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H: What is difference between “true in ZFC” and “true”?
Let $P$ be a claim. There are two questions.
Is it true in ZFC that $P$?
And the other is:
Is it true that $P$?
What is difference between these two questions?
AI: That depends entirely on what theoretical assumptions are silently being made (e.g., by being... |
H: Optimization Shape Question
What is the shape of the rectangle that will have the maximum area if a rectangle has a fixed perimeter equal to $S$.
Now, I have no clue how to solve it. I was told to use the area formula and perimeter formula and find the derivative.
AI: \begin{align}
S&=2l+2w\\
A&=(2l)\times(2w)\\
&=... |
H: Name for a certain class of groups that contains all the abelian groups
I cam across this type of groups. Is there a name for groups that satisfy this condition:
$$\forall x,y\in G[xyx^{-1}\in \langle y\rangle]$$
As mentioned in the title, it is easy to see that all the abelian groups satisfy this condition.
Thank ... |
H: Optimization Word Problem
A farmer is building a new cylindrical silo with a flat roof and an earthen floor that will hold $20,000 m^3$ of corn. What dimensions should the farmer construct his silo if he wants to use the least material for
construction?
I believe I have to use the surface area and area equations. C... |
H: Showing a function is contractive
This seems to simple of a question and thus I am doubting myself...
Show that the function $\dfrac{1}{2}x$ on $1\leq x \leq 5$ is contractive.
\begin{align}
|F(x) - F(y)| =& \left|\dfrac{1}{2}x -\dfrac{1}{2}y\right| \\
=& \dfrac{1}{2}|x -y| \\
\leq& \dfrac{1}{2}|x-y|
\end{align}
T... |
H: Solving for $n$ in a geomtric progression
Given the general term of geometric sequence: $a_n = \dfrac{x}{2^n}$
I would like to solve for the value of n that makes $a_n =1$.
My work so far:
\begin{align*}
a_n &= \frac{x}{2^n}\\
2^n &= \frac{x}{a_n}\\
n &=\log_2\left(\frac{x}{a_n}\right)
\end{align*}
Then I let $a_n ... |
H: Fields $k$ with subrings whose quotient field is $k$
Let $k$ be a given field, any field. Is there a subring $A$ of $k$ such that $k$ is the quotient field of $A$?
Let's restrict ourselves to fields I know anything about; subfields of $\mathbb{C}$, finite fields ($\mathbb{Z}/(p)$ and $(\mathbb{Z}/(p))[x]/(f)$ where... |
H: Whats the formula for the amount to scale up an image during rotation to not see the edges
I'm trying to figure out a formula... for how much a picture (rectangle) would have to be scaled up during a rotation (at any rotation amount) so that you don't see the edge of the picture in the square of the bounding box.
... |
H: MacLaurin powerseries and interval of convergence
Given the function $f(x) = 5/(6*x^2-x-1)$,
(a) Expand into MacLaurin powerseries the function $f$ up to order $3$.
(b) Find the interval of convergence of it.
(a) I will use the type of $$\frac{1}{0!}f(0) + \frac{1}{1!}f^{(1)}(0)x + \frac{1}{2!}f^{(2)}(0)x^2 + \frac... |
H: Faster way to compute the probability of $x$-th permutation being something
Say we have $12$ males and $24$ females
and we have to calculate the probability of the $2$nd person of being a male
we would do something like this
$(M)(M) + (F)(M)$
but what if we asked to calculate the probability of the $24$th person of... |
H: Find the length of the curve between two endpoints
Problem:
Find the length of the curve $\vec r(t) = \sqrt 2t \hat i + \sqrt 2t \hat j + (1 - t^2) \hat k$ from $(0, 0, 1)$ to $(\sqrt 2, \sqrt 2, 0)$
Solution Attempt:
First find the arc length function for this vector by using $s(t) = \int ||r'(t)|| dt$
Where $r'(t... |
H: Question about the proof that a countable union of countable sets is countable
Can One explain me a bit about the Hence
2.12 Theorem
Let $\{E_n\}$, where $n=1,2,3,...$, be a sequence of countable sets, and put
$S=\underset{n=1}{\overset{\infty }{\cup }}E_n$.
Then $S$ is countable.
Proof
Let every set $E_n$ be ... |
H: How do you make less mistakes in math?
