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H: How to solve this trigonometric equalities?
We have to solve $ \tan(x) = 2 \sin(x)$ on $0 \leq x \leq 2 \pi$.
I rewrote it to $\sin(x) = \sin(2x)$ using $2\sin(x)\cos(x) = \sin(2x)$.
But how can I now find the set of solutions by hand?
AI: You have already reached at $\sin2x=\sin x$
$\displaystyle\implies 2x=n\pi... |
H: What's wrong with my conjecture?
I was doing math homework, and I formulated the following conjecture from one of the questions:
If $f(x)$, $g(x)$ and $h(x)$ are continuous functions and the equations $f(x) = h(x)$ and $g(x) = h(x)$ both have only one root, then the equation $f(x) = g(x)$ has only one root. So can... |
H: Piecewise continuous function
I'm trying to prove this, and here what I got so far
Let f be defined on A, $a=kπ$ is cluster point of A
Part 1: if $a=kπ$ for some k∈Z then $f$ is continuous at $a$
Take ε>0, let $δ=sin^{-1}(ε)$ st $0<|x-kπ|<δ$. When $x$ is a rational
$|f(x)-f(a)|=|sin|x|-sin|kπ||=|sin〖|x||〗<ε$
... |
H: Finding numbers when their sum and LCM are given
The sum of two numbers is $2000$ and thier LCM is $21879.$ Find the numbers.
My attempt: Let two numbers be $x, 2000-x.$ Product of two numbers is equal to the product of their lcm and hcf. So, $x(2000-x)=21879*hcf.$
Now we have two variables and one equation. So I... |
H: Conditional probability problem in Elementary Probability Theory with Stochastic Processes
There are two kinds of tubes in an electronic gadget. It will cease to
function if and only if one of each kind is defective. The probability
that there is a defective tube of the first kind is .1; the probability
that ... |
H: "Preimage" of a binary relation
Consider the binary relation $R \subseteq X \times Y$. Is there a standard name and notation for the set $X' = \{x\ |\ (x, y) \in R\}$?
ProofWiki calls $X'$ the preimage of $R$, denoted as
$\operatorname{Im}^{-1}(R)$.
This site calls $R$ a correspondence, calls $X$ the predomain of ... |
H: Find the expressions for the common difference and common ratio
The first 2 terms of a geometric progression (first term $a$ and common ratio $r$) are the same as the first 2 terms of an arithmetic progression (first term $a$ and common difference $d$). The third term of the geometric progression is twice as big a... |
H: sequence of functions and its limit
Let $f_n (x) = 1 $ if $x \in [n, n+1] $. And $f_n(x) = 0$ otherwise. I want to find the
$$ \lim_{ n \to \infty } f_n $$
MY attempt: we know $ 0 \leq f_n \leq 1 $ for all $n$ So $f_n$ is bounded. IT is obviously monotone since $f_n \leq f_{n+1} $ So limit exists. But I have trou... |
H: building Matrix "echelon form"
given is
A = $ \begin{pmatrix} a & b & 0 & b \\ b & a & b & 0 \\ 0 & b & a & b \\ b & 0 & b & a \end{pmatrix} $
I need the echelon form for this matrix:
Whenever I begin calculating, I end up with an entry like "a/b", which isn't defined for b = 0.
However, I am supposed to name the ... |
H: Implication of $L^p$ convergence
Take $U$ as open subset of $\mathbb{R}^{n}$. If $u_{m} \rightarrow u$ in $L^{p}(U)$ then does it follow that $||u_{m}||_{L^{p}(U)} \rightarrow ||u||_{L^{p}(U)}$?
AI: Yes, by the triangle inequality (dropping the subscript on the norms):
$\|u\|=\|u-u_m+u_m\|\le\|u-u_m\|+\|u_m\|$, so ... |
H: Why is $t=\frac 3 2$ in $1+t=\sqrt{4+t^2}$?
I am confused about solving $1+t=\sqrt{4+t^2}$.
When I solve it per hand I come to the conclusion that $t$ can be everything.
$$\begin{align*}
1+t =\sqrt{4+t^2}& \qquad | \cdot^2 \tag{1} \\
1+t^2 = 4+t^2 & \qquad | -t^2 \tag{2} \\
1 = 4 \tag{3}
\end{align*}$$
However wol... |
H: Removable singularity when multiplied by linear factor
I was thinking about this situation: If $f(z)$ is holomorphic everywhere except at $a$ (where it has a pole), and has a removable singularity or a pole at $\infty$, does $(z-a)f(z)$ also have a removable singularity or a pole at $\infty$?
