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H: Road and candidiate problem
The city has n districts and n - 1 bidirectional roads. We know that from any district there is a path along the roads to any other district. Let's enumerate all districts in some way by integers from 1 to n, inclusive. Furthermore, for each road the residents decided if it is the proble... |
H: Find $a_{n+1}=\frac{a_n^2+1}{2}$ in terms of $n$.
I was trying to prove that for all $n\in \Bbb N$ there are integer numbers $\{a_1,a_2,\ldots,a_n,b_n\}$ s.t. $a_1^2+a_2^2+\dots+a_n^2=b_n^2$.
I founded that if $\{a_1,a_2,\ldots,a_n,b_n\}$ have the property then $\{a_1,a_2,\ldots,a_n,a_{n+1}=\dfrac{b_n^2-1}{2},b_{n+... |
H: Taking LCM in derivative question $\lim_{x\to 1} \left[ \frac{1}{1-x} - \frac{3}{1-x^3}\right]$
in my mathematics homework. I had a chance to pass through this question:
$$\lim_{x\to 1} \left[ \frac{1}{1-x} - \frac{3}{1-x^3}\right]$$
According to my understandings this should be converted into this:
$$\lim_{x\to ... |
H: Trigonometry gymnastics
The teacher is as usual jumping a million miles between steps, I appreciate if someone can break down how this step is done:
$$\frac{\partial r}{\partial s}\times\frac{\partial r}{\partial t}=\det
\begin{pmatrix}
i & j & k \\
-\sin(s)(2+\cos(t)) & \cos(s)(2+\cos(t)) & 0 \\
-\cos(s)\sin(t) & ... |
H: Fourier transform supported on compact set
Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Show that $$\hat{f}(y)=1_{[-\pi,\pi]}(y)\sum_{n=-\infty}^\infty f(n)e^{-iny}$$ in the sense of $L^2(\mathbb{R})$-norm convergence.
I know that $f$ must be continuous and going to $0$ at $\pm\in... |
H: Explain mathematical practice and axiomatization to non-mathematicians
I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P Smith's Introduction to Gödel's theorems, 28.6.,... |
H: What is the corresponding transition matrix for this DTMC?
there
I am a bit confused about forming the transition matrix for this inventory problem.
Suppose that $X(n)$ is amount of the item in the inventory at the end of the each period (day).
One thing that is quite straightforward is to consider $S=\{..,-3,-2,-1... |
H: Mathematical notation around the world
What are the differences in mathematical notation around the world? I know that in some other countries they write 1,2 meaning 1.2, but what else can be confusing in an academic environment (when people are doing math on a board or on paper).
AI: As seen here, in some countrie... |
H: How do I 'reverse engineer' the standard deviation?
My problem is fairly concrete and direct.
My company loves to do major business decisions based on many reports available on the media. These reports relates how our products are fairing in comparison to the competitor's offerings.
The latest report had these scor... |
H: Is every injective rational function $f:\mathbb Q\to\mathbb Q$ a polynomial?
I thought this might be quite easy to show, and then realized that the tools I know from real analysis aren't going to help here.
Suppose we have a rational function:
$$
f(X)=\frac{P(X)}{Q(X)}
$$
where $Q$ has no linear factors over $\m... |
H: Finding the GCD of $50!$ and $2^{50}$
I've been trying to figure out how $n!$ and $x^n$ are related (where x is an integer) for most of the morning - I know it must be the key to unlocking this problem.
Up to this point I've only used the Extended Euclidean Algorithm to find GCDs, but I know that's not going to wor... |
H: Change of Coordinate in Differential Equation
I'm sorry, it's probably a very simple question but I'm confused between change of variable and change of coordinate in a differential equation. To take a very simple example, let's start with this equation for the function $f(x,t)$:
$$
\frac{\partial f}{\partial t} = \... |
H: What will be set builder notation for this set?
Following set is neither of even nor of odd so how can I express this by using set builder notation?
{0,3,6,9,12}
AI: What you have there already is a form of set builder notation. If you’re asking how you can express it in set builder notation using some property or ... |
H: Approximation of $\frac{1+a}{1+b}$
I've found the following assertion on an economics book:
For $r$ and $g$ small enough, $\frac{1+r}{1+g}\approx 1+r-g$
(where $r$ is the interest rate and $g$ is the growth rate of the economy)
I would like to know why this is true. I've tried to find the solution by myself, but ... |
H: Set builder notation for this set
I need set builder notation for a set.
