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H: If $f\left(\frac{1-x}{1+x}\right)=x$ then find $f(x)$
If $f\left(\frac{1-x}{1+x}\right)=x$ then find $f(x)$
My attempt :-
Put $x= \tan^2(\theta)$
Then $f(\cos(2\theta))=\tan^2(\theta)$
After this ....
AI: You need to know how $x$ varies in terms of $\frac{1-x}{1+x}.$
Thus, you need to find the inverse of $g(x) ... |
H: Finding a nonexact closed $1$-form on a surface embedded in $\Bbb R^3$
Consider the subset $S=\{(x,y,z):x^2-y^2-z^2+1=0\}$ of $\Bbb R^3$. Defining a function $f:\Bbb R^3\to \Bbb R$ by $f(x,y,z)=x^2-y^2-z^2+1$, it is easily seen that $0$ is a regular value of $f$, so it follows that $S=f^{-1}(0)$ is an embedded subm... |
H: Definition of "barycenter"
I have the following definition given:
(From "Introduction to algebraic topology" by Joseph J. Rotman)
Is the definition really meant like this?
Or is $\frac{1}{m+1}(p_0+p_1+\dotso +p_m)$ meant?
For me it should read 1/(m+1) but the author (at least it seems to) writes "fractions" like t... |
H: Method to generate counterexample: An irreducible that is not prime.
I am a newcomer in ring theory.I like the method of studying by myself with as less help from the textbook as possible.I prove the theorems on my own and always look for counterexamples if the converse of a statement does not hold.Now while studyi... |
H: If $\lim_{x\to\pm \infty}f(x,c)=0$ $\forall c\in [a,b]$ then $\lim_{||(x,y)||\to\infty}f(x,y)=0$?
If $f:\mathbb{R}\times [a,b]\to \mathbb R$ is a continuous function with $$\lim_{x\to\pm \infty}f(x,c)=0$$ for all $c\in[a,b]$, does that imply that $$\lim_{||(x,y)||\to\infty}f(x,y)=0$$ (when taking values $(x,y)\in \... |
H: Why has this variable been used in Spivak's proof for Intermediate Value Theorem?
I am currently on chapter 8 of Spivak's Calculus and I'm struggling to understand the reason behind some of the proof of "Theorem 7-1" (IVM). Why has the variable $x_0$ been introduced (what is the utility of it)?
Could you not just ... |
H: Is there $f\in L^2$ with $\lim_{x\to\infty} f(x)^2\log(x)>0$?
I'm asking myself whether there is a function $f\in L^2(\mathbb{R}_+\to\mathbb{R})$ so that $\lim_{x\to\infty} f(x)^2\log(x)>0$. I think there is no such function. Here is my proof:
Let $f\in L^2$ with $\lim_{x\to\infty} f(x)^2\log(x)>0$. Then
$$\infty>\... |
H: Find all group homomorphisms $A_n \rightarrow \mathbb{C}^*$
Find all group homomorphisms $A_n \rightarrow \mathbb{C}^*$ for all integers $n \geq 2$
What I have up until now:
Define $f: A_n \rightarrow \mathbb{C}*$
Then by the first isomorphism theorem, we have that :
$A_n/ \ker(f) \cong f[A_n] \subseteq \mathbb{C... |
H: Combinatorics question where someone has to be at least one seat away from anyone else?
In a doctor’s waiting room, there are 14 seats in a row. Eight people are waiting to be seen.
There is someone with a very bad cough who must sit at least one seat away from anyone else. If all arrangements are equally likely, w... |
H: Are elements of a group are also elements of the quotient group?
I think the general answer to this question is no. What I struggle about the notation of a question in Thomas Hungerford's Abstract Algebra An Introduction Textbook. The question is the following;
Find the order of $\frac{8}{9}$ in the additive group ... |
H: Does homogeneous scaling minimize this integral quantity among all surjective maps?
Let $\lambda>1$ be a parameter. Let $\psi:[0,1] \to [0,\lambda]$ be a smooth surjective function, satisfying $\psi(0)=0$.
Question: Is it true that $$ E(\psi):=\int_0^1 \big((\psi'(r)-1)^2+(\frac{\psi(r)}{r}-1)^2\big) rdr \ge (1-\... |
H: Prove that $||A||_2 = \max_{||x||_2=1, ||y||_2 = 1} |y^TAx|$
Prove that $||A||_2 = \max_{||x||_2=1, ||y||_2 = 1} |y^TAx|$.
Definitions:
$$ ||A||_2=\max_{||x||_2=1}||Ax||_2=\sqrt{\rho(A^T A)}.$$
My attempt:
$$||A||_2 = \max_{||x||_2=1}||Ax||_2 = \max_{||x||_2=1} \left( \sum_i \left(\sum_j a_{ij}x_j\right)^2\right... |
H: The equation w.r.t integration of inverse function.
the question is as follows.
