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H: When proving that there is not rational number $m/n$ equal to $\sqrt{2}$, why does $m$ and $n$ must be not both even?
I've read this on Rudin's Principles of Mathematical Analysis:
1.1 Example We now show that the equation
$$p^2=2$$
is not satisfied by any rational $p$. If there were such a $p$, we could write $p... |
H: Maple: assign derivative to function
This is probably a basic Maple question. I'm trying to introduce $g$ as the derivative of $f$:
Somewhat puzzling, Maple now says $g$ is two times the function $x()$.
I've tried g := x -> diff(f(x),x) and g := diff(f,x), but no luck. Google tells me I can use subs(x=3,g) to eva... |
H: Prove or disprove: $ A^2 = I \Longrightarrow A=I \vee A=-I $
Linear Algebra/ Matrices
A is in a $n\times n$ matrix.
If
$$ A^2 = I $$
does this imply: $A=I \vee A=-I $
Thanks!
AI: No. Here are two counterexamples (with their interpretation in $\Bbb R^2)$:
Symmetry with respect to the $x$ axis: $$\begin{pmatrix}1&... |
H: China Girls Math Olympiad (CGMO) 2002
There are 3n girl students who took part in a summer camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the 3n students had been on duty on the same day exactly once.
(1) When n = 3, is there any arrangement sa... |
H: A measure having no point masses.
What does it mean for a measure to have no point masses? Is this sort if like saying that individual points have measure zero?
AI: Almost. A measure has no point masses if for every point, there is a measurable subset containing it that has measure zero. The difference plays only a... |
H: advanced coin tossing probability
Suppose that a fair coin is tossed until a head is obtained, and that this
entire experiment is then performed a second time. Find the probability that
the second experiment requires more tosses than the first experiment.
AI: Let $X$ and $Y$ represent the numbers of tosses. The prob... |
H: Dead presidents
I need to find out the probability that at least 2 of the presidents of the United States have died on the same day of the year.
I'm tempted to put 100% since it has actually happened (during the same year even!), but I am sure that is not the correct answer.
I have a suspicion that the math for thi... |
H: Show $\lim\limits_{n\to\infty} \frac{2n^2-3}{3n^ 2+2n-1}=\frac23$ Using Formal Definition of Limit
I want to show that $a_n=\frac{2n^2-3}{3n^ 2+2n-1}$ is convergent. So I did the following:
\begin{align*}
\left|a_n-\frac23\right|&=\left|\frac{2n^2-3}{3n^ 2+2n-1}-\frac23\right|\\
&=\left|\frac{-4n-7}{3(3n^2+2n-1)}\r... |
H: How do I write $e^{(-x/2)}$ as a summation?
I am new to power series. I know how to write $e^x$ as a summation, but i do not know how that helps me.
AI: You know how to write $e^x$ as a power series, meaning that you know how to write
$$\exp(y) = e^y = \sum_{n = 0}^\infty \frac{y^n}{n!}$$
You might know that this ... |
H: projection of inner products
Update of question
Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define our inner product to be:
$$
\langle p,q\rangle = p(1)q(1)+p'(1)q'(1)+p''(1)q''(1)+p'''(1)q'''(1).
$$
If we define $\langle\cdot,\cdot\rangle$ on the space of all... |
H: inequality with the Frobenius norm for matrices
Let $A\in M_n$. How can I show that $$\left|{\textrm{Tr}(A)\over\sqrt{n}}\right|\leq \Vert A\Vert_F$$
I tried it using the Cauchy-Schwarz inequality.
AI: Hint: Given $A\in\mathbb{C}^{t \times t}$ with entries, $a_{ij}$, we have $\mathrm{tr}(A) = \sum_{i=1}^n a_{ii}$, ... |
H: Question about 2^mersenne number
We are given that $2^n \equiv 2\ \pmod n $. If $m=2^n -1$, prove that $2^m \equiv 2\ \pmod m$
My first instinct is that we can somehow use fact that $2^n\equiv 1 \ \pmod m$ and use that, but I havent made any progress. Any suggestions?
AI: I would recommend tackling the equivalent $... |
H: IS ASA applicable on triangles on the sphere?
$ASA= \text{Angle-Side-Angle}$
I was wondering if $ASA$ still worked on triangles for the sphere. I have a pretty hard time visualizing triangles on the sphere because I know the sum of their interior angles can be more than $180^{\circ}$, which feels weird to me, since... |
H: Postage Stamp Problem with 3 stamp types
The Baker does not sell individual bear claws, but sells them in boxes
of 6, 9, and 20. Assuming an unlimited supply, what is the largest
number of bear claws that I cannot buy from the baker.