Set under consideration is:
$$\{m,n,o,p\}$$
What I suggest is:
$$\{x\colon x\in\{m,n,o,p\}\}$$
Any suggestions is it correct?
AI: Hint: How would you verbally describe those letters? What do they have in common? Given a verbal description, you... |
H: Definition of doubling measure
From the definition of a doubling measure on wikipedia:
"A measure on a metric space X is said to be doubling if the measure of any ball is approximately the measure of its double, or more precisely, if there is a constant $C > 0$ such that
$\mu(B(x,2r))\leq C\mu(B(x,r))$
for all x in... |
H: Question in probability and statistics
I'm terrible at questions involving probability, so I'm hoping you can help me!
Suppose there are eight people in a group, each person with a distinct first name. Each person has a card with their first name written on it.
If the eight cards are randomized and each person sel... |
H: BV function as left-continuous decomposition
A bounded variation (BV) function $f$ on the whole real line can be written as a difference of two monotone increasing function. This can be done by the construction
$$g(x) = \begin{cases} V[0,x] &\mbox{if } x\geq 0 \\
-V[x,0] & \mbox{if } x<0 \end{cases}$$ and $h(x)=g(x... |
H: If $X_n \rightarrow X$ in probability then $X_n \Rightarrow X$?
If $X_n \rightarrow X$ in probability then $X_n \Rightarrow X$?
Below are my thoughts, then my questions - I am interested in your feedback on this argument.
Suppose $X_n \rightarrow X$ in probability. Does $X_n \Rightarrow X$?
I know I can prove t... |
H: Matrix norm equivalence
If we define $ \|A\| = \max \{|A\cdot \mathbf{t}|:|\mathbf{t}|\leq 1\}.$ is it the same as defining it as $\max \{|A\cdot \mathbf{t}|:|\mathbf{t}|= 1\}$ ? If so, why? The book I'm following uses the first definition and while searching the web for some hints to my question, I encountered thi... |
H: First fundamental form question.
The question I posted;
$6.1.2\quad$ Show that and apply an isometry of $\Bbb R^3$ to a surface does not change its first fund. form. What is the effect of a dilation (i.e., a map $\Bbb R^3\to \Bbb R^3$ of the form $v\mapsto av$ for some constant $a\ne 0$.
And I posted its answer... |
H: Uniform prior distribution multiple results
When I have a simple Bernoulli trial with a certain variable taking, for instance, values 0 and 1, I have a constant prior distribution for the $\theta$ parameter, i.e. pdf $p(\theta) = 1$ between 0 and 1. That ends up meaning that the first moment of the distribution, wh... |
H: proofs of independence
If we have a first-order theory, do all independence proofs of a certain result in that theory need to use "outside" assumptions?
Cant we just enumerate all proofs in that theory and conclude that none of them leads to our result?
Consider following: we have the axioms that we can change MUI ... |
H: What is the difference between an indefinite integral and an antiderivative?
I thought these were different words for the same thing, but it seems I am wrong. Help.
AI: "Indefinite integral" and "anti-derivative(s)" are the same thing, and are the same as "primitive(s)".
(Integrals with one or more limits "infinity... |
H: Is it proper subset or not?
Consider following statment $\{\{\varnothing\}\} \subset \{\{\varnothing\},\{\varnothing\}\}$
I think above statement is false as $\{\{\varnothing\}\}$ is subset of $\{\{\varnothing\},\{\varnothing\}\}$ but to be proper subset there must be some element in $\{\{\varnothing\},\{\varnothin... |
H: Let $(R,m)$ be a Noetherian local commutative ring. And suppose that $m^{n} = 0$ for some $n \in \mathbb{N}$
Let $(R,m)$ be a Noetherian local commutative ring. And suppose that $m^{n} = 0$ for some $n \in \mathbb{N}$
Then I want to show that $m/m^2$ is a finite dimensional vector space over the field $R/m$. I am t... |
H: Complex Numbers Question?
I answered the first part of the question. But I'm having a trouble with the second part. I can only find the half-line at $2i$ and $\theta=\pi/6$.
Here's the solution guide:
AI: Geometrically $|z-3|=|z-3i|$ denotes all the points equally distant from $3$ and $3i$. This is exactly the mid... |
H: Let $f,g$ be differentiable with $f(0)=g(0)$ and $f'(x)
Let $f,g:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $f(0)=g(0)$ and $f'(x) < g'(x)$ for all $x$ belonging to the set of real numbers. Prove that $f(x)<g(x)$ for all $x>0$.