In the pole case, $f(z... |
H: Understanding of Analytic Branch
What does it mean for a function ''to have a branch that is analytic''? Can someone give me an example of it as well please?
thanks
AI: Some functions have multiple values, for example $\log z = \log |z| + i(\text{arg } z + 2\pi k)$, $k \in \mathbb{Z}$. Such a function "has an analy... |
H: Proof that the Riemann-Integral satisfies $\int_A \lambda f = \lambda \int_A f$
Suppose $A\subset\mathbb{R}^n$ is a closed rectangle and $f:A\to \mathbb{R}$ is Riemann-Integrable on $A$. I want to show that $\lambda f$ is integrable and that
$$\int_A \lambda f =\lambda\int_Af $$
My approach was the following: suppo... |
H: Determine $a<0$ such that $\int_a^0 f(x) dx = f(a)$
The function $f$ is given by
$$f(x)=\frac{e^{\frac 1 x}}{x^2}$$
where $x\ne 0$. Determine a number $a<0$ such that
$$\int_a^0 f(x) dx = f(a)$$
AI: Just go ahead and compute the integral. You have
$$
\int_a^0 f(x) dx=\lim_{\epsilon \uparrow 0} \int_a^\epsilon \frac... |
H: Directional Derivative (3 variables)
Find the derivative of the function $ u = xy + yz + zx$ at the point $M(2,1,3) $ in the direction from this point to the point $N(5,5,15)$
When it is 2 variables, I can easly find a vector that pass through the given points and is a unit vector. But how should I procedure in $\m... |
H: Example of a Subgroup That Is Not Normal
Can you kindly provide an example of a subgroup that is not normal? I have been told many times that, for coset multiplication to be defined, the subgroup must be normal. I have seen the proof and examples of quotient group multiplications. Now, I am trying to find out wh... |
H: Extend a linear bounded functional defined on dense subset of $C([0,1])$
Let $D \subseteq C([0,1])$ be a dense countable subset of the continuous functions from $[0,1]$ to $\mathbb{R}$. Let $L\colon D \rightarrow \mathbb R$ be a linear and bounded functional with norm $\leq 1$. How can I extend it to a continuous f... |
H: Limit of $\sum\limits_{i=1}^n\sin\left(\frac{i}{n^2}\right)$
Compute $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\sin\left(\frac{i}{n^2}\right)$$
Using Taylor expansion for $\sin x$, I know that this is $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)!}\left(\frac{i}{n^2}\right)^{2k+1}$$
... |
H: Find horisontal, vertical and slant asymptotes of this function...
$$y=\sqrt{x^2-1}$$
I found that this function has no asymptotes, but I have no idea if it's true.
AI: Hint: Presumably you're only defining the function $f(x) = \sqrt{x^2-1}$ for $|x|\geq 1$. There are no horizontal or vertical asymptotes for this f... |
H: How to evaluate the series: $ \frac {20} {(5-4)(5^2-4^2)} + \frac {20^2} {(5^2-4^2)(5^3-4^3)} + \frac {20^3} {(5^3-4^3)(5^4-4^4)} + \ldots $
How to evaluate this infinte summation?
$$ \dfrac {20} {(5-4)(5^2-4^2)} + \dfrac {20^2} {(5^2-4^2)(5^3-4^3)} + \dfrac {20^3} {(5^3-4^3)(5^4-4^4)} + \ldots $$
Telescopic cancel... |
H: Dirichlet's Approximation Theorem For Rationals
Dirichlet's Approximation Theorem implies that for all $x \in \mathbb{R} \setminus \mathbb{Q}$ we have
$$
\left|x-\frac{p}{q} \right| \leq \frac{1}{|q|^2} \quad \text{for infinitely many $(p,q) \in \mathbb{Z}^2$.}
$$
This fails for $x \in \mathbb{Q}$:
$$
\left|\frac{a... |
H: Cover information theory 7.21 tall, fat people
I am stuck on Thomas Cover information theory 2nd edition, problem 7.21 Fat, tall people. The problem is like following:
7.21 Tall, fat people. Suppose that the average height of people in a room is 5 feet. Suppose that the average weight is 100 lb.
(a) Argue that no m... |
H: Prove if prime can be written as $2^n+1$, $n = 2^k$
Prove that if prime can be written as $\ 2^n + 1$ then $n = 2^k$, $\;\;n, k \in \mathbb N$.
I am pretty new in this part of math.
AI: If n is odd, then $2^n+1 = (2+1)\cdot {2^n+1 \over 2+1} $ where the fraction evaluates to an integer, because this is true even f... |
H: Why is the empty set bounded?
