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H: Inverse Laplace Transform of $L^{-1} \Bigl(\frac{1}{s^2 (s+2)^2 }\Bigr)$
How can we find inverse laplace transform of this equation ?
$L^{-1} \Bigl(\frac{1}{s^2 (s+2)^2 }\Bigr)$
AI: Check $$\frac{1}{s^2(s+2)^2}=\frac{1}{4}\left (\frac{1}{s^2}-\frac{1}{s} + \frac{1}{s+2}+\frac{1}{(s+2)^2}\right)$$ use the fact that
... |
H: How to solve for a trigonometric inequality $2\cos(2x)\cos(x)+\cos(2x)\gt0$
Can someone please help me solve this inequality:
$2\cos(2x)\cos(x)+\cos(2x)\gt0$
I know how to get the values for $x$, but what I don't understand is how the answer is $0 < x < 45$ and $120 < x < 135$. How do you get these answers?
AI: Thi... |
H: Weak convergence of $n \min$ of $n$ iid Uniform random variables
Let $X_n$ be a sequence of iid $U[0,1]$ random variables. Let $Y_n = n \min_{1\leq i \leq n}X_i$. Show that $Y_n$ converges weakly to Exponential distribution with parameter 1.
I know that if $Z_n =\max_{1 \leq i \leq n}X_i$, then $n(1 - Z_n)$ conve... |
H: Help eliminating a term inside a square root
I have the somewhat ugly expression:
$((V_iV_j\lambda_2 + X_iX_j\lambda_1)^2 + \lambda_1\lambda_2(V_iX_j - X_iV_j)^2)^{\frac{1}{2}}$
Every single term here is a scalar over the reals.
The goal is to try to isolate that left term in the sum i.e I need to get either:
$((V_... |
H: understanding of the number relation for all n, n^3 mod 9 is 0,1, or 8
In the book, Elements of programming CASE ANALYSIS has been listed as one of the approaches to solve a problem.
As an example, the book states that
for all n , n^3 mod 9 would either be 0,1,8 and even splits this problem into following cases
whe... |
H: Ideals of the ring $\mathbb{Z}_3[x]/\langle x^4+x^3+x+1\rangle$
What are the ideals of the ring $\mathbb{Z}_3[x]/\langle x^4+x^3+x+1\rangle$?
I know that $\mathbb{Z}_3[x]$ is a PID. In addition, $(x+1)^4=x^4+x^3+x+1$ in $\mathbb{Z}_3[x]$. Therefore, our ring is nothing but $\mathbb{Z}_3[x]/\langle (x+1)^4\rangle$. ... |
H: Arnold on proof of uniqueness
In his proof of Uniqueness, Arnold mentions the integral approaches infinity as x3 approaches x2. How does he come about this conclusion?
AI: Assuming we know that $x_1 \lt x_3 \lt x_2$, then
$$\int_{x_1}^{x_3} \frac{d \xi}{k(\xi - x_2)}= \frac 1k \ln \frac{x_3-x_2}{x_1-x_2}.$$
If $x_3... |
H: On equicontinuity of a family of functions
Let $C[a,b]$ denote vector space of continuous functions on the closed interval $[a,b], (a,b\in\mathbb{R})$. Is it true that the family $\mathcal{A}:=\{\int_a^{x}f(t)dt \ | \ f\in C[a,b]\}$ equicontinuous ? I think this is not true. Can anybody provide a counter-example ?
... |
H: linear operators diagonalisation
Give an example of two diagonizable linear operators $f$ and $g \in \mathbb{R}^2$, for which $5f - 2g$ is not diagonizable.
AI: Hint Note that
$$\begin{bmatrix}2&1\\0&1\end{bmatrix}+\begin{bmatrix}-2&0\\0&-1\end{bmatrix} = \begin{bmatrix}0&1\\0&0\end{bmatrix}$$
is a decomposition of... |
H: Find $\sum_{n=1}^\infty\frac{z^{2n+1}}{1+(1+2n)^2},\ \ \ \text{where}\ \ \ |z|\leqslant1 $
Find the sum:
$$
S=\sum_{n=1}^\infty\frac{z^{2n+1}}{1+(1+2n)^2},\ \ \ \text{where}\ \ \ |z|\leqslant1
$$
I tried to factorize the denominator, but it didn't help at all. Could someone give me a clue to the solution of this ... |
H: Equivalence for a subspace to be bounded on a Banach space
I have done the following exercise , but I did not use one of the hypothesis so I wanted to make sure that everything was alright
Suppose that $X$ is a banach space then a subset $B$ is bounded if and only if $\forall f\in X' sup\{|f(b)|:b\in B\}< \infty$... |
H: Using residue theorem to calculate integral $\int\limits_0^{2\pi} \frac{dx}{10+6\sin x}$ - where is my mistake?