I'm not sure how to attack one of these problems when given three types.
AI... |
H: Prove $2^{(n+1)}>n^2$ by induction
Prove $2^{(n+1)}>n^2$, for all $n \in \mathbb{N}$.
I started by verifying the condition for $1$.
$$P(1):2^{1+1}>1^2$$
$$P(1):4>1$$
That is true.
Then I supposed $P(k)$ true, for some $k \in \mathbb{N}$. Now I've to prove $P(k+1)$:
$$2^{k+1+1}>(k+1)^2$$
$$2\cdot 2^{k+1}>(k+1)^2$... |
H: Properties of relation $R$ on $\mathbb{N} \times \mathbb{N}:\;(a,b)R(c,d) \iff a -c = b -d$
Still doing relation properties exercises, I'm now trying what seems to be a somewhat different type: now the relation is over a cartesian product $\mathbb{N} \times \mathbb{N}$.
I normally have no problems determining if it... |
H: Algebra question from Australia national olympiad 2013
Find all positive integers $n$ for which there are real numbers $x_1, \; x_2, \cdots,\; x_n$ satisfying $$(i) \; \; -1<x_i<1 \; for \; i=1,2, \cdots n$$ $$(ii) \; \; x_1+x_2+ \cdots +x_n=0 \; and$$ $$(iii) \;\; \sqrt{1-x_1^2}+\sqrt{1-x_2^2}+ \cdots +\sqrt{1-x_n... |
H: Condition such that $\langle{S}\rangle=S$
So let $S\subseteq{G}$ where $G$ is a group and $S$ an arbitrary subset. Let $\langle{S}\rangle$ be the subgroup in $G$ generated by $S$. What is the condition such that $\langle{S}\rangle=S$?
My thoughts on this; I'm thinking that $S$ must be a subgroup, or what I mean i... |
H: Taylor's theorem: $f'' + f = 0, f(0) = f'(0) = 0$.
I am having a hard time coming up with a solution to this problem.
Suppose that $f$ is twice differentiable and that $f'' + f = 0$. If $f(0) = f'(0) = 0$, use Taylor's theorem to show that $f = 0$.
The definition of Taylor's theorem we were given uses the Lagrange ... |
H: about the derivative of dirac delta distribution
Consider the delta dirac distribution $\delta (\varphi) = \varphi (0), \varphi \in \mathcal{S}(\mathbb{R}^n)$ (the Schwartz space). I know that $\delta ^{'} (\varphi) = - {\varphi }^{'} (0)$. How can I prove $\delta^{'}$ is not given by a measure, that is , doesn'... |
H: The set of real numbers and power set of the natural numbers
I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?
I have also learnt that there exists no set w... |
H: the max value of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$
What is the max value of the term $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$
If $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6$,
$a,b,c,d \in \mathbb{R}$
I tried to get from the equation to the term but it's too complicated.
Any idea?
Tha... |
H: How do you factor $(10x+24)^2-x^4$?
I tried expanding then decomposition but couldn't find a common factor between two terms
AI: Hint: $$a^2-b^2=(a+b)(a-b)$$
Work out what your $a$ and $b$ should be. |
H: The irrationality of the square root of 2
Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?
AI: This question has been asked before. Please search Math.SE for an answer to your question before asking.
Regardless, yes, there are lo... |
H: Least upper bound and greatest lower bound
Find the $\sup E$ and $\inf E$.
i) $E$=$(0,1]$
ii) $E$=$\{x \in Q : x^2 < 2\}$
i) $\sup E$ = $1$, $\inf E$ = $1$
ii)$\sup E$ = $\sqrt 2$, $\inf E$ = $DNE$
I got these answers using my intuition of sup and inf. Has it lead me astray?
AI: For $i)$, $infE=0$ because it is th... |
H: Inverse theorem on product of two convergent sequences
Suppose I have two sequences, $a_n$ and $b_n$. I know that:
$\lim_{n\to\infty} a_n=1$ and that $\lim_{n\to\infty} a_nb_n=c$.
Does this mean that $\lim_{n\to\infty} b_n$ converges?
If so, by algebra of limits does it mean that $\lim_{n\to\infty} b_n=c$?
AI: Yes... |
H: What's a good reference to study multilinear algebra?