Why is the empty set bounded below and bounded above? If it has no elements, how can you say that an upper or lower bound exists?
AI: Recall that implication has the property that when the assumption is false, the implication is true. In other words, if $P$ is false, then $P\implies Q$... |
H: Evaluation of a limit
Here is a question on limits. I would like to ask help. Here it goes: $$\lim_{N\to\infty}\left(\frac{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n+1}}{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n}}\right)$$ I do not know where to start but so far, I think I have to use L'Hopital's Rule, maybe because... |
H: Recurrence relation $g(n) = g( \lfloor {n/2}\rfloor) + \lfloor{log_2{n}}\rfloor $
$g(n) = g( \lfloor {n/2}\rfloor) + \lfloor{log_2{n}}\rfloor \\ g(0) = 0$
Series is like this:
$0,0,1,1,3,3,3,3,6,6,6,6,6,6,6,6,10,...$ and it's changes similar as $\lfloor{log_2{n}}\rfloor $
$0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,....$
Bu... |
H: What is the difference between Taylor series and Laurent series?
Can someone intuitively describe what is the difference between Taylor series and Laurent series? Also, what is the most general formula for both?
AI: Laurent series allows for terms with negative power. Intuitively, this allows for singularities to ... |
H: If A, B, C, D are non-invertible $n \times n$ matrices, is it true that their $2n \times 2n$ block matrix is non-invertible?
Is it true that
$ \left( \begin{array}{ccc}
A & B \\
C & D \\\end{array} \right)$
is non-invertible?
Assume that the matrix is over a field.
AI: Try the $2\times 2$ matrices that make up $$\... |
H: Tangents from a certain point, to a circle?
We have a circle with radius 2, centred on the origin. Find the equation of the lines passing through the point $(0,4)$ which are tangent to the circle.
So we have the circle $$x^2 + y^2 = 4$$
We need 2 lines $$y=ax+4$$
So if you fill that in the equation of the circle,... |
H: How to solve equations of this form: $x^x = n$?
How would I go about solving equations of this form:
$$
x^x = n
$$
for values of n that do not have obvious solutions through factoring, such as $27$ ($3^3$) or $256$ ($4^4$).
For instance, how would I solve for x in this equation:
$$x^x = 7$$
I am a high school stude... |
H: Using table of integrals to solve $\int y \sqrt{6+12y-36y^2}dy$
I'm supposed to use a table of integrals to solve the below equation:
$$\int y \sqrt{6+12y-36y^2}dy$$
I'm having trouble identifying the form to use because I guess my weakness in algebra shows in my inability to compete the square of the square root f... |
H: How many numbers from 1 through 60100 are divisible by none of the numbers from 2 through 6?
My thoughts on doing this problem:
total numbers is 60100
so from the total I subtract the numbers divisible by 2, 3, 4, 5, and 6.
Yet my answer
60100-30050-20033-15025-12020-10016
is a negative number. How to solve this pr... |
H: Dicrete Math Interesting question about Tromino
Prove that for a m$\times$n rectangle, if this rectangle can be covered completely by trominoes of the shape indicated in the picture, then mn is divisible by 3.
I came up with a tentative way to prove the statement above. There are two problems that are bothering m... |
H: The existence of "arbitrary large" connected compact sets in the plane
Studying some complex analysis I came up with the following hypothesis:
Let $\Omega \subseteq \mathbb C$ be a region (an open and connected set), and let $E \subset \Omega$ be a compact subset. There exists a connected compact set $F$ satisfyin... |
H: Complex formula that is equivalent to $f(x) = x$
In a computer you can't store any real number that you want to because in the $[0.0;1.0]$ interval there are infinite numbers and computer memory is finite. I want to show it in examples, which is why I need some formulas that do some complex calculations and return ... |
H: Induction proof of $a^r \ge 1$
I understand induction with one variable well, however I am not sure what to do when there are 2 or more variables.
The problem I came across is following:
Prove that $a^r \ge 1$, where $r \in \mathbb{N}$ and $a \in \mathbb{R} \wedge a \ge 1$
My solution which I am not sure whether is... |
H: Double factorial identity
Does anyone know a strategy for proving
$$
2\cdot(2k-3)!!=\sum_{i=1}^{k-1}(2i-3)!!(2(k-i)-3)!!\binom{k}{i}
$$
for $k\geq 2$? Note that $(-1)!!=1$. Hints would be most appreciated. Full solutions not so much.
I have considered induction but whereas the left hand side is multiplied by the ne... |
H: Prove convergence using $\varepsilon$-$N$ definition
This is an example presented by professor in class. I understand the idea behind this kind of definition, but I'm having trouble following my professor's thought process.
