text stringlengths 83 79.5k |
|---|
H: Vector Congruence (Beachy & Blair 2.2 - Equivalence Classes) proofs
I need some help with exercise 10 in Chapter 2.2 of Beachy and Blair's Abstract Algebra with a Concrete Introduction.
The question is as follows:
Let $W$ be a subspace of a vector space $V$ over $\mathbb{R}$ (that is the scalars are assumed to be r... |
H: Does Fubini's Theorem hold for the Polar Form of Double Integrals?
Simply put, does Fubini's Theorem hold for double integrals expressed as in polar terms? I.e., does the following hold given that all integration bounds are constants:
$$ \int_\alpha^\beta \int_a^b f(r, \theta) \, r \, \text{d}r \, \textrm{d}\theta ... |
H: Using Rouche's Theorem to find the number of solutions of $f(z)=z$ in the open unit disc
How many roots does the equation $f(z)=z$ have in the circle $|z|<1$ if for $|z|\leq 1$, $f(z)$ is analytic and satisfies $|f(z)|<1$?
My idea: I figured I could do this pretty easily using Rouche:
Consider $|z|=1$, and let $... |
H: Solve the equation $\frac{1}{x^2+11x-8} + \frac{1}{x^2+2x-8} + \frac{1}{x^2-13x-8} = 0$
Problem
Solve the equation $$\frac{1}{x^2+11x-8} + \frac{1}{x^2+2x-8} + \frac{1}{x^2-13x-8} = 0$$
What I've tried
First I tried factoring the denominators but only the second one can be factored as $(x+4)(x-2)$.
Then I tried sub... |
H: How do two conjugate elements of a group have the same order?
I'm reading group action in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
I have a problem of understanding the last sentence:
Since conjugation is an automorphism, any two conjugate elements have the same order.
Assume $x,y \in G$ are ... |
H: The same result for $\mathbb{C}$ is true for algebraically closed field?
The following result about polynomials is known:
Proposition: Let $K$ be a subfield of $\mathbb{C}$, $f(x) \in K[x]$ a polynomial with degree $n \geq 1$ and $\alpha \in \mathbb{C}$ a root of $f(x)$. Then
a) $\alpha$ is a simple root of $f(x)$ ... |
H: Proof of the Fundamental Theorem of Algebra: filling in some intermediate steps
I'm familiar with Rouche's theorem in the following form:
If $f, g$ are analytic on a domain $\Omega$ with $|g(z)| < |f(z)|$ on $\partial \Omega$, then $f$ and $f+g$ have the same number of zeros in $\Omega$.
I'm walking through how t... |
H: Denotation of the range of a function using its definition
Is this expression allowed in a strict sense?
Consider a function $f:[a,b] \rightarrow [f(a),f(b)]$ defined by $f(x) = x$.
What I mean by it is that
Let $g:[a,b]\rightarrow \mathbb{R}$ defined by $ g(x) = x $. Consider $f:[a,b]\rightarrow g([a,b])$ where $f... |
H: Caculate $\int_{-2}^{2}\ln(x+\sqrt{1+x^2})\ln(1+x^2)dx$
$\int_{-2}^{2}\ln(x+\sqrt{1+x^2})\ln(1+x^2)dx$
My work:
The origin $=-\int_{-2}^{2}\ln(-x+\sqrt{1+x^2})\ln(1+x^2)dx$, I think maybe we can exploit some integral properties related to odd function by playing with bounds but I don't see it.
EDIT:The origin $=-... |
H: Diameter of a ball in a metric normed space
Could you help me with the following please:
Prove that the diameter of a ball in a normed space it is twice its radius.
My attempt:
$diam(A)=\sup \{d(x,y): x,y\in A\}$
The first inequality is evident $diam(B_\epsilon(a))\leq 2\epsilon$, but for the second I have the fo... |
H: Optimize a multivariable function when the constraint is an inequality.
I know how to optimize functions with given constraints when those constraints are equalities, i.e. $g(x, y) = c$ with Lagrange multipliers. However, I have a problem where the constraint is an inequality and I'm wondering how (or if) I could d... |
H: prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected.
Define a set $A$ to be disconnected iff there exist nonempty relatively open sets $U$ and $W$ in $A$ with $U\cap W = \emptyset$ and $A = U\cup W.$ Define a set $A$ to be connected iff it is not disconnected.(there are... |
H: $|f(z)| \leq \frac{1}{1-|z|}$ implies $|f'(z)| \leq \frac{4}{(1 - |z|)^2}$
I have the following question:
Suppose that $f$ is analytic in the unit disk and $|f(z)| \leq \frac{1}{1-|z|}$. Prove that $|f'(z)| \leq \frac{4}{(1 - |z|)^2}$ in the open unit disk.
