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H: Is this a multiplication?
6x-13+4(-3)x=9+2x
I'm like, really dumb, I can't tell if if the 4(-3) bit is a multiplication
AI: $6x-13+4(-3)x=9+2x$
is the same as the equation $6*x-13+4*(-3)*x=9+2*x$
which is the same as $(6*x)-13+(4*(-3)*x)=9+(2*x)$
which is the same as $(6*x)-13+((-12)*x)=9+(2*x)$
which is the sa... |
H: What would be the right domain for a function that takes time as a parameter?
I want to define a function that takes a value which represents time and returns an integer.
So when using it, the function would look something like $f(t)$
What is the right (or commonly used) domain for such a function?
$f:T \rightarro... |
H: Solution Verification: Functions/Sets Question
Which of the following is a true statement? (Assume a finite domain.)
a. If a function is not a one-to-one correspondence, its domain must
contain more points than its image.
b. If a function is one-one, its domain and range are the same set.
c. If the domain o... |
H: Factorial lower bound: $n! \ge {\left(\frac n2\right)}^{\frac n2}$
A professor in class gave the following lower bound for the factorial
$$
n! \ge {\left(\frac n2\right)}^{\frac n2}
$$
but I don't know how he came up with this formula. The upper bound of $n^n$ was quite easy to understand. It makes sense. Can anyon... |
H: Equation with fractions
If $P=\frac{h}{1-h}$ then $h$ is equal to?
Answer is: $\frac{P}{1+P}$
I understand that $\frac{P}{1+P}$ is the right answer for when I replace $\frac{P}{1+P}$ for h the answer solves the equation, but what I can't do is find the answer by myself, how do I get to $\frac{P}{1+P}$?
AI: $$P = \f... |
H: Prove that every unitary matrix $U$ is unitarily diagonalizable
I just can't show that a unitary matrix $U$ is unitarily diagonizable. I know I need to show that $U$ is unitarily similar to a diagonal matrix, and this result is presumably a consequence of the spectral theorem.
EDIT: I was reading this wrong, and I... |
H: $H
Does anyone know of a counter example or a proof of the following proposition? If it doesn't hold in general are there any classes of groups for which it holds?
Let $G$ be a non-abelian finite group and let $H<K<G$ with $H$ a maximal subgroup of $K.$ If $gHg^{-1}<K$ then $g^{-1}Hg<K.$
Thank you
AI: Consider... |
H: Multiplication, What is It?
What is multiplication? Upon review logarithms, and square roots, I realized that I have no intuitive grasp of multiplication-well no more so than I have for addition. Is it simply another thing we need to memorize? I understand that things like $\sqrt2$ could just be memorize as the thi... |
H: is a plane smooth surface?
let f(u,v)=a + u.p + v.q : the equation of the plane
where p,q are unit vectors perpendicular to each other. a a point on the plane.
I do not understand how f can have partial derivatives of all orders, since derivative of wrt. u and v are p and q, respectively.
after this, aren't the de... |
H: weave of two sequences
Let x, y, z be three sequences of real numbers. z is said to be a weave (not "the weave", because weaves are not unique) of x and y iff x and y are disjoint subsequences of z that span all of z. For example, (0,0,1,0,1,0,0,1,0,1,...etc) is a weave of (0,0,0,...) and (1,1,1,..). My question is... |
H: Calculate modulo large number
How do I calculate 4^23 mod 31? I think it can be done using Euler's Totient Function, but I don't know how.
AI: This should be a relatively easy example. $4^{23}=2^{46}$. Now, since $2^5=32\equiv1\pmod{31}$,
$$2^{46}=(2^5)^9\times2=32^9\times2\equiv1\times2\equiv2\pmod{31}$$ |
H: How to integrate $\int e^{-x}\arctan(e^x) \, dx$
After trying this multiple ways, I give up. Here's the integral:
$$\int e^{-x}\arctan(e^x)\,dx$$
I have set
$u=\arctan(e^x)$ and $dv=e^{-x}d\,x$
and have obtained $du=\dfrac{e^x \, dx}{1+e^{2x}}$ and $v=-e^{-x}$
Using the integration by parts formula,
$$\int u\, dv... |
H: Function with a continuous domain but a discrete range
Does it makes sense for a function to have a discrete range even though the range is continuous? If yes how is it defined, and is it called something specific?
To explain what I mean if one had to model time against whether the light is on or off (to indicate w... |
H: Laplace transform of $t^2e^{at}$??
