text stringlengths 83 79.5k |
|---|
H: What is the necessary and sufficient condition for the matrix $A = I − 2xx^T $to be orthogonal?
Let $x$ be a non-zero (column) vector in $\mathbb{R}^n$. What is the necessary and sufficient condition for the matrix $A = I − 2xx^T$ to be orthogonal?
AI: Hint: $A$ is orthogonal if and only if $A.A^T=I$. Note that in ... |
H: Finding a polynomial $g$ such that $g^2=f$ for certain $f$ in $\mathbb{F}_{16}$
Let $L = \mathbb{Z}_2[x]/\langle x^4 + x + 1 \rangle$ and $\alpha := [x] \in L$.
I want to find $g \in L[y]$ with $g^2 = f$ and
$$f = (\alpha^2 + \alpha)y^8 + \alpha y^4 + \alpha^3y^2 + \alpha + 1 ∈ L[y].$$
My starting point is
$$ (a... |
H: How to solve this Real Analysis?
Given that $f(x)= \sin(x + \pi/4)$ is periodic with period $2\pi$. Find the complex Fourier series.
It's quite a moderate tough question. Can someone help me out. Thanks in advance
AI: Here is your Fourier series,
$ \sin(x+\frac{\pi}{4}) = \cos(\frac{\pi}{4})\sin(x)+ \cos(\frac{... |
H: Why does calculating matrix inverses, roots, etc. using the spectrum of a matrix work?
Suppose $A$ is a $n \times n$ matrix from $M_n(\mathbb{C})$ with eigenvalues $\lambda_1, \ldots, \lambda_s$. Let $$m(\lambda) = (\lambda - \lambda_1)^{m_1} \ldots (\lambda - \lambda_s)^{m_s}$$
be the minimal polynomial of $A$.
W... |
H: How to prove a set is a Dedekind Cut?
In my Real Analysis class we just started talking about Dedekind cuts and I'm very confused on how to prove this one homework assignment: Given the set $-A=\{-(a+b):a\in\mathbb{Q}^+, b\in\mathbb{Q}\setminus A\}$, how do I prove this is a cut? Please help, I'm so lost...
Thanks... |
H: Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist? Why?
Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist, given that $\mathbf A$ is a square and positive (and possibly singular) matrix and $\epsilon$ is a small positive number? I want to use this to regularize a sample covariance matrix ($\mathbf A = \... |
H: Probability of a person living at least $t$ years
Suppose the force of mortality of a person aged $x$ is $\mu_x = \frac{1}{200-x} + \frac{1}{100-x}$ where $x<100$. What is the probability that the person survives at least $t$ more years?
So would this be $$\exp \left(-\int_{x}^{x+t} \frac{1}{200-s} \ ds- \int_{x}^{... |
H: Basic Property of Normal subgroup
For a subgroup $N$ of $G$, if the product of two right cosets of $N$ in $G$ is again right coset of $N$ in $G$,then $N$ is normal subgroup of $G$.
AI: Say $aNbN = acN$. Then $Nb = cN$. Write $n_1b = cn_2$ for any $ n_1$ and some $n_2$ in $N$. Take $n_1 = e$ to get $b = cn_2$. Take ... |
H: A question about spaces having a locally finite refinement
I've been reviewing questions from previous exams I've had in my courses, and I noticed a question that I completely missed. I was wondering if anyone could help me out on this one.
Let $X$ be a topological space. Prove that every $\sigma$-discrete open cov... |
H: Abelian quotient group
I'm stuck on the following practice problem. Any hints would be appreciated.
Suppose $N$ is a normal subgroup of $G$ such that every subgroup of $N$ is normal in $G$ and
$C_{G}(N) \subset N$. Prove that $G/N$ is abelian.
I'm not sure how to use the fact that $C_{G}(N) \subset N$.
Thanks
AI... |
H: Find convergence of integral and compute it
Preparing for my calculus exam I found this exercise and I want to see if I'm on the right track
Find out the convergence of the following series and compute it.
$$\int_0^\infty \frac {\,dx} {(x+1)\sqrt {x^2 + x + 1}}$$
To find if the integral is convergent or divergent I... |
H: Closed sets of a curve
I am guessing that the closed sets of a curve (i.e. an algebraic variety of dimension 1) are finite . How would one go about proving this? Is it trivial?!
Thanks for any help.
AI: If $X$ is an (irreducible, which may or may not be part of your definition) variety over a field $k$ of dimension... |
H: Proof of the tangent addition theorem for complex numbers
How can the tangent addition theorem for complex numbers
$$\tan(z+w) = \frac{\tan z + \tan w }{1 - \tan z \tan w}$$
be proved?