I am not really sure where to start. Any help would be a... |
H: If $f$ is odd and periodic then a translation of $f$ is even?
Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a odd and periodic function, with period $L>0$. If we define
$$g(x):=f\left(x-\frac{L}{2}\right), \; \forall \; x \in \mathbb{R},$$
then $g$ is even?
I tried to prove it, as follows: let $x \in\mathbb{R}$... |
H: How to compute a simple sum
In the book I am reading the author left some excersises for the reader, I happend to be stuck at this sum
$$\sum_{n=1}^k{n!(n^2+n+1)}$$
So far I have tried to factorize the polynomial, and also tried to split the sum. I know how to compute $n\ n!$, but I have no idea on the other terms... |
H: How can I find a Fraisse's paper
I want to see Fraisse's "Sur certaines relations qui généralisent l’order des nombres rationnels". But at https://gallica.bnf.fr/ark:/12148/bpt6k3188h, the search returns nothing. I wonder how to find this paper.
AI: I believe it is here. (EDIT: as kimchi lover comments, the issue s... |
H: How do I integrate $\int\frac{x^3}{\sqrt{1+x^2}}$ by parts?
So I've just recently begun integration and now we're doing integration by parts. We've been told about ILATE (some sort of acronym to helps us remember which function we integrate and which to differentiate).
But that aside can we perform by part integrat... |
H: Proving a bound on the difference between expected value of a continuous random variable and the expected value sampled on all integers
I need to prove that for a random variable $X$ with
$$
X \geqslant 0
$$
it's true that
$$
\sum_{n=1}^{\infty} P(X \geqslant n) \leqslant E[X] \leqslant 1 + \sum_{n=1}^{\infty} ... |
H: Find the basis for column space $A=\left[\begin{smallmatrix}1&-1&3\cr 5&-4&-4\cr 7&-6&2\end{smallmatrix}\right]$
Find the basis for column space $$A=\begin{bmatrix}1&-1&3\cr 5&-4&-4\cr 7&-6&2\end{bmatrix}.$$
I'm quite confused because I thought there were two methods: using the transpose or not.
By not using the t... |
H: Which Greek letter is commonly used to represent a count?
Which Greek letter is commonly used to represent a count? For example, the Greek letter sigma ($\Sigma$) is commonly used to represent a sum.
AI: While your "count" is quite vague, here is a possibility:
The count of elements in a set:
What symbol gives the ... |
H: Let $f$ be a continuous function on $\mathbb{R}$ satisfying $\int_\mathbb{R}|f(x)|dx<\infty$. Can we conclude that $\sum_\mathbb{Z}|f(k)|<\infty$?
Let $f$ be a continuous function on $\mathbb{R}$ satisfying
$$\int_\mathbb{R}|f(x)|dx<\infty.$$
Can we conclude that
$$\sum_\mathbb{Z}|f(k)|<\infty?$$
Note: Continuity i... |
H: How can i find $\int _0^{\infty }\ln ^n\left(x\right)\:e^{-ax^b}\:dx$
I tried using certain substitutions like $u=ax^b$ but that lead to $\displaystyle\frac{1}{a^{\frac{1}{b}}b^n}\int _0^{\infty }e^{-u}\:\ln ^n\left(\frac{u}{a}\right)u^{\frac{1}{b}-1}du\:$
i tried to use special functions to evaluate this but that ... |
H: Converting between bound on probability measures and densities
Suppose that $P$ and $Q$ are two probability measures on the same probability space with $P(A) \leq c Q(A)$ for each (measurable) set $A$.
Is it true that $dP/dQ$ is then bounded by $c$ $P$-almost surely?
AI: Let $f$ denote the Random-Nikodym derivative... |
H: Can I determine if the result of a pairing function is the from the inverse of a given pair?
For example, say you had the following results:
f(a, b) = c
f(b, a) = d
Is there a pairing function that would allow for determining that c is sort of the "inverse" of d without de-pairing the results?
AI: For this pairing ... |
H: How can we find the crossing point of two lines with a prescribed angle?
In the 3D space, we have two given points of $P$ and $Q$.
Line $A$ passes through the point $P$ and whose angle with the x-axis is $\theta$ and with the z-axis $\phi$.
Line $B$ passes through the point $Q$ and has an angle of $\alpha$ with the... |
H: What's the proof for this summation with dependent variables?
Given $0\leqslant i<j\leqslant n$
$\sum_{i=0}^n\sum_{j=0}^n i = \sum_{i=0}^n(n+1)*i$
It seems to work like a nested loop and gave the right answer when substituting n for any number but I don't know how to derive it.