I'm trying to prove that $$\mathcal{L}\{t^2e^{at}\} = \frac{2}{(s-a)^3}.$$
I've gotten to the last integration by parts where
$$
\lim_{n\to\infty}\int_0^n\frac{1}{(a-s)^22e^{(a-s)t}}dt = \left. \lim_{n\to\infty}\frac{2}{(a-s)^3}e^{(a-s)t} \right|_0^n.
$$
Now what do I do? I can't ... |
H: Confused with finding C in economic integral
Question:
An automobile company is ready to introduce a new line of cars. They project that the sales will increase by:
$P'(t)=10-10e^{-0.1t}, 0\leq t\leq 24$
in t months after the campaign has started.
(A) - What will be the total sales S(t) t months after the beginning... |
H: Linear Functional on $V$
Need some help understanding step two.
Suppose $\varphi$ is a linear functional on $V$. Then there is a unique vector $v \in V$ such that $\varphi (u) = \left \langle u, v \right \rangle$ for every $u \in V$
We start off with $\varphi (u) = \varphi (\left \langle u, e_1 \right \rangle e_1 ... |
H: Conditional distribution of binomial random variables is hypergeometric
Let's say $X$ and $Y$ are binomial random variables with parameters $n$ and $p$ and $X+Y=m$. I want to show that the conditional distribution of $X$ if $X+Y=m$ is a hypergeometric distribution.
I'm thinking about putting these in terms of coin ... |
H: Seating $2n$ people around a table - Why $(2n - 1)!$ and not $(2n)!$?
There've been numerous questions about this so please let me know if this is a duplicate.
Page 12 in http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap1a.pdf says:
Let $A(i, r) =$ couple $i_r$ sit next to each other.
To compute the generic term... |
H: Did I do this Laplace transform correctly?
1) $w'' + w = t^2 + 2$; $w(0) = 1$, $w'(0) = -1$
2) $s^2W(s) - sw(0) - w'(0) = \frac{2 + 2s^2}{s^3}$
3) $s^5W(s) - s^4w(0) - s^3w'(0) = 2 + 2s^2$
4) $ W(s) = \frac{2 + 2s^2 - s^3 + s^4}{s^5}$
5) $W(s) = 2\left(\frac{1}{s^5}\right) + 2\left(\frac{1}{s^3}\right) - \frac{1}{... |
H: A convergent-everywhere expression for $\zeta(s)$ for all $1\ne s\in\Bbb C$ with an accessible proof
I'm looking for a way to define the Riemann zeta function $\zeta(s)=\sum_{n\in\Bbb N_0}n^{-s}$ on the whole complex plane, without having to use analytic continuation, or perhaps more accurately, in a way which can ... |
H: Definition of $H_\lambda$ (hereditary cardinality)
It seems to me that the definition of $H_\lambda$ (the set of sets of hereditary cardinality less than $\lambda$) on the web page at Cantor's Attic is not quite correct. From the page:
$H_\lambda=\{x: |\operatorname{trcl}(x)|<\lambda\}$ where $\operatorname{trcl}(... |
H: Data preprocessing
How would you preprocess 2 dimensional data to have 0 mean? Say you have a matrix $M $ that is $p \times q $. Would you calculate the mean of each row, get a vector of length $q $ and subtract each element of the vector from the corresponding column?
AI: If your data are 2-dimensional, then the m... |
H: Prove using Rolle's Theorem that an equation has exactly one real solution.
Prove that the equation $x^7+x^5+x^3+1=0$ has exactly one real solution. You should use Rolle’s Theorem at some point in the proof.
Since $f(x) = x^7+x^5+x^3+1$ is a polynomial then it is continuous over all the real numbers, $(-\infty,\... |
H: counting and probability question - help needed
I am stuck on how to start this exercise. Any help is welcome.
An instructor gives an exam with 14 questions. Students are allowed to choose any 10 to answer. Suppose the exam instructions specify that at most 1 of questions 1 and 2 may be included among the 10. How m... |
H: Infinite series for Euler-Mascheroni constant
Problem
Show that
$$
\gamma = \tfrac 12 \log 2 + {1 \over \log 2} \sum_{n=2}^\infty (-1)^n {\log n \over n}.
$$
Progress
I tried writing the terms $1/k$ of the harmonic sum in the definition of $\gamma$ as $\int_0^1 x^{k-1} dx$, and interchanging the order of summation ... |
H: How to find the following integral? $\int\tfrac{x}{\sqrt{1+3x^2}}\mathrm dx$
Find: $$\int\dfrac{x}{\sqrt{1+3x^2}}\,\mathrm dx$$
I can't fully integrate this, I get $1/x+\sqrt3 x$ and then I don't know what to do, not sure if I even started it correctly, thanks in advance.
AI: Check first (chain rule) that
$$\int\... |
H: What does the notation $G /^r H$ mean?