I am to calculate:
$$
\int\limits_0^{2\pi} \frac{dx}{10+6\sin x}
$$
We can set $\gamma(t)=e^{it}$ for $t \in [0, 2\pi]$ and then $z = e^{it}$, $\dfrac{dz}{iz}=dt$, $\sin t =\dfrac{1}{2i}(z-\frac{1}{z})$ s... |
H: Analysis question limit $\lim_{n \to \infty} (-1)^n$
Can someone please explain to me how we should evaluate this limit, I just know that it's indeterminate form and I tried to use L'Hospital's rule but I couldn't do it here what I have done
$$\lim_{n \to \infty} (-1)^n$$
$$=e^{n\ln(-1)}$$
But $\ln$ of $-1$ doesn't... |
H: Find the angle between the base and the side of a pyramid.
Been stuck with this problem for ages now. Please help.
The text reads:
Given a pyramid with a square base and the tip of the pyramid is above where the diagonals intersect (the sides are isosceles triangles), calculate the angle between the base and the si... |
H: Is a nonzero vector subspace of a nonzero NLS is compact?
This question is from topology of metric spaces by s.kumersean page-90 chapter-compactness .
Is a nonzero vector subspace of a nonzero NLS is compact?
Honestly I don't know how to show that because what i have read so far I don't find any link how to prove t... |
H: Using residue theorem to calculate $\int\limits_{-\infty}^{\infty} \frac{\cos(2x)\,dx}{(x^2+2x+2)^2}$
I want to calculate: $\int\limits_{-\infty}^{\infty} \frac{\cos(2x)\,dx}{(x^2+2x+2)^2}$
Firstly we notice that:
$\int\limits_{-\infty}^{\infty} \frac{\cos(2x)\,dx}{(x^2+2x+2)^2}=\operatorname{Re}\int\limits_{-\inft... |
H: Find $P(X=r, Y=s)$ from $P(X\le r, Y \ge s)$.
An urn contains balls numbered $1$ to $N$. Let $X$ be the largest number drawn in $n$ drawings when random sampling with replacement is used. Find the joint distribution of the largest and the smallest observation. (Hint: Calculate first $P(X \le r, Y \ge s)$.)
The hint... |
H: prove that every complete graph with 4 or more vertices has two spanning trees with disjoint edges
I read a possible proof of "every complete graph with 4 or more vertices has two spanning trees with disjoint edges" in the answer of another question.
That is, first claim that every complete graph with 4 or more ver... |
H: Derivative of $e^x$ after geometric transformation
Inverse function of $f(x) = e^x$ is of course $f^{-1}(x) = \ln{x}.$ We have, by definition, $\frac{d}{dx}e^x = e^x$. In other words, $e^x$ in some sense describes slopes of tangent lines on a curve given by outputs of $e^x$.
We can get $\ln{x}$ by reflecting curve ... |
H: How can I define a surjective ring homomorphism from $\mathbb{Z}_{22}$ to $\mathbb{Z}_7$?
How can I define a surjective ring homomorphism from $\mathbb{Z}_{22}$ to $\mathbb{Z}_7$?
AI: You can't. Not even a surjective group homomorphism.
The isomorphism theorem implies that image of a group homomorphism $G \to H$ of... |
H: Bijection between classes of natural transformations involving Kan extensions.
I am reading the chapter on Kan extenions in the Handbook of Categorical Algebra by Francis Borceux and I am a bit confused about some of the steps he takes.
My main question is a step in his proposition 3.7.4:
Let $\mathcal{A},\mathcal... |
H: Generating symmetric Positive Definite Matrix from random vector multiplication yield singular matrix
This might seem a very naive or ignorant question, but I am not able to give myself a satisfactory explanation.
Suppose i want to generate a random symmetric positive definite matrix. I was under the impression tha... |
H: Calculating poles and residues of given function
Let $$
f(z) = \frac{1}{(z+i)^7} - \frac{3}{z-i} = \frac{z-i-3(z+i)^7}{(z+i)^7(z-i)}
$$
This has pole of order 1 in $i$ and order 7 in $-i$. I can easily calculate the residue from pole at $i$:
$$
\text{Res}(f,i) = \lim_{z \to i} (z-i)\frac{z-i-3(z+i)^7}{(z+i)^7(z-i)}... |
H: Connectivity graph proof
Let $G=(V,E)$ be an undirected graph, with
$\text{deg}(v)=p>1$
for all $v$ of $V$ and $|V|=2p+1$.
I want to show that exists in the graph an Euler circle. I was able to prove that the graph is even, but I can't prove the graph is connected.
Thanks for the help
AI: Suppose that $G$ is not co... |
H: If we have the slope of $AB$ and $AC$. How can we determine the angle of $AB$ and $AC$?