AI: $ \sum\limits_{j=0}^{n}i=i(n+1)$ ... |
H: Conditions for symmetric, Toeplitz $\mathbf{M}$ with nonnegative elements to have inverse with nonnegative elements
Problem
Suppose we have symmetric, Toeplitz matrix $\mathbf{M}$ such that
$$
\mathbf{M} = \begin{bmatrix}
m_0 & m_1 & m_2 & m_3 & \cdots &m_{n-1} \\
m_1 & m_0 & m_1 & m_2 & \cdots & m_{n-2} \\
m_2 & ... |
H: Why can't the "chain rule" of derivatives be used to differentiate 3sin(x)?
My understanding is that functions of the form $f(g(x))$ can be differentiated using the "chain rule", where $$\frac{d}{dx}f(g(x)) = f'g(x) \cdot g'(x)$$
I was trying to apply that logic to the following equation:
$$\frac{d}{dx}3sin(x)$$
To... |
H: Showing a function is constant - Complex analysis
I am trying to solve the following problem.
$f(z)=u(x,y)+iv(x,y)$ is an analytic function in $D$ ($D$ is connected and open).
If $u, v$ fulfill the relation $G(u(x,y), v(x,y))= 0 $ in $D$ for some function ($G:\mathbb{R^2}\to\mathbb{R}$) with the property
Show that ... |
H: Write $f(x) = \int_{-2}^{x}t|t-1|\,dt$ without the sign of integral.
Here is my attempt:
$\displaystyle \int_{-2}^{x}t|t-1|dt$
=$\begin{cases} \displaystyle\int_{-2}^{x}t^2-tdt, \; t \ge 1 \\\displaystyle \int_{-2}^{x}-t^2+tdt, \; t <1\end{cases}$
=$\begin{cases} \displaystyle\dfrac{x^3}{3}-\dfrac{x^2}{2}-\dfrac{(-... |
H: Simplifying $\frac{1}{\sqrt{x-i}}\left\{1-2 \sum_{n=1}^{\infty}(-1)^{n-1} \exp \left(-\pi n^{2}\left(\frac{1}{x-i}+i\right)\right)\right\}$
This is related to another recent question of mine. Considering that
$$
\psi(x)=\sum_{n=1}^{\infty} \exp \left(-n^{2} \pi x\right)
$$
has the finctional equation
$$
\frac{1+2 \... |
H: Proving that $\vec{r'}(t)$ is orthogonal to $\vec{r''}(t)$
With a given nonzero vector $\vec{r}(t)$, how do I that $\vec{r'}(t)$ is orthogonal to $\vec{r''}(t)$? The length ($||\vec{r'}(t)||$ is constant.)
This is what I have tried so far.
Let $\vec{r}(t)= <f(t),g(t),h(t)>$. Then, $\vec{r'}(t)$ is $<f'(t),g'(t),h'(... |
H: Rather easy arctan limit (without L'Hôpital)
It's not an abomination of a limit, but I can't wrap my head around it. This is a factor of a bigger limit that was plausible enough, but this little bit kept me stuck for too much time. Here it is:
$$\lim_{x\to0^+}\frac{x}{\pi-3\arctan{\frac{\sqrt{3}}{1+x}}}$$
I would r... |
H: Root test for complex series and cancelling powers with absolute values
The root test for convergence of a complex power series is given as
$$\lim_{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|} = L$$
If $a_n = \frac1{(1+i)^n}$ then I read that when applying the root test I can just remove the powers since they... |
H: Example of dependent random variables $X,Y$ such that there is no measurable $f$ with $Y=f(X)$
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $X,Y$ be real-valued random variables on $(\Omega,\mathcal A,\operatorname P)$.
If $X$ and $Y$ are dependent, can we always find a Borel measurable $f:\... |
H: Getting a probability curve (Central Limit Theorem)
in a game I play there's a chance to get a good item with 1/1000.
After 3200 runs I only got 1.
So how can I calculate how likely that is and I remember there are graphs which have 1 sigma and 2 sigma as vertical lines and you can tell what you can expect with 90%... |
H: In how many ways you can make up 20 pence using 20p, 10p, 5p, 2p and 1p coins
I would like to ask in how many ways you can make up 20 pence using 20p, 10p, 5p, 2p and 1p coins. What I was thinking is that I have counted there are 11 ways to make up 10 pence using 10p, 5p, 2p and 1p coins. So, the number of way to m... |
H: If $X_n$ converges to $X$ in $L_p$ and $Y_n$ converges to $Y$ in $L_p$ then $X_n + Y_n $ converges to $X + Y$ in $L_p$
I want to show that if $X_n \xrightarrow{L^p} X$ and $Y_n \xrightarrow{L^p} Y$ then $X_n + Y_n \xrightarrow{L^p} X + Y$ ($p \geq 1)$.
