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H: Finding points of intersection between a line and a curve
I know this is easy but I am unsure about how to answer a multivariable-calculus question as a volunteer tutor, I left a couple of students with a method but a lot of others needed help today so I had to move on, I am hoping someone can confirm my method or ... |
H: How can I solve this Initial Value Problem using the Euler method?
My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using the Euler method. I have a given Interval of $[1,2]$ and a given step size $h$ of $h=0.1$
My Approach was: ... |
H: Probability of an event happening N or more times
I need to determine the probability of an event happening N or more times over M iterations. I know the probability of the event in question happening and its likelihood does not change over time.
For example, say I am rolling a six-sided die and I want to know what... |
H: Please explain me how can I show that the last limit does not exist?
I posted my answer with its question.
But how can I show that the last limit -on the second page- does not exist?
That is,
$\mathbf{\lim_{(h_1, h_2)\to (0,0)}\frac{\sqrt {|h_1.h_2|}}{\sqrt {h_1^2,+ h_2^2}}}$ does not exist. How?
11.2.3. Pro... |
H: What is the cardinality of $X$?
Let $X=\{x\in D^\mathfrak{c}:
0<|\{\xi<\mathfrak{c}:x(\xi)=1\}|\le\omega_1\}$, where $D=\{0,1\}$.
What is the cardinality of $X$? I think it is $\mathfrak c$, however I'm not sure. Also I don't know how to prove it. Thanks for your help.
AI: $|X|=\mathfrak{c}^{\omega_1}=(2^\omega)^{\... |
H: Sum of the first n Prime numbers
Let $P_i$ denote the i-th prime number. Is there any formula for expressing
$$S= \sum_{i=1}^m P_i.$$
We know that there are around $\frac{P_m}{\ln(P_m)}$ prime numbers less than or equal to $P_m$. So, we have:
$$S\le m\times P_m\le \frac{P_m^2}{\ln(P_m)}.$$
I want to know, if there... |
H: Field homomorphism respects arbitrary arithmetic expression
A field homomorphism $f:A \to B$ respects the fields' binary operations, in the sense that $f(x + y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$.
When you have an explicit expression like $expr = x^3 + 15x - 32/y$, it's easy to verify using the normal order of o... |
H: Interesting Differentiation Technique
@HansEngler Left the following response to this question regarding "bad math" that works,
Here's another classical freshman calculus example:
Find $\frac{d}{dx}x^x$.
Alice says "this is like $\frac{d}{dx}x^n = nx^{n-1}$, so the answer is $x x^{x-1} = x^x$."
Bob says "no, ... |
H: Zorn's lemma exercise
Exercise: Let $L$ be a lattice in which every chain has an upper bound. Prove that $L$ has a unique maximal element.
I said:
p1) Suppose $L$ has two maximal elements. (Proof by contradiction is where I'm headed)
p2) Since $L$ is a lattice, then every two elements in the set have a least up... |
H: Two numbers are randomly selected from the set {0,1,2,3,4,5,6,7} without replacement
What is the probability that the sum of the two numbers is 7?
[what i did]
0+7
1+6
2+5
4+3
my ans: 4/8
Is this right or wrong? Does without replacement mean we can't use the same number twice? IE: 4+2 AND 2+4?
Thanks for the he... |
H: Why are the limits of integration of $\Gamma (z)$ the way they are?
$$\Gamma (z)=\int_0^\infty t^{z-1}e^{-t} dt$$ To make things easier, let's call this $G(z)$. If we take the derivative with respect to $t$ of both sides, we get $\frac {dG(z)}{dt}=\frac {d}{dt} [\int_0^\infty t^{z-1}e^{-t} dt]=0$ . The reason why I... |
H: How determinate the torsion subgroup and the normal maximal $\pi$-subgroup of $\mathbb{R}/\mathbb{Z}$?
Let $\mathbb{R}$ and $\mathbb{Z}$ the real numbers and the integers, respectively. Consider $G =\mathbb{R}/\mathbb{Z}$. Find
$O_{\pi}(G)$ - ie, the (unique) normal maximal $\pi$-subgroup of $G$;
$T(G)$ - ie, the t... |
H: Can different functions of a random variable be independent?
Let $X$ be a random variable. $f$ and $g$ are two non-trivial (e.g. non-constant) measurable functions defined on the range of $X$. Can $f(X)$ and $g(X)$ be independent? Thanks!
AI: $f$ and $g$ can be independent if they are simple functions that attain f... |
H: Extension of the chain rule to multiple variables?
