Are they related to the definition of channel capacity in terms of mutual information and the other in terms of SNR and bandwidth?
|
Suppose $A$ is a $m \times n$ matrix and $V$ is a $m \times 1$ matrix with both $A$ and $V$ having rational entries and suppose the system $AX=V$ has a solution in $\mathbb{R}^n$. Then the equation has a solution with rational entries.
Is the above statement true or false?
|
Let $K\subset \mathbb{C}$ be a subfield and $f\in K[t]$ an irreducible polynomial. Show that $f$ has no multiple roots in $\mathbb{C}$.
If I understand this question correctly, I must show that there is no $a \in \mathbb{C}$ such that $(t-a)^n|f$ in $F[t]$ with $n>1$. So suppose $(t-a)^2|f$ and $f=(t-a)^2h$. Then we ... |
Forgive the undoubted ignorance of this question. I am out of my element both in probability and in nonstandard analysis.
A mathematically curious layperson friend recently had a conversation with me where he wanted a notion of "pick a random integer" (meaning uniformly random). I told him that there isn't room for thi... |
$\newcommand{\Z}{\mathbf Z}$
Problem. Let $G$ be the free group generated by three symbols $a, b$ and $c$, and let denote $G$ by writing $F(a, b, c)$. Let $N$ be the normal subgroup of $G$ generated by the words $aba^{-1}b^{-1}$ and $cbc^{-1}b^{-1}$. I want to find out what is $F(a, b, c)/N$.
I asked this of my frien... |
When answering this question,
In a model category, is the full subcategory of fibrant objects a reflective subcategory?
I realized that I wasn't even sure what the correct definition of a model category was! When I first learned about the basics of model category theory, I learned from Mark Hovey's book "Model categori... |
I'm having trouble solving for k1 in this equation:
ln(0.286/k1) = (100000/8.314)(1/500 - 1/490)
The right side should equal 0.491, which I can calculate just fine, but then the left side gives me trouble. Here's my approach:
ln(0.286/k1) = 0.491
ln(0.286) - ln(k1) = 0.491
- ln(k1) = 0.491 - ln(0.286)
- ln(k1) = 1.74... |
We have a set
$$ M=\{f:\mathbb{R} \rightarrow \mathbb{R}\mid f(1)>0\}\;.$$
I have never encountered this kind of set before.
I assume it is correct to say that $M$ is a subspace of a mapping space?
That this subspace is open if it maps to open sets like in this
case and that it is closed if it maps to closed sets?
An... |
Not all matrix with positive eigenvalues is positive definite, i.e. $\mathbf{x}^\mathsf{T}A\mathbf{x}>0$ for all non zero vector $\mathbf{x}$. For example consider matrix
$$A = \begin{bmatrix} 1 & -3 \\ 0 & 1 \end{bmatrix}.$$
How to prove that if we add symmetry into hypothesis then the assertion is true? That is, a sy... |
Not getting the right answer for this, can someone point me to where I'm going wrong?
|
I tried to determine the number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$.
Please could someone tell me if this is correct?