If we have the slope of $AB$ and $AC$. How can we determine the angle of $AB$ and $AC$?
I searched the internet but I don’t understand. Please help! Thank you very much.
AI: Let $m$ be the slope of AB and $n$ the slope of AC. ... |
H: how to find independent path?
I confused while find independent path.
What should be $\lambda$ for the path independent of Moreover evaluate the integral for this value of $\lambda.$
$$\int_{(2,4)}^{(1.2)} (\frac{xy+\lambda}{y})dx+(\frac{2y\lambda-x}{y^2})dy$$
Exactly Differential and
it could be check out conserva... |
H: Series Dependencies
If i have a series
$$
a_n = \sum_{n=0}^\infty \frac{x^{2n}}{b^nn!}(n+c)^2
$$
and another series
$$
c_n = \sum_{n=0}^\infty \frac{x^{2n}}{n!}(n+c)^2
$$
Is there some way to find some $f(x)$ so that
$$
a_n = f(x,b)c_n
$$
Thank you
AI: First note that what you called $a_n$ actually does not depend ... |
H: IB HL Math, proving that a function is greater than $1$, for all $x>0$
The function $f$ is given by $$f(x)= \frac {3^x + 1}{3^x - 3^{-x}}$$ for $x>0$.
Show that $f(x)>1$ for all $x>0$
Hi all, I think I have solved this question but was having trouble proving this in a succinct and intuitive way. I would appreciate ... |
H: Fast modular exponentiation for $60^{53} \text{ mod } 299$
I'm trying to find the modular exponentiation for $60^{53} \text{ mod } 299$. I know it is $21$, but I would like to to show the answer step by step so that a normal calculator (with no modulo function) would be able to follow the solution steps.
I calculat... |
H: matrix multiplication expression
How to define the following expression as matrix multiplication?
$$-\sum_{i=1}^n\left[\mid e_i\mid-\exp(\sum_{j=0}^p \gamma_j X_{ij})\right]\exp(\sum_{j=0}^p \gamma_j X_{ij})X_{i.}$$
X is a matrix, e and γ are vectors
$$X_{n,p+1}$$
$$e_{n,1}$$
$$\gamma_{p+1,1}$$
AI: For typing conve... |
H: What's the partial derivative of $\partial _j h(a)=\partial_j(a)$?
Let $h : \mathbb{R}^n\to \mathbb{R}^n, h(a)=a$. What's $\partial _j h(a)$?
$\partial_jh(a)=\partial_j(a)=\partial_j(a_1,...,a_j,...,a_n)=(0,...,1,...,0)$
is there a better way to notate this case?
AI: Since $h_k=a_k$, $\partial_jh_k=\delta_{jk}$ so ... |
H: Proving $\frac{\cos(3A)-\sin(3A)}{1-2\sin(2A)}=\cos(A)+\sin(A)$
Can someone please show me step-by-step how to prove this trigonometric identity?
$$\frac{\cos(3A)-\sin(3A)}{1-2\sin(2A)}=\cos(A)+\sin(A)$$
This is what I've done so far:
Using the LHS,
$$\frac{cos(2A+A)-sin(2A+A)}{1-4sin(A)cos(A)}$$
$$=\frac{cos(2A)... |
H: How to determine the biggest interval where f(x) is invertible and f(x)^-1 passes through P
I need some help here.
$$f(x) = \sin{(3x)} - 1$$
$$P = (-1,0)$$
AI: In order to make $\;\sin x\;$ we restrict its definition domain to $\;\left[-\frac\pi2,\,\frac\pi2\right]\;$ (though this is not the only interval that can ... |
H: Last two digits of $[(\sqrt{5}+2)^{2016}]$
I was trying to find the last two digits of the largest integer $\left\lfloor\left(\,\sqrt{\, 5\, }\, +\, 2\,\right)^{2016}\right\rfloor$
less than or equal to $\left(\,\sqrt{\, 5\, }\, +\, 2\,\right)^{2016}$.
My idea is to relate $\left\lfloor\left(\,\sqrt{\, 5\, }\, +\,... |
H: Looking for divisibility by 29 and a general proof( if any)
Is there any elegant way to prove that 28C14 -1 is divisible by 29? Also, is this kind of a result a theorem or a generalisation? If so please do help...
note i do mean 28 choose 14
AI: I presume you mean $\binom{28}{14}-1$?
Note that
$$14!\binom{28}{14}=2... |
H: How can we form the categories of $R$-modules from that of unital rings?
I know the fundamental facts about rings and modules over rings and I understand how they form their respective categories. What I was wondering about is the relationship between $\operatorname{R-mod}$ and $\operatorname{Ring}$.
It is clear to... |
H: Umbrella Term for Multiplication and Division
Is there an umbrella term or name for both Multiplication and Division? Do these two operator types fall under one specific family with a particular name?