Say there is a function $F(x,y,z)$, and $x$, $y$, and $z$ are all functions of a single variable $t$. That means that $F$ is by proxy a function of a single variable $t$. Is there an extension of the chain rule to multiple variables that would make it possible to s... |
H: Why do we multiply in tree diagrams?
In probability, we always lay out the events through tree to see what depends on what. Then we were taught to "multiply" through that branch to get the probability of that event.
Why do we "multiply"? I've noticed we have the same sort of rule in calculus too (chain rule)
EDIT... |
H: verifying differential equation solution with sage
I solved the linear ODE system of equations:
\begin{equation}
x' = \begin{pmatrix}3&0&4\\0&2&0\\0&0&-3\end{pmatrix}x
\end{equation}
Skipping the details I got the following eigenpairs:
\begin{align}
\lambda_0=3, x_0=\begin{pmatrix}1\\0\\0\end{pmatrix}&\\
\lambda_1=... |
H: What's the mean of all real numbers?
At first, I had thought the average must be zero, since for every positive number there's an equal magnitude negative number to cancel out the positive number's effect on the average, leaving only zero to set the average.
But you can make a similar argument about any number, for... |
H: For a convex function, the average value lies between $f((a+b)/2)$ and $(f(a) + f(b))/2$
Suppose that $f\in C^2$, $f''(x)\geq 0$ $\,\,\,\forall x \in [a,b]$. I want to show that $$\frac{1}{2}(b-a)(f(a)+f(b))\leq \int_a^bf(t)\,dt\leq (b-a)f\left(\frac{a+b}{2}\right).$$If we divide by $b-a$, we see that the left term... |
H: Trigonometry Functions/equations
Given that $\sin A=\dfrac{8}{17},\;$ find the possible values of $\cos A$ and $\cot A$.
Can someone please explain this question? I' new to the topic, and I'm very unsure as to how its done.thanks
AI: Well, $\sin^2A+\cos^2A=1$; this is the Pythagorean theorem.
So you know two possib... |
H: Fourteen indviduals are to be divided into comittee of three, four and seven members
How many ways can the individuals be grouped?
$${14\choose3}\cdot{11\choose4}\cdot{7\choose7}$$
Is this the right way to approach this?
AI: Yes, your answer is fine. It reduces to the multinomial coefficient
$$\binom{14}{3,4,7}=\fr... |
H: Question on induction-1 is the least positive integer
Question on induction
prove: 1 is the least positive integer.
proof:
Let $A=\left\{x\geq 1\left|x\in Z^+\right.\right\}$, and then $1\in A$, if positive integer $n\in A$, then $n\geq 1$,
Since $1\leq n<n+1$, so $n+1\in A$. By induction, we have $A=Z^+$. So for... |
H: Given that $\cos x =-3/4$ and $90^\circ
Given that $\;\cos x =-\frac{3}{4}\,$ and $\,90^\circ<x<180^\circ,\,$ find $\,\tan x\,$ and $\,\csc x.$
This question is quite unusual from the rest of the questions in the chapter, can someone please explain how this question is solved? I tried Pythagorean Theorem, but no lu... |
H: Limit of ${x^{x^x}}$ as $x\to 0^+$
Can you please explain why
\begin{align*}
\lim_{x\to 0^+}{x^{x^x}} &= 0
\end{align*}
AI: Ley $y=x^{x}$. Then taking log on both sides, $\ln y=x\ln x$. Taking $x\to0^+$, we can use L'Hopital on $\displaystyle{\frac{\ln x}{\frac{1}{x}}}$, as both the numerator and denominator approa... |
H: Given that $\csc A=-\frac{5}{2}$ and $\sin A< 0$, find $\cot A$
Trig question, Given that $\csc A=-\frac{5}{2}$ and $\sin A< 0$, find $\cot A$
Ive done this so far, although my answer is abit off, where the correct answer is $-\frac{1}{2}\cdot\sqrt{21}$ while my answer is missing the negative. Can you please spot t... |
H: Evaluating $\mathbf{\lim_{(h_1, h_2)\to (0,0)}\frac{\sqrt {|h_1.h_2|}}{\sqrt {h_1^2+ h_2^2}}}$
I was looking at Peter Tamaroff's answer to this question. The question was to show that the following limit did not exist: $$\mathbf{\lim_{(h_1, h_2)\to (0,0)}\frac{\sqrt {|h_1.h_2|}}{\sqrt {h_1^2+ h_2^2}}}$$
In the answ... |
H: Viewing $R/I$ as $R/I$-module
While doing exercise 4 of Chapter 6 in Atiyah & Macdonald's Introduction to Commutative Algebra, I got stuck at this step:
I have shown that $R/I$ is a Noetherian $R$-module. Here $R$ is a commutative ring with $1$ and $I$ is some ideal of $R$. How can I (no pun intended) conclude
f... |
H: Basic probability confusion
A machine has six switches. The probability that any particular switch works properly is $0.98$. Assuming independent operation of the switches, calculate the probability that at least one switch fails to work properly.