$f$ is differentiable on $[a,b]$ and $f'$ is continuous. And $\forall x\in[a,b]$, f'(x) is not 0.
Then, show that $$\int_a^b f(x)\,dx+ \int_{f(a)}^{f(b)} f^{-1}(x) \,dx = bf(b)-af(a)$$
I take the $$\int_{f(a)}^{f(b)} f^{-1}(x) \,dx = \i... |
H: Minimizing by Linear Programming
I have a linear program which I can't write its equations. The problem is
An office furniture company has two plants that produce lumber used in the manufacturing of a line of desks and computer tables that the company sells. In one week, plant $A$ can produce the lumber required t... |
H: Prove that the balanced hull of a compact subset $K$ of a Hausdorff TVS $E$ is compact
Let $E$ be a topological vector space Hausdorff (over $\mathbb{C}$) and $K \subset E$ compact. I want to prove that: if $M$ is the balanced hull, that is, $M$ is smallest balanced set containing $K$, then $M$ is compact.
But I di... |
H: Invertibility of an element in a Banach algebra (Gelfand's formula)
In Folland's A Course in Abstract Harmonic Analysis, Theorem 1.8 states that for a unital Banach algebra (with unit $e$), the spectral radius of an element $x$ is given by $\lim_{n \to \infty} \|x^n\|^{1/n}$. In the proof, Folland writes:
We have $... |
H: How do you solve $2^x-x=3$?
Maybe it's a simple question, but I can't figure it out. How do you solve $2^x-x=3$? Using logarithms? I could write $\log_2(x-3)=x$, but then what?
Thank you!
AI: Here is a step-by-step Lambert W solution (see Robert's answer).
Recall the definition: $ae^a = b\Longleftrightarrow a = W(b... |
H: integration of a function $\int$ $x^2$/($x^2$+$R^2$)$^{3/2}$dx.
the function is $\int$ $x^2$/($x^2$+$R^2$)$^{3/2}$dx.
I substituted x=Rtan$\theta$ and got $\int$cos$\theta$tan$^2$$\theta$d$\theta$. And here i am stuck
AI: Use $\tan^2\theta=\sec^2\theta-1$ to write that as$$\int(\sec\theta-\cos\theta)\mathrm{d}\the... |
H: Does a prime ideal contains an irreducible element?
The context. Let $R$ be an integral domain. It is known that a domain $R$ is a UFD if and only if any nonzero prime ideal contains a prime element.
It is also known that $R$ is a UFD if and only if any non zero element has a decomposition as a product of a unit an... |
H: Condition making a Compact Operator have to be a finite rank operator
I have tried to the following exercise :
Let $T$ be a compact operator, X an infite dimension Banach space and suppose that there is a closed subset $M$ such that $X= Im T\bigoplus M$, then $T$ is a finite rank operator.
Now my first approach t... |
H: Divisibility in a Euclidean Domain
Let $R$ be a Euclidean Domain. I am working on showing that
$$ \text{If } a \, | \, bc \text{ with } a,b \neq 0 \text{ then } \frac{a}{(a,b)} \, \bigg| \, c. $$
Note that the first part of this problem is to show that it $(a,b) = 1$ and $a \, | \, bc$ then $a \, | \,c$. I had n... |
H: Closure and interior of set of functions.
Given is the set: $C([0,1]) = \{f: [0,1] \rightarrow \mathbb{R} \mid f \text{ is continuous} \}$ with the following metric: $d_\infty(f,g) = \sup\{ |f(x) - g(x)| \, | \, x \in \, [0,1] \}$.
Find the closure and interior of the following set: $N = \{ f: [0,1] \rightarrow \... |
H: Group element that normalizes finite subgroup that is generated by a subset of $G$
Let $N$ be a finte subgroup of a group $G$, and assume $N = \langle S \rangle$ for some subset $S$ of $G$. Prove that an element $g \in G$ normalizes $N$ if and only if $gSg^{-1} \subset N$.
My question is about the forward direction... |
H: Cycle Notation for a Permutation Group
Can anyone thoroughly explain how you would arrive at this answer? I'm very confused with how you would do this problem.
AI: Let's do $\sigma_2$ for example. Since $\sigma_2(a)=2a \bmod{7}$, it maps $1\bmod{7}$ to $2 \bmod{7}$, $2 \bmod{7}$ to $4\bmod{7}$, and $4\bmod{7}$ to ... |
H: Quadratic Function from the Taiwan IMO TST 2005
Lately, I came across the Team Selection Test for the IMO 2005 Taiwan Team. One of the Question is stated as follow:
Set $f(x) = Ax^2+B^x+C$ and $g(x)=ax^2+bx+c$, with $A \times a \neq 0$, $ A,a B,b, C,c \in \mathbb{R}$ satisfies:
$|f(x)| \ge |g(x)| \forall x \in \mat... |
H: How to find geometrical figures areas?