My idea is to use the following facts (whose proofs I won't giv... |
H: how to integrate $\int\frac{1}{x^2-12x+35}dx$?
How to integrate following
$$\int\frac{1}{x^2-12x+35}dx?$$
What I did is here:
$$\int\frac{dx}{x^2-12x+35}=\int\frac{dx}{(x-6)^2-1}$$
substitute $x-6=t$, $dx=dt$
$$=\int\frac{dt}{t^2-1}$$
partial fraction decomposition,
$$=\int{1\over 2}\left(\frac{1}{t-1}-\frac{1}{t+1... |
H: Local trivialization of two vector bundles?
Let $M$ be a smooth manifold, and $E_1\xrightarrow{\pi_1}M,E_2\xrightarrow{\pi_2}M$ are two vector bundles of rank $m$ and $n$. By definition, there exist two open cover $\{U_i\}_{i\in I}$ and $\{V_j\}_{j\in J}$ such that $\pi_1^{-1}(U_i)\cong U_i\times \mathbb{R}^{m}$ an... |
H: Can an autonomous differential equation be nonhomogeneous?
When a differential equation $\dfrac{dy}{dt}=f(y)$ does not depends on $y$ then it is autonomous, and it can't be non-homogeneous?
Can anyone explain this? Thanks!
AI: $\frac{dy}{dt}=f(y)$ is an autonomous differential equation irrespective of whether $f$ d... |
H: How can I differentiate ($2x+1$)($x-5$) by expanding then differentiating?
The question is to differentiate ($2x+1$)($x-5$) by expanding then differentiating each term
But I am running into problems and am running into the wrong answer
I end up with
$4x-15$
but answer is $2x$$^2$($2x-3$)
How?
AI: Both answers are... |
H: Did my math textbook make a typo?
My textbook defined connectedness in graphs in the following way:
A graph G(V, E) is said to be connected if for every pair of vertices u and v there is a path in G from u to v.
The textbook then asks the reader to complete the following exercise:
Show by giving an example tha... |
H: Prove a function is Lebesgue measurable
Problem
$f$ is defined on a measurable set $E$, $D$ is a dense subset of $\mathbb{R}$ ($\overline{D}=\mathbb{R}$), prove $\forall\ r\in D$, if the set $\{x\in E:f(x)>r\}$ is measurable then $f$ is measurable on $E$
Since $D$ is dense in $\mathbb{R}$, $\forall\ a\in\mathbb{R... |
H: Proving two different expressions of non-centrality parameters are equivalent
I am stuck in proving $$\sum_{i=1}^{K}\xi_i(\mu_i - \bar{\mu})^2 = \sum_{i,j}\xi_i\xi_j(\mu_i - \mu_j)^2,$$
where $\bar{\mu} = \sum_{i=1}^{K}\xi_i\mu_i$ and $\sum_{i=1}^{K}\xi_i = 1$.
I am not sure whether it is true or not.
AI: Let $P = ... |
H: Diamond Distribution in system K (Garson Modal Logic exercise 1.8)
I want to prove $\Diamond (P \lor Q) \Rightarrow \Diamond P \lor \Diamond Q$
It was a biconditional, but I have proved the other one. Thanks for the answer. Please use Garson's method. Thanks. I am stuck on this, any help. Please.
I don't have any w... |
H: Joint density of $(V,Z)$
The joint density of $(X,Y)$ is $f_{XY}(x,y)=k(1-\sqrt{\frac{y}{x}}),0<x<1,0<y<x$.
Find the value of $k$ and say if $X$ and $Y$ are independent or not.
$\rightarrow k=6 \Rightarrow f_{XY}(x,y)=6(1-\sqrt{\frac{y}{x}})$; $f_X(x)=2x$ and $f_Y(y)=6(1+y-2^{\frac{1}{2}})$, so $f_X(x)f_Y(y)\neq ... |
H: Show $A=(I-S)(I+S)^{-1}$ is an orthogonal matrix if $S$ is a real antisymmetric matrix
I am trying to show that if $S$ is a real antisymmetric matrix ($S^T=-S$), then $A=(I-S)(I+S)^{-1}$ is an orthogonal matrix. $I$ is the identity matrix.
To show that $A$ is orthogonal, i.e. $A^T=A^{-1}$, I first calculated $A^T... |
H: How I can prove this?
I have to prove the next proposition and I think I have to do it reducing it to the absurd but I don't know how to do it.