This semester I'm taking a course in linear algebra and now at the end of the course we came to study the tensor product and multilinear algebra in general. I've already studied this theme in the past through Kostrikin's "Linear Algebra and Geometry", but I'm no... |
H: Let $Tx = 1+\log(1+e^x)$. Show that $T$ has no fixed points.
Let $Tx = 1+\log(1+e^x)$. Show that $T$ has no fixed points.
This is what I have:
We say that $T$ has a fixed point if $Tx=x$.
$$Tx = 1+\log(1+e^x) = x$$
$$\log(1+e^x) = x-1$$
$$1+e^x = e^{x-1}$$
$$e^{x-1}-e^x - 1 = 0.$$
How do I argue mathematically th... |
H: Evaluate $\lim_{x \to 0} \frac{1-\cos(\sin(4x))}{\sin^2(\sin(3x))}$ without L'Hospital
$$\lim_{x \to 0} \frac{1-\cos(\sin(4x))}{\sin^2(\sin(3x))}$$
How can I evaluate this limit without using the L'Hospital Rule? I've expanded $\sin(4x)$ as $\sin(2x+2x)$, $\sin(3x) = \sin(2x + x)$, but none of these things worked.
... |
H: Definition for $\lim(s_n)$ and $\limsup(s_n)$
Can someone provide me the definition of a (finite ) $\lim (s_n)$ and how it correlates to the definition of $\limsup(s_n)$?
$\lim(s_n)=+\infty$ if $\forall M>0, \exists N=N(M)\in \Re$ s.t.$\forall n>N $ we have $s_n >M$.
I believe this is the definition for the infin... |
H: Prove that $\nabla\langle Ax,Ax\rangle = 2A^TAx$
Prove that $\nabla\langle Ax,Ax\rangle = 2A^TAx$.
My book uses this property to prove the $2-norm$ of a matrix $A$ is the square root of the spectral radius of $A^TA$. That is
$$||A||_2 = \sqrt{r(A^TA)}$$
where $r(A)$ is the spectral radius. How would one go about p... |
H: Is there integrable function sequence which is uniformly converges to not integrable function?
Is there any example (Riemann) integrable function sequence which is uniformly converges to not integrable function?
AI: This is not possible; if $f_n$ is a sequence of Riemann-integrable functions that converge uniformly... |
H: Find the radius of convergence and interval of convergence
Seems like you are suppose to do the root test to come up with the answer. but the 2x-5 in the numerator concerns me. the (-5) part. The root test says that the series has to have positive terms. With the - 5 in there. It makes me confused.
Please help!
AI... |
H: Help with a trigonometric limit
Find the limit and determine if the function is continuous at the point that is being approached:
$$\lim_{y \to 1}\;\; \mathrm{sec}\;(y\;\mathrm{sec}^2y \;-\;\mathrm{tan}^2y\;-\;1)$$
My try: I just rewrote it so that it reads
$$\lim_{y \to 1}\;(\frac{1}{\mathrm{cos}}) (\frac{y... |
H: Complement of A or B
I have a small general question..
Let's say we have two events $A$ and $B$. Is the probability that $A$ or $B$ will happen, the complement of the event that the complement of $A$ and the complement of $B$ will happen?
I'm sorry if that's hard to understand, I hope it makes sense..
AI: Yes, this... |
H: prove that this operator is not compact
Let $g\in C[0,1]$ be a continuous function and $g\ne 0$. Let $G:C[0,1]\to C[0,1]$ the operator defined by: $G(f)(x)=f(x)g(x)$. I proved that the operator is linear and continuous. I want to prove that $T$ is not a compact operator. But I don't know how. Please help me with th... |
H: using Taylor's formula in a proof
Prove that $1+\frac{1}{n} < e$ for all $n$ in the natural numbers. How does this connect to Taylor's formula? I know that $e^x > 1+x$ for $x>0$, but then where does Taylor's formula come in to play?
AI: I think what you meant to ask is something along the lines of why $\left(1+\fra... |
H: If $f(z)$ is entire, and it is constant in $\{z:|z|\le 1\}$, then $f$ is constant in $\mathbb{C}$?
If $f(z)$ is entire, and it is constant in $\{z:|z|\le 1\}$, then $f$ is constant in $\mathbb{C}$?