Find the area of the region that lies inside the circle $r = 1$
and outside the cardioid $r=1-cos\alpha$
We know that area can not be negative value(at least basic calculus).
I wonder where I made a mistake,I tried to show equations.
$$ 1=1-cos\alpha $$
$$\alpha = \pi/2,3\pi/... |
H: Quasilinear PDE $\left\{\left(x+y\right)\frac{\partial }{\partial x}+\frac{\partial }{\partial z}\:\right\}u\left(x,y,z\right)=0$
$\left\{\left(x+y\right)\frac{\partial }{\partial x}+\frac{\partial }{\partial z}\:\right\}u\left(x,y,z\right)=0$
This means that
$\left(x+y\right)\frac{\partial u}{\partial x}+\frac{\pa... |
H: Permutation Question with two independent conditions
In how many ways can $8$ boys $(B_1,B_2,...,B_8)$ and $5$ girls we arranged linearly such that $B_1$ and $B_2$ are NOT together and exactly four girls are together?
I can solve when either of the conditions is imposed, the former would give $13!-2!\cdot12!$ and... |
H: An ideal containing a unit
In a ring $R$ and $I$ is an ideal of the ring, we have a result that if $I$ contains a unit then the ideal is equal to the whole ring and since the moment the ideal contains a unit it won't be just an ideal which means there does not exist an ideal with only an unit?
I am not able to expr... |
H: What is the name of this theory?
What is the name of the theory which states that even if the chances of something happening are so low (like 1%) but said event, if it were to happen, it is so disproportionate in character that even a 1% chance is still to be considered?
Like taking a gamble to which the chances ar... |
H: Prove that there are at least $2005$ pairs of $(x, y)$ of non negative integers x and y that satisfy $x^2+y^2 = N,$ for some positive integer $N$.
Prove that there exists a positive integer $N$ such that there are at least $2005$ ordered pairs $(x,y),$ of non-negative integers $x$ and $y,$ satisfying $x^2 + y^2 = N... |
H: If $f(x)=x^2+x+2$ and $g(x)=x^2-x+2$, how to prove that there is no function $ h:\mathbb R \to \mathbb R $ such that $h(f)+h(g) = g(f)$?
I tried to substitute $f(x)$ & $g(x)$ in their places but didn't find a relation;
The function beginning bijective or surjective etc have nothing to do with our case I believe
$g(... |
H: How to find limit of $\lim_{n\to \infty} \frac{n \sin\frac{x}{n}}{x(x^2+1)}$ without L'Hospital's rule?
How to find limit of $$\lim_{n\to \infty} \frac{n \sin\frac{x}{n}}{x(x^2+1)}$$ without L'Hospital's rule?
I thought rewriting $\sin(\frac{x}{n})$ using Taylor expansion would work but it didn't help. I solved it... |
H: $Show: a\in \mathbb{Z} \Rightarrow 6\mid a^3 -a$
$Show: a\in \mathbb{Z} \Rightarrow 6\mid a^3 -a$
My attempt:
NTS: $6\mid a^3 -a$, so $(2\mid a^3 -a) \land (3\mid a^3 -a)$
Assume that $a\in \mathbb{Z}$, therefore I have 2 cases:
a is even $\Rightarrow a=2k, k\in \mathbb{Z}$
a is odd $\Rightarrow a=2k+1, k\in \math... |
H: Number of loops in permutations
From the famous prisoners problem there are n labelled boxes and inside each the relative number. These are randomly mixed so that you don’t know which number is inside each box. How can I compute the distribution of the loops lengths given n? Where a loop is defined as the path tha... |
H: Bijection between real and natural numbers.
I know, I know, this question has been asked several times, but I feel mine is a little bit different.
Imagine a correspondence between $[0,1]$ and natural numbers in the following sense:
$$ 0.12 \longleftrightarrow 21 $$
$$ 0.443 \longleftrightarrow 344 $$
$$0.12345 \lon... |
H: determine the power series 1/P(x) for a power series P(x)
We have a power series $P(x)=\sum_{n=0}^\infty(-1)^n2^n(n+1)x^n$ and now have to determine the power series $\frac{1}{P(x)}$. I am at a total loss here, maybe one of you can help.