$G /^r H$
I saw this notation in an answer to a question and am not sure what it means.
The exact context is as follows:
$G=Sym(5)$ acts on the set $G/^r H$ of all right cosets of $H$ in $G$.
AI: Space of right cosets. When $H$ is normal, you'd just interpret $G/H$ as the quo... |
H: Evaluating the integral, $\int_{-\infty}^{\infty} e^{-x^2/a}\ln\left(1 + be^{-cx^2}\right)dx$
I recently got stuck on evaluating the following integral, $$\int_{-\infty}^{\infty} e^{-x^2/a}\ln\left(1 + be^{-cx^2}\right)dx$$ where $a>0$, $b>0$. I don't know if there is an effective substitution to use.
AI: First ass... |
H: Number of strings consisting of k ones and n zeros such that no two ones are adjacent
What is number of strings consisting of k ones and n zeros such that no two ones are adjacent?
I already know the answer from wikipedia $\binom{n+1}{k}$ but i want to know its source.
AI: Write down the $n$ $0$'s in a row, with a ... |
H: How to approach proving $\lim\limits_{n \to \infty}(\int^{b}_{a}(f(x))^ndx)^{\frac{1}{n}}=\max\{f(x):x\in[a,b]\}$?
Let $f:[a,b] \to \mathbb{R}$ be positive and continuous. Let $M = \max\{f(x):x \in [a,b]\}.$
Could anyone advise me on how to establish the following result:
$$\lim_{n \to \infty}(\int^{b}_{a}(f(x))^nd... |
H: How to find this Differential Equation ?
Find differential equation of a circle in $XY-Plane$ such that it passes through $(-1,1)$ and $(1,1)$
I need some clue please. I am not able to find the general equation. Is it a trick problem ?
AI: Hint: Find the equation of the circle. The equation of a circle with centre ... |
H: Can any one tell me the books for power series?
Can any one tell me the books for power series? I want to find the power series for sqrt(x). I surf on the internet but there is no success.
So please tell me the name of the book where I can find the power series of sqrt(x)/
AI: A few sites where you can find useful ... |
H: how to find accumulation point of $z_n=e^{in}$
How can I find accumulation point of
a) $z_n=e^{in} $
b)$z_n=i^n$
c)$z_n=(1-\frac{1}{n})+(-1)^ne^{\frac{1}{n}}i$
I tried at b) $\lim_{n\to\infty}(i^2)^{\frac{n}{2}}=(-1)^{\frac{n}{2}}=-1,+1$
at c) $\lim_{n\to\infty}(1-\frac{1}{n})+(-1)^ne^{\frac{1}{n}}i=lim_{n\to\inf... |
H: Prove that if $F : A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is Injective
Statement: if $F : A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is $1-1$
Proof: If $F$ is not $1-1$, then there exist $x_{1}, x_{2} \in A$ where $x_{1} \neq x_{2}$ and $F(x_{1}) = F(x_{2})$. Therefore, $F^{-1}(y) \neq F^{... |
H: A question of straight lines
If the straight lines $x+y-2=0$, $2x-y+1=0$ and $px+qy-r=0$ are concurrent, then what is the slope of the member of family of lines $2px+3qy+4r=0$ which is farthest from origin?
I wrote the coefficients of the variables of the given lines in a determinant, equated it to $0$ and got $p+5... |
H: Finding the limit $\lim_{n \to \infty}{\frac{\Sigma_{0}^{n}(1/n)}{\ln(n)}}$
Let
$$
\lim_{n \to \infty}{\frac{
\sum_{1}^{n}(\frac{1}{n})}{\ln(n)}}
$$
Please provide some hint or a solution.
Thanks!
AI: Using Stolz–Cesàro theorem you get that
$$\lim\limits_{n\to\infty} \frac{\sum_{k=1}^n\frac1k}{\ln n}
=\lim\limits_{... |
H: Factorise $13$ into a product of irreducibles in $\Bbb Z[i]$
I need to factorise $13$ into a product of irreducibles in $\Bbb Z[i]$ but I'm having trouble factorising it at all.
So far I have
$13=(a+bi)(c+di)$
$13=ac + cbi + adi - bd$
So
$bc = -ad$
$ac-bd=13$
I don't know how to go any further for working out the f... |
H: Find all matrices that satisfy $\mathrm B \mathrm A = \mathrm I_2$
Given the matrix $$A=\begin{pmatrix}1&8\\3&5\\2&2\\ \end{pmatrix}$$ find all $2 \times 3$ matrices in $B \in M_{2 \times 3}(\mathbb R)$ with $BA=I_2$.