Any help? Im so confused :P
AI: Set $ \displaystyle h(x):= f(x) - g(x), \... |
H: Euclidean geometry (triangles) - homework
I am struggling with a geometry question that our noble professor gave to us. He wants us to prove that given an acute triangle ABC, if we construct an equilateral triangle that shares a side CA with the acute triangle (and has an orientation into the acute triangle) then t... |
H: Show: $\lim(a_n+b_n-c_n)=\lim(a_n)+\lim(b_n)-\lim(c_n)$
Consider sequences $(a_n), (b_n), (c_n)$ in $\mathbb{R}$ with $\lim_n a_n=a<\infty, \lim_n b_n=b<\infty$ and $\lim_n c_n=\infty$.
1.) Show that
$$
\lim(a_n+b_n-c_n)=\lim(a_n)+\lim(b_n)-\lim(c_n).
$$
2.) Show the same statement with $\liminf_n$ instead.
How c... |
H: The number of ways each husband can sit to the left of his wife
There are 5 husband/wife couples. What is the number of ways we can sit the couples around a circular table with each husband sitting to the left of his wife?
AI: I'm assuming you mean that the husband sits directly to the left of his wife, in which ca... |
H: Two questions on trigonometric identities
Develop a formula for $\sin \left( \frac{x}{2} \right)$ in terms of $x$.
Use a double angle formulae to develop a formula for $\sin (4x) $ in terms of $x$.
I have absolutely no idea how to do this. Could someone please explain what the question is asking, how to solve ... |
H: Probability that at least 1 person receives its letter for a distribution of n letter to n people (paired 1 to 1)
I'm trying to solve an elementary probability problem, but I don't get the right answer and can't find the flaw in my reasoning. The problem and my (wrong) solution goes as follows.
Somebody distributes... |
H: Zorn's Lemma and Injective Modules
In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective modules as being those that have no proper essential extension 3) the structure t... |
H: When does a left Haar measure on a locally compact group restrict to a left Haar measure on a locally compact subgroup?
Given a locally compact group which is not $\sigma$-compact, there exists a $\sigma$-compact subgroup $H$ of $G$ which is open and closed.
A Remark in Folland's, A Course in Abstract Harmonic Anal... |
H: Zagier's proof of the prime number theorem.
In Zagier's paper, "Newman's Short Proof of the Prime Number Theorem", (link below) his theorem ${\bf (V) }$ states that,
$$ \int_{1}^{\infty} \frac{\vartheta(x) - x}{x^2} dx \text{ is a convergent integral.} $$
Note: $\vartheta(x) = \sum_{p \le x} \log(p)$, where $p$ is ... |
H: The probability of being dealt a 2 pair 5 card hand...
Given a standard playing card deck of 52 cards, what is the probability of being dealt a 2 pair 5 card hand consisting exactly of one pair of face cards and one pair of NOT face cards is?
AI: There are $\binom{52}{5}$ hands, all equally likely.
For the number ... |
H: Maximizing slope of a secant line
Two points on the curve $$ y=\frac{x^3}{1+x^4}$$ have opposite $x$-values, $x$ and $-x$. Find the points making the slope of the line joining them greatest.
Wouldn't the maximum slope of the secant line be with the max/min of the curve?
So $x=3^{1/4}$ and $x=-3^{1/4}$?
AI: The slo... |
H: Limit calculation
$\lim_{n\to\infty}((\frac94)^n+(1+\frac1n)^{n^2})^{\frac1n}$
Here's what I did:
$\lim_{n\to\infty}((\frac94)^n+(1+\frac1n)^{n^2})^{\frac1n}\\
=(\lim(\frac94)^n+\lim((1+\frac1n)^{n})^n)^{\frac1n}\\
=(\lim(\frac94)^n+\lim e^n)^{\frac1n}\\$
Any hints on how to continue?
PS: no logs/integration/deriva... |
H: How do you solve the diference quotient for $f(x) = 23 \sqrt{x}$
How do I solve the difference quotient for $f(x) = 23 \sqrt{x}$? I know how to plug it in but I don't understand how to simplify.
AI: Hint: rationalize the numerator
$$
\begin{align}
\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} &=
\lim_{h\to 0}\frac{23\sqrt{x+h... |
H: Using the result of a derivative, find the integral?