AI: Division can be rewritten as multiplication. Indeed, suppose that we want to evaluate $a:b$. Then, we can write... |
H: Find $x_0\in \mathbb{Q}$ s.t. $(x_n)_{n}$ is convergent.
Let $(x_n)_{n\geq 0}$ a sequence of real numbers given by the relation $2 x_{n+1}=2x_n^2-5x_n+3$, for every $n\geq 0$. Find $x_0\in \mathbb{Q}$ s.t. $(x_n)_{n}$ is convergent.
An easy remark is that the limit of the sequence is $3$ or $\dfrac{1}{2}$, but now ... |
H: A triangular grid of side $n$ is formed from $n^2$ equilateral triangles with sides of length $1$. Determine the number of parallellograms.
So here is Question :-
A triangular grid of side $n$ is formed from $n^2$ equilateral triangles with
sides of length $1$.Determine the number of parallellograms.
First of all ,... |
H: Find a and b such that the line $10ex + 10y = 0$ will be a tangent of the curve $ y = ae^{1/(x-1)} - b$ at $x = 2$?
How do I solve this, it seems very simple, yet when I try it turns impossible because I have no idea how to continue:
$$
10ex+10y=0 \\
ex+y=0\\
y=-ex\\
f(x) = -ex\\
f(2)=-2e
$$
And now? I am not sure ... |
H: Is it true: $\mathbf{a.B}=\mathbf{B^{T}.a}$
Important notion
I know that this question may seem too simple for you mathematicians, but I know nowhere than here to ask it.
Question
Is it true:
$$\mathbf{a.B}=\mathbf{B^{T}.a}$$
Assumptions
Assuming $\mathbf{a}$ as a Cartesian vector and $\mathbf{B}$ as a second-order... |
H: Finding the number of solutions to $\sin^2x+2\cos^2x+3\sin x\cos x=0$ with $0\leq x<2\pi$
For $0 \leq x<2 \pi$, find the number of solutions of the equation
$$
\sin^2 x+2 \cos^2 x+3 \sin x \cos x=0
$$
I have dealed the problem like this
$\sin ^{2} x+\cos ^{2} x+\cos ^{2} x+3 \sin x \cos x=0$
LET, $\sin x=t ;\quad... |
H: Using the Taylor-Maclaurin series and differentiation/integration calculate the infinite sum
Using the Taylor-Maclaurin series and differentiation/integration calculate the infinite sum n/((n+1)(2^n)) from n=1 to infinity. I have tried to write it as ∑ 1/2ⁿ - ∑ 1/((n+1)*2ⁿ) but still cannot solve the second sum and... |
H: Specific differential equation with initial conditions
I have diff. equation : $ y'' +2y' = (y')^2 e ^x , y(0)=3, y'(0)=1 $, and i have problem with solving that. I used substitution $ u(x)=y' $ and i got bernoulli's diff. equation. I solved that and got $ y'=[(1/2)*e^x +D*e^{2y'}]^{-1} $ and that is basical... |
H: Proving a lemma for Taylor series proof
I've recently learnt Taylor series, and am trying to prove very simple lemma by induction.
I need to prove that every polynomial $Q(x)$ can be written as
$Q(x)=\sum_{k=0} ^{n} c_k (x-a)^k$
When $n$ is the polynomial degree, $a$ is constant and $(c_k)_{k=0} ^m$ is a sequence o... |
H: number of permutations of $[2n]$ that move exactly k elements from $\{1,...,n\}$ to $\{n+1,...,2n\}$
I'm trying to figure out the size of
$S^{(k)}:=\{\sigma\in S_{2n}\ :\ \sigma \text{ moves exactly }k\text{ elements from }\{1,\dots,n\} \text{ to }\{n+1,\dots,2n\}\}$ for $k=0,\dots,n$, where $S_d$ is the permutatio... |
H: Can a circle have a negative radius?
I am working on a problem that asks if a given sphere intersects the zx-plane.
The equation of the sphere is $(x-2)^2+(y+6)^2+(z-4)^2=5^2$
Can someone please explain to me how the sphere does not intersect the zx-plane because the radius of the circle is said to be a positive q... |
H: Set of rational sequences is countable and dense in $l_2$
Let
$$D=\{(a_n):\quad (\forall n\in \mathbb{N}\space a_n\in \mathbb{Q})\land (\exists p\in \mathbb{N}:\space \forall n\geq p\space a_n=0) \}.$$
I want to show that $D$ is countable and dense in $l_2$. I proved there is an injection $h:\mathbb{Q}\rightarrow D... |
H: Trying to understand conditional probability
So let's suppose we have 2 red balls, 3 blue balls, and 4 green balls in a cup and we take 3 balls without replacing them.