How do you make less mistakes in math? Do you try to be more alert, do you take your time more, or what? Usually I don't make that many mistakes, but sometimes (like now) I do math as I imagine I would do it if I was ever drunk. I just did a couple of problems and I'm confusin... |
H: Integral representation of cosh x
On Wolfram math world, there's apparently an integral representation of $\cosh x$ that I'm unfamiliar with. I'm trying to prove it, but I can't figure it out. It goes \begin{equation}\cosh x=\frac{\sqrt{\pi}}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \frac{ds}{\sqrt{s}}\,e^{s+\... |
H: Graph theory maybe relate with Planar graph
Girls and Boys in the same school, each girl know 10 boys and each boy knows 10 girls, prove that the number of boys and the number of girls in the school is the same (know is symmetric = if girl G is know the boy B, then B is know G)
I think this question is related wi... |
H: A sequence raised to a sequence
I was wondering if the following is true...
If $(a_{n})$ is a sequence of positive terms converging to $a$ and $(b_{n})$ is a real sequence converging to $b$, then the sequence $(a_{n}^{b_{n}})$ converges to $a^{b}$.
This is what I did
$$ a_{n}^{b_{n}} = e^{b_{n}\ln{a_{n}}}$$
Now as ... |
H: How many faces does the $n$-dimensional cube $I^n$ have?
I am taking an online Coursera Calculus course, and this question popped up as one of the challenge problems, reproduced below. I had a difficult time understanding the answer, which was $I^n$ has $2n$ faces.
This problem concerns the boundary operator $\pa... |
H: The union of a countable set of countable sets?
Let $A$ be an countable set, and let $B_n$ be the set of all $n$-tuples $\left(a_1,\ldots,a_n\right)$
$B_n$ is the union of a countable set of countable sets.
This question maybe about the English.
Is my rephrase right?
$B_n$ is a countable set as the union of coun... |
H: What does it mean to "identify" points of a topological space?
I was recently reading about circle rotations (a basic example in dynamical systems) and got confused by some notation. It said consider the unit circle $S^{1} = [0,1]/{\sim}$, where $\sim$ indicates that $0$ and $1$ are identified.
What does "identify... |
H: Trigonometry Equations.
Solve for $0 \leq X \leq 360$, giving solutions correct to the nearest minute where necessary,
a) $\cos^2 A -8\sin A \cos A +3=0$
Can someone please explain how to solve this, ive tried myself and no luck. Thanks!
AI: The double angle identities mentioned by Avatar give a good approach. But ... |
H: Question about Lemma 7.7.1 from Hirschhorn's Model Categories and Their Localizations
The Lemma states the following.
Let M be a model category. If $g:X\rightarrow Y$ is a weak equivalence between cofibrant objects in M, then there is a functorial factorization of $g$ as $g=ji$ where $i$ is a trivial cofibrations a... |
H: Parametric equations, Exponential Function.
Consider the curve defined by the parametric equations $x=t^2 +t-1$ and $y=te^{2t}$
i) Show that $dy/dx =e^{2t}$
ii) Hence show that the tangent to the curve at the point on the curve where $t= -1$ passes through the origin.
I'm sorry to bug you guys, but I'm clueless and... |
H: Would a number that bounds the convergent sequence always be a bound for its limit?
Given a sequence of real numbers $\left\{a_n\right\}$, that $a_n \to a$ and $a_n \leq c$. Does it follow that $a\leq c$?
I'm just perplexed by a line in the proof of the Intermediate Value Theorem in 4th Ed. of Royden. In the last ... |
H: How could I see that a strongly monotonically monolithic space $X$ is first countable?
Let $A$ be a subset of $X$. A family $\mathcal B$ of subsets of $X$
is called an an external base of $A$ in $X$ if all elements of
$\mathcal B$ are open in $X$ and for any $x\in A$ and any open set
$U$ in $X$ with $x\in U$, there... |
H: How to understand the words "More algebraically"
How to understand the words More algebraically
When seeing one proof about the theorem that rational set is countable, I saw this
More algebraically , but less clearly, an explicit one-to-one onto map
$f:N\times N\to N$
is
$f(m,n)=\frac{(n+m-2)(n+m-1)}{2}+m$.
so, ... |
H: How does knowing a function as even or odd help in integration ??
So, I am learning Fourier Series and it involves integration. I am not too good at integration.
Now, the resource I use is videos by Dr. Chris Tisdell. In the opening video he says that knowing whether the function as even or odd will greatly simplif... |
H: Why is $\sin^2x + \cos^2x = 1$ important?