Why isn't it this: P(not working) = $1 - 0.98 = 0.02$
There are six ... |
H: Convergence of Cantor series
Show that for every nonnegative real number $x$, there is a series (called Cantor series) with $$x=\sum_{k=1}^\infty\frac{c_k}{k!}$$ where $c_k$ are integers such that $0 \leq c_{k+1} \leq k$ for all $k \in \mathbb{N}\setminus\{0\}$
Note: By defining $c_1$ to be the largest integer $\le... |
H: Topologies coinciding at a point or a set.
Consider a set equiped with two topologies. What does it mean to say that the two topologies coincide at a point in the set? Is it meaningful to talk about this concept in general. Is it meaningful in the context of metric spaces?
p.s. I know what the relative topology is ... |
H: Ratio Test and L = 1
Consider the series $$\sum\limits_{n=1}^\infty\frac{(2n)!}{a^n(n!)^2}$$ with $a > 0$. Determine if the series converges for:
i) a > 4
ii) 0 < a < 4
iii) a = 4
For i) and ii), I will use the ratio test, $$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\frac{4}{a}$$
So, if $a > 4$, the series ... |
H: How can I use the Heun's method to solve this first order Initial Value Problem?
My Problem is this given Initial Value Problem: $$y^{\prime}=\frac{3x-2y}{x}\quad y(1)=0$$ I am looking for a way to solve this problem using Heun's method. I have a given Interval of $[1,2]$ and a given step size of $h=0.1$
The exampl... |
H: Holomorphic function with zero derivative is constant on an open connected set
I was wondering about this fact, as I do not know how to prove it correctly. I tried with Cauchy-Riemann, but since they are PDE's I found it hard to show that this is the only thing that can cause this zero derivative
AI: By integration... |
H: Random vector with uniform distribution.
Let $(X,Y)$ be a random vector with uniform distribution at $0 \leq x \leq 1$, $x \leq y \leq x+h$ with $0<h<1$.
Find $E(X)$ and $E(XY)$.
What i did:
(1) Find densities:
$f_X(x) = \left\{
\begin{array}{lr}
1 & : 0 \leq x \leq 1 \\
0 & : otherwise
\e... |
H: If $\tan A=2$, find the possible values of $\csc A$
If $\tan A=2$, find the possible values of $\csc A$.
Can someone please show me thorough steps as to how to do this question?
AI: The $\cot(A) = 1/2$, then use $\csc^{2}(A) = 1+ \cot^{2}(A)$ to get $\csc(A)= \pm \sqrt{\dfrac{5}{4}}$. Depending on increasing level ... |
H: a basic problem about lagrange multipliers
Find the max and min values of the function $f(x,y)=x^2+y^2$ under the restriction $g(x,y)=\frac{x^2}{2}+y^2-1=0
$
Note that we can use Lagrange Multipliers Theorem , since $grad(g(p))\ne 0$ $\forall p$ in the ellipse.
Then we put the system of equations $$
\left( {2x,2y} ... |
H: Prove that $\cot(A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B}$
The question is:
Prove that:
$$ \cot(A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B} $$
I have tried expanding it as $\dfrac{\cos(A+B)}{\sin(A+B)}$ and $\dfrac{1}{\tan(A+B)}$.
AI: Witout using the identity for $\tan(A+B)$:
$$\cot(A+B)=\frac{\cos(A+B)}{\sin(A... |
H: a problem using Lagrange multipliers
Prove that $
\frac{{n!}}
{{n^{\frac{n}
{2}} }}
$ is the max of the function $
f\left( x \right) = \prod\limits_{i = 1}^n {x_i }
$ under the restriction $
g\left( x \right) = \sum\limits_{i = 1}^n {\frac{{x_i ^2 }}
{{i^2 }} - 1 = 0}
$.