(I'm asking this because I need to prove that a function is constant, so I wonder if I'm already done if I showed that... |
H: Extraneous solutions to simple equations
I had an interesting thought during my procrastination: is it legal to take an equation, say
$3 = a * b * c$
and do the following:
$3 = abc$
$0 = abc - 3$
$0 / a = bc - 3/a$
$0 / b = c - 3/a/b$
$0 / c = -3/a/b/c$
$0 = -3/a/b/c$
But this is not true since no q... |
H: Determinant of matrix $A^3 + 2A^2 - A - 5I$ Given the eigenvalues of A
So A is a 3 by 3 matrix with eigenvalues -1, 1, 2. And I have to find the determinant of $$A^3 + 2A^2 - A - 5I$$
Let $u$ be the eigenvector for the eigenvalue -1. Let
$S = A^3 + 2A^2 - A - 5I$ then
$Su = \lambda u$.
$=(A^3 + 2A^2 - A - 5I)u\\
=A... |
H: On why the Vitali Covering Lemma does not apply when the covering collection contains degenerate closed intervals
I believe I have a fundamental misunderstanding of the concept of the Vitali Covering Lemma.
Definition - A closed bounded interval $[c, d]$ is said to be nondegenerate provided $c < d$.
Definition - A ... |
H: Evaluate of $\lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n}$
Evaluate the limit
$$
\lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n}
$$
My Attempt:
$$
\lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n} = \lim_{n\rightarrow \inft... |
H: How Would You Translate This Sentence To Predicate Logic?
"There exists an Apple such that for every person, he loves that apple."
I believe the translation is:
$$\exists x(\forall y((\text{Apple}(x) \wedge \text{Person}(y)) \to \text{Loves}(y,x)))$$
would that be correct?
Thanks!
AI: Close but no cigar. The correc... |
H: Proofs that there is no $f(z)$ such that $\exp f(z) = z$ for all $z \in \Bbb{C}\setminus\{0\}$
When I first learned about this result I was completely stunned that there is no holomorphic function $f(z)$ on $\Bbb{C}\setminus\{0\}$ such that $\exp f(z) = z$. What are some interesting proofs of this? Here are two I k... |
H: Finding points of continuity on piecewise function
For what values of $a$ and $b$ is the function continuous at every $x$?
$$\displaystyle f(x)=\begin{cases}
-1
& \text{if }\;\; x \leq -1\\ ax+b & \text{if }\;\; -1<x<3\\ 13 & \text{if} \;\;\;x \geq3 \end{cases}$$
The answers are: $a=\frac{7}{2}$ and $b=-\f... |
H: How To Simulate Mirrors/Reflection?
If light is hitting a Parabolic Trough defined by $y=x^2$ at a 60 degree angle from vertical so that the effective cross-section of the modified parabola is paramaterized by: x=t, y=t^2, z=tcot(60).
Then how would parallel light rays interact with this new paramaterized shape? W... |
H: Properties of Congruences
For context:
The question/answer is given as follows:
Show by induction that if n is a positive integer, then $4^n ≡ 1 + 3n (\text{mod } 9)$.
For the base case, $4 ≡ 1+3 (\text{mod } 9)$.
For the induction hypothesis, assume that $4n ≡ 1+3n (\text{mod } 9)$ for some positive integer n.
T... |
H: For which values $a$ does the improper integral $\int_0^{\infty}\frac{\ln(1+x^2)}{x^a}dx$ converge
Find the values $a$ s.t. the integral
$$\int_0^{\infty}\frac{\ln(1+x^2)}{x^a}dx$$
converges.
I tried some values of $a$ by programming, it seems that for $a=2$, the integral converges, and for $a=3$, it diverges. But ... |
H: Continuous inverse functions.
I am asking this question because I am having somewhat of a difficult time finding a direct answer searching online.
I know that a function $f: X \rightarrow Y$ is continuous if given an open set $O$ in $Y$, $f^{-1}(O)$ is open in $X$.
Let's say I have an open set in $X$, call it $R$,... |
H: Show that the function $g(x) = x$ can intersect $f$ no more than once.
I was wondering how I can use either Rolle's theorem or the Mean Value theorem to do this question:
Let $f:\mathbb{R}\rightarrow \mathbb{R}$. Suppose that ${f}'(x)> 1$ for all $x$. Show that the function $g(x) = x$ can intersect $f$ no more tha... |
H: Arctan Identity, impact of (-1)
I am slightly unsure of a basic trig identity operation.