$$ \text{Aut}(\mathbb Z_{720}) \cong U(720) \cong U(9) \oplus U(16) \oplus U(5) \cong \mathbb Z_{6} \oplus \mathbb Z_{8} \oplus \mathbb Z_{4}$$
We want elements of orders $i,j,k$... |
Give $\Bbb R^I$ the uniform metric, where $I = [0, 1]$. Let $C(I, \Bbb R)$ be the subspace consisting of continuous functions. Show that $C(I, \Bbb R)$ has a countable dense subset, and therefore a countable basis. (Source: Munkres Question 15 Article -30)
My try:
Consider the set of all polynomials $\mathscr{P}=\{a_0+... |
Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$
we have $f(\overline{A})\subset \overline{ f(A) }$
give an example where $f(\overline{A})\ne \overline{ f(A) }$
Outline of my ideas for proof:
First, I would pick some $x... |
I read in an undergraduate real analysis textbook that the set of the natural numbers $\mathbb N$ is defined as the intersection of all inductive subsets of $\mathbb R$. However, I'm having trouble understanding what this means, specifically what an inductive subset of $\mathbb R$ is referring to. Are they sets such ... |
Given the Collatz function and its iterates:
$$
T(n) =
\begin{cases}
n/2, & \text{if $n$ is even} \\
(3n+1)/2, & \text{if $n$ is odd}
\end{cases}
$$
and $$ T^{k}(n) = T(T^{k-1}(n)).$$
Question. How does one apply iterates to large integers such as
$$
n = \big(3756801695685 \; \times \;2^{666669}\big) – 1\: ?\label{... |
I just spent more time than I care to admit pushing through the algebra of this question and, although I arrived at the answer, I'm curious to know if there is some simplification that could be done earlier on that would reduce the amount of algebra involved or otherwise improve the method (without introducing entirely... |
Let $M_0,M_1,\dots$ be a martingale with respect to $X_0,X_1,\dots$ and $T$ be a stopping time with respect to $X_0,X_1,\dots$ Define $T_n=\min\{n,T\}$ and let $M_{T_n}$ be the stopped martingale.
By Optional Stopping Theorem, I know that $EM_{T_n}=EM_0=EM_n$. However, my question is: how might I show $E|M_{T_n}|\leq E... |
I am trying to prove the Poincare Inequality on a n-dimensional box. That is a domain $ \Omega = (0,1)^n$ for $f(x) \epsilon H^{1}_{0}(\Omega) $, show there exists a constant $C$ such that $||f||_{L^{2}(\Omega)} \leq C||\nabla f||_{L^{2}(\Omega)} $ . I am not sure how to proceed.
Can I use the Divergence theorem here?... |
If $A\in L(X)$ then prove that $A^{-1}$ is linear and invertible.
Proof: Since $A$ is invertible then $A$ is injective and surjective. We know that $A^{-1}$ defines by $A^{-1}(Ax)=x$.
Remark: Also we can prove that $A(A^{-1}x)=x$. Indeed, if $x\in X$ and $A$ is surjective then exists $y\in X$ such that $x=Ay$. Hence $... |
I tried to solve the following exercise:
Let $p,q$ be odd primes and $n,m$ positive integers. Explain why $U(p^m) \oplus U(q^n)$ is not cyclic.
I solved the question as follows:
We have $U(p^m) \oplus U(q^n) \cong U(p^m q^n)$. It is a theorem that $U(n)$ is cyclic if and only if $n=2,4,p^k,2p^k$. But $p^m q^n$ is not ... |
Once upon a time, I memorized the following formula out of laziness.
Let $k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}$. Then $k'(x)$ is as follows.
$k'(x)=\frac{j(x)(g(x)h(x)f(x)^{g(x)-1}f'(x)+f(x)^{g(x)}(h(x)log(f(x))g'(x)+h'(x)+i'(x))-j'(x)(h(x)f(x)^{g(x)}+i(x))}{j(x)^2}$
(as confirmed by wolframalpha)
This was because ... |
$f$ has exactly one discontinuity.
$f$ has exactly two discontinuities.
$f$ has infinitely many discontinuities.
$f$ is continuous.
I know theorems related to connectedness and continuity but I don't seem to incorporate them here in solving the problem. Can you suggest me how to approach this problem?
|
I am looking at the classical proof of uniqueness for the heat equation in Evans.