Well Using LaGrange multiplier I have the... |
H: Unknown symbol '#' in set
I am reading a text on Complexity theory. There is a set whose notation I cannot understand:
"Let $\sum$ = {0,1,#}"
From the context, and given that the book is used computer science courses, it seems like $\sum$ is the set which contains all combinations of bit strings up to length '#'.
... |
H: "Nice" application of the fundamental group
I'm looking for an example of a topological result which is easy to prove using the fundamental group, but hard or impossible elementarily.
First I thought about something like $\mathbb{R}^2\not\cong\mathbb{R}^n$ for $n>2$, but this should be possible to show without usin... |
H: Black and white balls in the urn. What is the probability of...?
There are $100$ balls in the urn - $50$ black and $50$ white. A person randomly chooses a batch of $50$ balls and replaces it with a batch of inversely colored balls (i.e. for example a batch of $34$ black and $16$ white balls would be replaced by a b... |
H: Expected number of shots for a game to end
A basketball player plays a shooting game. He gets +1 point if he scores a basket and -2 points if he misses. He starts with 0 points. The game ends when the player reaches +10 or -10. What is the expected number of shots taken for a game to end, given a player scores a ba... |
H: I solved the question. But I am asking a little bit. $\det\big(D(f\circ g)(a)\big)=?$
$\color{red}{\text{b)}}$ If $n=m$ and $f:B_r\big(g(a)\big)\to\mathbb{R}^n$ is differentiable at $g(a)$, prove that $$\det\big(D(f\circ g)(a)\big)=\det\big(Df(g(a))\big)\det\big(Dg(a)\big)$$ Solution: $$D(f\circ g)(a)=Df(g(a))Dg(a... |
H: Parameter integrals - show continuity
One little question concerning the proof of
Let $I=[a,b], D=[c,d]$ compact ranges and $f\in C^0(I\times D)$ with values in $\mathbb{R}$ or $\mathbb{C}$. Then
$$
F(t):=\int_a^b f(x,t)\, dx
$$
is continious for all $t\in D$.
Proof: $I\times D$ is compact in $\mathbb{R... |
H: Diameter of finite set of points is equal to diameter of its convex hull
Let $M\subset \mathbb{R}^2$ be a finite set of points, $\operatorname{C}(M)$ the convex hull of M and
$$\operatorname{diam}(M) = \sup_{x,y\in M}\|x-y\|_2$$
be the diameter of $M$
What I want to show now is, that it holds
$$\operatorname{diam}... |
H: Use the compound angle identity to prove $\sin(90^\circ +A)=\cos A$
Use the compound angle results to prove
$$\sin(90+A)=\cos A.$$
I know this is rather easy question, but I cant wrap my head around it. So can you please show me how to do it. thanks
AI: I'm not sure what you call "compound angle results" to, but we... |
H: proving a sum of binomial coefficients
How can i prove that $\displaystyle\sum_{k=0}^{n}{2n\choose 2k}=2^{2n-1}$
I tried using induction and pascal's identity but it didn't help me.
AI: We know:$$ (1+x)^{2n}=\sum_{j=0}^{2n}{2n \choose j}x^j$$
Now put $x=1$ and and $x=-1$ in turn and add,
$\displaystyle 2^{2n}=(1+... |
H: Can I change the order of two terms when factoring: $x^2(x^2-4-3x)$ to $x^2(x^2-3x-4)$?
I'm doing homework and I'm stuck on this assignment:
$$x^4 - 4x^2 - 3x^3$$
I figured this would equal
$$x^2(x^2-4-3x)$$
Now I know if I would change the order to
$$x^2(x^2-3x-4)$$
I can factorise it again, but my question is if ... |
H: Solving the multiple integral $\int \int_D \sqrt{x^2+y^2+3}dx dy$, D:{$1\le x^2+y^2\le 4, y\le \sqrt3x, y\ge \frac x{\sqrt 3}$}
$\iint_D \sqrt{x^2+y^2+3}\, dx dy$, $D=\left\{(x,y) \in \mathbb{R}^2 \mid 1 \leq x^2+y^2\le 4, y\le \sqrt3x, y\ge \frac x{\sqrt 3} \right\}$.
So I've started by drawing two circles, One wi... |
H: Simple groups of order 300
I am aware of No simple group of order $300$.
It is said that "there would be 6 Sylow 5-groups, one of which will have an index of 6", but why does one have index 6? If we write $|G|=p^k m$ where $p\nmid m$, then by Lagrange's theorem any $p$-Sylowgroup has index $m$, which in this case i... |
H: Does $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge?