Being $V$ a finite vectorial space and $v_1,v_2\in V$ ($v_1 \neq v_2$). Prove that $\exists \phi\in V^*$ where $\phi(v_1)\neq\phi(v_2)$
AI: Obviously we have to assume tha... |
H: Example of Devil's nested radicals
Let the following nested radical :
$$\sqrt{665+2x}=x$$
There is a hidden quadratic equation and the result is :
$$x=\sqrt{666}+1$$
So we see the number $666$ appear .
My question :
Do you know (more or less trivial) nested radical where the devil's number appear ?
Can we build it ... |
H: Finding a basis given two matrix representations
Given $T:\mathbb{R^3}\rightarrow \mathbb{R^3}$ defined by: $$ [T]^E_E=\begin{pmatrix}
-2 & 4 & 5 \\
-8 & 12 & 12 \\
8 & -11 & -10 \\
\end{pmatrix} $$
I need to find a basis $B=(\vec b_1,\vec b_2,\vec b_3)$ such that: $$ [T]^B_B=\begin{pmatrix}
0 &... |
H: Writing an integral in terms of Lebesgue measure
I have seen the following identity, but I think that it is not correct:
Given a positive measure $\mu$ in $\mathbb{R}^N$, we have that
$$\displaystyle\int_{\|x-y\|^{-\gamma}\leq R}\|x-y\|^{-\gamma}d\mu(y)=\displaystyle\int_0^R \mu\{y\in \mathbb{R}^N:\|x-y\|^{-\gamma... |
H: How to determine when the following sum will be prime?
I was playing around with dates in my head and thought of the following prime number problem.
Problem:
The following (numerical) days of the month are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
From this, we know the following (numerical) months are prime: ... |
H: How to prove $f$ is an even function?
Suppose $g$ is any odd function, and we have $\int_{-1}^{1} f(t)g(t)dt=0$ and $\int_{-1}^{1} f(-t)g(t)dt=0$, $f$ and $g$ are continuous on $[-1,1]$, how to show that $f$ is an even function, that is $f(t)=f(-t)$?
I considered the integral $\int_{-1}^{1} (f(t)-f(-t))g(t)dt=0$, a... |
H: Deriving formula for cross-product.
It is given on pg. #106, 107 in the book by: Thomas Banchoff, John Wermer; titled: Linear Algebra Through Geometry, second edn..
Consider a system of two equations in three unknowns:
$$a_1x_1 + a_2x_2 + a_3x_3 = 0$$
$$b_1x_1 + b_2x_2 + b_3x_3 = 0$$
We set $A = \begin{bmatrix} a_1... |
H: iven $f:U\to\mathbb{R}^n$, $U\subset\mathbb{R}^n$ U is convex and $||D_f(x)||<1$ for every $x\in U$. $g$ is diffeomorphism.
Given $f:U\to\mathbb{R}^n$, $U\subset\mathbb{R}^n$ U is convex and $||D_f(x)||<1$ for every $x\in U$.
Setting $g(x)=x+f(x)$ prove $g$ is diffeomorphism.
I've manged to prove the $g$ is one to... |
H: Identity proof with binomial coefficients
I have to prove the following identity and have to idea how to start.
$ { {-n} \choose k} = (-1)^k {{n+k-1} \choose k} $
I know that $ \sum_{k=0}^{n} (-1)^k{n \choose k}=0 $
but I cannot see a way, in which it could help prove the mentioned above.
Thanks for your help
AI: W... |
H: Unable to derive an expression in study of Linear Transformations
While self studying Linear Algebra from Hoffman and Kunze I am unable to derive this following expression.
My question is in 5 th line after the underlined line in blue ie the line " The only terms which survive in this huge sum are the terms where... |
H: Maximum number of possible disjoint subsets of unequal size
Given a set $S$ of size $n$, there is a maximum number of disjoint subsets where the size of each of the subsets is larger than zero and different for each of the subsets. Which is obtained by counting the number of times $(k)$ we must add up the sequence ... |
H: An alternate definition of Limit
Let's say we were to define the concept of limit as follows:
$\displaystyle{\lim_{x \to c}}f(x)=L$ means that for every $x$ in the domain of $f$, there exists an $x_0 \neq x$ in the domain of $f$ such that:
$$|x-c|>|x_0-c|$$and$$|f(x)-L|\ge|f(x_0)-L|$$
I have two questions:
Does th... |
H: $\int\frac {dx}{\sqrt{2ax−x²}}=a^n \sin^{-1}(\frac {x}{a}-1)$
The question is telling us to find n
The options are:
a) 0
b) -1
c) 1
d) none of these
I have tried to solve this by using some general formula of integration yet Iam unable to find the answer
AI: Hint:$$\sqrt{2ax-x^2} =\sqrt{a^2-(x-a)^2}$$
Also, $$\int\... |
H: Graph Theory - Prove complement graph has a diameter of at most $3$
Given a graph $G$ with a diameter of at least 3. Prove that the diameter of $\bar{G}$ (Complement graph) is at most 3.