Here's what I did:
$$\begin{pmatrix}a&b&c\\d&e&f\\ \end{pmatrix} \begin{pmatrix}1&8\\3&5\\2&2\\ ... |
H: Computing $[T]_\beta$
Let $V=\mathbb R^2$, $T(a,b)=\begin{pmatrix} 10a-6b\\17a-10b\\ \end{pmatrix}$ and $\beta$={$(1,2),(2,3)$}.
Where $T$ is a linear operator on $V$ and $\beta$ is an ordered basis. I have to compute $[T]_\beta$ and determine whether $\beta$ is a basis consisting of eigenvectors for $T$.
I know ... |
H: Evaluating series by contour integration, the residue theorem, and cotangent
I'm trying to understand this section in Tristan Needham's book Visual Complex Analysis about what he says is a standard method for evaluating series via a contour integral. My specific question is about the computation of the residues of ... |
H: calculus antiderivative of the function. Find the position of the particle
A particle is moving with the given data:
$$a(t)=\cos{t}+\sin{t},\, s(0)=8,\, v(0)=5.$$
Find the position $s(t)$ of the particle.
I don't understand the problem. What do those symbols stand for?
AI: If we assume $s(t)$ to be position functio... |
H: Let $a_1 > 1$, and for $n \in N$, define $a_{n+1} = 2 - 1/a_n$.
Let $a_1 > 1$, and for $n \in N$, define $a_{n+1} = 2 - {1\over a_n}$.
Show that the sequence $a_n$ is monotone and bounded. Find $lim_{n\to\infty} a_n$
So far I have it set up like this (not sure if it's the right way)
monotonic: $a_n < a_{n + 1}$ (in... |
H: composition of two uniformly continuous functions.
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ are two uniform continuous functions. Which of the following options are correct and why?
$f(g(x))$ is uniformly continuous.
$f(g(x))$ is continuous but not uniformly continuou... |
H: Relatively Prime problem
If $a$ and $b$ are relatively prime integers then $b$ and $a$ plus some multiple of $b$ are also relatively prime. I can see how it works for concrete examples but can't prove it. i.e. $(a,b)=1$ implies $(b,a+kb)=1$.
AI: If $d$ divides $b$ and $a+kb$
$d$ will divide $a+kb-k\cdot b=a$ as $k$... |
H: Help with simple limit calculation
I have tried searching the site for an answer but I couldnt find any even though it's a simple calculation.
$$ \lim_{n \rightarrow \infty} \left(\frac{4^n + 7^n}{4^{n-1} + 7^{n-1}}\right) $$
Thanks , Danny.
AI: $$\lim_{n \rightarrow \infty} \left(\frac{4^n + 7^n}{4^{n-1} + 7^{n-1}... |
H: Are all continuous bijective translations isometries?
Let $(M, d)$ be a metric space. I define a translation on $M$ to be a function $f$ from $M$ to $M$ such that $d(x,f(x))=d(y,f(y))$ for all $x$ and $y$ in $M$. In a previous question, I asked if every translation was an isometry in the post Are all metric transla... |
H: Volume of a solid with base of circular disk, parallel crosssections perpendicular to base are squares.
Working on a problem of volume using integration:
The problem is this:
The base of is a circular disk with radius . Parallel crosssections
perpendicular to the base are squares.
I already have an idea how the... |
H: Missing step in rearrangement
Can someone explain the missing step in the following rearrangement ?
From $ \displaystyle a^2 + b^2 + \frac {a^2y}{x} + \frac {b^2x}{y} \geq (a+b)^2 $ to $ \displaystyle \frac{(a+b)^2}{x+y} \leq \frac{a^2}{x} + \frac{b^2}{y} $ ?
AI: We have
$$ \left(\frac{a^2}{x} + \frac{b^2}{y}\right... |
H: Which one is bigger? $e^{\pi} $ or $\pi^e$
$e^{\pi}$ or $\pi^e$,
Can we find which one is bigger by using calculus?
Thanks.
AI: $$e^\pi>\pi^e\iff\pi>e\log\pi\iff\frac{\log e}e=\frac1e>\frac{\log\pi}\pi$$
Now look at the function
$$f(x):=\frac{\log x}x\;,\;\;x\ge e\implies f'(x)=\frac{1-\log x}{x^2}\le0\implies f(x... |
H: How do I simulate a simple pendulum?
I have the equation of motion of a simple pendulum as
$$\frac{d^2\theta}{dt^2} + \frac{g}{l}\sin \theta = 0$$
It's a second order equation.