Clearly, we differentiate under the integral sign of the square of $w$. A very basic question is, why are the assumptions of $u$, $Du$ and $\Delta u$ in $L^2$ not imposed here? When do we need them to be imposed in this problem?
|
I proved for a bounded set $\Omega$ and $1 \leq p \leq q \leq \infty$ that $L^{q}(\Omega) \subset L^{p}(\Omega)$. What is an example that would show strict inclusion, $ p<q$, and false if $\Omega$ is not bounded?
|
The following is a discussion on the following second differential equation
$$ \frac{dy^2}{dx} - y = 0 $$
So, let us introduce the following, convention and definition, represent the derivative operator by
$$ D = \frac{d}{dx} $$
So that our first equation can be represented as
$$ \left( D^2 - 1 \right) y(x) = 0 $$
My ... |
Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem
$$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$
to form a telescoping series. As a better approximation, taking one more term of this series, $$\pi^2\approx 9.9$$ was obtained.
... |
I have a question where I am asked to find the amount of terms required in a Maclaurin polynomial to estimate $\cos(1)$ to be correct to two decimal places.
So far what I have done is used Taylor's Theorem to get the follow:
$$|R_n(x)| = (|f^{(n)}*x^n|)/n! < (x^n)/n! < 0.005$$
I think so far this is my best attempt but... |
Question:
Consider n independent tosses of a $k$-sided fair dice. Let $X_i$ be the number of tosses that result in $i$.
What is the covariance $\mathrm{cov}(X_1,X_2)$ of $X_1$ and $X_2$.
\begin{align}
\mathrm{cov}(X_1,X_2) = \mathbf{E}[X_1X_2] - \mathbf{E}[X_1]\mathbf{E}[X_2]
\end{align}
I get a different $\mathbf{E}... |
Please, can someone help me? I have the following problem:
Let $X$ be a normed space, $Y \subset X$ a linear subspace of $X$. I have to prove that the function $$d_{Y}(x)=inf\{||x-y||:y \in Y\}$$ is a seminorme;
I tried to prove that, if $a,b \in X$: $d_{Y}(a+b)\le d_{Y}(a)+d_{Y}(b)$;
I denoted by $m=d_{Y}(a+b)$, $m_{1... |
I'm confused about the following exercise:
Prove that $U(144)$ is isomorphic to $U(140)$.
Here are my thoughts:
$$U(144) = U(12^2) = U(3^2)\oplus U(2^4) = \mathbb Z_{6} \oplus \mathbb Z_{8}$$
and
$$ U(140) = U(2^2 *7*5) = U(2^2) \oplus U(5) \oplus U(7) = \mathbb Z_{2} \oplus \mathbb Z_{4} \oplus \mathbb Z_{6}$$
And I h... |
Given the following definite integral
$$\int_0^4 \left[\left(1/2x^2 - 2x +8\right)-\left(1/4x^2+x\right)\right]\;\mathrm dx$$
I have done in the following process.
$$\int_0^4 \left[\left(1/2x^2 - 2x +8\right)-\left(1/4x^2+x\right)\right]\;\mathrm dx$$
$$\implies \int_0^4 [1/4x^2 -3x+8)]\;\mathrm dx\\ \implies {{{{1\ove... |
My text book does not provide much about counting measures and integration. So I decided to setup integration on space $(N , P(N) , \mu_c ,R)$ myself imitating the construction of Lebesgue integral. All functions are measurable since we are considering power set.I started with simple functions which are basically seque... |
I need help on problem based on integration calculus.
Q: how to integrate
$$\int\frac{dx}{1+\sin(x)\tan(x)}$$
Wolfram and integrate calculator does not help me.
|
Let $X$, $Y,$ and $Z$ be random variables. (There are no restrictions on these variables, but you may assume that these are continuous random variables if you want.) Suppose that $X$ and $Z$ are independent, and also suppose that $Y$ and $Z$ are independent. Does it follow that $\mathrm{cov}(XY,Z)=0$? (I understand tha... |
Find the jordan form of the matrix
$$A = \begin{pmatrix}
1 & 1 & 2 & 2\\
1 & -2 & -1 & -1\\
-2 & 1 & -1 & -1\\
1 & 1 & 2 & 2 \\
\end{pmatrix}$$
Hint: check that the matrix A has a single eigenvalue, and $trace(A) = 0$.