Sorry for the oversimplified question, but does the series $\sum_{n=1}^{\infty} \frac{1}{n+1}$ converge? The ratio test of it gives the result of "1".
Thanks a lot.
AI: If the ratio test gives $1$ then we are in a situation where we can't conclude wether the series... |
H: DFT of binomial coefficients
I want to compute the DFT of a vector whos entries are binomial coefficients, i.e.
$$(v)_k = \binom{N-1}{k},$$
where $k$ runs from 0 to $N-1$.
I use the DFT matrix
$$F_{k,j} := \frac{1}{\sqrt{N}}e^{-\frac{2\pi i k j}{N}}.$$
So I want to compute
$$(Fv)_\ell = \frac{1}{\sqrt{N}}\sum_{k=0}... |
H: Chain rule question
Use chain rule to calculate $\frac{\partial f}{\partial u}$ and $\frac{\partial f}{\partial v}$ when:
$f(x,y) = e^{xy}, x(u,v) = u^2 + v^2, y(u,v) = u^2 - v^2$
So I said:
$\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial f}... |
H: Set contains a proper infinite subset
Consider that we define an "infinite" set as one which contains some proper subset that is in bijection with itself. Now suppose that $B$ is a set and $A \subset B$ is an infinite proper subset of $B$. Can we prove using just that definition and elementary facts about functions... |
H: Proving a Lambert series identity from Ramanujan's Collected Papers
While studying Ramanujan's Collected Papers I came across an identity $$q(1 + q + q^{3} + q^{6} + \cdots)^{8} = \frac{1^{3}q}{1 - q^{2}} + \frac{2^{3}q^{2}}{1 - q^{4}} + \frac{3^{3}q^{3}}{1 - q^{6}} + \cdots$$ which I am unable to establish (althou... |
H: prove：$\left|x+\frac{1}{x}\right|\geq 2$
prove：
$\left|x+\frac{1}{x}\right|\geq 2$
Can I just use
$\left|\left(\sqrt{x}\right)^2+\left(\frac{1}{\sqrt{x}}\right)^2\right|\geq 2$
and
$\left|\left(\sqrt{-x}\right)^2+\left(\frac{1}{\sqrt{-x}}\right)^2\right|\geq 2$?
AI: Hint: Prove that $$\left(x+\frac{1}{x}\right)^2 ... |
H: Descartes' theorem: Find the midpoint of the $4$th circle
I'm thinking about Descartes' theorem "Wikipedia".
I understood how to find the radius with algebra. Now I'm trying to use ruler and compass to find the midpoint of the 4th circle.
I thought about using the midpoints of $k_1-k_3$ to get a triangle and the ce... |
H: Using simple linear algebra for encryption?
e.g.
the character $a = 97$ (it's computer decimal format, commonly known)
and then using a pattern/key like $y = 31 x + 5$ to get $3012$ (substitute $97$ into $x, y$ is now the encrypted code).
How easy/hard would it be for someone to crack a code like this? 3012 3197 3... |
H: An exercise in Fulton's Algebraic Curves
I'm doing Fulton's Algebraic Curves [Ex 1.34]
Let $R$ be a UFD. $(a)$ Show that a monic polynomial of degree two or three in $R[X]$ is irreducible if and only if it has no roots in $R$. $(b)$ The polynomial $X^2 - a \in R[X]$ is irreducible if and only if $a$ is not a squar... |
H: prove that $f(t)$ is orthogonal to $f'(t)$ for som all $t \in I$
Suppose that $I$ is nonempty open interval and that $f: I \to \Bbb R^m$ is differentiable on $I$
If $f(I) \subseteq \partial B_r(0) $ for some fixed $r>0$, prove that $f(t)$ is orthogonal to $f'(t)$ for som all $t \in I$
I guess I need to use the cha... |
H: Why the sum of the squares of the roots of the $n$th Hermite polynomial is equal to $n(n-1)/2$?
How to prove that the sum of the squares of the roots of the $n$th Hermite polynomial is $\frac{n(n-1)}{2}$?
I tried with Vieta formulas, but it's hard. I appreciate a proof or reference to it.
An idea is to use the de... |
H: Can we check whether a Cantor set is self-similar or not?
Given a Cantor set $C$ on the real line, do we have some ways to determine whether it is self-similar or not? In particular, how can we check that $C$ is not self-similar?