I'm just a little bit unsure of this question:
"Show that: $d/dx (\sin x)^3=3\cos x(\sin x)^2$ and hence use this result to find the integral (limits of $pi/2$ and $0$) of $(\sin x)^2\cos x.$"
Is it simply putting one third in front of the integral and multiplyin... |
H: Laplace boundary value problem
I came across the following Laplace bvp:
$u_{xx}+u_{yy}=0\ $ for $\ 0<x<1,\ 0<y<2$
$u(x,0)=u(x,2)=0$
$u(0,y)=0$
$u(1,y)=y(2-y)$
I didn't have any problems solving it. It was quite straightforward, following the general theory. However, the next exercise asked to solve the following bv... |
H: Proving $\lim_{x \rightarrow a} f(x) = \lim _{ x\rightarrow a}g(x)$
Suppose there is a $\delta > 0 $ such that $f(x)=g(x)$ when $0 < | x-a| < \delta$
Prove: $$\lim _{x \rightarrow a} f(x) = \lim_{x \rightarrow a} g(x)$$
I defined each limit like this: $0 < | x-a| < \delta _1$ then $|f(x)-L|< \frac{|L-m|}{2}$ and f... |
H: Minimizing area of a triangle
Consider the function $g(x)=1-x^2$
.For $x>0$, the tangent line to $g(x)$ forms a right triangle with the coordinate axis. Find the point of the curve such that the right triangle has the smallest possible area.
When I try to minimize the derivative all I get is X=0 (assuming I set th... |
H: Specifying a topology by connected subspaces
Let $\Gamma_1$ and $\Gamma_2$ be topologies on a set $X$ such that:
for every subspace $A \subseteq X$, $A$ is connected in $\Gamma_1$ iff it is connected in $\Gamma_2$
Are $\Gamma_1$ and $\Gamma_2$ homeomorphic?
AI: No.
Let $X=\{a,b\}$ and let $\Gamma_1$ be the indisc... |
H: Summation questions
I got stuck on this:
(a) Find $N_1 \in \mathbb{N}_+$ such that $n \geq N_1 \implies \sum_{k=n}^\infty\frac{1}{k^2}\leq10^{-20}$
(b) Find $N_2 \in \mathbb{N}_+$ such that $n \geq N_2 \implies \sum_{k=n}^\infty\frac{1}{2^k}\leq10^{-20}$
part (b) I think I managed to do because it is a geometric se... |
H: Question about Notation. What does this means? $f[0]=1, f[0,1]=-1$
Question about Notation. What does this means?
$f[0]=1, f[0,1]=-1, f[0,1,2]=2$ (The values are exact, which is pretty confusing too, if they are refering to intervals)
This question is from a numerical analysis paper about using polynomials to appr... |
H: Having trouble reading a set theoretic equation
I've just begun an introductory text on probability. In the first chapter there is a preview/review of set theory, which I am not familiar with. One of the examples has me a little confused.
I do not understand "how to read" the mathematics due to my unfamiliarity wit... |
H: What is the method for coming up with these limits?
From Baby Rudin,
I see that for $inf$ he chose $a_{n+1}$ to be $\frac{1}{8}$ and $a_n$ to be $1$ because that's the smallest ratio. For $sup$ he chose $\frac{1}{8}$ to be $a_n$. Can someone verify this?
I also do not know how the third limit is reached.
AI: You h... |
H: How to integrate $\int\frac{\sqrt{1-x}}{\sqrt{x}}\ \mathrm dx$
I'm having a bit of trouble solving this integral: $$\int\frac{\sqrt{1-x}}{\sqrt{x}}dx$$
Here is my attempt at a solution:
I multiplied the numerator and the denominator of $\frac{\sqrt{1-x}}{\sqrt{x}}$ by $\sqrt{x}$, yielding $$\int\frac{\sqrt{x-x^2}}... |
H: Weak convergence in $H^1_0(U)$ implies convergence in $L^2(U)$
Recall some definitions :
$ H^1_0(U) =W_0^{1,2}(U)$ and $u\in W^{k,p}_0(U)$ i there exists $u_m\in C_c^\infty(U)$ s.t. $u_m\rightarrow u$ in $W^{k,p}(U)$
question For $1\leq p \leq \infty$, $W^{k,p}(U)$ is Banach. How can we prove that $H^1_0(U)$ is Hi... |
H: Sizes of kernels of homomorphisms
I have a problem that I have been stuck on for two hours. I would like to check if I have made any progress or I am just going in circles.
Problem: Let $\alpha:G \rightarrow H, \beta:H \rightarrow K$ be group homomorphisms. Which is larger, $\ker(\beta\alpha)$ or $\ker(\alpha)... |
H: Orthogonality of Laguerre polynomials...