We must prove that the $\displaystyle \lim_{n\to\infty} \frac{2n-1}{n^2} = 0$ using the $\va... |
H: Heat Equation: Initial value boundary value problem
Given the initial-boundary value problem
$$ u_t −u_{xx} = 2, \ \ \ \ \ \ x ∈ [−1,1], t ≥ 0,$$
With initial and boundary conditions
$$u(x,0) = 0$$
$$ u(−1, t) = u(1, t)=0$$
Claim: the solution, $u(x,t)$, is such that
$$ u(x, t) ≤ −x^2 + 1$$
... |
H: How to find laurent series of $\exp(1/z)$
I want to find the laurent series of $f(z)=\exp(1/z)$.
I started with the formula for laurent series: $f(z)=\sum_{0}^{\infty} a_n (z-z_0)^n +\sum_{1}^{\infty} b_n (z-z_0)^{-n}$, but I don't know how to apply it.
Can someone help me find the series?
AI: (Posting a comment a... |
H: Find the limit of $\sum\limits_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$
$$\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$$
Note that $\forall x\ge 0, \sqrt{x}-1\le\sqrt{1+x}-1\le x$
Then
$$\sum_{k=1}^n\left(\sqrt{\frac{k}{n^2}}-1\right)\le S_n\le \sum_{k=1}^n\frac{k}{n^2}=\frac{1}... |
H: Are strictly upper triangular matrices nilpotent?
An $n\times n$ matrix $A$ is called nilpotent if $A^m = 0$ for some $m\ge1$.
Show that every triangular matrix with zeros on the main diagonal is nilpotent.
AI: Its characteristic polynomial is $T^n$, so by Cayley-Hamilton, $A^n=0$. |
H: Star-shaped set
Definition.- A set $S\subset \mathbb{R}^n$ is called star-shaped if there exists a point $z_0$in $S$ such that the line segment between $z_0$ and any point $z$ in $S$ is contained in $S$, $z_0$ is called a center of $S$.
Let $S\subset \mathbb{R}^n$ be a star-shaped open set , I have a problem is t... |
H: A question of H.G. Wells' mathematics
H.G Wells' short story The Plattner Story is about a man who somehow ends up being "inverted" from left to right. So his heart has moved from left to right, his brain, and any other asymmetries belonging to him. Then H.G Wells' goes on a slight metaphysical exposition:
There i... |
H: Is $\mathbb{Q}(5^{1/3})$ a Galois extension over $\mathbb{Q}$?
I am trying to prove or disprove that the simple extension $\mathbb{Q}(5^{1/3})$ is Galois over $\mathbb{Q}$.
I suspect that this extension is not Galois, because an extension if Galois over $\mathbb{Q}$ if and only if it is the splitting field of an ... |
H: We have $T(n) \leq T(\lceil \frac{n}{5} \rceil) + T(\lceil \frac{7n}{10} \rceil)$. Show that $T(n) < c'n$.
We have $T(n) \leq T(\lceil \frac{n}{5} \rceil) + T(\lceil \frac{7n}{10} \rceil)$. Show that $T(n) < c'n$ for all $n$ and for some constant $c'$.
Looks straightforward enough, but suprisingly I get stuck at th... |
H: How do you factor this using complete the square? $6+12y-36y^2$
I'm so embarrassed that I'm stuck on this simple algebra problem that is embedded in an integral, but I honestly don't understand how this is factored into $a^2-u^2$
Here are my exact steps:
$6+12y-36y^2$ can be rearranged this way: $6+(12y-36y^2)$ and... |
H: Write $y=-2\sin (x- \frac{\pi}{3})$ in the form $y= a\sin k(x-\beta)$, where a and k are nonnegative
I would like some help with this question please:
Write the following in the form $y= a\sin k(x-\beta)$, where a and k are nonnegative
$$y=-2\sin (x- \frac{\pi}{3})$$
I am confused over the effect the negative ... |
H: Logically speaking, why can variables be substituted?
Suppose that
$$a^2+a+1=b$$
Suppose also that $a=5/4$. What makes it valid to substitute $5/4$ into the first equation? Is it because equality is transitive?
AI: It's just a basic principle of first-order logic with equality that if $a = b$ and $P(b)$ for a formu... |
H: Give a push down Automata for this language: the length of is odd and it's middle symbol is 0
Give a push down automaton for this language:
{w| the length of w is odd and it's middle symbol is 0}
Here is the CFG I wrote for this language:
S --> 0|0S0|0s1|1s0|1s1
This what I have done for odd length part (I'm not ... |
H: Show that $\dot{n_s}=-\kappa_s t$
I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it.