I am trying to simulate it using a SDL library in C++. I know how to solve first order differential equation using Runge-Kutta method. But... |
H: Cartesian Product - compute the number of unique combinations
given the number of sets
and given the number of elements in each set
What mathematical equation efficiently computes the number of unique combinations
Example:
lets say there are 3 sets {1,2,3},{1},{1,2}
and we know that the 3 sets have 3,1,2 elements ... |
H: Let $x > 0 \in \mathbb R$. Find sequences $\{a_n\}_{n=0}^{\infty}$, $\{b_n\}_{n=0}^{\infty}$ such that $a_n + b_n = n$ and $a_n/b_n \rightarrow x$
Let $x > 0 \in \mathbb R$. Find sequences $\{a_n\}_{n=0}^{\infty}$, $\{b_n\}_{n=0}^{\infty}$ of natural numbers ($a_n, b_n \in \mathbb N$) such that $a_n + b_n = n$ and ... |
H: No. of $t,s \in [0,1]$ such that $f(t)=f(s)$ for a continuous function $f:[0,1]\rightarrow [0,1]$
Question is :
Let $f:[0,1]\rightarrow [0,1]$ be a continuous function such that $f(0)=f(1)$. Let
$$A=\{ (t,s) \in [0,1]\times [0,1] : t\neq s ; f(t)=f(s)\}$$
The number of elements in $A$ is...... ?
As $f(0)=f(1)$ ... |
H: Is there a proof that $n^xm^x = (n^x)^{(\log(mn)/\log(n))}$?
This isn't a homework question, just something I'm curious about, but you can treat it that way if you like.
So the other day I was playing with my calculator and I noticed that
$$
2^x10^x = (2^x)^{(\log(20)/\log(2))}
$$
I tried it out with some other nu... |
H: Sum to $n$ terms.
Evaluate the following expression:
$\sum_{i=1}^n \frac{i \cdot 2^{i}}{(i+2)!}$
I can't find a way to use telescoping sums here. What else could be done?
AI: $$\frac{i\cdot 2^i}{(i+2)!}=\frac{(i+2-2)\cdot 2^i}{(i+2)!}=\frac{2^i}{(i+1)!}-\frac{2^{i+1}}{(i+2)!}$$
Observe that if we set $\displaystyle... |
H: Which sentences survive the passage from $X$ to the set of all functions $I \rightarrow X$?
Suppose $X$ is a mathematical structure with a single underlying set which we will also denote $X$, equipped with some functions and relations. Letting $I$ denote an arbitrary non-empty set, we see that the set of all functi... |
H: Finding where f is increasing/decreasing
$$f'(x)= (x+1)^2(x-4)^5(x-2)^4 $$
For the critical points, I got $x = -1,2,$ and $4$.
According to my professor's answer key, the interval is increasing on $(4, \infty)$ and decreasing on $(-\infty,4)$. I got increasing on $(-1,2)$ and $(4,\infty)$ and decreasing on $(-\inft... |
H: Conjecture similar to Fermat's Theorem.
I was wondering about a problem which i could reduce to asking the following
Does there exist a set $a,b,c$ of prime numbers such that $$a^a+b^b=c^c$$
Is it really a tough problem or do you think it can be solved with some amount of work .
Thank you for your ideas and help ... |
H: Is the compact interval $[0,1]$ in the usual topology compact in this new topology?
Let $\mathbb{R}$ be a topological space with topology consisting of the sets $A \cup B$, where $A$ is open in the usual topology, and $B \subseteq \mathbb{R} \setminus \mathbb{Q}$. Is the interval $[0,1]$ compact in this topology?
I... |
H: Prove that $\{(x,y) \in \mathbb R^2 | y = x^2 \}$ is not compact
I know I need to choose an open cover and then show it has no finite subcover.
If I use $((-n,-n^2),(n,n^2)) \forall n \in \mathbb N$ does this work?
AI: Hint : prove that the given set is not bounded. |
H: Laplace transform of the following function
find the laplace transform of the function :
$$f(t) =\begin{cases}
t^2, & 0<t<1 \\
2\cos t+2, & t>1 \\
\end{cases}$$
My attempt:
$$L\{f(t)\}=\int_{0}^{1}e^{-st} \ t^2 \ \text{d}t+\int_{1}^{\infty}e^{-st} \ (2\cos t+2) \ \text{d}t$$
Now,
$$\int_{0}^{1}e^{-st} \ t^2 \ \te... |
H: How to evaluate $\displaystyle \lim_{n\to \infty}\left[a^n+b^n+c^n\right]^{1/n}$
What is the $\displaystyle \lim_{n\to \infty}\left[a^n+b^n+c^n\right]^{1/n}$, assuming that $0<a<b<c$?
I think that, as 1/n tends to zero, the limit 1. Is this correct?