Edited:
Definition: Let $\{f_i\}_i$ be a family of contraction maps, i.e. $|f_i(x)−f_... |
H: Möbius transform calculation, over an annulus
I started learning about Möbius transformations in my Complex Analysis textbook. This question appeared as an exercise (no solutions are provided, sadly):
Let's say you have a Möbius transform that maps the annulus $r<|z|<1$ to a region bounded by two circles (for the s... |
H: number of binary sets - combinatorics
Just ran into this question:
let $f(n,m)$ be the number of binary strings where there are at most $n$ 1's and at most $m$ 0's.
the empty string also counts as a string.
show that $f(n,m)=\binom{n+m+2}{n+1}-1$.
thanks in advance,
Yaron.
AI: This is a pretty direct proof. One wo... |
H: The variational derivative
I'm reading about variational derivative from the following pdf-file:
http://www.cs.iastate.edu/~cs577/handouts/variational-problems.pdf
On the very first page, the author takes the partial derivative of $J(y_1, ..., y_n)$ w.r.t $y_k$ on formula $(1)$ and gets the formula $(2)$ from it. C... |
H: Proving a language is not regular using pumping lemma
I had an exam today and the professor gave us the following problem:
Let $L = \{a^nb^m : n|2m \}$. Prove that $L$ is not regular.
Ok this sounds easy. Here is my solution: Assume opposite -- $L$ is regular. Then by the pumping lemma there exist decomposition $xy... |
H: Interpolating a linear transformation
I'm experimenting with some rudimentary ideas for data encryption (I've never formally taken a cryptology class). An idea that I had for an encryption was to use matrices. So I treat a data set as vector and apply a matrix (non-singular of course) to the vector.
So my question ... |
H: Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$
Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$
example1: $|x-1|+|x-2|\geq 1$
my solution：(substitution)
$x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$
square,
$t^2-2t+1\geq 1-2|t|+t^2,\text{Since} -t\leq -|t|,$
so proved.
question1 : Is my proof right? Alternatives... |
H: Finding an inner product
Question:
Given 2 vector spaces $U=sp(1,1), W=sp(2,0)$.
How do I find an inner product in $\Bbb R^2$ s.t. $U=W^{+}$ (orthogonal)
I would love an explanation for the algorithm really, more than this specific question.
Thanks.
AI: Given two non-collinear vectors $u$ and $v$ you can get an inn... |
H: For $f,g~(f0$ let $\{h\in\mathcal C[0,1]:t-c
For $f,g~(f<g),t\in\mathcal C[0,1],c>0$ let $\{h\in\mathcal C[0,1]:t-c<h<t+c\}$$=\{h\in\mathcal C[0,1]:f<h<g\}.$ I want to show that $t-c=f,~t+c=g.$
$$t-c<t<t+c\text{ and } \\f<\dfrac{f+g}{2}<g.\\\text{Then }t-c<\dfrac{f+g}{2}<t+c\text{ and } f<t<g.$$
I don't know how ... |
H: Are there any algorithms or methods to compute Landau function $g(n)$?
The details about Landau function can be found in A000793.
Maybe there is some methods in A000793, but I don't understand what it says. If possible, can someone illustrate the method?
AI: I considered Michael Somos' pari script and added some co... |
H: convergence test : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $
I have to check if $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $ is convergent or divergent.
My approach was to integrate the function , hence : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx=-\lim_{x \to \infty} 1/\ln(x)+ \lim_{x... |
H: Inverse image presheaf
Let $f:X\rightarrow Y$ be a continuous map of topological spaces, and $\mathscr{G}$ a sheaf on $Y$. So far I failed to come up with a simple example where the presheaf $f^{-1}\mathscr{G}$ on $X$ obtained via the direct limit
$$f^{-1}\mathscr{G}(U):=\lim_{f(U)\subset V}\mathscr{G}(V)$$
is not ... |
H: prove that $\operatorname{Ker}T=\operatorname{Im}(S)^\perp$ for some transformation
Question
Let $V$ be an inner product space of finite dimension. Given linear transformations $T,S\colon V \to V$ such that $\langle T(v),w\rangle=\langle v,S(w)\rangle$, for all $v,w \in V$. Show that $\operatorname{Ker}T=(\operator... |
H: $\mathbb A^n$ is not a complete quasiprojective variety
Suppose that $\mathbb K$ is an algebrically closed field. In the course of Algebraic Geometry I've attended we gave the following definition (in what follows every space is to be intended on the field $\mathbb K$):
Definition. A set $X \subset \mathbb P^N$ ... |
H: Volume of revolution $y = x^2 + 2$
$$y = x^2 + 2, 0 \leq x \leq 2$$
I know what this looks like, I was suppose to find it for a revolution around the x axis for which I used the disk method and around the y axis which I used shell so I never had to change variables.