I got stuck at the very beginning.. why those specific numbers? I mean, I would understand if the graph $H$ would have a diameter... |
H: On a 6th degree polynomial
Say $p(x)$ is a 6th degree polynomial. We know that $p \geq 1, \forall x \in \mathbb R$ and that $$p(2014) = p(2015) = p(2016) = 1$$ while $p(2017)=2$. What is the value of $p(2018)$?
My first approach was to define $q(x) = p(x)-1$ such that its factorization is something like
$$q(x) =... |
H: Identifying extrema of a functional
I'm new on Calculus of variations and I don't figure out how to find a minimum (or maximum) for the following functional
$$ J(f) = \int_{-3}^{-2}(f^2(t)+f'(t)) ~dt . \tag{1}$$
I have tried to use the Euler-Lagrange equation
$$\frac{\partial\mathcal{L}}{\partial f}-\frac{d}{dt}\le... |
H: Integration by parts but not in terms of dx
I have a question regarding the following integration. How do we integral
$\int_0^\infty x d(e^{-sx})$.
The answer is
$xe^{-sx}|_0^\infty - \int_0^\infty e^{-sx}dx$.
I am not sure how to solve this. I can only solve integrals in terms of $dx$ so this one really confuses m... |
H: Does Fréchet derivative needs continuity?
I want to ask a question that I feel everyone acts as if it is a common knowledge. I am really new to functional analysis. How does Fréchet differentiability imply continuity? In the definition I have only continuity of the derivative itself,A, is mentioned:
A mapping F : U... |
H: Solving $\int_{-1}^{1}8x^3-5x^2+4dx$
I've done the following so far:
$$\left.\int_{-1}^{1}\left(8x^3-5x^2+4\right)dx=\left(\frac{8}{4}x^4-\frac{5}{3}x^3+4x\vphantom{\int}\right)\right|_{-1}^{1}$$
$$=\left(\frac{8}{4}-\frac{5}{3}+4\right)=\frac{13}{3}$$
However, I double-checked on wolfram alpha and the solution is ... |
H: Proving that every constructible number is algebraic
I am starting to self study Galois Theory from J.S. Milne's notes on the subject. He claims that
If $\alpha$ is a constructible number then $\alpha$ is algebraic over $\mathbb{Q}$ and $[\mathbb{Q}[\alpha],\mathbb{Q}]$ is a power of $2$.
Now to prove this he use... |
H: Given the equation $\alpha \mathbf{v} + \mathbf{v}\times\mathbf{a} = \mathbf{b}$, solve for $\mathbf{v}$.
I'm reading a textbook at the moment that provides the following linear equation,
$$
\alpha \mathbf{v} + \mathbf{v}\times\mathbf{a} = \mathbf{b},
$$
and asks to solve for $\mathbf{v}$. The form of $\mathbf{v}$ ... |
H: If every element of a subset is strictly smaller than the ones of another, what is the relation between their bounds?
Suppose S is a set with the least-upper-bound property and the greatest-lower-bound property.
Let $X$ and $Y$ be subsets where every $x \in X$ and every $y\in Y$ is such that $x < y$.
I'm ... |
H: Quantum-mechanical Schwarz inequality: Proving $\langle \psi \mid \phi \rangle^* \langle \psi \mid \phi \rangle \ge 0$ for 1D case.
I am currently studying the textbook Mathematical methods of quantum optics by Ravinder R. Puri. When discussing the postulates of quantum mechanics, the author introduces the quantum-... |
H: Calculate a meromorphic function on $\mathbb{C}$ s.t. $g(z) = \sum_{n\geq0}(n+1)T^n$
I have the following power series $f(T)=\sum_{n\geq0}(n+1)T^n$ and I have to calculate a meromorphic function $g$ on $\mathbb{C}$ such that $\forall z \in B_1(0), g(z)=f(z)$
I have tryed the following:
$f(z) = \sum_{n\geq0}nz^n + ... |
H: numerical instability of integer programming
Let us assume the objective function $f$ of some IP looks as follows
$$ f = \sum x_i + \varepsilon \sum y_i.$$
With $\varepsilon$ being very small ($\approx 0.00001$) and $x_i$ and $y_i$ some variables.
There may also be some constraints, but let us not focus on them.