To start off, I understand the proof behind this identity, and I can visualize it in my head with the unit circle.
But I read this quote:
They only need to remember three facts – that $\sin 30^\circ = ½$ , that $\tan 45^\circ =1$, and that $\sin^2x + \cos^2x =1$ . Just ab... |
H: Basic question about field homomorphism
I have a question about the definition on this page.
I feel like conditions 1 and 2 imply 3, because a group homomorphism maps identity to identity.
Am I missing something? The proof of the lemma on that page uses condition 3 to assert that $1_F$ is not in the kernel of $\psi... |
H: Please let me know how to do this step by step. I had tried , but no solution yet....
A store has an introductory sale on 12 types of candy bars. A customer may choose one bar of any five different types and will be charged no more than $1.75. Show that although different choices may cost different amounts, there m... |
H: Find volume of the body $V = \{ z = \sqrt{6-x^2-y^2}, z = x^2 + y^2 \}$
Find volume of the body $V = \{ z = \sqrt{6-x^2-y^2}, z = x^2 + y^2 \}$
Now what I said is:
$$V = \iint_{D} {\sqrt{6-x^2-y^2} - x^2 - y^2 dxdy}$$.
But when I wanted to get what $D$ is, I intersected the two $z$ functions to get $\sqrt{6-x^2-... |
H: Is it always true that $\det(A^2+B^2)\geq0$?
Let $A$ and $B$ be real square matrices of the same size. Is it true that
$$\det(A^2+B^2)\geq0\,?$$
If $AB=BA$ then the answer is positive:
$$\det(A^2+B^2)=\det(A+iB)\det(A-iB)=\det(A+iB)\overline{\det(A+iB)}\geq0.$$
AI: If $A= \left( \begin{matrix} 1 &1 \\ 0 &1 \end{mat... |
H: Quadratic Bezier curves representation as implicit quadratic equation
A quadratic bezier curve from points P1=(x1, y1) to P3=(x3, y3) with control point P2=(x2, y2) can be represented as parametric quadratic curve P(t) where t is in [0, 1].
$$P(t) = (P_1t + P_2(1 - t))t + (P_2t + P_3(1 - t))(1 - t)
= P_1t^... |
H: Proof for the total derivative of a function
I'd be interested to understand why is the total derivative of a function $f(t,x,y)$, where $x = x(t)$ and $y=y(t)$ defined as:
$$\frac{df}{dt} = \frac{\partial f}{\partial t}\frac{dt}{dt} + \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{... |
H: Closed form for $a_{n+1} = (a_n)^2+\frac{1}{4}$
I've been given the following sequence:
\begin{align*}
&a_0 = 0; \\
&a_{n+1} = (a_n)^2+\frac{1}{4}.
\end{align*}
I also have to prove that whatever I come up with is correct, but that will likely be the easy part.
Here are the first few values:
\begin{align}
&a_0 = 0... |
H: Help understanding a derivation in a book
I'm reading about Pattern recognition and when I read the appendix on my book I came across with the following derivation:
$J(\theta)$ is cost a function with parameter $\theta = (\theta_1, ..., \theta_d)$. If $J(\theta) = c$ then:
$$dc = 0 = \frac{\partial J(\theta)^T}{\pa... |
H: Get the number of digits from a number
I'm looking for a function $f$ that would give me the following results :
For any $x$ such as $ x \in \mathbb {N^*}, x>0 $
$f(x) = 1 $ when $1\leq x < 10$
$f(x) = 2 $ when $10\leq x < 100$
$f(x) = 3 $ when $100\leq x < 1000$
and so on ...
So far , I have tried the floor fu... |
H: Probability of getting green
If a bag contains x black balls, y green balls and z yellow balls , now a ball is drawn :
1. It's yellow -> try again
2. it's green -> stop
3. it's black -> stop
What is the probability of getting a green ball?
I am a newbie so please forgive me if the question is too amateurish . Thank... |
H: Why does $\int\limits_0^1 {\dfrac{{x - 1}}{{\ln x}}} \;\text{d}x=\ln2$?
I have found that
$$\int\limits_0^1 {\dfrac{{x - 1}}{{\ln x}}} \;\text{d}x=\ln2$$
but I can't prove it.
Any hint?