Is the below true?
$$
\arctan \Big(\frac{-y+u}{-x+c}\Big) = \arctan \Big(\frac{y-u}{x-c}\Big)
$$
Or is it instead this...?
$$
\arctan \Big(\frac{-y+u}{-x+c}\Big) = -\arctan \Big(\frac{y-u}{x-c}\Big)
$$
Or is it instead somethin... |
H: An equation to map values from one range to another
I need a formula (preferably something I can write with JavaScript, so simple arithmetic) that will map my value that is between 0.5 and 1, to a new value between 0 and 1. My brain is dead, this seems trivial, yet I can't say I ever learned how to do this. I'm s... |
H: Proving that a function is 1-1 to show that it is invertible
I want to prove that $h(x)=x^3 +2x+1$ is a $1-1$ function to show that it is invertible on all of $\mathbb{R}$.
This my attempt: Let $x_1,x_2\in \mathbb{R}$ where $x_1\neq x_2$.
Suppose for contradiction $h(x_1)=h(x_2)$.
Then $h(x_1)=x_1^3 +2x_1+1$ and $... |
H: Pattern of (1201, 2121, 3142, 4253, 5342)
What could be a part of the following set of numbers: $1201, 2121, 3142, 4253, 5342$
$a) 1317$
$b) 2315$
$c) 2573$
$d) 3456$
Differences between the numbers are $920, 1021, 1111, 1089.$ It's not making sense. Increasing then decresing. So difference of difference too won'... |
H: Compact image is compact?
Let $f: X \to Y$ be a function between metric spaces $X, \phi$ and $Y, d$ such that $f^{-1}(U)$ is open in $X$ for every subset $U$ open in $Y$. Prove that if $C$ is a compact subset of $X$, then $f(C)$ is a compact subset of $Y$.
My attempt: Let $C$ be a compact subset in $X, \phi$. Let... |
H: Moment generating function for a gamma distribution
I have a PDF:
$$f_y(y) = \frac{\lambda^n}{\Gamma(n)} (y-n\tau)^{n-1}e^{-\lambda(y-n\tau)}$$
I want to find the moment generating function for it: (I believe I made a mistake somewhere?)
$$\begin{aligned}
M(t) = E[e^{Yt}] &= \int_0^\infty e^{yt} \frac{\lambda^n}{\... |
H: Find $m$ such that the roots of this polynomial are greater than -1.
The polynomial $x^2 + 2mx + 3m+4$.
I know that the discrimate must be greater or equal to zero, otherwise it would have complex roots, and complex numbers are "measureable", don't know how else to explain, against -1. Because complex numbers hav... |
H: inverse functions
If ${h^{-1}}$$(y)$ is the inverse function to $h$, find the values of $(h^{-1})'(y)$ at the points corresponding to $x=0$, $x=1$, and $x=-1$.
I know that $h(x)=x^3+2x+1$ and $h'(x)=3x^2+1$. Generally, I can find the inverse of a function by writing in terms of $x$ and then switching the $y$ and $x... |
H: How find this $\lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{i}{n}\right)^n$
How find this $$\lim_{n\to\infty}\sum_{i=1}^{n}\left(\dfrac{i}{n}\right)^n$$
I think this answer is $\dfrac{e}{e-1}$
and I think this problem have more nice methods,Thank you
AI: For each fixed $x$, Bernoulli's inequality
$$ (1 + h)^{\alpha} ... |
H: Number of simple directed graphs
How many simple directed graphs are there on the vertex set $\{1,\ldots,n\}$?
I know there are $2^\binom{n}{2}$ simple undirected graphs, but I am confused as to where to go on this problem. I believe that it is simply two times this answer because on each graph you now have to cho... |
H: Are closed, properly embedded manifolds of co-dimension 1 in $\mathbb{R}^n$ orientable?
I have been trying to figure this out as it would seem that it should be so. I have been search though, and the only solution seems to treat the compact case with homology beyond what I know. I believe that it is true, but I cou... |
H: Fixed Points: Intermediate Value Theorem
For a function $f:D\rightarrow \mathbb{R}$, a solution of the equation
$\hspace{150pt}$$f(x)=x$, for $x\in D$
is called a fixed point of $f$. A fixed point corresponds to a point at which the graph of the function $f$ intersects the line $y=x$. If $f:[-1,1]\rightarrow \mat... |
H: Limit of $(1+5/n+6/n^2)^n$ when $n$ goes to infinity
Find $$\lim_{n \to \infty} \left(1+\frac{5}{n}+\frac{6}{n^2}\right)^n$$
AI: Note that this factors as $$\displaystyle\lim_{n\to \infty}\left(1+\frac{2}{n}\right)^n\left(1+\frac{3}{n}\right)^n=e^2\cdot e^3=e^5$$
Interestingly, this is the same answer as we would h... |
H: Any continous function satisfies this? (Stone-Weierstrass)
I found this question: Prove that if f in $C(X \times Y)$ then there exists functions.