Laguerre polynomials is a kind of orthogonal polynomials whose inner product is zero. (Is this correct?)
To show that two Laguerre polynomials $L_n(x)$ and $L_m(x)$ are orthogonal, they must satisfy the integral
$\int\limits_0^\infty e^{-x} L_m (x) L_n (x)dx=0$
with respect... |
H: Show that $a
At first I thought it was obvious, but one implication is giving me a hard time. I would appreciate if one the implications demonstrated were to be revised as well help or hints with the other implication.
I have to prove for $a,b\in \mathbb Q^+$ that $a<b \Leftrightarrow a^n<b^n$ with $n\in\mathbb N$.... |
H: Find all primes $p$ such that $14$ is a quadratic residue modulo $p$.
I want to find all primes $p$ for which $14$ is a quadratic residue modulo $p$. I referred to an example that was already posted for finding all odd primes $p$ for which $15$ is a quadratic residue modulo $p$, but I am getting stuck.
This is what... |
H: Formula for Multiple of $23$
For any non-negative integer, why is $$(3^n) \cdot (2^{3n})-1$$ always a multiple of $23$? I'm thinking of pulling out the $n$ and doing something with mod.
AI: $3^n2^{3n}-1=24^n-1$
Now, Since $24\equiv 1\pmod {23}\implies 24^n\equiv ?\pmod{23}$ |
H: Prove: The series $\sum_{n=1}^\infty x_n$ converges $\iff$ the series $\sum_{n=1}^\infty a_nx_n$ converges.
I could use hints for this assignment.
Let $(x_n)_{n\in\mathbb{N}}$ and $(a_n)_{k\in\mathbb{N}}$ be (real) sequences with $\lim_{n\to \infty} a_n = a \not= 0$.
Prove that $$\sum_{n=1}^\infty x_n \text{ conv... |
H: Find $x$ if $\sinh(x)=2$
I want to know how to find $x$ if $\sinh(x)=2$. I already know that $\sinh(x) = \dfrac{e^x -e^{-x}}{2}$. Hence,
$$\frac{e^x -e^{-x}}{2} = 2 \implies e^x -e^{-x}=4$$
but I don't know what should I do then.
AI: The answer is $x = \sinh^{-1}(2)$. Note that it can also be written as follows: We... |
H: Puzzle in a Calculation
I want to calcalate $a$ from
$$\frac{d a}{d u} = \frac{\sqrt{2}}{4\pi} \oint_\alpha \frac{dx}{\sqrt{(x-1)(x+1)(x-u)}}$$
where $\alpha$ is the contour only around cut between -1 and 1. First I want to integrate $u$ and I find some trouble.
Directly integrating $u$ gives
$$ a= \frac{-2\sqrt{2... |
H: conditions for LU on a $2 \times 2$ matrix
I have
$$A=
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
$$
I want to do LU factorization on it, so I need to find the elementary matrix $E$ such that $EA=U$ and thus $L=E^{-1}$. So I know I need the row operation $aR_2-cR_1\rightarrow R... |
H: No. of integer ordered pairs $(x,y,z)$ in $x^2+y^2+z^2 = 2012$
[1] No. of integer ordered pairs $(x,y,z)$ in $x^2+y^2+z^2 = 2012$
[2] No. of integer ordered pairs $(x,y,z)$ in $x^2+y^2+z^2 = 1980$
$\bf{My\; Try}::$ for (1) one ::
$x^2+y^2+z^2 = 2012 = 4\times 503$
means $x,y,z$ must be an even integer. So $x=2x^{'}... |
H: Adjust Saturation in CIE L*a*b* space.
Given a color in CIE L*a*b* space, how does one change the saturation? This is what I know...
$$\mathrm{chroma} = \sqrt {(a^*)^2 + (b^*)^2}$$
$$\mathrm{hue} = \arctan \left( a^* \over b^* \right)$$
$$\mathrm{sat} = {\frac{\mathrm{chroma}}{\sqrt{(L^*)^2 + \mathrm{chroma}^2}}}$$... |
H: Uniqueness/Existence ODE question
here's my question
Consider the differential equation:
$t\frac{dg}{dt} = 2g.$
I got that the general solution is $g = ct^2$. However, I don't understand how to answer these questions:
What is the unique solution for the IVP $g(-1) = -1$ on the interval $-2 < t < 0$. What is the l... |
H: Probability that a five-card poker hand contains exactly two aces if it has exactly one face card
We are dealt a five-card poker hand. What is the probability that this hand contains exactly two aces if it has exactly one face card?