AI: \begin{align}
P(x) &= 1\cdot1 - 2\cdot2x + 4\cdot3x^2 - 8\cdot4x^3 + \cd... |
H: subfields of a finite subfield
I'm trying to work out all the subfield of a finite field with $3^7$ elements. So I've said that since every subfield of that field if of the form $3^n$ where $n$ is a positive divisor of $7$. so I got the finite subfields as the finite field with $3^1$ and $3^7$ elements. just wanted... |
H: Every Radon measure defines a distribution
I have read that given a Radon measure $\mu$ in $\mathbb{R}^d$, the operator $T_{\mu}(\phi)=\int_{\mathbb{R}^d} \phi d \mu$, $\phi \in C_c^\infty(\mathbb{R}^d),$ defines a distribution.
However, when I try to prove this result I find the following difficulties related to... |
H: Evaluate $\int \frac{1}{\sqrt{1-\sin^4{x}}}dx$
$$f(x) = \int \frac{1}{\sqrt{1-\sin^4{x}}}dx$$
I tried this by breaking the denominator as $\sqrt{(\cos^2x)(1+\sin^2x)}$ and then trying to make the integral in forms of $\sec x$ and $\tan x$. But I couldn't succeed.
Can somebody please help me out?
AI: $$I=\int \fr... |
H: Statement regarding Goldbach's Conjecture?
Question
I think using elementary (but twisted) means I can prove an interesting statement and was curious how a number theorist would prove the same.
Let us we want to find $2$ primes which satisfy:
$$p_1 + p_2 = 2 m$$
and $m$ is not a prime. Then I can show:
$$ m = n_1 ... |
H: What's the number of subgroup of order 17 in $Z_{17}\times Z_{17}$?
It seems like there is a method to solve this kind of problem. But I can't figure it out.
AI: Any group of order 17 is cyclic. So, in order to find subgroup of order 17 in $\mathbb{Z}_{17} \times \mathbb{Z}_{17}$ we need to look at the elements... |
H: Reforming a difference equation
I am trying to solve a difference equation, but once I get a certain formula, I do not know how to reform it. There is a solution, but it does not explain how to reform the equation.
The image of the problem and the solution
I get to this point $ (\alpha-\beta)*Y_{t+1} = -\beta Y_{t}... |
H: $f(x)=\exp(-x^{-1})$ infinitely differentiable, induction?
$f:\mathbb{R}\rightarrow{\mathbb{R}}; f(x)=\exp(-x^{-1})$ if $x>0$ and $f(x)=0$ if $x\leq 0$.
Show that you can differentiate $f$ on $\mathbb{R}_{>0}$ as often as you want. And that for every $n \in \mathbb{N}$ polynomals exist so that $f^{(n)}(x)=\frac{p_n... |
H: Entire relation, projective object and choice object, and the axiom of choice
I was reading on the axiom of choice and I came across these few statements in nLab:
Projective object: $P$ is projective if for any morphism $f: P \rightarrow B$ and any epimorphism $q: A \rightarrow B$, $f$ factors through $q$ by some ... |
H: Vector Projection explanation
Please can someone explain why this is? I think I understand projection when it is a comparison of two vectors but below has 3.
Im revising and I am so slow.
The projection of the vector $$\begin{pmatrix} 3\\\ -2\\\ -1\end{pmatrix}$$ onto the plane spanned by the vectors
$$\begin{pmat... |
H: The anti definite integral
Is there such a possibility as defining an anti (definite integral)?
For instance:
$$
S(a,b)=F(b)-F(a)=\int_a^b f(x)dx
$$
The question is what operation $D$ on $S$ produces $f(x)$:
$$
D[S(a,b)]=f(x)
$$
For the simple case of $f(x)=x$, it is pretty obvious that, say $D[b^2-a^2]=x$ because... |
H: deduction proof
Prove $p \wedge \neg p \vdash q$ for any propositional variables $p$ and $q$ without using disjunctive syllogism or excluded middle or $\neg$-elimination.
I can prove this easily using $\neg$-elimination: assume $p\wedge \neg p$ and $\neg q$. Then by $\wedge$-elimination, we have $p$ and again by ... |
H: Estimation of the standard deviation for Power Law distribution
I've understood everything in the picture (source) below except for this equality:
$$\hat{\sigma} = \frac{\hat{\alpha} - 1}{\sqrt n}$$
Can someone please explain where does it come from?
AI: The likelihood is a Pareto distribution with shape $\alpha-1$... |
H: Determine every polynomial with real coefficients such that $P(P(x))=[P(x)]^k$
I have doubts wrt my solution for the following problem. Determine every polynomial with real coefficients such that $P(P(x))=[P(x)]^k$
My guess is that $P(x)=x^k$. I started by saying let $r_n \in \mathbb{C}$ such that $P(r_n)=n$. Hence... |
H: 1.You are trying to guess a three-letter password that uses only the letters A, E, I, O, U, and Y.