It... |
H: Natural logarithm calculation in RGR example
I'm studying the book Statistical Methods for Research Workers by R.A. Fisher (1934). I'm in chapter 2, called diagrams. It contains an example of Relative Growth Rate of babies’ weight growth in their first 13 weeks of life. I understand how RGR is calculated, but tryin... |
H: Uniqueness of homomorphism -universal property
If $\varphi: R\to A$ is a homomorphism such that for a subset $S$, $\varphi(s)$ is inevertable for every $s\in S$ then there is a unique homomorphism $\varphi^\prime:S^{-1}R\to A$ such that $\varphi = \varphi^{\prime}\circ \pi $. where $\pi:R\to S^{-1}R$ which is defin... |
H: How to prove $a^x\times a^y=a^{x+y}$ for cardinals?
How can I prove this: $$a^x\times a^y=a^{x+y} $$
when $card(A)=a$ , $card(X)=x$ and $card(Y)=y$.
AI: Assuming the axiom of choice, and assuming that $a$ is an infinite cardinal and that $x,y>0$: use the fact that if $\kappa$ and $\lambda$ are infinite cardinals th... |
H: Finding the plane where the line lies on
Having a line and a normal orthogonal to it, how do I find the plane in which the normal will come out from and the line will lie on?
AI: In the three-dimensional affine space $\mathbb{A^3}_{\mathbb{R}}$ (I'm assuming $\mathbb{K}=\mathbb{R}$) a line is defined, in its cartes... |
H: Finding Radon-Nikodym derivative $d\mu/dm$ where $m$ is the Lebesgue measure on $[0,1]$, $f(x)=x^2$, and $\mu(E)=m(f(E))$
Consider the function $f:[0,1]\to \Bbb R$, $f(x)=x^2$. Let $m$ denote the Lebesgue measure on $[0,1]$ and define $\mu(E)=m(f(E))$. Since $f$ is absolutely continuous and nondecreasing, $f$ maps ... |
H: Online resource recommendation for learning about vector analysis
In the last semester I had Vector Analysis lecture. We have seen about some basic geometry of sphere, then we got into basics if vectors that we seen in first semester in mechanics lecture but this was more like linear algebra. Then I learned about l... |
H: Proof that in a field $x=0$ is equivalent to $x=-x$
Let $F$ be a field and $x\in F$. If $x=-x$ and $1\neq -1$, then $$0=\frac{x+x}{1+1}=\frac{1+1}{1+1}x=x.$$
This means that the statement in the title is true if and only if $1\neq -1$.
But how do we know that $1\neq -1$ for an arbitary field?
AI: But how do we know... |
H: Distribution of sum of two independent random variables
Here's a small problem I tried to solve. 2 dice are thrown, let $X$ denote the result of the first dice and let $Y$ denote the result of the second dice. I'm asked to describe the law of $Z=X+Y$.
I tried to solve this problem using the law of total probabiliti... |
H: Prove that triangle $\triangle ABC \cong \triangle G H I$ . Explain each step.
My question:
Prove that triangle $\triangle ABC \cong \triangle G H I$ . Explain each step.
Here are my triangles
I proved that $\triangle ABC \cong\triangle DEF$ because the first sign of equality.
angle $ABC = $angle $DEF$
$AB = D... |
H: Find a suitable polynomial function for the data points: $(-1,1),(0,1),(1,3),(2,1)$.
I'm working through examples on different kinds of interpolation methods, and I came across this video, with the following question:
Find a polynomial equation that best fits the following data points:
$$(-1,1),(0,1),(1,3),(2,1)$$... |
H: Prove that a group has injective homomorphism into direct product of quotients
Herstein problem 2.13.10
Let $G$ be a group, $K_1,..,K_n$ normal subgroups of $G$. Suppose that $K_1\cap K_2\cap...\cap K_n=(e)$. Let $V_i=G/K_i$. Prove that there is an isomorphism of $G$ into $V_1\times V_2\times .. \times V_n$
I tri... |
H: Why $\frac{1}{m}\frac{dm}{dt}= \frac{d}{dt}\left ( \log_{e}m \right )$ is true?
In the explanation of relative growth rate calculation, in chapter 2 of R.A. Fisher's Book Statistical Methods for Research Workers, it is shown the following equality:
$$\frac{1}{m}\frac{dm}{dt}= \frac{d}{dt}\left ( \log_{e}m \right )$... |
H: Find every equation of the line that passes through the point $(5,13)$
"Find every equation of the line that passes through the point $(5,13)$ and passes both axis at non-negative, whole values."