And I was going to make a comment there but since is too old, I don't belive I'll get an answer.
The only answer in the question, says that you have to use Stone-Weierstr... |
H: Probability - Is my answer ok?
Each item in a computer parts catalogue is given a unique code consisting of two distinct
uppercase letters followed by four distinct digits. For example, the code for a particular
keyboard is XY1702.
a. How many different item codes are available (keeping in mind that repetition of
l... |
H: Integral $\int_{-\infty}^{\infty}\frac{\mathrm dx}{(ax^2+2bx+c)^{\alpha}}$
let $a>0,ac-b^2>0,\alpha>\dfrac{1}{2}$
show that
$$I=\int_{-\infty}^{\infty}\dfrac{\mathrm dx}{(ax^2+2bx+c)^{\alpha}}=\dfrac{(ac-b^2)^{\frac{1}{2}-\alpha}}{a^{1-\alpha}}\dfrac{\Gamma{(\alpha-\dfrac{1}{2})}}{\Gamma{(\alpha)}}\sqrt{\pi}$$
This... |
H: Choosing two sets with k mutual elements
I am struggling with the following question:
We have N balls. We first draw n of them and write their numbers.
We then put them all back and draw another m balls. What is the sample space?
What is the probability of drawing two sets with exactly k elements?
So here's what I... |
H: Prove there is a solution to the equation $f(x)=x$ for all $x\in \mathbb{R}$
Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and that its image $f(\mathbb{R})$ is bounded. Prove there is a solution to the equation $f(x)=x$ for some $x\in \mathbb{R}$.
We want to find a solution to the equation $f(x)... |
H: Probability Question with Expected Values
I have a question as such:
Class A has 45 students in it, and class B has 30 students in it. In class A, every student attends any particular lecture with probability 0.7 independent of the other students. For class B, two thirds of lectures are attended by everyone, with ... |
H: Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
Suppose that the function $f:[a,b]\rightarrow \mathbb{R}$ is continuous. For a natural number $k$, let $x_1,\cdots,x_k$ be points in $[a,b]$. Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{... |
H: When do improper integrals converge?
I am studying for an exam, and I am getting myself confused about how to tell if an improper integral is convergent or not. I know that if a function $f$ is unbounded on $[a,b]$ then $f$ is not integrable on $[a,b]$. However, when you consider improper integrals, this rule is th... |
H: Trying to show $\mathbb{Q}(\cos(2\pi/n)+\sin(2\pi/n)i)\supset \mathbb Q(\cos(2\pi/n),\sin(2\pi/n )i)$
I am not sure if the following is true, $\mathbb{Q}(\cos(2\pi/n)+\sin(2\pi/n)i)\supset \mathbb Q(\cos(2\pi/n),\sin(2\pi/n )i)$.
My attempt is to look at powers of $\cos(2\pi/n)+\sin(2\pi/n)i$ and to see if I can ob... |
H: Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$
How can I find a closed form for the following sum?
$$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$
($H_n=\sum_{k=1}^n\frac{1}{k}$).
AI: EDITED. Some simplifications were made.
Here is a solution.
1. Basic facts on the dilogarithm. Let $\math... |
H: Gluing maps on closed subspaces
$\textbf{PROBLEM}$
Let $X$ and $W$ be topological spaces and suppose $W = A \cup B$. with
$A,B$ closed subsets of $W$. Suppose $f: A \to X$ and $g: B \to X$ are
continuous functions such that $f(w) = g(w) \; \; \forall w \in A \cap
B $. Then $h : W \to X $ defined by
$$ h(w) = ... |
H: Finding the integral of $x^2 \tan^{-1}x$
I am given the following integral:
$\int x^2\tan^{-1}x\space dx$
I have tried to solve it the following way, using integration by parts and substitution:
$$\int x^2\tan^{-1}x\space dx = \frac{x^3}{3}\tan^{-1}x - \frac{1}{3}\int\frac{x^3}{1+x^2}\space dx$$
Now, focusing solel... |
H: if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$
if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $
then $(A_n+B_n) \longrightarrow \infty$.
How do you prove it?
AI: Assume not. Than there is a $\text{M}$ such that $\forall n \in \mathbb{N},... |
H: The boundary of an $n$-manifold is an $n-1$-manifold
The following problem is from the book "Introduction to topological manifolds".