I got two different numbers after about an hour o... |
H: question about functions (asymptotic)
This is right?
$f=\Omega(g)\Rightarrow2^f=\Omega(2^g)$?
If not I'd like to get a Counter-example.
Thank you!
AI: No.
Let $f(x) = x$ and $g(x) = 2x$. Then $f(x) = \Omega(g(x))$ since $f(x) = \frac{1}{2} g(x)$ but for any $c>0$ it will not be true that $2^{f(x)} = 2^x \geq c 2^{... |
H: Find the value of $\space\large i^{i^i}$?
Is $\large i^{i^i}$ real ? How to find it?
Thank You!
AI: $i^i=e^{i\log i}$
Now on principal branch,using $i=e^{i\pi/2}\implies \log i=i\pi/2$ gives $i^i=e^{-\pi/2}$
Therefore, $i^{i^i}=i^{e^{-\pi/2}}=e^{e^{-\pi/2}\log i}=e^{i(\pi e^{-\pi/2})/2}=\cos\left(\pi \frac{e^{-\p... |
H: Is the point $(3, 2, −1, 4, 1)$ in the open ball $B^{5}$ $((1, 2, −4, 2, 3), 3)?$
This is question 11 from the text here.
Is the point $(3, 2, −1, 4, 1)$ in the open ball $B^5 ((1, 2, −4, 2, 3),
3)$?
In my attempt to solve this question, I took the point, subtracted each corresponding component from the ball's ce... |
H: Induction proof, help please?
I have a problem that I need to prove using induction. Prove that a surjective function has at least as many members in its domain as it does in its codomain.
Do I begin by using the axiom of choice?
Thanks.
AI: If you need to use induction then it is probably that your function is bet... |
H: Example of a restriction of sheaf not being a sheaf
Let $\mathscr{F}$ be a sheaf on $X$, and $Y\subset X$ a subset. Define a presheaf $\mathscr{F}|_Y$ on $Y$ via the direct limit
$$\mathscr{F}|_Y(V):=\lim_{V\subset U}\mathscr{F}(U),$$
where $V$ is an open subset of $Y$, and $U$ is an open subset of $X$. Clearly $\m... |
H: Vector spaces and finite dimensions related problem.
Please can you help me whit this problem.
For $1.$ I did it as it's classical.
What I am having trouble with are the other questions. Hints would be good but if you can explain that would be great.
Let $E, E'$ be two vector spaces over $\mathbb{K}$, $\phi$ is a l... |
H: Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$.
Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$
How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).
Any guid... |
H: Hypothetical probability question
I found this question on an online University of Washington course assignment related to Bayesian Probabilities:
You've lost contact with your safari leader and now you find yourself confronted by a legion of 100 marmosets. Your training tells you that you must give a present to th... |
H: Calculating this Riemann sum limit
Calculate the limit $$\lim_{n\to \infty} {\sum_{k=1}^{n} {\left(\frac{nk-1}{n^3}\right) \sin\frac{k}{n}}}$$
How exactly do we calculate this limit of the Riemann sum? I am never able to find what is the partition. I know that our $f(x)$ is $\sin(x)$.
AI: Rewrite the sum as
$$\fr... |
H: How many distinct pairwise difference multisets exists?
For all $K$-point integer-valued sets whose elements are between $1$ and $N$ and also distinct, how many distinct pairwise difference multiset exists?
Note that the difference is evaluated modulu N.
for example: for $(N,K)=(5,3)$ we have:
$\color{green}{\{1,2... |
H: simple deriving of a function - preparing for hesse-matrix
$ f(x,y) = 2x^2-2xy^2+y^2$
I want to prepare this function for a hesse-matrix. But I'm stuck at deriving.
I get $\frac{\partial f^2}{\partial^2 x} = 4 $
$\frac{\partial f^2}{\partial y^2} = 4x + 4$
$\frac{\partial f^2}{\partial x \partial y} = 4x - 4y + ... |
H: A question about differentiability
So, this is the last subject I have to study for my exam on Friday, and I still can not comprehend how to prove that a function is differentable.
$$f(x,y) = \begin{cases}
(x^2 + y^2) \cdot \sin \left(\frac{1}{\sqrt{x^2 + y^2}}\right) & \text{ if }(x,y) \neq (0,0),\\
0& \text{ if ... |
H: locally compact Hausdorff space which is not second-countable
I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't find anything.