AI: Let A be the event of exactly 2 aces and let B be the event of exactly one face... |
H: how to solve factorial involving multiplication
I am trying to solve this question but not able to find any helpful material. It involves factorial with multiplications,
$$\frac{8!}{5!}\cdot \frac{7!}{7!10!}$$
I tried crossing 8 and 5 and 7 with 7 but it's not giving me right answer
AI: $\dfrac{8!}{5!}\cdot \dfrac{... |
H: Find the random variable, value function, and value you would pay to break even...
In a game you receive three cards, $\omega$ , from a well-shuffled deck. You then receive $10 if the hand contains at least two face cards. In order to determine how much you would be willing to pay, per game, to play a large number ... |
H: Partial fraction when $N^r$ and $D^r$ are quadratic and cubic polynomials
I need to find the nth derivative of the following function
$$y=\frac {x^2+4x+1}{x^3+2x^2-x-2}$$
The trouble is I don't know how to break a fraction like the above one. How do I break it into partial fractions? Or is there any other way to ca... |
H: Prove that $f$ is a constant function
Let $f:\mathbb{R}\to\mathbb{R}$ be a function. Suppose:
$$\left|\sum_{k=1}^{n}3^k(f(x+ky)-f(x-ky))\right|\leqslant 1\quad\forall n\in\mathbb{N}\quad\forall x,y\in\mathbb{R}$$
Show that $f$ is a constant function.
I don't even know where to start and what is the possible approac... |
H: Find the derivative of y when y= ln (arccosh x)
I want to know how to find the derivative of y when y= ln (arccosh x)
I know arccosh x = 1/[x^2 -1]^(1/2)
So
1/[(arccosh x)^[2] [x^2 -1]^(1/2)]
But the right answer is
1/[(arccosh x)^[2] [x^2 -1]^(1/2)]
Why?
Please help
Thanks all
AI: You have : y = ln[arccosh(x)]... |
H: The sum of the points obtained by rolling a perfect die
A perfect die is rolled n times. The sum of all points should follow a multinomial distribution, and the pmf is
$$P(S_n=k)=\sum_{a_1+...+a_n=k}{n\choose{a_1,a_2,...,a_n}}(\frac{1}{6})^n$$
So when n=100, how can I find $P(330\leq S_n \leq 380)$? It seems that I... |
H: Sum $\sum_{n=2}^\infty\frac{a^{n+1}}{n(n-1)}z^{n}$
Given the power series $\sum_{n=2}^\infty\frac{a^{n+1}}{n(n-1)}z^{n}$ , where $a>0$, find the radius of convergence and the sum of the series.
The radius is $\frac{1}{a}$ , but what about the sum?
AI: Hint: Integrate $\sum_2^\infty a^{n+1}z^{n-2}$ twice term by ter... |
H: polynomials factorization over rings and finite fields
Any nonzero polynomial over a subring $R$ of $\mathbb{C}$ is a product of irreducible polynomials over $R$. And for any subfield $K$ of $\mathbb{C}$, factorization of polynomials over $K$ into irreducible polynomials is unique up to constant factors and the ord... |
H: A way to show that if $y''(x) +y(x)=0$ then $y(x)=\cos (x)$
How can I show that if $y''(x)+y(x)=0$ then $y(x)=\cos(x)$. I found this out by intuition but is there a algebraic way to show this?
AI: Multiply by $2y'$ and integrate:
$$
2y'y''+2yy'=0\implies y'^{\,2}+y^2=a^2
$$
for some constant $a$. Separate variables... |
H: Given three numbers $a, b$, and $n$, is it possible to find a number $k$ such that $ka\equiv b\pmod{n}$?
I came up across this problem working on some (purely academic) attack on cryptographic schemes.
Given any three integers $a,b,n$ such that $a<n,b<n$, I am interested in an integer $k$ which satisfies $$ka \equi... |
H: linear algebra foundation of Riemann integrals
Let $V$ be the vector space of real functions $f\colon [a,b]\to \mathbb R$ and let $X$ be the set of characteristic (indicatrix) functions of subintervals: $X=\{\mathbb 1_I\colon I\subset [a,b] $ interval $\}$. We define $T\colon X \to \mathbb R$ as $T(\mathbb 1_I) = |... |
H: How do you solve this trig equation, I've tried what I know..