How can I check that the matrix has a single eigenvalue without using the determinant?
That's what... |
What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Murray-von Neumann equivalent to $e^{*}$?
|
For any $i \in \{1,2,3\}$, let:
$w_i \in [0,1]$ is an unknown number such that $\sum_{i \in \{1,2,3\}} w_i = 1$.
$t$ is a known number in $[0,1]$. Suppose that $t = 0.8$.
$f_i$ is also a known number in $[0,1]$. Suppose that $f_1 = 0.2$, $f_2=0.6$, and $f_3=0.5$.
Which are related by the equation: $t = \sum_{i \in \{... |
Definition- A groups $G$ is called Frobenius if it has a proper nontrivial subgroup $H$ such that $H \cap H^g=1\ \forall\ g\in G-H$.
Do we have a structure or classification theorem for (finite) Frobenius groups that classify all the groups which are Frobenius. I dont think so.
Then how can I get a lot of examples of F... |
Let $Q$ be the quaternion group. Find all homomorphisms $\phi: Q \rightarrow \mathbb{Z}_8$
What I get into is one big ifology:
Of course $\phi(1) = 0$, then $0 = \phi(1) = \phi(-1 \cdot (-1)) = 2 \phi (-1)$, so $\phi(-1) = 0 \lor \phi(-1) = 4$. If $\phi(-1) = 0$, then $ 0 = \phi(i^2) = 2\phi(i)$, so $\phi(i) = 0 \l... |
It is well-known that Banach space is $\sigma$-compact iff it is finite dimensional. I'm looking for examples of infinite dimensional normed $\sigma$-compact spaces.
|
The above project is financed by a loan. The company pays 6.25% on the money borrowed and earns 4% on money invested in its deposit account.(Spare funds can't be used to repay the loan at any time)
Cashflow can be understood as :
(1) At the start i.e at t=0 , the loan amount is 80,000 , at t=1 , income is 10,000 and... |
Say I have the differential equation
$$y' = \frac{3t^2 - 2ty}{4 - t^2} \text{, where }y(1)=-3$$
Clearly the equation is undefined at $t = \pm2$, and a solution exists at $t = 1$. Can I conclude from this that the solution must at least be defined on the interval $(1,2)$? Or could there be singularities in the solution ... |
If $a_1, a_2,\cdots a_n$ be all positive rationals such that $S_n=a_1^m+a_2^m+\cdots +a_n^m$, $P_m=\sum a_1a_2\cdots a_m$ (the sum of products m taken m at a time). Prove that $$(n-1)!S_m\geq (n-m)!m!P_m.$$
Here one has to apply the laws of Inequalities.
$AM\geq GM$ on m number of terms given by $a_1^m, a_2^m, \cdo... |
Let
$$ f_n(x) = \left\{ \begin{array}{ll}
n & \mbox{if $0 \lt x \lt \frac{1}{n}$};\\
0 & \mbox{otherwise}.\end{array} \right. \ $$
I have to show that
$\lim_{n \to \infty}f_n(x)$ exists in terms of distributions
$\lim_{n \to \infty}f_n^2(x)$ doesn't exist in terms of distributions.
$\lim_{n \to \i... |
Consider $g$ a continuous function on $[a,b]\times\mathbb{R}$, and let $z_0\in\mathbb{R}$. Define the (nonlinear) operator on $C[a,b]$: $$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$ for $x\in[a,b]$. Prove that:
(i) $M$ is continuous and
(ii) maps bounded set in $C[a,b]$ to relatively compact set (closure is compact) in $C[a,b]... |
If $y'+y=|x|$ and $y(-1)=0$, what is $y(1)$?
I calculated the integrating factor to be $e^x$.