Show that, if $\gamma$ is a unit-speed plane curve, $$\dot{\mathbf{n}}_s=-\kappa_s\mathbf t.$$
I know that
$$\dot t =\kappa_s n_s$$ and ... |
H: Prove why this algorithm to compute all list permutations works
Note: this is not homework or for a class, as I'm no longer in school.
Let's say I have a list of characters {1,2,...,N} and I want to generate all permutations.
For example, if I had {1,2,3} the permutations would be
{1,2,3}
{1,3,2}
{2,1,3}
{2,3,1}
{3... |
H: Why does "imread" generate 3 images?
The MATLAB help for imread says:
A = imread(filename, fmt)
The return value A is an array containing the image data. If the file
contains a grayscale image, A is an M-by-N array. If the file contains
a truecolor image, A is an M-by-N-by-3 array.
I thought it would generate... |
H: Is $P( A \cup B\, |\, C) $ the same as $P(A | C) + P(B | C) $ ( $A$ and $B$ are mutually exclusive)
Is $P( A \cup B \,|\, C)$ the same as $P(A | C) + P(B | C)$ ?
Here $A$ and $B$ are mutually exclusive.
AI: Yes.
Remember:
$$
\begin{align*}
P(A\cup B\mid C)&=\frac{P((A\cup B)\cap C)}{P(C)}\\
&=\frac{P((A\cap C)\cup... |
H: A student must answer five out of $10$ questions on a test, including at least two of the first five questions.
A student must answer five out of $10$ questions on a test, including at least two
of the first five questions. How many subsets of five questions can be answered?
$$\binom52\binom53+\binom53\binom52+\bin... |
H: Is this derivative wrong?
Given the following in WebAssign: $\ y = 8x + {{7} \over {x }} $
Why is it that the derivative is shown to be: $\ {dy \over dx}= {8x^2 - 7 \over { x^2}} $
Yet, when I find the derivative the quotient rule being used on the $\ 7 \over x $ makes my derivative come out to: $\ {dy \over dx}= ... |
H: Some explinations on the proof of a lemma from (Gamelin-Topology) book
In the following lemma, I need to understand why $g$ is continuous and where he used the property $|h(t)|\to \infty$ as $|t|\to \infty$.
My attempt: Let $h(t_n)\to h(t)$, then $g(h(t_n))=t_n-^?\to t=g(h(t))$. I'm not sure in which cases we can r... |
H: Finding the smallest positive integer $ n $ satisfying a modular identity.
Is there any good way of finding the smallest positive integer $ n $ such that
$$
3^{n} \equiv 1 \pmod{1000000007}?
$$
AI: First to make things clear, we are trying to find the multiplicative order of $3$ modulo $1000000007$
Now use the fact... |
H: Generating Functions and Catalan Numbers
The task is to evaluate
$$\sum_{k=0}^{n}\dfrac{1}{k+1}{2k\choose k}{2(n-k)\choose n-k} $$
This is what I've gotten so far.
$$\begin{align*}
A(x)&=\sum_{n\geq 0}\left(\sum_{k\geq 0}^{n}\dfrac{1}{k+1}{2k\choose k}{2(n-k)\choose n-k}\right)x^{n}\\
&=\sum_{n\geq 0}C_{n}x^{n}\s... |
H: Simple Polynomial Interpolation Problem
Simple polynomial interpolation in two dimensions is not always possible. For example, suppose that the following data are to be represented by a polynomial of first degree in $x$ and $y$, $p(t)=a+bx+cy$, where $t=(x,y):$
Data: $f(1,1) = 3, f(3,2)=2, f(5,3)=6$
Show that it is... |
H: Why not keep $\epsilon$ in Proof : Any Convergent Sequence is Bounded
I am trying to understand the proof of the proposition:
Any convergent sequence is bounded.
In my textbook, the author uses the definition of convergence for a sequence $\{a_n\}\to l$ and fixes $\epsilon=1$ so that there is a natural number $N$... |
H: expected value of $f(U)$ where $U$ has a uniform distribution on $[0,1]$ and $f$ is a measurable function on $L^1[0,1]$
Let $ (\Omega, F , P)$ a probability space. And let $(\mathbb R,B(\mathbb R))$ the real numbers and the borel sets. Let's consider $ U: (\Omega, F , P)\to (\mathbb R,B(\mathbb R))$ be a random va... |
H: 4 Element abelian subgroup of S5.
I have a homework question from my intro to group theory class.