Letter can be used more than once. Find the probability that you guess the correct password :You
AI: There are $6$ possible letters, meaning that there are
$$
6 \times 6 \times 6=216
$$
possible combinations. If you r... |
H: Non-isomorphic graphs with 2 vertices and 3 edges
Are there any non-isomorphic graphs with 2 vertices and 3 edges? From my understanding of what non-isomorphic means, I don't think there are any, but I'm not sure.
AI: Does this image answer your question? |
H: Prove that $\lim_{x \to 0^{+}}f(x) = \infty$ iff $\lim_{x \to \infty}f(\frac{1}{x}) = \infty$ Explanation of solution and concept.
Prove that $$\lim_{x \to 0^{+}}f(x) = \infty \\ \text{iff} \\ \lim_{x \to \infty}f\bigg(\frac{1}{x}\bigg) = \infty$$
I'm confused by the reasoning behind a solution that I found for thi... |
H: Writing complex number $c$ in form $e^{iz}$
Let $c\in \mathbb C$. Then we can write $c = r\cdot e^{i\varphi},$ where $\varphi\in [0,2\pi)$ and $r\in [0, \infty)$.
My professor stated, however, that if $z \in \mathbb C$, then $c = e^{i\cdot z}$ holds as well, i.e., there exists a representation for $c$ in the "pure... |
H: Solve the following equation in integers $x,y:$ $x^2+6xy+8y^2+3x+6y=2.$
Question: Solve the following equation in integers $x,y:$ $$x^2+6xy+8y^2+3x+6y=2.$$
Solution: For some $x,y\in\mathbb{Z}$ $$x^2+6xy+8y^2+3x+6y=2\\\iff x^2+2xy+4xy+8y^2+3x+6y=2\\\iff x(x+2y)+4y(x+2y)+3(x+2y)=2\\\iff(x+4y+3)(x+2y)=2.$$
Now if $(... |
H: Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.
Give an algebraic and a combinatorial proof for the following identity:
$$ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}.$$
For the combinatorial argument, use the analogy of $n$ party guests, where $m$ of them describe them... |
H: "Natural" equivalence of categories?
Let $\mathbf{C}$ be a category and $F,G:\mathbf{C}\to\mathbf{Cat}$ be category-valued functors on $\mathbf{C}$. Suppose there is a family of equivalences of categories
$$(\Phi_C:FC\simeq GC)_{C\in\mathbf{C}}\tag{1}$$
such that for all $f:C\to C'$ in $\mathbf{C}$, there is a natu... |
H: How to find limit of a sequence of functions $f_n(x)=\frac{x^n e^x} {n+1}$?
How to find limit of a sequence of functions $f_n(x)=\frac{x^n e^x} {n+1}$? $$\lim_{n\to \infty} \frac{x^n e^x}{n+1}$$
I have no idea how to evaluate this limit. I thought maybe I should rewrite $e^x$ using $\sum_{n=0}^{\infty} \frac{x^n}{... |
H: Proof of dominated convergence theorem
I was going through the proof of the Dominated Convergence Theorem.
Now if we have that ($f$$_n$) is a sequence of measurable functions such that $\lvert f_n\rvert$ $\le$ $g$ for all n where g is integrable on $\Bbb{R}$.
And if $f$ = $\lim_{n}$$f_n$ almost everwhere.
We can s... |
H: Are infinite subsets of the rationals definable?
This is really two questions in one. Consider the structure $(\mathbb{Q},<)$. We adjoin to it a subset $S$ of $\mathbb{Q}$. Is there a first-order formula $F$ in the expanded language such that $F$ is true precisely when $S$ is an infinite subset of $\mathbb{Q}$? If ... |
H: Tensor product of two direct factors is a direct factor of the tensor product
Let $A$ be a ring, $E$ a right $A$-module, $F$ a left $A$-module, $M$
a submodule of $E$ and $N$ a submodule of $F$. Suppose that $M$ is a
direct factor of $E$ and $N$ is a direct factor of $F$. Then the
canonical homomorphism $M\o... |
H: Recalculate Normal Vector without rotation matrix
I have 3 points in 3 dimensions (P0, P1, P2) and a normalised vector N1, that lies on the plane constructed by those points and is perpendicular to the line P0-P1.
I want to find the normal vector N2 perpendicular to the line P0-P2, lying on the same plane and facin... |
H: Are morphisms in a slice category $c/C$ inherited from the original category $C$?
Going through Emily Riehl's Category Theory in Context and something keeps tripping me up.