Suppose $M$ is an $n$-dimensional manifold with boundary.
Show that the boundary of $M$ is an $(n-1)$-dimensional manifold (without boundary) when endowed with the subspace topology.
... |
H: Bigger than and equals rewritten in normal distribution question
So it is correct to say that $P(482\le x \le 510) = P(x \le 510) - P(x < 482)$ where x is a random variable in a normal distribution? Thanks!
AI: In general, $P(A\setminus B)=P(A)-P(B)$ whenever $B\subseteq A$. Now if $a<b$, then
$$
P(a\leq X\leq b)=P... |
H: Path connectedness is a topological invariant?
$\textbf{PROBLEM}$
Path-connectedness is a topological invariant
MY try: we can show that the image of a path connected space $X$ under a continuous mapping is path connected
Suppose $X$ is path connected space. let $\gamma : X \to Y$ be continuous bijective map. Tak... |
H: Implicit derivative - Graphic
Find $\left(\Large\frac{dy}{dx}\right)_{x=1}$ and $\left(\Large\frac{d^2y}{dx^2}\right)_{x=1}$, if
$$x^2 -2xy +y^2 +x+y -2 = 0$$
Using the obtained results, show aproximately the proportions of the given curve in the neighbourhood of $x=1$
Obtained Results:
$\left(\Large\frac{dy}{dx}\r... |
H: A Veronese map is a morphism?
My question is really simple, I'm beginning to study Algebraic Geometry and I'm still struggling to get the basic concepts.
I would like to know if a Veronese map $v_{n,d}:\mathbb P^n\to \mathbb P^N$ is a morphism.
Thanks
AI: A morhism $\varphi:X \to Y$ between two projective varietie... |
H: Why is this true:$ \nabla \cdot (\vec V \otimes \vec V)=(\vec V\cdot \nabla ) \vec V +\vec V(\nabla\cdot \vec V) \;\;? $
Can someone help me why the following is true: $$ \nabla \cdot (\vec V \otimes \vec V)=(\vec V\cdot \nabla ) \vec V +\vec V(\nabla\cdot \vec V) \;\;? $$
I've thought of the following relation t... |
H: How to show $\langle{[a]}\rangle=\mathbb{Z}_n$ iff $(a,n)=1$
If I want to prove the above statement, I need two directions.
$\Leftarrow$ Let $(a,n)=1.$ Then $ax+ny=1, x,y\in\mathbb{Z}$ Thus, $1-ax=ny \rightarrow \frac{1-ax}{n}=y$. Therefore, $ax\equiv{1}\mod{n}$. Under addition, then $[ax]n=[n], $ but since $[... |
H: Prove/Disprove: $vwvw=vvww$ iff $\{v\}^*\{w\}^*=\{vw\}^*$
Let $\Sigma$ be an alphabet and $v,w\in \Sigma^*$.
I'm trying to prove that:
$$vwvw=vvww\quad\text{iff}\quad\{v\}^*\{w\}^*=\{vw\}^*.$$
I tried to do it by induction, with no success. Any help will be greatly appreciated.
AI: Let us assume (Edit: assumption ... |
H: Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$?
I'm having a hard time developing abstract thinking to solve problems regarding a relation's properties. I've spend quite an absurd amount of time on this one, but I think I finally... |
H: $\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$
Solve $\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$,
such that $x_0 = 0, x_{20} = 5$ and $x_{k+1} - x_k = u_k$.
I think I know how to solve problems like these recursively, but I don't know how I should attack this specific problem as it invo... |
H: Formula simplification when working with parameters
There's a pragraph in my books which states the following;
$3x_2 - 5 \Lambda = 6$
Equal to
$x_2 = \frac{5}{3} \Lambda + 2$
So $5/3$ and $6/3$, which makes sense because we stopped multiplying $x_2$ by 3, but where did the minus sign go? Why did it become a plu... |
H: Proving that limits exist and the derivative is continuous
I need to check whether the following functions are differentiable at 0, and if so if the derivative is continuous at 0.
$f(x) = x^2\sin(1/x)$ if $x\not = 0$, $f(0) = 0$
$f(x) = (1/x)\sin(x^2)$ if $x\not = 0$, $f(0) = 0$
In (1), $\lim_{x\to 0} {x^2\sin(1/... |
H: How to prove that every simple left $R$-module is isomorphic to a minimal left ideal of $R$
We know that:
$T$ is a simple left $R$-module $\Longleftrightarrow T\cong R/M$, where $M$ is a maximal left ideal of $R$.