Thank you in advance
AI: $$t>-1:$$
$$\begin{aligned}f(t)=\int_0^1 \frac{x^t-1}{\ln x}\,dx\;\Rightarrow\; f'(t) &=\int_0^1 x^t\,dx ... |
H: If $f_n\to f$ uniformly on [a,b] and f is continious on [a,b] then $f_n$ is continious in [a,b]
Yesterday I wrote a test in calculus and had to answer the following question:
Prove or contradict: if $f_n\to f$ uniformly on $[a,b]$ and f is continious on [a,b] then $\exists n_0\in\mathbb N$ s.t $\forall n>n_0,f_n$ ... |
H: Congruence in rings
Let $R$ be a commutative (and probably unitary, if you like) ring and $p$ a prime number. If $x_1,\ldots,x_n\in R$ are elements of $R$, then we have $(x_1+\cdots+x_n)^p\equiv x_1^p+\cdots+x_n^p$ mod $pR$. Why is this true? I tried to show that in $R/pR$ their congruence classes are equal, but wi... |
H: Question regarding positive definiteness decompositions
The definition of positive definteness that I'm working with is:
"A real n by n matrix $\mathbf{A}$ is positive definite if for all conformable, non-zero vectors $\mathbf{x}$ (n by 1), the following is satisfied: $\mathbf{x^T}\mathbf{A}\mathbf{x}>0$"
So I have... |
H: Prove that $\gcd(a, b) = 1 ⇒ \gcd(a^2, b^2) = 1$
I have a question from a sample exam I find difficults to solve:
Prove that if $\gcd(a, b) = 1 ⇒ \gcd(a^2, b^2) = 1$ .
I don't have any idea how to start. I'd like to get helped. thanks!
AI: The "trick" way is to solve $ax+by =1$ then cube both sides to get a solutio... |
H: Angle in a triangle with bisectors
The line segments AD, BD and CD are bisectors in $\triangle ABC$ where D is the intersection point of the bisectors. $\angle BDC$ is $128^\circ$. How big is $\angle BAC$ ?
The answer is $76^\circ$. But how to get that?
AI: Hint: $1.$ Because $\angle BDC$ is $128^\circ$, $\angle DB... |
H: Step in this proof unclear
I have a question about the following proof that is given here: Proofwiki
Why does the intersection of all $B_x$ only contain $\{x\}$? So how do I see that there is nothing else in this intersection? Does anybody of you know the exact argument?
AI: As David points out in the comments abov... |
H: Finding probability of finding alternative balls
A bag contain $5$ white balls and $4$ black balls. One ball is drawn at a time .
What is the probability that the balls drawn will be alternately white and black ?
My try :
i though $p(W)$ is white ball and $p(B)$ is black ball
This event is not independent , its... |
H: filtered modules (LNAT, Davis & Kirk)
In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written:
Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a filtration of $A$ with $Gr(A)=0$ despite $A\neq0$.
Q2: What is the precise statement o... |
H: What is the limit of $\frac {x^4 +y^4}{x^3 +y^3}$ as $(x,y) \to(0,0)$
What is the limit of $$\lim_{(x,y)\to(0,0)} \dfrac {x^4 +y^4}{x^3 +y^3}$$
if it exists?
I have tried to solve it by converting it to polar system $(x,y)=(r\cos a,r\sin a)$ and another settings. However I could not find the limit and not to show ... |
H: Textbook confusion in 8-bit representation
A textbook I'm following says the following in the section 8-Bit Representation of a Number:
"Now consider the two’s complement of an integer n that satisfies the inequality $1 \leq n \leq
128$.
Then,
$$− 1 \geq −n \geq− 128$$
and,
$$2^8 − 1 \geq 2^8 − n \geq 2^8 - 128 $... |
H: finding probability finding good fuse
There are $8$ good and $4$ bad fuses in a box. If $3$ are drawn at random, what is the probability that all will be good.
How i can solve this question when $3$ are drawn at random ?
AI: there are exactly $\binom{8}{3}$ ways of choosing three good fuses while there are $\binom... |
H: The identity of a polynomial sum
I am wondering if there is a recursive formula to calculate
$$S=1^{k}+2^{k}+3^{k}+\dots+n^{k}$$
Where $k$ and $n$ are natural numbers.
AI: I believe this should do,
$$f[k][n]=f[k][n-1]+(f[k-1][n]-f[k-1][n-1])\cdot n$$
where $f[k][n] = 1^k+2^k+\dots+n^k$. so we get $n^{k-1} = (f[k-1]... |
H: Mathematical word for geometrical object?