AI: Hint:
Assuming that $0\le a\le b\le c$, then
$$(c^n)^{\frac{1... |
H: How find this equation solution $2\sqrt[3]{2y-1}=y^3+1$
find this equation roots:
$$2\sqrt[3]{2y-1}=y^3+1$$
My try: since
$$8(2y-1)=(y^3+1)^3=y^9+1+3y^3(y^3+1)$$
then
$$y^9+3y^6+3y^3-16y+9=0$$
Then I can't.Thank you someone can take hand find the equation roots.
AI: Let $f(y) = \frac12 (y^3+1)$, we have
$$2\sqrt[3... |
H: Drawing phase planes
I have to draw the phase plane of Y'=$
\begin{pmatrix}
-3 & 1 \\
1 & -3 \\
\end{pmatrix}
$Y
The general solution of the system was,
Y(t)=C$_1$$
\begin{pmatrix}
1 \\
1 \\
\end{pmatrix}
$e$^{-2t}$+C$_2$$
\begin{pmatrix}
... |
H: Positively non-positive (from Brilliant.org) Whats wrong with my method?
(taken from brilliant.org)
For how many positive integers $N$ between $3$ and $1000$ (inclusive) is the following statement true:
If $\{a_i\}^N_{i=1}$ is a set of $N$ (not necessarily distinct) real numbers such that
$a_1+a_2+…+a_N=0$,
then w... |
H: Indefinite intergral of $\int { \sqrt{ x^2-a^2} \over x } \mathrm dx $
I need to integrate $$\int { \sqrt{x^2-a^2} \over x } \mathrm dx $$ using substition, and show it equals $$ \sqrt{x^2 - a^2} - a(\operatorname{arcsec} ({x\over a })) +c $$
I've tried $x=a\sin t$ but I couldn't finish it out.
Thank in advance fo... |
H: estimate on $| \nabla (u |u|^2) - \nabla(w|w|^2)|$ for $u,w \in H^1$
suppose $u, w \in H^1 (R^2)$. I'd like to know where does the following inequality come from (it appears in a proof I've been reading and I can't figure it out)
$$ | \nabla (u |u|^2) - \nabla(w|w|^2)| \leq C | \nabla(u-w)| \cdot(|u|^2 + |w|^2)
+ ... |
H: Correct interpretation of Kleene (Intro to Metamathematics) symbol $\vdash^x$ in Predicate Calculus
Rif. S.C.Kleene, IM (1952) : which is the correct interpretation (or the "modern equivalent") of the "x" used as exponent of the "turnstile" as in:
$$A(x) \vdash^x \forall xA(x)$$
[see Derived Rules (Th.2, pag.98)] ?... |
H: How to prove(or disprove) $\begin{vmatrix} A&B\\ B&A \end{vmatrix}=|A^2-B^2|$
Let $A$ and $B$ be square matrices of the same size.
(1) If $f$ is not invertible and $AB=BA$, show that
$$\begin{vmatrix}
A&B\\
B&A
\end{vmatrix}=|A^2-B^2|.$$
(2) If $A$ is invertible and $AB\neq BA$, then do we have
$$\begin{vmatrix}
A&... |
H: Limit with Stolz-Cesàro theorem: $\lim\limits_{n\to \infty} \frac{1+2\sqrt2+3\sqrt3+\ldots+n\sqrt n}{n^2 \sqrt{n}}$
$$\lim_{n\to \infty} \frac{1+2\sqrt2+3\sqrt3+\ldots+n\sqrt n}{n^2 \sqrt{n}}= \text{?}$$
Book has no answers.
It's on Stolz-Cesàro theorem lesson, if that helps. Can't find a solution.
AI: This is the ... |
H: What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?
For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq W^{1,p}(\mathbb{R}^n) $.
Any hints or par... |
H: Finding $\lim_{x\to+\infty} (3^x-x)^{1/(4x)}$
I have to find the limit of $(3^x-x)^{1/(4x)}$ as $x\to+\infty$ without using de l'Hôpital's method or Taylor series.
I've tried to use some notable limits as $(1+1/t)^t$ or other but the problem is the fact that $x$ goes to infinity, then I tried to use substitution bu... |
H: Conditional Probability coin tossing
I was going through some exercises on probability and came across a question. Two people A and B are tossing a fair coin with A tossing first. The process is repeated till someone gets a heads. What is the probability of A winning.