Thank you for any help.
AI: Let $Y$ be an ... |
H: Evaluate $\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$
Evaluate $$\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$$ by any method. In short I am interested in any method that overcomes the lack of convergence of this integral and gives an "number" to it.
EDIT
As I'm getting answers regarding convergence test, this sh... |
H: Roots in Banach algebras.
I'm studying positive functionals on normed algebras and I got stuck in the following problem:
Let $A$ be a unital Banach algebra, and $x\in A$ be such that $\Vert x\Vert <1$. Then the series
$$\sum_{k=0}^\infty\dfrac{1}{k!}\left(\dfrac{1}{2}-0\right)\cdots\left(\dfrac{1}{2}-k+1\right)x... |
H: Topology and commutative algebra.
I don't know both of these subjects, but I was wondering if there was any topology in commutative algebra. I don't need any detailed answer (since I don't know any of them yet)...So would it be helpful to know topology when studying commutative algebra, or is there no connection be... |
H: Krull dimension of a ring which is finitely generated module over a finitely generated $k$-algebra
Let $k$ be a field and $R=k[y_1,\cdots,y_d]$ where $y_i$ are algebraically independent over $k$. Suppose that $S=k[x_1,\cdots,x_d]$ is a subring of $R$ such that $R$ is a finitely generated $S$-module. It is well know... |
H: finding a basis for $W^\perp$ and understanding it.
Given
$$
w_1 = \begin{bmatrix}
1 \\ -1 \\ 1 \\ 1
\end{bmatrix},w_2= \begin{bmatrix}
0 \\ 1\\2\\3
\end{bmatrix}
$$
let $W$ be the subspace spanned by the given vectors. Find a basis for $W^\perp$ Now my problem is, how do envision this? They do the following:
... |
H: Quadratic expression that generate primes
I recently learned that there exist quadratic expression that generate some primes and some of these equations generate more primes than others. In the following video, the person shows the following expression
$$4x^2 -2x+1$$
that generates some primes. So I was wondering, ... |
H: How many different expressions can you get by inserting parentheses into: $x_{1}-x_{2}-\cdots-x_{n}$?
I ran into this question and I am finding it very difficult to solve:
How many different expressions can you get by inserting parentheses into:
$$x_{1}-x_{2}-\cdots-x_{n}\quad ?$$
For example:
$$\begin{align*}
... |
H: Lagrange Multipliers restriction equation problem
There's a straight line going from B to C in the first quadrant of x,y coordinate system.
B is $(0,s)$, C is $(t,0)$
Let A $(3,3)$ be a point on the line going from A to B.
Find the equation of the line so the distance between B and C will be minimal.
Well, I've st... |
H: Do we need measure theory to prove length of diameter and circumference is constant is a circle?
In high school I was taught that $\pi$ is the ratio of the length of circumference and diagonal of a circle. But is it necessary to use some measure theory machinery to define the length of circumference of a circle? Or... |
H: Finite dimensional subspace of Hilbert space and basis
Let $H$ be infinite-dimensional Hilbert space with basis functions $b_i$.
Let $B_n = \text{span}\{b_1, ...,b_n\}$.
So $\text{dim}(N) = n$.
Let $c_i$ be another basis for $H$. Is it true that
$$B_n = \text{span}\{c_{j_1}, ..., c_{j_n}\}$$
for some indices ${... |
H: Numerical Analysis, build a contractive function
I have a question regarding Numerical Analysis. I've never been asked these sorts of questions before and don't even know where to begin.
The goal of this exercise is to find a value alpha such that:
$$f(\alpha)=0$$
When $f$ is:
$$f(x) = x-0.2\sin(x)-0.5$$
Do the fol... |
H: How to show that this limit is identical to..
Suppose we have expressions
$$f_1=\frac{x(y-1)}{x(2y-1)-y}$$
and
$$f_2=\frac{xw/y+(1-x)w/(1-y)+4}{8}-\sqrt{\left(\frac{xw/y+(1-x)w/(1-y)+4}{8}\right)^2-1/4(1+xw/y)}.$$
Moreover, we can assume $w>0$, $x,y\in[0,1]$ and $x\ge y$. As you can probably guess, $f_2$ is a solut... |
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