Is there a mathematical word to designate the concept of a geometrical object like:
square
cube
tesseract
N-dimensional cube
circle
sphere
hypersphere
regular and non-regular polygons
regular and non-regular polyhedrons
...
What is the common word to designate all these o... |
H: Number of integer distance grid points in a cubic grid
Given an $n$-dimensional integer lattice $\mathbb Z^n$, how does the number of grid points $N(k)$ at integer distance $\leq k$ from a given grid point scale asymptotically? I'm mostly interested in $n = 3$ (I'm a physicist :) ), but having also the results for ... |
H: Prove that the maximum number of edges in a graph with no even cycles is floor(3(n-1)/2)
The question is in the title. I can see why the bound is sharp (for example, a lot of triangles sharing one common vertex if n is odd, or the same but with one spare edge hanging out if n is even). But I can't prove why the bou... |
H: How can I construct a square using a compass and straight edge in only 8 moves?
I'm playing this addictive little compass and straight edge game:
http://www.sciencevsmagic.net/geo/
I've been able to beat most of the challenges, but I can't construct a square in 8 moves.
To clarify a move is:
Drawing a line
Drawing... |
H: An inequality on holomorphic functions
Let $D := \{z \in \mathbb{C}: |z| < 1\}$ and $f\colon D \rightarrow \mathbb{C}$ be holomorphic. Suppose $\lvert f(z)\rvert \leq 1$ on $D$, show that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1 - |f(0)||z|} \ \forall z \in D.$$
I have tried using... |
H: A set with members allowed to appear more than once
I'm looking for a definition for a set which its members could be appeared more than once!
for example: $$D=\{1,1,2,4,6,6\}$$
Could we call this a set?
AI: This object is variously called a "multiset" or a "bag". It is not, however, a set. Sets are characterized... |
H: Value of this infinite sum
I want to compute the value of
$$\sum_{n=1}^{\infty} \frac{1}{((2n)^2 - 1)^2}.$$
I have tried evaluating the first couple partial sums, but can't find any recurrence. I don't have too many tools in my toolbox to proceed... Can anyone see how I could go on here?
AI: Hint:
$$\frac{1}{((2n)^... |
H: Trigonometric Equations
Eliminate B from each pair of equations:
x=sinB -3cosB and y=sinB+2cosB
I've tried solving this simultaneously just as the textbook has guided me through, but it still doesn't work. My initial working out was moving sinB to the left hand side and everything else to the right and solve... |
H: Proving that permutations $f$ and $f^{-1}$ are of the same type
Proving that permutations $f$ and $f^{-1}$ are of the same type.
Can someone give me like a hint to start proving this?
Thanks!
AI: Reversing every cycle of $p$ would result in $p^{-1}$, thus they're the same type. |
H: distribution of $\cos(\omega_0 n)$ where n are integers?
Assume we have the sequence $\,x[n]=\cos(\omega_0 n),$ where $n$ are integers.
If we suppose these are realizations of a random variable, what would be the p.d.f. of that random variable?
AI: So the point of this answer is not so much to answer the question ... |
H: average/weighted average within Excel
5 star 128
4 star 17
3 star 10
2 star 2
1 star 14
Average stars 4.42
How many more 5 stars would I need to get Average Stars to 4.45 or 4.47 etc.
So what I tried was using SUMPRODUCT in Excel but I am approaching this wrong. First of all I cannot achieve the cu... |
H: Show that the matrix $A+E$ is invertible.
Let $A$ be an invertible matrix, and let $E$ be an upper triangular matrix with zeros on the diagonal. Assume that $AE=EA$. Show that the matrix $A+E$ is invertible. WLOG, we can assume $E$ is Jordan form. If $A$ is Jordan form, it's trivial. If $A$ is not Jordan form, how ... |
H: Mappings from $S^n$ to $S^n$ with odd degree
Let $f:S^n\rightarrow S^n$ be of odd degree, i.e. $f^*(1)$ is odd where $f^*:H_n(S^n)\rightarrow H_n(S^n)$ is the induced map on homology. Prove that there exists an $x\in S^n$ with $f(-x)=-f(x)$.
I tried to imitate the proof of Borsuk-Ulam theorem, but with no achieveme... |
H: Prove that the longest decreasing subsequence of $p$ consists of $d$ elements.
Let $p$ be a permutation and let $d$ be the smallest integer so that $p$ can be written as the union of $d$ increasing subsequences. Prove that the longest decreasing subsequence of $p$ consists of $d$ elements.