a. What is the probability that 3 balls of the same color are chosen
b. What is the Conditional Probability of choosing 3 green balls from the sampl... |
H: Establishing set equivalence: Munkres exercise 1.7
An exercise in Munkres asks to write this set in terms of $\cup$, $\cap$, and $-$. The set I cannot figure out is:
$$F = \{x \mid x \in A \text { and } (x \in B \implies x \in C)\}.$$
Here is what I have done so far.
\begin{align*}
x \in A \text{ and } (x \in B \im... |
H: Integral $\int_0^{\infty} \frac{\sin^3{x}}{x} \; dx$
I need help with this integral, please: $$\int_0^{\infty} \frac{\sin^3{x}}{x} \; dx$$
I know the Dirichlet integral $\int_0^{\infty} \frac{\sin{x}}{x} \; dx=\frac{\pi}{2}$ but what about with the cube of $\sin{x}$ in the numerator? I tried using Feynman's method... |
H: Projections on Normed Spaces
Let $\mathbb{E}$ be a (real or complex) Banach space (complete normed space). Let $P$ be a projection ($P^2=P$) from $\mathbb{E}$ into itself.
Is it necessarily that the norm of $P$ equals to $1$?
AI: This is not true in general (e.g. with $P=0$). However, if the projection is orthogona... |
H: If $(a \in A) \land (b \in B) \implies (a \in B) \land (b \in C)$ is true, then is $a \in A \implies a \in B$ also true?
Let A,B and C be sets.
If $(a \in A) \land (b \in B) \implies (a \in B) \land (b \in C)$ is true, then is $a \in A \implies a \in B$ also true?
Thanks!
AI: Let $X = A \cup B \cup C$.
Assume that ... |
H: What is $\lim_{n \to \infty} \sqrt{\frac{2^n+20^n-7^{-n}}{(-3)^n+5^n}}\;? $
I'm struggling to find following limit:
$$\lim_{n \to \infty} \sqrt{\frac{2^n+20^n-7^{-n}}{(-3)^n+5^n}} $$
Could anyone help me with a little explanation of that?
AI: $$\lim_{n\rightarrow +\infty}\sqrt{\frac{2^n+4^n-7^{-n}}{(-3)^n+5^n}} = \... |
H: Limit of an expression to $e$
Can you explain why its $e$?
I know limits like:
$$
(1+\frac{1}{x})^x \to e
$$
And similar limits.
But i dont understand how they got to $e$ there.
Help?
Thanks.
AI: Observe that
$$\frac{\sin\frac1{n^2}}{\cos\frac1n}n^2=\frac{\sin\frac1{n^2}}{\frac1{n^2}}\frac1{\cos\frac1n}\xrightarro... |
H: Finding $\mathbf{u} \cdot \nabla \mathbf{u}$ in cylindrical coordinates
Evaluate $\mathbf{u}\cdot\nabla\mathbf{u}$ (the directional derivative of $\mathbf{u}$ in the direction of $\mathbf{u}$)in cylindrical coordinates $(r, \phi,z)$, where $\bf{u}=e_{\phi}$.
The textbook that I am reading uses a purely vectorial ap... |
H: Prove that if $x^n + a_{n-1}x^{n-1}+ \dots + a_0 = 0$ for some integers $a_{n-1}, \dots, a_0$, then $x$ is irrational unless $x$ is an integer.
This is an exercise from Spivak's "Calculus" 4th edition.
18.a. Prove that if $x$ satisfies
$$x^n + a_{n-1}x^{n-1}+ \dots + a_0 = 0$$
for some integers $a_{n-1}, \dots, a_... |
H: What is the probability that 8 unique and randomly selected two digit numbers will be in ascending order?
A computer will randomly generate 8 unique (no repeated numbers), two-digit numbers (10-99) and assign them a position from 1-8. I'm trying to find out the probability that the numbers will be in ascending orde... |
H: Sum of two indpendent Poisson Distribution graph
Let $X$ and $Y$ be independent. I know that if $X$ ~ $Poisson (a)$ and $Y$ ~ $Poisson (b)$ then $X + Y$ ~ $Poisson (a+b)$, but I don't fully understand why. I know how to develop the equation for it, but i'm lacking an intuitive sense of how this changes from a graph... |
H: Inverse trigonometric equation and finding value of given expression
If $\arctan(4) = 4 \arctan(x)$ then value of $x^5-7x^3+5x^2+2x+9870$ is?
I used $2\arctan(x) = \arctan(2x/1-x^2)$ twice for RHS of the equation which gave me $x^4+x^3-6x^2-x+1=0$ and I am clueless how to proceed after that.