I came across a solution where it says since A ... |
H: Compact Operators and Complete Metrics Spaces
I have a couple of questions about compact operators and compactness in complete metric spaces:
1.I have the following implications: Let $Y$ be a metric space with $A$ a subset of $Y$. $A$ is precompact iff $\bar{A}$ is sequentially compact iff any sequence in $A$ has ... |
H: Given $\begin{pmatrix} a & b\\ c & d\end{pmatrix}∈GL_2^+(R)$ , $\beta(w)=\frac{aw+b}{cw+d},\Im(w)>0$.Is $\beta$ bijective?
Given any matrix $A=\begin{pmatrix} a & b\\ c & d\end{pmatrix}∈GL_2^+(R)$, we can define a function $\beta:H\to{ \mathbb{C} }$ by $$\beta(w)=\frac{aw+b}{cw+d},w∈H$$,where $H$ is the upper compl... |
H: Find the eigenvalues and eigenvectors of A geometrically
I am really confused with this question:
Find the eigenvalues and eigenvectors of A geometrically: $$ A = \begin {pmatrix} 0 & 1 \\ 1 & 0 \end {pmatrix} $$^ reflection in the line $y=x$.
Thanks.
AI: I am trying to understand the question. Forgive me if I a... |
H: Dual of the space of all convergent sequences
I need to find what it wrong with my logic and Ii will be glad if someone can told me what I do wrong.
Define $C$ be the subspace of $ l^{\infty} $ that consists of convergent sequences and let $C_0$ be the subspace of $C$ that consists of sequences that converge to 0.... |
H: Convert 59 to octal
I'm reading Discrete Mathematics by Kevin Ferland and I'm stuck with exercise 33 of Chapter 0: Write the octal number equivalent of 59.
I'm following this procedure:
59/8 | 3
7/8 | ??
By using an online calculator, the result should be 73, however how can the reminder of 7/8 be 7?
AI: When you... |
H: Continuity of a multivariable absolute value function
The function is as follows:
$$f(x,y)=\sqrt{\left |xy \right |} $$ I have to check whether it is continuous, differentiable and has defined partial derivatives at $(0,0)$. My attempt is as follows:
Function is discontinuous at the origin.
Not differentiable ... |
H: Need help understanding the factorial formula $n!=n(n-1)(n-2)\cdots(3)(2)(1)$
The caption says the following:
If $n$ is an integer such that $n \ge 0$ then $n$ factorial is defined as,
$$n!=n(n-1)(n-2)\cdots(3)(2)(1)$$
if $n \ge 1$ by definition.
I'm really just confused by the $(3)(2)(1)$ in the formula, and if $n... |
H: Expected value and variance on exponential distribution
The length of time Y necessary to complete a key operation in the construction of houses has an exponential distribution with mean 10 hours. The formula C = 100 + 40Y + 3Y^2 relate the cost C of completing this operation to the square of the time to completion... |
H: Are these disjoint/dependent?
Given $P(A) = 0.7, P(B) = 0.6, P(A^c | B^c) = 0.25$, are:
I) $A$ and $B$ disjoint?
II) $A$ and $B$ dependent?
So, what I said:
$I)$ Since $P(A) + P(B) = 1.3 > 1$ then $P(A \cap B) \neq \emptyset$, thus $A, B$ are non-disjoint.
II) $P(A^c | B^c) = \frac{P(A^c \cap B^c)}{P(B^c)} = 0.2... |
H: Generating points from a standard Gaussian
I'm new to Gaussian distributions and I'm trying to generate say, $ N$ points from a $ M$ dimensional standard gaussian. What does this mean? How would I do this in matlab?
AI: here's how in Matlab. In general, you can generate a uniform random variable on [0,1] then feed ... |
H: Parametric Equations: Find $\frac{\mathrm d^2y}{\mathrm dx^2}$.
Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$, as a function of $t$, for the given the parametric equations: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$
$\displaystyle\dfrac{\mathrm d^2y}{\mathrm dx^2}=\ldots$
I don't really understand th... |
H: How many inverse relations
How many inverse relations are there for an n-element set?
I know that $R \circ R^{-1}=R^{-1} \circ R$ where $R$ is an invertible relation, but that's as far as I can get.
AI: Every relation has an inverse relation, so this question is really asking how many relations there are on some gr... |
H: proving a sequence has exactly four limits points
Given the sequence \begin{align*}
x_n:=\begin{cases}
1,&\textrm{if }n\equiv0 \mod 4\\
2,&\textrm{if }n\equiv 1\mod 4\\
3,&\textrm{if }n\equiv 2\mod 4\\
4,&\textrm{if }n\equiv 3\mod4
\end{cases}
\end{align*}
I want to show that the limit points are exactly $1,2,3,4$
... |
H: Prove that the symmetric derivative of a function exists whenever the derivative exists.