Question:
Find a 4 element abelian subgroup of $S_5$. Write it's table.
This is where I've gotten so far, but I don't even know if I'm on the right mental track.
We know $S_5 = \{1,2,3,4,5\}$.
Abelian means the subg... |
H: An open set that contains $\mathbb{R}\setminus\mathbb{Q}$
If $A\subseteq\mathbb{R}$ is an open set such that $\mathbb{R}\setminus\mathbb{Q}\subseteq A$, then $A=\mathbb{R}$?
AI: Let $X\subseteq\Bbb Q$ a closed subset of $\Bbb R$, for example a singleton or $\Bbb Z$ or even far more complicated sets.
Then $A=\Bbb R\... |
H: Convergence of improper integrals in $\mathbb{R}^n$
This is probably an elementary result, but every time I need it I'm always confused.
I also looked for the solution but I could not find it, so I think this will serve well as a reference for the future!
Here are the problems: consider $A_n = B(0,1) \subset \mathb... |
H: Prove a formula in terms of n:
$1+5+9+...+(4n+1)$
I HAVE to use induction, but I am new to induction, so I am a bit confused...
I believe I have to use the base case first: so $n=1$ is $4(1)+1=5$, but i get the second term in the sequence instead of the first term? is this ok? Next I tried to substitute k+1 in the ... |
H: limsups and liminfs
Suppose we have a sequence $(x_n)$ in $\mathbb{R}$ such that $x_n \leq X $ for all $n$, then we have $$ \limsup x_n \leq X $$
and if $X \leq x_n$ for all $n \implies X \leq \liminf x_n $.
My try. Since $x_n \leq X$, then $X$ is an upper bound for the set $\{ x_n \}$. In particular $ \sup \{ x_n... |
H: Fermat's Little Theorem and congruences.
I am taking number theory and have hit a roadblock taking the next logical step in one of my proofs. I am told that $n=195=3 \cdot 5 \cdot 13$. I am asked to show that $a^{n-2} \equiv_n a, \; \forall a \in \mathbb{Z}.$
The previous part of the problem was relatively straight... |
H: Normal distribution
I have this question:
A normal distribution is such that 16% of it is smaller than 13, and 2.5% of it is larger than 22. What's the mean of this normal distribution?
I know I should be using the 68-95-99.7 rule, but I have no clue how to start without the standard deviation being given. Any hint... |
H: Let $a,b$ nonnegative, $a\ge cb$ for every $c$ in $(0,1)$, Should $a \ge b$ be true?
In the proof of monotone convergent theorem, The above technique is used. I really appreciate if anybody can explain it to me.
AI: Suppose that $a<b$. Then choose $c=\frac{b+a}{2b}$ and clearly we have $c<1$. Moreover:
$$
cb=\frac{... |
H: Use the relation of Laplace Transform and its derivative to figure out $L\left\{t\right\}$，$L\left\{t^2\right\}$，$L\left\{t^n\right\}$
If $F(s) = L\left\{f(t)\right\}$, then $F'(s) = -L\left\{tf(t)\right\}$
Use this relation to determine
$(a)$ $L\left\{t\right\}$
$(b)$ $L\left\{t^2\right\}$
$(c)$ $L\left\{t^n\right... |
H: easy calculus thing that i forgot
Suppose $(s_n)$ is a non- negative sequence in $R$. Suppose $m > n $, then is that true that we must have $ (s_n) \subseteq (s_m ) $ and hence we can conclude $ \inf s_n \geq \inf s_m $. but we know that $h_n = \inf_{ k \geq n } s_k $ is a non-decreasing sequence. What am i doing w... |
H: How many 9-digit numbers are there with twice as many different odd digits involved as different even digits.
How many 9-digit numbers are there with twice as many different odd digits
involved as different even digits (e.g., 945222123 with 9, 3, 5, 1 odd and 2, 4
even).
(5c1)(5c2)3^9 +(5c2)(5c4)6^9+(5c3)(5c6)9!
ev... |
H: Does "Big Data" Have a Ramsey Theory Problem?
I'm erring on the side of conservatism asking here rather than MO, as it is possible this is a complex question.
"Big Data" is the Silicon Valley term for the issues surrounding the huge amounts of data being produced by the global IT structure. Advanced mathematics is... |
H: A statistical approach to the prisoners problem
Two days ago, I found this problem on reddit (I didn't have access to reddit when I did the math, so I did it with 24 instead of 23, and I decided the warden picked someone every day, not "whenever he feels like it"):
A prison warden tells 24 prisoners he has a "game... |
H: Proving any product of four consecutive integers is one less than a perfect square
Prove or disprove that : Any product of four consecutive integers is one less than a
perfect square.