I just need a hint, plea... |
H: Calculating $a_0$ in Fourier Series
I am using this YouTube video to learn Fourier Series. The question can be clearly seen in the picture. The instructor calculates $a_0$ as the area under the triangle which is fine. Nothing wrong with it, I get it.
However, I was trying to calculate is as a sum of integration... |
H: Prove that $\sum _{k=0}^{n}{{(-1)}^{k}{\binom n k}}=0$
Prove that:
$$\sum _{k=0}^{n}{{(-1)}^{k}\binom n k}=0$$
I tried with induction and failed.
A solution explain would be greatly appreciated.
AI: The Binomial Theorem says $$(a+b)^n=\sum_{0\le r\le n}\binom nr a^{n-r}b^r$$
for real or complex $a, b$, and non-neg... |
H: How can I find the value of this limit
$$\lim _{ { x }\to { 0 } }{ \frac { \sin x-\arctan x }{ {x }^{ 2 }\log(1+x) } }$$
this log is natural logarithm
AI: Using Series Expansion
$$\frac{\sin x-\arctan x}{x^2\log(1+x)}$$
$$=\frac{x-\frac {x^3}{3!}+O(x^5)-\{x-\frac {x^3}3++O(x^5)\}}{x^3}\cdot\frac1{\frac{\ln(1+x)}x}... |
H: Coefficients of this Taylor series are bounded
Let $f(z)$ be a function that's analytic in the open unit disk, and also in a region containing the closed unit disk with the exception of a few simple poles (i.e. the poles have degree 1) that lie outside the open unit disk. Show that the coefficients of the Taylor ex... |
H: Asymptotic equality and inequalities
If $f$ equals $g$ asymptotically, i.e., $f(x)/g(x) \to 1$ (as $x \to \infty$), and $h \leq f$, does that mean $h \leq g$ for sufficiently large $x$? It seems to be true because the relative error, $|f/g - 1|$, tends to zero. I, however, think it is not true in general, that is, ... |
H: Finding probability at least one male child
A family has three children. what are the probabilities that it has no male child and at least one male child if we consider the order of birth.
My Try:
I didn't understand the complete question. It's confusing me while solving.
AI: There are different ways to solve the... |
H: Problem about a process with bins of balls
A friend of mine give me this problem for fun:
Given $\frac {n(n+1)}{2}$ balls, first we divide arbitrarily these balls in baskets, after that we make another basket with one ball of each basket e do this procedure infinitely.
I want to prove that one time this stabilizes ... |
H: What "is" a matrix in the context of a vector space?
I'm familiar with the definition of a vector space $V$ over a field $F$
I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space $W$.
The Wiki article on Vector Spaces says this:
Matrices are... |
H: Propriedades do MDC (Properties of Greatest Common Divisor)
Estou com uma grande lista de exercícios de PROPRIEDADES DO MDC (MÁXIMO DIVISOR COMUM), e não estou conseguindo entender quais os passos que tenho que seguir nas demonstrações, e gostaria muito de aprender este conteúdo, alguém me ajuda em uma questão para... |
H: Solving Bessel integration
What would be the solution of the bessels equation,
$$b=k A(t)\int_0^{\infty} J_0 (k \rho) e^ \frac{-\rho^2}{R^2} \rho d \rho$$
Can I sove that by using this formulation?
$$c= \int_0^{\infty}j_0(t) e^{-pt} dt= \frac{1}{\sqrt{1+p^2}}$$
AI: No, the Laplace transform of a Bessel does not app... |
H: How many dice needed to make rolling at least $3$ sixes probable ($p>0.5$)
I'm wondering how to solve a questions of the example form:
How many dice are needed to make rolling at least 3 sixes in a single throw probable (p>0.5)
I know how to solve the question by graphing out all of the binomial probabilities for... |
H: The Derivative of $y=x^2$ with respect to $\ln x$
Find the derivatives of $y$$=$$x^2$ with respect to $\ln x.$
AI: The function $f(x)=x^2$ takes you from$ x$ to $x^2$. However you want a function that takes you from $ln(x)$ to $x^2$. You can find this function by taking the composition of two functions.
To do this ... |
H: How to quickly tell that a quadratic only has a single root?