How do I solve this question?
$$2\sin{x} + \cos{x} = 0$$
I tried taking $\cos{x}$ from each side, and dividing through to make $-2\tan{ x }= 0$, but then I got stuck. I probably worked it out incorrectly, can someone please show me how to solve a questio... |
H: Product with exponential converges absolutely and uniformly
Prove that the product $$\prod_{n=1}^\infty \left(1+\frac{z}{n}\right)e^{-z/n}$$ converges absolutely and uniformly on every compact set.
What can I transform this product into? It's $\lim_{k\rightarrow\infty}\prod_{n=1}^k \left(1+\frac{z}{n}\right)e^{-z/n... |
H: Notation and terminology for functions, interpreting $f(y)$
It seems to me there are two different interpretations of a symbol $f(y)$.
I will explain what I mean:
Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example).
Also suppose I have a dependence between $x$ and $y$ which is a... |
H: Possible solution for Sipser 1.63
Sipser's question 1.63:
Let A be an infinite regular language. Prove that A can be split into two
infinite disjoint regular subsets.
Is my solution correct?
Since $A$ is infinite and regular, then the pumping lemma holds. We have a pumping length $p \geq 0$ for this language. We... |
H: Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$
Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$
I've been thin... |
H: Prove that $ f(A) \subseteq B \implies A \subseteq f^{-1}(B) $
Let $ A,B ⊆ \mathbb R \ $ and $f\colon\mathbb R \to \mathbb R$ be a function. Then $ f(A) \subseteq B \implies A \subseteq f^{-1}(B) $.
AI: Take any $a\in A$. Since $f(a)\in f(A)\subseteq B$, we have $a\in f^{-1}(B)$.
You do not need the continuity - on... |
H: The limit of $\ln(1+\ln(2+\ln(3+...+\ln(n)))...)$
Does this limit:
$$\lim_{n\to\infty}\ln(1+\ln(2+\ln(3+...+\ln(n)))...)$$
exist ? And if yes, which value does it have ?
AI: You can get a proof of convergence along these lines: Show by induction on $k$ that, for all $n$,
$$\ln(n+\ln(n+1+\ln(n+2+\cdots+\ln(n+k))))\... |
H: problem with understanding logarithm
Could anyone explain why this is true?$$\log_{a}{b^{\log_{a}c}} = \log_{a}{c}\log_{a}{b}$$
I think it's related to $a^{\log_{a}{b}}=b$ but I can't link it ...
AI: Hint:
$$M\log_{a}{x}=\log_{a}{(x^M)}$$ |
H: Seeking for construction s.t. every intersection contains at least 3 lines
In Euclidean geometry, is there some set of lines in s.t. there are at least 2 intersections, but every intersection contains at least 3 lines, and no lines in the set are parallel?
I tried for a long time to construct this by hand but could... |
H: when convergence in measure implies convergence almost surely
If I have a discrete measure space $(\Omega,\mathcal{A},\mu)$, that is when $\Omega$ is countable and $\mathcal{A}$ is the $\sigma$-field over $\Omega$ containing all the subsets of $\Omega$, then convergence in measure over this measure space implies co... |
H: Geodesics Examples
Can someone provide me exemples of connected Riemannian manifolds containing two points through each there are : (i) infinitely many geodesics (up to reparametrization) and (ii) no geodesics.
Thank you
AI: Ok, so for (i) we can consider $S^{2}$ and for (ii) we can consider $\mathbb{R}^{n}\setm... |
H: Solving Second order ODE with variable coefficients?
$m\ddot{x} + c(x)\dot{x} + k(x)x = 0$
where
$\dot{x} = \dfrac{dx}{dt}, \ddot{x} = \dfrac{d^2x}{dt^2}$ and $k(x), c(x)$ are functions of $x$. I saw some methods to solve variable coefficient ODEs but they had functions of $t$ as coefficients rather than functions ... |
H: A simpler proof of evaluating the limsup of a standard normal sequence
Let $\{X_n\}$ be a sequence of i.i.d. random variables following the standard normal distribution. We need to prove that $$\limsup_n\left(\frac{X_n}{\sqrt{2\log n}}\right)=1$$
This is quite easy to show if you use Stirling's approximation and th... |
H: Integral $\int_{0}^{i}z\sin{z} \operatorname dz$
Find this complex integral
$$I=\int_{0}^{i}z\sin{z}dz=\dfrac{5\pi^2}{96}?$$
where $i^2=-1$
My try: use
$$\sin{z}=\dfrac{e^{iz}-e^{-iz}}{2i}$$
so
$$I=\dfrac{1}{2i}\int_{0}^{i}z(e^{iz}-e^{-iz})dz$$
My idea is true? Thank you
Thank you Christian Blatter help.and I ha... |
H: Solve the ODE $yy'' + (y')^2 = 0$
I am asked in a book to solve the following ODE:
$$y\dfrac{d^2y}{dt^2} + \left(\dfrac{dy}{dt}\right)^2 = 0$$
One solution for the ODE above is $y = 0$.