OK so I start with $n(n+1)(n+2)(n+3)$ which can be rewritten $n(n+3)(n+1)(n+2)$
After multiplying we get $(n^2 + 3n)(n^2 + 3n + 2)$
... |
H: Generating Function for the adjusted Fibonacci numbers
The task is to find another relation for the adjusted Fibonacci numbers. I've found there genertaing function
$$A(x)=\dfrac{1}{1-x-x^{2}}$$
Furthermore I've created the generating function in a different way and now want to grab the correct coefficient.
Since... |
H: Find the value of $\lim_{n\rightarrow\infty}\frac{a_n}{1\cdot 2}+\frac{a_{n-1}}{2\cdot 3}+\ldots +\frac{a_1}{n(n+1)}$
$\displaystyle \lim_{n\rightarrow\infty}\frac{a_n}{1\cdot 2}+\frac{a_{n-1}}{2\cdot 3}+\ldots +\frac{a_1}{n(n+1)}$ if $\lim_{n\rightarrow\infty}a_n=a$
Note that $\displaystyle\sum_{k=1}^n\frac{a_{n-k... |
H: Integration of $(5^{3/x}-\sqrt[3]{x^8}+9)/(3x^2)$
Problem: evaluate $$\int\frac{5^{3/x}-\sqrt[3]{x^8}+9}{3x^2}\mathrm dx$$
Please provide me a hint. I tried to separate to 3 parts but don't know how to integrate $\frac{5^{3/x}}{3x^2}$.
AI: Hint: Let $u = \frac{3}{x}$, and then $du = \frac{-3}{x^2} dx$. Then
$$\int ... |
H: Show that $\lim_{n \to \infty} \frac{k^n}{n!} =0$ for all $k$ in $\mathbb{R}$
$$\lim_{n\to\infty} \frac{k^n}{n!}=0 \, \forall\:k\in \mathbb{R}$$
I have tried to find an $N$ in term of epsilon in the definition of limit, but to no avail. I've tried log, but $\log (n!)=\log(1)+\log(2)+...+\log(n)$ does not seem to he... |
H: Discrete Math Combinatorics Homework Help
Find the value of
${n\choose0} + 3{n\choose1} + 9{n\choose2} + {27}{n\choose3} + \dots + 3^n{n\choose n}$
I know that ${n\choose0} = 1$, ${n\choose1} = n$ so $3{n\choose1} = 3n$, and ${n\choose n}=1$ so $3^n{n\choose n}=3^n$. But I don't realize the pattern going on here... |
H: Differential Equation $y(x)'=(y(x)+x)/(y(x)-x)$
can someone give me some tips on how to solve this differential equation.
I looked at the Wolfram solution which substituted $y(x)=xv(x)$. I'd know how to solve from there, but I have know idea why they did it in the first place, well why the algorithm did it in the f... |
H: How can one show that $ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$?
Let $f:[0, 1] \rightarrow \mathbb{R}$ a continuous function. If $a>0$, show that:
$$ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$$
Tried usi... |
H: Compute $\int_0^1\int_0^1...\int_0^1\lfloor{x_1+x_2+...+x_n}\rfloor dx_1dx_2...dx_n$
Compute $\int_0^1\int_0^1...\int_0^1\lfloor{x_1+x_2+...+x_n}\rfloor dx_1dx_2...dx_n$ where the integrand consists of the floor (or greatest integer less than or equal) function.
The case $n=1,2,3$ all can be solved geometrically. A... |
H: How can I evaluating limits? $\lim_{x\rightarrow0^+}(xe^{2x}+1)^{5/x}$ and $\lim_{x\rightarrow{\pi / 2}}(1+\sec(3x))^{\cot(x)}$
Can someone please give me an idea how to do these two limits. I guess it is the L'Hopital's rule, not sure.
$$\lim_{x\rightarrow0^+}(xe^{2x}+1)^{5/x}$$
$$\lim_{x\rightarrow{\pi / 2}}(1+\s... |
H: How can I get more intuition about CX?
Let $X$ be a topological space and define $CX$ as the quotient space $X \times I / X \times \{0\}$. From my understanding $X \times \{0\}$ defines some sort of equivalence relation, but I am unsure how this can be described. Is this assumption correct? If so, what is a better ... |
H: Evaluate $\oint_C\frac{dz}{z-2}$ around the circle $|z-2| = 4$
I don't completely understand how to approach these questions. I suppose the notation $\oint_C$ is something I'm not used to.