Also I am not sure abou... |
H: Calculating $\int_0^{2\pi} \frac{1}{3 + 2 \cos(t)}dt$ using Residue Theorem
My complex analysis has the following exercise in the end of the Residue Theorem chapter:
Evaluate the integral $$\int_0^{2\pi} \frac{1}{3 + 2 \cos(t)}dt$$
Because this is the first exercise on the residue theorem they gave the following ... |
H: Deriving Law of Cosines from Law of Sines
How to eliminate $\alpha$ from the Law of Sines of plane trigonometry
$$ \dfrac{a}{\sin \alpha}= \dfrac{c}{\sin \gamma} =\dfrac{b}{\sin (\gamma+\alpha)} =2R $$
in order to arrive at the Law of Cosines
$$ c^2= a^2+b^2-2 a b \cos \gamma \;?$$
Starting to isolate $\alpha$
$$ ... |
H: Galois group of $f := X^6 - 6 ∈ \Bbb Q[X]$
A quick sanity check:
A splitting field for $f$ over $\Bbb Q$ is $L := \Bbb Q(\zeta_3, \sqrt[6]{6})$. It is of degree 12 over $\Bbb Q$, so Gal$(f)$ will be a group of order 12. An automorphism of $L$ must send $\sqrt[6]{6}$ to $± \zeta_3^k \sqrt[6]{6}$ for some $k ∈ \{0,1,... |
H: Modulus operation to find unknown
If the $5$ digit number $538xy$ is divisible by $3,7$ and $11,$ find $x$ and $y$ .
How to solve this problem with the help of modulus operator ?
I was checking the divisibility for 11, 3:
$5-3+8-x+y = a ⋅ 11$ and $5+3+8+x+y = b⋅3$ and I am getting more unknowns ..
AI: From modulus... |
H: Find outline of connected points in a plane
Please note that this is very similar to this question but it is not the same question.
My situation is as follows:
I have an arbitrary amount of points on a 2D plane that are connected by an arbitrary amount of lines, following three rules:
The number of lines connected... |
H: Find $F'(x)$ of $F(x) = \int_{0}^{x^2}e^{t^2}dt$
For
$$
F(x) = \int_{0}^{x^2}e^{t^2}dt
$$
I need to find $F'(x)$.
The answers say its:
$$
F'(x) = 2xe^{x^4}
$$
I need help understand how they got to this. I try to find the integral of $\int e^{t^2}dt$ but its not working, i tried integration by parts and substitutio... |
H: Complex numbers, set of values for which z will be purely real or imaginary.
A complex number z is given by $ z = \frac{a+i}{a-i}, a∈R$.
Determine the set of values of a such that
(a). z is purely real;
(b). z is purely imaginary.
(c). Show that |z| is a constant for all values of a.
Hi all,
I solved the question... |
H: Does a dense subset of a projective variety has the same dimension as the variety?
I know that an open subset of a projective variety, which is dense, has the same Krull dimension as the projective variety and I also know that, in general, a dense subset may not have the same Krull dimension as the set. Therefore I... |
H: Calculating the arc length of a radical function
I am very new to calculus and StackExchange so I'm sorry if I make any mistakes. I want to work out the arc length of:
$y = \sqrt{5x} - 2.023, [0.075, 0.58]$.
I have used the definition of a definite integral and got
$\int_{0.075}^{0.58} \sqrt{1+\left(\frac{\sqrt{5}}... |
H: $\overline{A}=A\cup \{p\}$
Let $X$ be a Hausdorff space and suppose that $x_n \rightarrow p$. Let $A$ be the set $\{x_1,x_2,\dots\}$, how to prove that $\overline{A}=A \cup \{p\}$.
The inclusion $A \cup \{p\} \subset \overline{A}$ is trivial. The other inclusion is easily proven in the case where $X$ is a metric sp... |
H: Prove among some irrational numbers, there exists some numbers in that set s.t. each pairwise sum is irrational.
This question is basically a weaker statement of the question found here: We have $2n+1$ irrational numbers, then exists $n+1$ of them such that every subset of this set with $n+1$ elements has the sum a... |
H: Dimension of a subspace of a vector space
Let $V$ = $P_{n}(\Bbb{R})$ be a vector space of polynomials with real coefficients up to degree $n$.
Let $W = \{ p(x)\in V\mid p(a) = p'(a) = p''(a)=\ldots=p^{(r)}(a) = 0 \}$
What is the dimension of $W$?
I can notice that if $p(x)$ belongs to $W$ then $x-a$ will be a fact... |
H: Can an uncountable union of distinct finite sets be countable or even finite?
Does there exist an uncountable set of distinct finite sets such that their union is countable, or even finite?