Hello good math wizards, I'm trying to figure out why the following equation has at most one root:
$$f (t) = \textbf{x} \cdot \textbf{x} + \textbf{x} \cdot t\textbf{y} + t\textbf{y} \cdot \textbf{x} + t\textbf{y} \cdot t\textbf{y} = \left \| \textbf{x} \... |
H: Is it possible to have multiple decimal points in a number?
Is it ever possible to have multiple decimal points in a number? If so, how?
For example is the value 1.1.2 possible?
This is a question posed purely out of curiosity.
AI: Mathematical notation, like other aspects of human language, is a human creation - w... |
H: Creating a cover of closed balls for a compact set
No idea how to start thinking about this..any pointers would be appreciated! :]
Let $K \subseteq \mathbb R^n$ be a compact set.
Let us assume $K$ has a cover of $n$ open balls with radii $r_1,...r_n$.
Show that there exist $0<t_i<r_i$ for every $0\le i \le n$ such ... |
H: Incorrect notation in math?
Does math have an incorrect notation / syntax? I don't mean writing misaligned notation (google), but when you take something like a number to powers to powers to powers, $${{2^2}^2}^3$$ (I was told this is incorrect notation by a teacher). Is it really incorrect, or does it just need to... |
H: Express $f(x) = x^2 \cos(2x) + \sin^2(x)$ as a power series
Express $f(x) = x^2 \cos(2x) + \sin^2(x)$ as a power series
What I know:
I know that $$x^2\cos(2x) = x^2 \cdot \sum_{n=0}^{\infty} {(-1)^n \cdot \frac{(2x)^{2n}}{(2n)!}} = \sum_{n=0}^{\infty} {(-1)^n \cdot \frac{2^{2n} \cdot x^{2n+2}}{(2n)!}}$$
But how... |
H: What does “unconfirmed or conditional” mean on this site?
On the home page of the “Bounded gaps between primes” polymath project, there are listed bounds for $H$ in Zhang's proof of prime gaps. For example:
What does “unconfirmed or conditional” mean on this page and what is the difference between results $H=12... |
H: $f$ integrable but $f^2$ not integrable
At this point in Bartle, $X$ is a nonempty set, $\mathcal{X}$ is a $\sigma$-algebra of subsets of $X$, and $\mu$ is a measure on $\mathcal{X}$.
$f\in L(X,\mathcal{X},\mu)$ means:
$f:X\to R$ is measurable.
$\int f^+\,d\mu<+\infty$ and $\int f^-\,d\mu<+\infty$.
$\int f\,d\mu... |
H: Prove that $f_n(x) = 1 - \cos(\frac{x}{n})$ is uniformly convergent
Prove that $f_n(x) = 1 - \cos(\frac{x}{n})$ is uniformly convergent in $[0,2\pi]$.
So I said this, and my professor said it is wrong:
$|1-\cos(\frac{x}{n})| \leq 2, $ then we take some $\epsilon >2 $, and we have for every $n \in \mathbb{N}$ and... |
H: n-correlation function.
So I was thinking of generalization of notions in statistics, like auto-correlation or cross-correlation (auto-correlation is a specific example of cross-correlation where we take the same proccess).
The problem is how do I generalize for example the notion of WSS (wide-sense-stationary)?
Ta... |
H: Optimal and maximal codes
Is there an example of a code which is maximal but not optimal?
Definitions.
An $(n,M,d)$ code $C$ over $\mathbb{F}_q$ is maximal if $C$ is not contained in an $(n,M+1,d)$ code.
An $(n,M,d)$ code $C$ is optimal if $|C|=M=A_q(n,d)$, where $A_q(n,d)=\max \{M\mid\exists (n,M,d)\text{ code}... |
H: Triple integral over a region.
I'm trying to find the volume of $D=\{(x,y,z): \displaystyle\frac{x^2}{a^2} + \displaystyle\frac{y^2}{b^2} + \displaystyle\frac{z^2}{c^2} \leq 1\}$.
I use the change of variables $x=ap\cos\theta\sin\phi$, $y=bp\sin\theta\sin\phi$, $z=cp\cos\phi$ and i have that the region of integrati... |
H: Probability of winning.
There is a game with three types of cards.
Win cards, If you get one of these, you win.
Lose card, if you get this, you lose.
Revive card: If you get one of these, you draw another card.