So far, I have $\oint_C\frac{dz}{z-2} = \log(z-2)$. From here, I suppose I could utilize $|z-2| = 4$ and $z-2 = re^{i\theta}$ ... |
H: Help with simple algebra
Can someone please explain what to do with x in the following equation, I have not done algebra for a long time so a bit lost:
$$f(x)= -5 x^5 + 69 x^2 - 47$$
AI: Given:
$$f(x)= -5 x^5 + 69 x^2 - 47$$
We have:
$f(0) = -5(0)^5 + 69 (0)^2 - 47 = -47$
$f(1) = -5(1)^5 + 69 (1)^2 - 47 = 17$
... |
H: Partial derivatives of second order
Find all functions $f:\mathbb{R}^2\rightarrow \mathbb{R}$ of class ${\cal C}^2$, such that:
$\frac{\partial^2f}{\partial x\partial y} = 0$
$\frac{\partial^2f}{\partial x^2} = \frac{\partial^2f}{\partial y^2}$
(Separate questions)
For the first one I prove that $f(x,y) = h(x)+... |
H: Giving presents to 8 employees out of 10
Each of 10 employees brings one (distinct) present to an office part. Each present is given to a randomly selected employee by Santa (an employee can get more than one present). What is the probability that at least two employees receive no presents? How would you approach t... |
H: Determine the maximal ideals of $\mathbb{R}^2$.
Determine the maximal ideals of $\mathbb{R}^2$.
Well for any real number that is not divisible by another number other than 1 and itself generates a maximal ideal for $\mathbb{R}$. Is that right? Would it be the same as for $\mathbb{R}^2$?
AI: Not quite, since ther... |
H: Conditional Joint Density Function $(Y | X) \sim \mathcal{N}(x, 1)$ and $X \sim \mathcal{N}(0,1)$ whats $f(y | x)$
Conditional Joint Density Function: $(Y | X) \sim \mathcal{N}(x, 1)$ and $X \sim \mathcal{N}(0,1)$ whats $f(y | x)$
Since $X \sim \mathcal{N(\mu, \sigma^2)}$ then $f(x) = \frac{1}{\sigma \sqrt{2 \pi}} ... |
H: Hint is required for an integral
$$\int\frac{\mathrm dx}{\sqrt x \cos(1-\sqrt x)}$$
please provide a hint about the substitution. The website gives a long answer See here.
Wondering if there is a simplification.
AI: Hint:
$$u = 1 - \sqrt{x} \implies du = -\frac{1}{2 \sqrt{x}} dx$$
so the integral can be written as
... |
H: What does $\text{poly}$ stand for in $O(\log^{10.5}n \cdot \text{poly}(\log \log n))$?
I posted this question on cstheory and found that "poly(f(n))" is shorthand for "polynomial in f(n)" or $f(n)^{O(1)}$, hence poly(log log n) is shorthand for $(log log n)^{O(1)}$. However, I don't understand the notation where th... |
H: Trigonometry - 'Find all solutions to the equation'
I have 3 questions to solve for trigonometry, and I'm not sure of the process to solve them. They are:
Find all solutions to the equation:
$4 \sin 2x = 2.4$ $\forall$ $0 ≤ x ≤ 2\pi$
$2 \cos 2x = 1.3$ $\forall$ $0 ≤ x ≤ 2π$
$3 \tan 2(x + \frac{\pi}{3}) = ... |
H: Probability on drawing colored balls
Here is another question from the book of V. Rohatgi and A. Saleh. I would like to ask help again. Here it goes:
An urn contains $r$ red and $g$ green marbles. A marble is drawn at random and its color noted. Then the marble drawn, together with $c > 0$ marbles of the same color... |
H: Tallest bubble tower induction proof
A hemispherical bubble is placed on a spherical bubble of radius $1$. A smaller hemispherical bubble is then placed on the first one. This process is continued until $n$ chambers, including the sphere, are formed. (The figure shows the case $n = 4$.) Use mathematical induction t... |
H: Proving that there is no values of z such that cosz = isinz
Here is my attempt:
Assume there are values of $z$ such that $\cos z = i \sin(z)$
Then
$\cos(z) = \dfrac{e^{iz} + e^{-iz}}{2}$
and
$i\sin(z) = \dfrac{i(e^{iz} - e^{-iz})}{2i}$
and so after doing some algebra we get
$4e^{-iz} = 0$
and this will never happen... |
H: algebraically determining if a number is irrational or not
Is it possible to use an algebraic formula, equation, concept, or principle to determine with perfect accuracy (or high precision, if not perfect) whether or not a number is rational?
An example number I have in mind is $\sqrt{937}$.
AI: I think you need to... |
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