I will use the following substitution:
$$v = \dfrac{dy}{dt}$$
Therefore, $y''$ will become, in terms of $v$:
$$\dfrac{d^2y}{dt^2} ... |
H: Prove that there is no integer a for which $a^2 - 3a - 19$ is divisible by 289
Prove that there is no integer a for which $a^2 - 3a - 19$ is divisible by 289.
Not got any clue. Please help.
AI: Assume that $a^2 -3a -19 = 289n$ for some integers $a$, and $n$.
Then: $a^2 - 3a -19 - 289n = 0$, taking "delta"
$$\Delta... |
H: Is there a logic for recursion rules of divisibility?
I knew the divisibility rule for 7, but my sir told me that these methods are known as recursion rules for divisibility. My sir also told them for 11, 13,17,19. But is there any logic behind it? Or is it just out of the blues?
AI: Of course there is logic behind... |
H: A vector space of linear transformations that cannot be spanned by $8$ elements.
Let $S$ be a set of matrices all of which represent linear transformation from $\Bbb R^3$ to $\Bbb R^3$. All linearly independent subsets of $S$ with $8$ elements will not span $S$. What is $S$?
AI: As egreg already pointed out, a line... |
H: Question about homomorphism of groups
$G,H$ are groups, $\varphi:G\to H$ is homomorphism.
How do I prove: $$\ker\varphi=\left\{{e_G}\right\} \Leftrightarrow \varphi\;\text{is injective} $$
I have problem with $\Rightarrow$ direction.
Thank you.
AI: Assume $\varphi(x)=\varphi(y)$, we are going to prove that this imp... |
H: No isomorphism between $(\mathbb R,+)$ and $(\mathbb R^*, \times)$
My goal is to disprove the existence of an isomorphism between $(\mathbb R,+)$ and $(\mathbb R^*, \times)$.
I proceeded by contradiction. Suppose $f$ is such a map.
Then $$f(0-0)=f(0)f(-0)=-f(0)^2$$
$$ f(0+0)=f(0)^2$$
$$f(0-0)=f(0+0)=f(0)$$
$$-f(0)^... |
H: How natural is the associativity law, or are there any real world non-associative examples
Are there any interpretations and applications of algebraic structures which are not associative?
AI: What about subtraction? We have $x-(y-z) \neq (x-y)-z$ when $z \neq 0$.
Example from real world ;-): You have 5 apples, and... |
H: Simplify $(6\sqrt{x} + 3\sqrt{y})\cdot(6\sqrt{x} - 3\sqrt{y})$.
This expression $(6\sqrt{x} + 3\sqrt{y})\cdot(6\sqrt{x} - 3\sqrt{y})$ equals: ?
Simplify the expression.
I'm tried multiplying but I'm not getting the right answer.
AI: Just use $a^2 - b^2 = (a+b)(a-b)$ and get the answer as $36x-9y$ |
H: Lemma of Bézout
Let $A$ be a PID. By the Lemma of Bézout I mean the statement that for elements $a_1,\ldots,a_n\in A$ we have $(a_1,\ldots,a_n)=((a_1,\ldots,a_n))$ where $(a_1,\ldots,a_n)$ denotes a greatest common divisor of $a_1,\ldots,a_n$. Is the following proof correct? Proof: as $A$ is a PID we must have $(a_... |
H: Prove that column vectors of a matrix $A $ {span} $(\mathbb{R^m})$
Prove the following statement: Given an $m \times n$ matrix $A$, if $Ax=b$ is consistent for all $b$ then the column vectors of $A$ span $\mathbb R^m$.
AI: If you can solve the system $Ax=b$ for every $b$ then the application $f: \Bbb{R}^n \to \Bbb... |
H: Derivatives of second order
Consider a real function $f$ of one variable. Suppose the second order derivative exists. To find the second order derivative of $f$, I usually derivate $f$ two times. I start with $f$, and derivate to get the function $\frac{d}{dx} f(x)$ of one variable, which I then derivate again. I ... |
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