AI: There does not. Suppose that $\mathscr{F}$ is a family of finite sets, and let $X=\bigcup\mathscr{F}$. If $X$ were countab... |
H: Transfinite induction
Is it true that while using transfinite induction we dont need to prove the zero case?
because, if we want to prove some property $ \psi $ , we assume that for any $ x\in A $ if for any $ y\leq x $ it follows that if $ \psi\left(y\right) $ then also $ \psi\left(x\right) $ holds. the minimum ob... |
H: Strictly more holomorphic functions on annulus than on punctured disc
Consider the punctured disc $D=\{z\in\mathbb{C}:0<|z|<R\}$ and the annulus $A=\{z\in\mathbb{C}:r<|z|<R\}$. It is clear that every function holomorphic on $D$ is also holomorphic on $A$. But I need to show that there are strictly more functions ho... |
H: Sum of first K primes is triangle number
I was reading something this morning and came across the fact that 28 is both the sum of the first five prime numbers and of the first seven natural numbers. Naturally, I then tried to find other numbers U such that for some integers n and k
$$U=\sum_{a=1}^{n}a=\sum_{a=1}^{... |
H: How to make inverse to work here?
I have this equation:
$$\sqrt{5 - x} = 5 - x^2$$
My current approach is - I note that if I will let: $f(x) = \sqrt{5 - x}, g(x) = 5 - x^2$ then I will have $f(g(x)) = g(f(x)) = x$ Or, in other words, $f(x) = g^{-1}(x)$ (they're inverse) which means that if they intersect, then they... |
H: Two circles of different radii are cut out of cardboard...
Two circles of different radii are cut out of cardboard. Each circle is subdivided into $200$ equal sectors. On each circle $100$ sectors are painted white and the other $100$ are painted black. The smaller circle is then placed on top of the larger circle,... |
H: For a module $M$ over a ring $R$, the class of modules generated by $M$ is closed under direct sums.
Let $M$ be a module over a ring $R$, we define $Gen(M)$ as the class of all modules over $R$ which are generated by $M$, that means that if $N \in Gen(M)$ there is an epimorphism $f:M^{(X)} \twoheadrightarrow N$ whe... |
H: Bounded linear operator of $\ell^{2}(\mathbb{N})$ which is normal but not self-adjoint
My question is : Does there exist a bounded linear operator $T:\ell^{2}(\mathbb{N})\rightarrow\ell^{2}(\mathbb{N})$ which is normal but not self-adjoint?
Just to be clear, if $H$ is an Hilbert space, then a bounded linear operato... |
H: How to computer $f(\frac{1}{2})$ given $f(f(x)) = x^2 + \frac{1}{4}$?
I have observed that $f(f(\frac{1}{2})) = \frac{1}{2}$ and $f(f(f(x))) = f(x^2 + \frac{1}{4})$, and when $x = \frac{1}{2}$, we have $f(\frac{1}{2}^2 + \frac{1}{4}) = f(\frac{1}{2})$. But I don't know how to proceed, or if any of these observation... |
H: What is the domain of the function $\ln\left( \sin \frac{x}{x+1}\right )$
I have to calculate the domain of the function $$\ln \left(\sin \frac{x}{x+1}\right).$$
So I have to put $$\sin \frac{x}{x+1}>0 \implies 2k \pi < \frac{x}{x+1} < \pi (1+2k).$$
I should isolate $x$ from this couple of inequalities but I'm afra... |
H: Convergence of $s_{n+1}=\sqrt{1+s_n}$
Does the sequence $s_{n+1}=\sqrt{1+s_n}$ always converge, no matter what the initial value of $s_1$ is?
Is this sequence always increasing and bounded? I think so, but what's throwing me off is that to find what the sequence converges to, we just solve $s^2=1+s$ to get $s=\frac... |
H: Can I Construct Anything (Within Reason) When Building a Proof?
I apologize if this is a silly question, but I'm curious about this. Is it acceptable to claim anything (as long as it's logically sound) during construction when building a geometric proof? For example, let's say I have $\triangle$ ABC and $\triangle$... |
H: Are the average payoffs in the convex hull?
This is an exercise in the Steve Tadelis An Introduction to Game Theory book:
(10.12) Folk Theorem Revisited: Consider the infinitely repeated trust game described in Figure 10.1
(a) Draw the convex hull of average payoffs.
So, this is pretty easy:
The vector of payoffs ... |
H: Prove or disprove if bijectivity is kept after set difference with a countable set
I ran into some claims of my own while doing an exercise that I would like to be true for my solution to be correct. Are the following true or false, how to prove or disprove it?
Let $ \varphi : \mathbb{N} \setminus I \rightarrow \m... |
H: Is a field of characteristic zero where -1 is a square algebraically closed?
Let $F$ be a field of characteristic zero where $-1$ is a square. Must $F$ be algebraically closed?
AI: Unless you give further assumption: no. Consider $F=\Bbb Q(i)$. This is a field extension of $\Bbb Q$ and thus of characteristic $0$ wi... |
H: First-order logic equivalence proof
I have a question on how to prove
$$(\neg \forall x \, P(x)) \rightarrow (\exists x \, \neg P(x)) $$
with a natural deduction proof, where $P$ is a predicate.