Now Initially we have a deck of cards of all three types. p Win cards, q Lose cards, and r revive cards.... |
H: Summing Finitely Many Terms of Harmonic Series: $\sum_{k=a}^{b} \frac{1}{k}$
How do I calculate sum of a finite harmonic series of the following form?
$$\sum_{k=a}^{b} \frac{1}{k} = \frac{1}{a} + \frac{1}{a+1} + \frac{1}{a+2} + \cdots + \frac{1}{b}$$
Is there a general formula for this? How can we approach this if ... |
H: Irreducibility of a topological space
Let $X$ be a topological space and $X=X_1 \cup X_2$ with $X_1, X_2$ nonempty open irreducible subsets. Then $X$ is irreducible iff $X_1 \cap X_2 \ne \emptyset$.
The easy part: if it were $X_1 \cap X_2 = \emptyset$ then we would have
$$
X = (X \setminus X_1) \cup (X \setminus... |
H: Volume of a rotated region $y = x^2 + 2$
I have no idea what the best method to do this is but I chose the shell method.
$y = x^2 + 2$
about $y = -2$
So I convert to f(y) since it rotates on a horizontal axis.
$x = \sqrt{y-2}$
I know the bound are 0 to 4 for either x or y, it doesn't really matter.
I know that the ... |
H: If $m \equiv n \pmod{A}$, then $s^m \equiv s^n \pmod{A}$?
I'm kind of stuck with the following assignment:
Prove: If $m \equiv n \pmod{A}$, then $s^m \equiv s^n \pmod{A}$
I tried $m = k_1 \times A + r$ ,
and $n = k_2 \times A + r$ ,
then $s^m = s^{k_1 \times A + r}$,
but not sure how to proceed ...
Really apprec... |
H: Name of the $(-1)^n$ function?
Does the function $f\left(n\right)=\left(-1\right)^n, n \in \mathbb{Z}$ used in a lot of mathematical formulas have a special name ?
EDIT: The context of this question is that I need a name for this function in a software.
AI: Not really. One just writes it out like that, for example,... |
H: How to write boolean expressions as linear equations
I want to convert a set of boolean expressions to linear equations. In some cases, this is easy. For example, suppose $a, b, c$ $\in$ {0,1}. Then if the boolean expression is: $a$ $\ne$ b, I could use the linear equation $a + b = 1$.
To give a more complicated e... |
H: How to construct a vector field under these conditions?
Task: Construct a vector field $v:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that all the circles that "touch" y-axis in the origin (i.e. derivative in $(0,0)$ is in form $(0,a)$ for some $a$) are field lines of this field, and this vector field has to be cont... |
H: Are there infinitely many $n$ such that $n$ and $2n+1$ are both prime numbers?
It seems that there are many such $n$. For example, the following are such $n$'s under $4000$.
2 3 5 11 23 29 41 53 83 89 113 131 173 179 191 233 239 251 281 293 359 419 431 443 491 509 593 641 653 659 683 719 743 761 809 911 953 1013 1... |
H: Finding the inverse of a matrix using elementary matricies
Can somebody help me understand what exactly is being asked here? I understand how to construct elementary matrices from these row operations, but I'm unsure what the end goal is. Am I to assume that $Y$ is built from these row operations?
Let $Y$ be the $... |
H: Maximization of $x^2e^{-(x^4+y^2)}$
My textbook proposes to maximize $f(x, y)=x^2e^{-(x^4+y^2)}$. It begins by showing that it tends to $0$ as $r=\sqrt{x^2+y^2}$ tends to $\infty$.
First it remarks that $x^2e^{-x}\rightarrow 0$ as $x\rightarrow\infty$, and that $e^{-x^4}$ is eventually smaller than $e^{-x}$, theref... |
H: Elements $a,b\in L$ of degree $3$ over $K$ such that $a+b$ has degree $6$ over $K$
A finite field extension $L/K$ and elements $a,b\in L$ of degree $3$ over $K$ such that $a+b$ has degree $6$ over $K$. Can you give me an example about such field extension? Thanks.
AI: Hint: Let $K=\mathbb{Q}$ and let $L$ be the spl... |
H: Partial fraction $\int \frac{dx}{2x^2 - 3}$
$$\int \frac{dx}{2x^2 - 3}$$
I think I have to factor this but I don't know how to. If I don't know how to is it valid to do
$$\int \frac{dx}{2x^2 - 3} \quad =\quad \int \frac {Ax+ B}{2x^2 - 3}\,dx$$
AI: $$\int \frac{dx}{2x^2 - 3} = \int \frac{dx}{2(x^2 - 3/2)}\,dx = 1/2... |
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