The notation for slice categories, which reminds me of factor group notation in group theory, indicates to me that the morphisms in $c/C$ are i... |
H: Expected number of matching pairs from a random list
Question
A random list of length $n$ (even number) consists of $n_1$ stars $\star$ and $n_2$ squares $\square$. Suppose we randomly put these shapes into $n/2$ pairs, denote:
$X_1$ to be the number of matching pairs of $\star$
$X_2$ to be the number of matching p... |
H: $\{x\in\mathbb{R}:m(E\cap(x-k,x+k))\geq k, \forall k>0\}$ is Lebesgue measurable
Consider a Lebesgue measurable set $E\subset\mathbb{R}$. Prove that the set $\{x\in\mathbb{R}:m(E\cap(x-k,x+k))\geq k, \forall k>0\}$ is Lebesgue measurable.
I am just a bit confused on where to begin. It looks like I can just appl... |
H: Finitely generated projective resolution
Let $K$ be a field, $A$ be a finite dimensional $K$-algebra and $M$ be a finitely generated $A$-module. Is it true that $M$ admits a projective resolution by finitely generated projective $A$-modules?
AI: As $A$ is a finite algebra over $K$ it is noetherian. As $M$ is finite... |
H: Prove that a function $u: u= \ln\|x\|{_{2}}$ has $\Delta u = 0$.
I had a similar case some time ago and following the advices there I tried to solve this one too.
I tried to find its first partial derivative and I got:
$\frac{\partial}{\partial x_{i}}=\frac{1}{2\cdot \|x\|_{2}^{1/2}}$
Now i have to find the second ... |
H: Is there a simpler way to solve the differential equation $y''+2xy'+(x^2-1)y=0$
A student asked me to solve this differential equation
$$y''+2 x y'+(x^2-1)y=0$$
Is there a method simpler than power series?
AI: We want to express this as the second derivative of some function of $x$ and $y$. Via the ansatz introduct... |
H: Solve $X^3 = A$ in $M_2(\mathbb{R})$ where the matrix $A$ is given.
Consider the matrix:
$$A =
\begin{pmatrix}
3 & -2 \\
6 & -4 \\
\end{pmatrix}$$
I have to solve the equation:
$$X^3 = A$$
where $X \in M_2(\mathbb{R})$.
First, I tried using the notation:
$$X =
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}$$
wh... |
H: $\frac{1}{4} (a^2+ 3 b^2)$ is of the form $(c^2+ 3 d^2)$
If $ 2 \mid (a^2+ 3 b^2)$ and $(a,b)=1$ then $4\mid (a^2+ 3 b^2)$. How can I show $\frac{1}{4} (a^2+ 3 b^2)$ is also of the form $(c^2+ 3 d^2)$?
Here, clearly $ a$ and $b$ are both odd.
Let $a=2m+1$ and $ b=2n+1$
$\implies\frac{1}{4} (a^2+ 3 b^2)= m^2 + m+1... |
H: finding convergence of integral having exponential and cosine terms
Finding whether the series $$\int^{\infty}_{1}e^{x}\cos (x)\cdot x^{-\frac{1}{2}}dx$$ converges or diverges.
What i try::
$$I=\int^{\infty}_{1}e^{x}\cos(x)\cdot x^{-\frac{1}{2}}dx\leq \int^{\infty}_{1}e^{x}\cdot x^{-\frac{1}{2}}dx$$
Now put $x=t^... |
H: Question about generator of $K[x]$ using simple extension.
Let $K \subset \mathbb{C} $ be a field, and $K[x]$ the polynomial ring, $\alpha$ algebraic over $K$ and $L=K(\alpha)$.
I am reading this book and it states that, since any element of $L$ is of the form $\frac{f(\alpha)}{g(\alpha)}$ with $f,g \in K[x]$ and $... |
H: Question Cauchy Riemann equations
Let $f(z) =zRe(z)$. Determine all points $z_0$ for which the complex derivative $f'(z_0)$ existst.
I wrote $f(z)$ as $f(z)=f(x+iy)=(x+iy)Re(x+iy)=x^2+(xy)i:=u(x)+v(x,y)i$.
So we get the partials $u_x=2x$, $v_y = x$, $u_y=0$, $v_x=y$.
Now the CR equations, $u_x=v_y$ and $u_y=-v_x$... |
H: Injectivity and Surjectivity of two different functions
a) If there is a function f: A-->B where there are two distinct elements a, b that are in A such that f(a) ≠ f(b), does this make f injective?
I think the answer is true because if a and b don't equal each other, then f(a) ≠ f(b).
b) If there is a function g: ... |
H: Question about Spivak's proof of how to use u-substitution when the derivative of the inner function does not appear in the integral
Spivak (3rd edition) proposes solving the integral $$\int \frac{1+e^x}{1-e^x} dx$$ by letting $u=e^x$, $x=\ln(u)$, and $dx=\frac{1}{u}du$. This results in the integral $$\int \frac{1+... |
H: Suppose two estimators are unbiased, what is the intuition behind the preference of the estimator with the less variance?