Then $e^x y'+ e^x y=e^x |x|$ hence $\frac {d(e^x y)}{dx}=e^x |x|$ hence $d(e^x y)=e^x|x|dx $
Integrating both sides,
$e^xy=e^x \int |x|dx- \int [(\int |x|dx)(e^x)]dx +c.$
From here how to proceed?
Can I write $\int |x|dx=-2 ... |
Consider the function $x : \mathbb{R} \to \mathbb{R}$ given by
$$x(t) = \frac{1}{30000} \frac{1}{\mathrm{e}^t}+ \frac{2}{30000} \mathrm{e}^{\frac{t}{2}} \cos \left(\frac{\sqrt{3}}{2}t\right), \quad t\in \mathbb{R}.$$
Arrange the roots of this function $x$ in increasing order. Let $t_n$ denote the $n$th root of the fun... |
The inflation index of a country $A$ in $1995$ relative to $1990$ was $5$. Meaning that the ratio of dollars spent during $1990$ for goods compared to dollars spent for same goods is $1:5$. In country $B$ the inflation index is $9$ for the same period. In country $C$ the inflation index is $1$3. A pair of shoe cost $ \... |
I just want to be sure that I understand the connection between "Naive Comprehension", the Axiom of Separation, Russell's Paradox, and the existence of a universal set. Is the following correct?
The Naive Comprehension Axiom has as an instance the following:
$$
\exists y\forall z(z\in y\leftrightarrow z\notin z)
$$
Th... |
Let $f(x)=(x+3)^2+\cfrac{9}{4}$ for $x\ge -3 $.Compute the shortest possible distance between a point on the graph of $f$ and a point on the graph of $f^{-1}$.
My effort
Let $P,Q$ be points on the graph of $f$ and $f^{-1}$ ,respectively,such that $PQ$ is the shortest distance possible.
So $P$ has coordinates $(a,f(a))... |
$\lim_{(x,y)\to(0,0)}\frac{x^2y^2}{\sin(x)\cos(y)}$ is it allowed to split a multi-variable limit into its component variables as in the next step?
$= (\lim_{x\to0}\frac{x^2}{\sin(x)})(\lim_{x\to0}\frac{y^2}{\sin(y)})$ this is an indeterminate form and now I use L'Hopital
$=(\lim_{x\to0}\frac{2x}{\cos(x)})(\lim_{x\to0... |
So, been up all night working on getting the velocity/angle of arrow simulations perfect. Running into an issue with the physics engine I'm using is ever so slightly off (probably a rounding issue)
The Velocity is a fixed 55 "Units"
Correction Required at ranges (Range being the variable):
Range | Correction
30 0.... |
In a Post I found it says:
Whenever ${\rm E}(X)$ exists (finite or infinite), the strong law of large numbers holds. That is, if $X_1,X_2,\ldots$ is a sequence of i.i.d. random variables with finite or infinite expectation, letting $S_n = X_1+\cdots + X_n$, it holds $n^{-1}S_n \to {\rm E}(X_1)$ almost surely. The infi... |
Let $X$ be a locally comapct and Hausdorff space. We say a positive Radon Measure on $X$ is faithful if
$$0\leq f ~~~,~~~\int fd\mu=0\rightarrow f(x)=0 ~~\forall x\in X$$
Q: True or false: If there is a faithful positive Radon measure on $X$ then $X$ has a countable dense subset ?!
|
There are $n$ seats in a room. If $n$ people come to the room, what is the probability that $j$ specified people occupy $j$ specified seats? ($j$ names were tagged on the $j$ seats)
$n$ people can occupy $n$ seats in $n!$ ways.
I can't get my head around how to go forward from here.
Any help would be appreciated.
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I tried searching for a related thread to this, so please don't roast me too hard if one already exists.