Here's my attempt:
Finding first two equations, with $k=\pm1$ is fairly simple. After that, plugging in the $x=5$ and $... |
H: Sum of reciprocals of odd prime numbers equal to one
I was wondering if there is any known sum of reciprocals of distinct odd prime numbers such that $$\sum_{k=1}^{n}\frac{1}{p_k}=1$$ Could someone give an example of one, or tell if there is none known? Or maybe it is impossible to find one, then it would be great ... |
H: How can I prove this derangement question
I need to prove this equation while using combinations but I have no idea how to proceed. I just need a hint. Thank you
$n!=D_n{n\choose 0} + D_{n-1} {n\choose 1} + D_{n-2}{n\choose 2} +... D_0{n\choose n} $
$D_0=1$
$D_n$ is the number of derangements of an $n$-element set... |
H: Example of a Graph
What is the smallest simple Graph with all but one nodes having degree 3. The last node having degree 2?
I have tried looking for relevant Graph Theory books but couldn't find how to proceed.
AI: The graph must have at least 4 vertices, for any to have degree three.
Since the sum of the degrees i... |
H: Mandelbrot Set Main Shape
What is the exact shape of the main component of the Mandelbrot set? I’m referring to the heart-shaped area centered at at the origin. Is there a simple way to express this shape in Cartesian or polar coordinates?
AI: It's a cardioid. The Wikipedia article on the Mandelbrot set explains t... |
H: Points at infinity of $x^2-6xy+9y^2-3z+1=0$ (in the projective space $\mathbb{P^3}$) (Have I done it well?)
We have to calculate the points at infinity of the following curve:
$x^2-6xy+9y^2-3z+1=0$ (in the projective space $\mathbb{P^3}$)
I know how are done this type of exercise but in this one exactly, I've got ... |
H: definite Integral with limit approches $\infty$
Evaluation of $$\lim_{n\rightarrow \infty}\int^{\infty}_{0}\bigg(1+\frac{t}{n}\bigg)^{-n}\cdot \cos\bigg(\frac{t}{n}\bigg)dt$$
What i Try: put $\displaystyle \frac{t}{n}=u.$ Then $dt=ndu$
$$I_{n}=\lim_{n\rightarrow \infty}\int^{\infty}_{0}n(1+u)^{-n}\cos(u)du$$
By u... |
H: Eigenvalues from the eigenvectors
$T\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$ is a linear transformation such that $T^{3}(v)=T(v)$. I know the matrix $[T]$ in the canonical basis has trace and determinant both equal to zero.
Also
$$[T]=[Q][D][Q]^{1}$$
such that
$$[Q]=\begin{bmatrix} 1/\sqrt{2} & -1/\sqrt{3} & -1/\sqrt... |
H: Is every sufficiently smooth function on a compact manifold approximated by a linear combination of a 'few' Laplacian eigenfunctions?
Let $M$ be some smooth, compact manifold. Let $\Delta$ be the usual Laplacian, and $f_0, f_1, \ldots, $ its eigenfunctions in order of increasing eigenvalues.
I know that minimizing ... |
H: How many different fractions can be made up of the numbers
How many different fractions can be made up of the numbers 3, 5, 7, 11, 13, 17 so that each fraction contains 2 different numbers? How many of them will be the proper fractions?
AI: The numbers of fraction which have two different numbers according to @Benj... |
H: is linear projection sufficient for capturing all extreme points?
Given a set $X \subset R^n$ with $m$ points. We can find it's Convex Hull and together with set of extreme points $E(X)$. And none of any points are linear multiplier of each other.
Under a linear projection of $f: R^n \to R^{n-1}$, we can find extr... |
H: Is the derivative of a periodic function always periodic?
True or False :
The derivative of a periodic functions is always periodic.
I thought it to be true , as everything about a periodic function repeats itself at regular intervals, and so should it's derivative . But , to my surprise it is given false , which... |
H: hermitian forms are related by linear transformations
Suppose $(-,-)$ and $[-,-]$ are two positive deinifite hermitian forms on an $n$-dimensional vector space, show that there exists an invertible linear transformation $\phi$ such that $(u,v) = [\phi(u),\phi(v)]$.
Attempt: I tried to write the hermitian forms in m... |
H: The distance from the center to the perimeter of a square, given an angle theta
Given a square with some width w, and an angle theta, what's the distance d from the center to the perimeter?
Letting $r=w/2$, we clearly have $d=r$ at $\theta = \frac{n\pi}{2}$, and $d=\sqrt{2}r$ at $\theta=\frac{n\pi}{2}+\frac{\pi}{4}... |
H: The winning probability in a card game
Suppose that I am playing a card game with my friend - a $1$ vs $1$ card game. All cards in standard card deck (52 cards) are shuffled randomly, then two cards are drawn to each person respectively. (without replacement) Each player is required to play one of these cards. The ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.