Suppose there are two unbiased estimators that we can use to estimate a parameter $\theta$, why do we often prefer the one with less asymptotic variance?
The question is rather simple and perhap... |
H: Orbit under group of automorphisms is finite.
Let $a,b\in\mathbb{C}$, and $\sigma$ be an automorphism of $\mathbb{C}$ such that $b=\sigma(a)$. My question is: Why if the set
$$\{\sigma'(a)\mid \sigma' \text{ is an automorphism of }\mathbb{C}\}$$
has at most $n$ elements then $b$ is an algebraic number of degree at ... |
H: Trig Question with angles of elevation
A UFO is flying above two people standing on the ground at points A and B.
A and B are 300m apart.
The angle of elevation of UFO from A is 30 degrees
The angle of elevation of the UFO from B is 23 degrees
Find height of UFO above ground.
I got something like 480m but apparent... |
H: How to find $d\phi(I)$ where $\phi(A)=AA^T$
I'm trying to show that $so(3) = \{A\in M(3; \mathbb{R}): A = -A^T \} $ is the lie algebra of $SO(3)$. For this i am using the following fact:"The tangent space at the identity to a Lie subgroup of $GL(n,\mathbb{R})$, endowed with the matrix commutator, is isomorphic to i... |
H: How to determine the integer k as lambert branch to knowing the solutions of an equation?
Actually this is my first time to self-study about lambert w function, and i have interest on it, so forgive me if this question sounds stupid.
I can derive it manually with algebra if the solution of
$$\text{$x^x=2$ is $x... |
H: Identity for sum of binomial coefficients derived from convolution of negative binomial random variables
Let $X\sim NB(r,p)$ and $Y\sim NB(s,p)$ be negativ binomials (I use the variant where we count all trials until we reach r successes). Further, they are independent and I am interested in the distribution of $X+... |
H: Is $\lim \limits_{n\rightarrow \infty} n^{1-\ln((1+\frac{1}{n})^n)} = 1 $
could you help me understand if this statement is correct?
$$ \lim \limits_{n\rightarrow \infty} n^{1-\ln((1+\frac{1}{n})^n)} = 1 $$
Its easy to see that $ \lim \limits_{n\rightarrow \infty} 1-\ln((1+\frac{1}{n})^n) = 0 $, but since this expr... |
H: Proving that $2+\sqrt{2}$ is irreducible in $\mathbb{Z}[\sqrt{2}]$.
I'm asked to show that $x=2+\sqrt{2}$ is irreducible in $\mathbb{Z}[\sqrt{2}]$ by using the norm map $$N:\mathbb{Z}[\sqrt{2}]\rightarrow \mathbb{Z}^+:a+\sqrt{2}b\mapsto |a^2-2b^2|$$
Now, if $x=yz$, then $2=N(x)=N(y)N(z)$ forcing wlog $N(y)=1$. I'm ... |
H: evaluation of volume of solid obtained by rotating the curve $y=x^2,y=x+2$ about $x$ axis
Evaluation of volume of solid obtained by rotating the regin enclosed by the curves $y=x^2$ and $y=x+2$ about $x$ axis,is
What i try:
Sokving $y=x^2$ and $y=x+2$. we get $x=-1, x=2$
So volume of solid obtained by rotating ... |
H: Why does the Boolean equation A.B' + B = A + B hold?
I have pretty good amount of knowledge in Boolean algebra. However, I struggled with the equality $(1)$ more than I should have.
$$x'z' + z = x' + z\tag{1}$$
How is it that this holds, algebraically? I can assure you that I've tried it enough. I just cannot get i... |
H: Please help me to find what did I wrong in the question(Probability and distribution)
$Q)$ There is rolling- a dice game following the rule.(Here the dice is average dice having sides $1$~$6$ )
Game rule
Getting the number $6$ in one trial, We get $5$ points.
Otherwise getting the other numbers in one trial, We... |
H: Use Clairaut's Theorem to find an effecient way to solve $f_{yyzzx}$.
I just asked a similar question, but is there any other way the $f_{yyzzx}$ could be rearranged to get a number other than $0$ in the end?
When $x^2\sin(4y)+z^3(6x-y)+y^4$.
I have $f_y=4x^2\cos(4y)-z^3+4y^3$, then $f_{yy}=-16x^2\sin(4y)+12y^2$
A... |
H: Coordinates of circumcentre of a triangle in terms of triangle point coordinates
For a triangle $ABC$, its circumcentre is the intersection of perpendicular bisectors of its three sides. Now for a particular triangle I can draw the bisectors and find the circumcentre. However I need a general formula to solve the p... |
H: Find all integers $x$ of the form $x\equiv _5 3$, $x\equiv _8 6$.