Let $f$ be a function defined on an interval $(a,b)$ and let $c \in (a,b)$. The symmetric derivative of $f$ at $c$ is defined by $f'_s(c)=lim_{h\to 0} \frac{f(c+h)-f(c-h)}{2h}$ provided that the limit exists. Prove that $f'_s(c... |
H: Do densities of invariant distributions satisfy the Fokker Planck equation?
Suppose that $\{X_t\}_{t\in[0,\infty)}$ is a $\mathbb{R}^n$ valued homogenous diffusion process with drift vector $b$ and diffusion matrix $A$. Is it ever true that if the process has an invariant distribution with a density $\pi(x)$, then ... |
H: Prove that $\alpha$ is algebraic over $K$.
Let $\alpha$ be a transcendental element over a field $E$, and $F=E(\alpha)$. Prove that for any subfield $K$ of $F$ containing $E$ as a proper subset, $\alpha$ is algebraic over $K$.
Can anyone give me some hint? I have no idea how to start.
AI: Since $K$ is a subset of $... |
H: FoxTrot Bill Amend Problems
So I found this on the Wolfram website today:
So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed:
$ \lim_{x \to +\infty} \dfrac {\sqrt{x^3-x^2+3x}}{\sqrt{x^3}-\sqrt{x^2}+\sqrt{3x}} $
$ \displaystyle\sum... |
H: Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$
I need some help, showing that the left hand side is equivalent to the right hand side. I tried but I get stuck, I am not sure if I am on the right path. Here is my attempt:
$C(2n,n+1) + C(2n,n)$
Equivalent to this
$\frac{(2n)!}{(n-1)!... |
H: time taken to complete work
A can do a piece of work four hours faster than B. They worked together for two hours and then the remaining part of the work was done by A in an hour. How many hours would B take to complete the job if he were to work alone?
AI: let t be the time A takes in hours, then A does $\frac{1}{... |
H: Proof for the length of the shortest 4-connected path and 8-connected path on a chessboard
I have a chessboard with a square marked with A as in the following figure:
$$\begin{array}{|c|c|c|} \hline
8&1&2\\ \hline
7&A&3\\ \hline
6&5&4\\ \hline
\end{array}$$
The $4$-connected neighbors squares of A are marked with $... |
H: Beautiful Mathematical Images
My Maths department is re-branding itself, and we've been asked to find suitable images for the departmental sign. Do you have a favourite mathematical image that could be used for the background of an A1-sized sign?
AI: I produced the following images, which I personally like ;-)
A pa... |
H: Probability with 13 trees
A company is planting trees and we know that 90% of the trees survive. What is the probability that from 13 trees:
1, at most $10$ survive
2, at least $10$ survive
3, exactly $10$ trees survive.
AI: Here is a similar problem. Suppose we flip 10 coins. What is the probability that at leas... |
H: Notation of double-sided infinite sum
The notation $\sum_{k=1}^\infty a_k$ always means $$\lim_{n\rightarrow\infty}\sum_{k=1}^n a_k.$$
What about $\sum_{k=-\infty}^\infty a_k$, such as in the Laurent series? Does it always means $$\lim_{n\rightarrow\infty}\sum_{k=-n}^n a_k,$$ or does the meaning depend on the cont... |
H: My Odometer, Speedometer, and the Time
I was driving from home to university earlier this week when I glanced at my dashboard just soon enough to notice the odometer tick up while simultaneously registering my speed and the time. I thought to myself,
"How might I find the probability that the odometer will tick up... |
H: Can you think of a good approximation to this integral?
So, I have
$$\int_{0}^{\arcsin\left(\frac{r}{g}\right)}\left(g\cos\theta-\sqrt{r^{2}-g^{2}\sin^{2}\theta}\right)^{2}d\theta$$
but the integral, which one can evaluate with mathematica, has a singularity at the upper limit to the integral. Can anyone think ... |
H: Conditional expectation by $\sigma (G_n,Y)$ when $Y$ is $G_\infty$-measurable
Let $G_n$ be a filtration (an increasing sequence of sigma-algebras), $Y$ a random variable that is $G_\infty$-measurable, and $X$ a random variable. Is it true that in $L^2$-norm,
$$ \mathbb{E}[X \mid \sigma (G_n,Y)]- \mathbb{E}[X \mid G... |
H: Using partial fraction for $\cot \pi z$ to compute infinite sum
I want to compute the values $\sum_{n=1}^\infty \dfrac{1}{n^2}$ and $\sum_{n=1}^\infty \dfrac{1}{n^4}$ and $\sum_{n=1}^\infty \dfrac{1}{n^6}$ by comparison to the partial fraction development of $\cot \pi z$.
First, I note that $$\pi z\cot(\pi z)=1+2\s... |
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