I especially have problems with what to do about the negation in front of $\forall$.
In general, how should I go about th... |
H: What does the notation (<,,>)=<,,> mean?
I came across notation of (<,,>)=<,,>
I'm not sure what "<" mean.
Does it mean:
\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
Which would transform to:
\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}
AI: The "$\langle \, \rangle$" is just the notation for a vector. Though ... |
H: Galois group of $f := X^6 - 6 \in \Bbb F_5[X], \Bbb F_7[X]$
In $\Bbb F_5[X]$
What we know
We have that $f = X^6 - 1 = (X-1)(X+1)(X^4 + X^2 +1)$, so $Ω^f_{\Bbb F_5} = Ω^{X^4 + X^2 + 1}_{\Bbb F_5}$. Evaluating $f$ in all elements of $\Bbb F_5$ shows there are no other roots of $f$ there. I don't believe that allows u... |
H: Why is $\frac{ab}{c}=\frac{a}{c}b$
Simple question. Why is the following true?
$$\frac{ab}{c}=\frac{a}{c}b$$
AI: Dividing by $1$ does not change a number, so $b=\frac b1$
When multiplying two fractions, simply multiply the numerators and multiply the denominators.
So
$$\frac acb=\frac ac\frac b1=\frac{a\times b}{c\... |
H: Convergence test for an integral of bounds $0$ to $1$
How to prove that the integral: $\int_{0}^{1} \frac{dx}{2\sqrt{x}(x+1)}$ converges using the convergence test?
I know that $\int_{0}^{1} \frac{1}{x^{\alpha}} d x$ converges $\iff \alpha < 1$. But in my case, the denominator does not look like that, and I find it... |
H: For graph G and integer k, exist coloring $V\to [k]$ s.t. $(1-1/k)$ of every vertex get different color.
For any simple graph $G$ and positive integer $k$, there exists a coloring $c: V(G) \rightarrow[k]$ such that for every vertex $v$ of $G,$ at least $(1-1/k)$-fraction of its neighbors get different colors.
I am ... |
H: Integrating $ \int\frac{5}{\ 16 + 9\cos^2(x)}\,dx $
I am trying to integrate the following:
$$
\int\frac{5}{\ 16 + 9\cos^2(x)}\,dx
$$
I have applied the following substitution:
$$
x = \tan^{-1}u
$$
I have simplified the denominator through using the following trig identity:
$$
\cos^{2}x = 1/(1 + \tan^{2}x)
$$
$$
16... |
H: How many partitions of $n$ different objects into equinumerous parts are there?
How many partitions of $n$ distinct objects are there, given that all parts are equinumerous? (Let us not consider the empty partition and the identity partition.)
The cases when $n$ is the unit, or when $n$ is prime, are trivial. In th... |
H: In an algebraically closed field, does every nonzero element have n distinct n-th roots?
Let $F$ be an algebraically closed field. This certainly implies that every non-zero element $x$ of $F$ has at least one $n$-th root, for each positive integer $n$. Does it in fact imply the condition that every non-zero elemen... |
H: Given polynomial $P(x) = x^2 + ax + b$ and that there only exists one $c$ such that $P^2(c) = c$. Calculate the minimum value of $a + b + c$.
Given polynomial $P(x) = x^2 + ax + b$. Knowing that there only exists one value of real $c$ such that $P^2(c) = c$, calculate the minimum value of $a + b + c$.
Notation: ... |
H: Understanding the Big-O of this summation
I understand the Big-O for this summation is $O(n^3)$.
I tried to break it down algebraically and I seem to be getting $O(n^3):$
SUMMATION:
\begin{align}
\sum\limits_{i=1}^n i(n-i)
&= \sum_{i=1}^n i \cdot \sum_{i=1}^n (n-i)\\
&= \sum_{i=1}^n i \cdot (\sum_{i=1}^n n- \sum_{... |
H: Determinig when a Trig function $\cos(\sqrt{x+3})$ is negative
So I have this trig. function:
$\cos(\sqrt{x+3})$
I want to know when this function will be negative.
I know that cos is negative in 2nd and 3rd quadrants but I'm not able to think in terms of quadrants here?
I would also like to know the range of this ... |
H: Weibel 1.2.7: Existence of exact sequences of complexes.
If $C$ is a complex, show that there are exact sequences of complexes:
$$
0 \longrightarrow Z(C) \longrightarrow C \stackrel{d}{\longrightarrow} B(C)[-1] \longrightarrow 0;
$$
$$
0 \longrightarrow H(C) \longrightarrow C / B(C) \stackrel{d}{\longrightarrow} Z... |
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