Anyways, if I have a set $A = \{a, b, c\}$ then $\{a, b, c\}$ would not be considered a partition of $A$ while $\{\{a, b, c\}\}$ would be a partition, why is this? Also, would $\{a, b, c, ∅\}$ be considered a partit... |
It is known that a $T_2$ space $X$ is $KC$, i.e. every compact subset of $X$ is closed. The space $[0, \omega_1]$ is $T_5$ but not $T_6$ and the subset $[0, \omega_1)$ is sequentially compact (and thus countably compact) but not closed.
Is there an example of a $T_6$ space with a sequentially compact (or countably comp... |
Let $T_n:X:=C([0,1])\rightarrow C([0,1])=:Y$ be a sequence of linear operators.
Suppose:
$T_n$ is bounded
$(Tf)(x):=\lim\limits_{n\to\infty}(T_nf)(x)$ exist for every $f\in C([0,1])$ and $x\in [0,1]$
$Tf\in C([0,1])$ for every $f\in C([0,1])$
Show that $T$ is linear and bounded. It is true that the last assumption ... |
Suppose I have three random variables, $X,Y,Z$ with $X$ independent of $Z$, $Y$ independent of $Z$.
Which transformation can I apply to $X,Y$ to that the result is again a random variable independent of $Z$? Or better, for which $f(x,y)$ is $f(X,Y)$ independent of $Z$?
For example $f(x,y) = xy$ does not work because in... |
I know that if you have a non-arithmetic or geometric progression, you can find a sum $S$ of a series with the formula $S=f(n+1)-f(1)$ where the term $u_n$ is $u_n=f(n+1)-f(n)$. Then you can prove that with induction.
What I don't understand is how I should go about finding the function $f(n)$. For example if I want to... |
Suppose we have a box containing $n$ balls numbered $1, 2,\dotsc,n$. A random sample of size $k$ is drawn without replacement and the numbers on the balls noted. These balls are than returned to the box and a second random sample of size $r$ is then drawn without replacement. $(r + k < n)$ Find the probability that tw... |
In a category $\mathcal C$ with terminal object $1$ and objects $A$, $B$, $C$ we have
$\quad$ $A\times (B \times C) \cong (A\times B) \times C$;
$\quad$ $1 \times A \cong A \cong A\times 1$;
$\quad$ $A\times B \cong B\times A $.
where $\times$ is a (binary) product defines on objects of the category (in the case t... |
$$I=\frac {1}{2\pi i} \int_{\gamma} z^7 \cos \frac 1{z^2} dz=?$$
The function $\cos \frac 1{z^2}$ is neither analytic at $z=0$ and nor it has a pole at $z=0$. By Cauchy Integral Formula can I get the right answer? Or there is some other method?
Since $\cos \frac 1{z^2}$ does not have pole at $z=0$, can I say that the... |
This part is taken from differential equations with applications and historical George simmons.
According to the given information , there are another integral transformation.I wonder why is the Laplace transform ($a=0 , b=\infty , K(p,x)=e^{-px}$) begin used for solving ODEs.Are there any advantages or other particul... |
I have this set
$A=\left\{\frac{1}{n}|n\in\mathbb{N}\right\}$
I need to show that it is neither open or closed in $\mathbb{R}$.
And that the union
$B=\left\{\frac{1}{n}|n\in\mathbb{N}\right\}\cup\{0\}$
is closed in $\mathbb{R}$.
My book has a Lemma:
Let $(X,d)$ be a metric space. A subset $F\subseteq X$ is closed if an... |
I came across such an exercise:
Let $V$ be a linear space over $K$ such that $\dim V = n$. Show that for any $\alpha_1, \alpha_2, \dots, \alpha_m$ with $ m > n + 1$ there exist
$a_1, \dots, a_m \in K$ such that
$\sum_{i=1}^m a_i \triangleright \alpha_i = 0_V$
and
$\sum_{i=1}^m a_i = 0_K$
But then the answer ... |
Verify $(H_1 \cap H_2)g=H_1g \cap H_2g$ where $H_1,H_2\leq G$ and $g\in G$.