So please tell me how to prove that every simple left $R$-module is isomorphic to a minimal left id... |
H: How prove this $(p-1)!\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\right)\equiv 0\pmod{p^2}$
Show that
$$(p-1)!\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{p-1}\right)\equiv 0\pmod{p^2}.$$
Maybe use this
$$\dfrac{1}{k}+\dfrac{1}{p-k}=\dfrac{p}{k(p-k)}$$
and then I can't. Can you help me to prove i... |
H: Number of assigning a label to m ordered items, from n labels, so that any later item doesn't get a lower label
I have a set of $m$ points $(p_1,p_2,\ldots,p_m)$, that are to be taken two at a time as one of the edges of a triangle. It is given that consecutive edges are formed by $p_1 - p_2$ and $p_2-p_3$, etc - t... |
H: Finding the closure of $\mathbb{Z}$ and $\mathbb{Q}$ in $\mathbb{R}$
Let be $A$ subset of a metric space $(X,d)$
Definiton. Point $x\in X$ is adherent point (it can also have any other definition but sorry and forgive me if I wrong) of set $A$ if $$T(x,r)\cap A\neq \phi, $$ for all r>0.
Set of all adherent points o... |
H: $X \times Y$ path connected implies $X$ and $Y$ path connected.
$$\textbf{PROBLEM}$$
Suppose $X$ and $Y$ are topological spaces. If $X \times Y$ is path
connected, then $X$ and $Y$ are path connected.
$$\textbf{ATTEMPT}$$
IF $X \times Y$ is path connected, then we can take a path $\gamma$ from $[0,1]$ to $X \ti... |
H: Show that inclusion-exclusion principle applies to finding max.
Show, that : $\text{max} \{x_1,x_2,...,x_n\} = x_1+x_2+...+x_n-\text{min}\{x_1,x_2\}-...-\text{min}\{x_{n-1},x_n\}+\text{min}\{x_1,x_2,x_3\}+...\pm \text{min}\{x_1,x_2,...,x_n\}$
In a way I'm supposed to prove, that the inclusion-exclusion principle so... |
H: Infinite Series $\sum\limits_{n=1}^\infty\frac{H_n}{q^n}$
How can I prove that
$$\sum_{n=1}^{\infty}\frac{H_n}{q^n}=\frac{q}{q-1}\log(\frac{q}{q-1})$$
($H_n=\sum_{k=1}^n\frac{1}{k}$, $|q|>1$).
AI: Hints:
Look at the partial sums of $\sum_{n=1}^{\infty}\frac{H_n}{q^n}$ as fractions.
Your equation fail if $|q| > 1... |
H: sufficient condition for a polynomial to have roots in $[0,1]$
Question is to check :
which of the following is sufficient condition for a polynomial
$f(x)=a_0 +a_1x+a_2x^2+\dots +a_nx^n\in \mathbb{R}[x] $ to have a root in $[0,1]$.
$a_0 <0$ and $a_0+a_1+a_2+\dots +a_n >0$
$a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dots +... |
H: Is the disjoint union of two disjoint subsets always homeomorphic to their union?
Suppose that $A$ and $B$ are disjoint subsets of some topological space, equip $A,B$, and $A\cup B$ with the subspace topology. Is it always true that $A\cup B$ is homeomorphic to $A\sqcup B$? My guess is no, consider $X=\{a,b,c\}$ wi... |
H: calculating various cuts of a circle
im trying to find some sort of formula to calculate lines within a circle.
I need to find the length of the various lines within the circle from which I only know the diameter. Is there some sort of formula that uses the dropoff of a circles side from the center?
Thanks,
Martij... |
H: square formed by the quadratic equations.
Question:A Square is Formed By The Straight Lines $x^2-8 x+12 = 0$ And $ y^2-14y+45 = 0$. What are the coordinates?
How do I solve it? Providing a basic intiution will do the job.
Also the graphs of the equations have been referred to as straight but shouldn't they be para... |
H: left- and right limit calculation (decide positive or negative absolute value)
I have this assignment:
$$\lim_{x \to 0}\frac{|x-2|}{x-2}$$
I could probably do the calculation by just setting x to zero:
$\frac{|0-2|}{0-2}$ = $\frac{|-2|}{-2}$ = $\frac{2}{-2}$ = -1, which is correct (there is just one limit).
But, be... |
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