I'm asked to find all integers $x$ of the form $x\equiv _5 3$, $x\equiv _8 6$. It turns out that the set of all such integers is $$\{ x\in \mathbb{Z} : x=38+40t, \ \text{for some} \ \ t\in \mathbb{Z}\}$$
yet I haven't been able to get closer to such ... |
H: Finding LU factorization and one number is off
I'm trying to find LU of this 2x2 matrix, but when I check my work, the bottom right number is 31 instead of 1.
A is the matrix at the top.
What am I doing incorrectly?
AI: Considere $A=\begin{pmatrix}
2 & 8 \\
4 & 1 \\
\end{pmatrix} $
Then we do
$A(-2) _{1,2}$ the e... |
H: Which is greater $\frac{13}{32}$ or $\ln \left(\frac{3}{2}\right)$
Which is greater $\frac{13}{32}$ or $\ln \left(\frac{3}{2}\right)$
My try:
we have $$\frac{13}{32}=\frac{2^2+3^2}{2^5}=\frac{1}{8}\left(1+(1.5)^2)\right)$$
Let $x=1.5$
Now consider the function $$f(x)=\frac{1+x^2}{8}-\ln x$$
$$f'(x)=\frac{x}{4}-\fra... |
H: modular forms with complex multiplication
I would like the definition of a modular form with complex multiplication and if possible a reference.
Thank you !
AI: A newform $f=\sum_{n=1}^\infty a(n)q^n$ of level N and weight k has complex multiplication if there is a quadratic imaginary field K such that $a(p)=0$ as... |
H: Given a chart $(U,\phi)$ find a chart $(V,\psi)$ such that $(U,\phi)$ and $(V,\psi)$ are $C^\infty$-compatible and $\psi(V)=\mathbb{R}^n$?
On page 4 of the book "Differential Topology" (written by Amiya Mukherjee) the following is written:
[...] observe that the charts $(U,\phi)$ and $(U,\alpha\circ \phi )$,
whe... |
H: Show $\sum_{k=0}^n\frac{c_k}{(n-k)!}=1$ where $\sum_{n=0}^{\infty}c_nx^n=\frac{e^{-x}}{1-x}$
Show $$\sum_{k=0}^n\frac{c_k}{(n-k)!}=1$$ for each $n\geq 0$, where $$\sum_{n=0}^{\infty}c_nx^n=\frac{e^{-x}}{1-x}.$$
Since $$e^{-x}=\sum_{n=0}^{\infty}\frac{(-x)^n}{n!}$$ and $$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$ on $[-... |
H: Proving That $\sum^{n}_{k=0} \bigl(\frac{4}{5}\bigr)^k < 5$
Using induction, prove that $$\sum_{k=0}^n \biggl(\frac 4 5 \biggr)^k = 1+\frac{4}{5}+\bigg(\frac{4}{5}\bigg)^2+\bigg(\frac{4}{5}\bigg)^3+\cdots +\bigg(\frac{4}{5}\bigg)^n<5$$ for all natural numbers $n.$
What I have tried is as follows.
Consider the sta... |
H: Convergence of Glaisher-Kinkelin Constant Limit Definitions
The Glaisher-Kinkelin constant $A$ is given by the limits
$$\begin{align}
A&=\lim_{n\rightarrow\infty}\frac{H(n)}{n^{n^2/2+n/2+1/12}e^{-n^2/4}}\\
&=\lim_{n\rightarrow\infty}\frac{(2\pi)^{n/2}n^{n^2-1/12}e^{-3n^2/4+1/12}}{G(n+1)}
\end{align}$$
where $H(z)$ ... |
H: Prove there are infinitely many positive integers which cannot be represented as a sum of four non-zero squares.
Prove there are infinitely many positive integers which cannot be represented as a sum of four non-zero squares. Every positive integer can be written as the sum of four squares. But not all necessarily ... |
H: How to distinguish between nonlinear and linear
I ve two questions.
First . How to distinguish between nonlinear and linear
The difference between linear and nonlinear as I know is whether it is proportional to the result or not.
(that is linear is Linear equation )
second .
Is it nonlinear to use an index as a ... |
H: Can any Hilbert space be expressed as countable union of unit balls?
I was going through functional analysis text by J.Conway, and have encountered with next claim (2.4.6) :
Let $T\in \mathcal{B}_0(\mathcal{H},\mathcal{K})$ for two Hilbert spaces $\mathcal{H},\mathcal{K}$. Since $\text{cl}[T(\text{ball } \mathcal... |
H: A Question About The Tower Of Hanoi
I've been reading through an inductive proof on why the minimum number of moves in a Tower of Hanoi with n disks is $2^n -1$. The proof is based on the fact that the minimum amount of moves for $k+1$ disks is $2T(k) + 1$: $T(k+1) =2T(k)+1$.
I understand that this is because you ... |
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