Let $S_1,S_2\leq G,g\in G$
What i had done, $x\in (S_1 \cap S_2)g$. Then $x=sg,s\in S_1\cap S_2$. So clearly, $x\in S_1g$ and $x\in S_2g$ hence $x\in S_1g\cap S_2g$.
But for the reverse inclusion, I let $x\in S_1g\cap S_2g$. Then $x\in S_1... |
How do I prove or disprove if $\{1, \cos x, \cos 2x,..., \cos nx\}$ is linearly independent?
I tried solving the problem using the definition of linear independence,
$\sum_{k=0}^n a_k\cos kx = 0$
$\Rightarrow a_k =0 $
but I am not able to prove/disprove it.
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Notation:
$p$ - a prime integer,
$\Bbb{Z}_p$ - set of $p$-adic integers,
$\Bbb{Q}_p$ - set of $p$-adic rationals,
$\Bbb{Q}$ - set of rationals,
$\Bbb{R}$ - set of reals.
While reading up on $p$-adic numbers I came to know that $\Bbb{Z}_p$ is both open and closed in $\Bbb{Q}_p$. Since $\Bbb{Z}_p$ is properly contained... |
Use Fourier Transform to solve the BVP:
\begin{cases}
u_t + a u_x - b u_{xx} = 0, & \mbox{for } x \in [-1,1] \\
u(x,0) = f(x) \\
u(x+2,t) = u(x,t)
\end{cases}
I solved the problem (attached); however, I do not know how to apply the periodic boundary condition. I know how to do it with separation of variable, but not f... |
Consider the variable coefficient, real valued wave equation
$$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$
where $c, q \geq 0$ depend only on $x$. Then we define the total energy at time $f$ of a $C^2$ solution $u$ as
$$ E(t) = \frac{1}{2}\int_\Omega (u_t^2 + c(... |
$p$ is a finite Markov chain where $p(i,j)>0$ for all $i,j$. Prove a reversible stationary distribution exists for $p$ if
$p(i,j)p(j,k)p(k,i)=p(i,k)p(k,j)p(j,i)$ for all $i,j,k$
This question is from Durrett's Essentials of Stochastic Processes. There is a hint provided that says: Hint: fix $i$ and take $π_j = cp(i,j)... |
Let $(X_{n})_{n\geq1}$ be i.i.d. Ber$\left(\frac{1}{2}\right)$. I want to show that $$\sum_{{n\geq1}}\frac{2X_{n}}{3^{n}}$$ converges almost surely to a random variable $X$, without saying that this random variable distribution is Cantor-Distribution. I guess we should use Borel–Cantelli somehow - but I am not sure abo... |
Can the cardinality of a power set ever be odd? If it can, what conditions must be met?
|
Assume that we have a moving average like this:
$E_t = a*S_{t-1}+(1-a)*E_{t-1}$
where $E$ would be an estimate we are interested in, and $S$ is a sample we take at each point in time.
Now, if we knew say three samples (e.g. at $t = 0,1,2$ and currently $t = 3$), we could get the current $E$ by calculating $E$ at each p... |
I am having some trouble with part (b) and part (c) of this:
(b) I know that I have to differentiate it and I get $\cos (\frac{\pi}{x})$ and by using the definite integral I get $\cos (\pi n)-\cos (\pi (n+1))$
What I said was if n is even, it can be expressed as 2k; and if it's odd, it can be expressed as 2k+1
But... |
It is well known that for a standard brownian motion, the time spent above $0$ follows an arcsine distribution (whose density function is U-shaped). Can anyone tell me how to generalise this result to 'time spent above $x$'?
I imagine it is some beta distribution (given that the arcsine distribution is a special case),... |
How to find the linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satistfying $4x_1-3x_2+x_3=0$ is
a) Null space of $T$
b) Range of $T$
I'm not able to approach this problem.
Any help would be appreciated.
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