For real numbers, the wikipedia page says one can use Euler's formula
$$e^{ix} = \cos x + i\sin x.$$
But I dont see if and how this... |
H: Category with zero morphisms
Let ${\bf C}$ be a category with zero morphisms, i.e., for each $X,Y\in {\bf C}$, there is a morphism $0_{XY}:X\rightarrow Y$ satisfying certain properties ($0_{XY}$ composes with every morphism from $Y$ or to $X$ to give another such). According to Wikipedia, "the collection of $0_{XY}... |
H: Spivak's proof of Inverse Function Theorem
I am having trouble with Spivak's proof of the Inverse Function Theorem in his Calculus on Manifolds:
2-11 Theorem (Inverse Function Theorem). Suppose that $f: \mathbb{R}^n\to\mathbb{R}^n$ is continuously differentiable in an open set containing $a$, and det $f'(a)\neq 0$.... |
H: Simple Maclaurin Series $e^{\tan(x)}$
In a multichoice online test that I did the other day, I was required to select the Maclaurin series for $e^{\tan(x)}$. It was necessary for me to find the first four terms in order to establish which answer was correct. In the end of year exam, I will have reference to a Usefu... |
H: continuous constant function general topology
I've read that if $X$ and $Y$ are topological spaces, and if the constant function $f: X \rightarrow Y$ mapping all of $X$ into the single point $y_{0}$ of $Y$, then $f$ is continuous. (Munkres page 107)
Munkres gives the following brief proof: Let $f(x) = y_{0}$ for... |
H: Triple "Pythagorean identity"
It's not hard to find multiple trigonometric functions of period $2\pi$ that added to self shifted by some constant offset result in a constant.
In classic pythagorean identity, you have
$$F(x)+F\left(x+\frac{\pi}{2} \right) = 1 $$
where $F(x)=\sin^2 x$
Or you can use symmetry of the ... |
H: Is the point $P$ that minimizes $PA+PB+PC$ the orthocenter of $\Delta ABC$?
I have a question about a triangle in ${\bf R}^2$. Let $\Delta ABC$ be a triangle with
vertices $A(x_1,y_1)$, $B(x_2,y_2)$ and $C(x_3,y_3)$.
Question : Let $P$ be a point in the interior of $\Delta ABC $ which minimizes $PA + PB + PC$. He... |
H: Hölder condition : Simple use case
This may sound a newbie question,
I would like to hear a simple example of usefulness of Hölder condition i.e. how it helps in researching the function's properties.
AI: I am not sure what you are specifically looking for, but here are some scenarios where the the Hölder condition... |
H: nonstandard example of smooth function which fails to be analytic on $\mathbb{R}$
When I teach second-semester calculus I usually discuss the function $f$ defined by
$$ f(x)=e^{-1/x^2} $$
for $x \neq 0$ and $f(0)=0$. Or, almost the same example, $g$ defined by
$$ g(x)=e^{-1/x^2} $$
for $x>0$ and $g(x)=0$ for $x \in... |
H: homology relative to homotopy equivalent subspaces
let $A$ and $B$ are two subspaces of a topological space $X$. Is it true that if
$A$ and $B$ are of the same homotopy type then the relative homology
$H_*(X,A;\mathbb Z)\cong H_*(X,B;\mathbb Z)$. More generally, if $A$ is a subspace of the top space $X$ and $B$ is... |
H: Showing that $\cos (x - \frac{\pi}{2}) = \sin x$
Show that$$\cos(x-\frac{\pi}{2})=\sin x .$$
I understand I have to use$$\cos(A+B)=\cos A\cos B-\sin A\sin B .$$
Am I wrong to let $A=x$ and $B= -\frac{\pi}{2}$? Can someone please explain to me in detail how to solve this?
Thank you.
AI: You're on exactly the right... |
H: If $A,B$ are $n \times n$ complex matrices, then is it possible that $ABA-BAB =I$?
If $A,B$ are $n \times n$ complex matrices, then is it possible that
$ABA-BAB =I$?
I was able to solve a related problem "Can $AB-BA =I$?" using a trace argument, but that argument does not seem to extend to this problem
AI: Putting... |
H: How to Determine what these vectors are.
So I'm not sure how to solves these problems.
The Question asks to Determine whether the given vectors are orthogonal, parallel or neither.
1) a = <-5, 3, 7> and b = <6, -8, 2>
2) a = <4, 6> and b = <-3, 2>
So I understand that I have to find the angle between the two vect... |
H: Homology groups of a tetrahedron
I have been solving a lot of questions lately in preparation for an exam which is in about 3weeks. Well, I came across this question but I have no clue. I think I would need a lot of help.
I am asked to compute the homology groups of a tetrahedron (i.e a triangulation of the 2-spher... |
H: Inverse Laplace transform - using the table
I am trying to find the inverse Laplace transform $(g(t))$ of
$$ G(s) = \frac{2s}{(s+1)^2+4}$$
I know about the inverse transforms $e^{a t}\cos(\omega t)$ and $\mathrm{e}^{at}\sin(\omega t)$ however I am trying to get the inverse transforms without these, as these are no... |
H: how does this simplify?
I am really stumped as to how it simplifies. I should know this, but it has been a very, very, long summer.
AI: Pull out a factor of $2$ and cancel:
$$
\begin{align*}
\frac{4-2\sqrt{2}}{4} &= \frac{2(2 - \sqrt{2})}{2 \cdot 2}\\
&= \frac{2}{2}\cdot \frac{2-\sqrt{2}}{2}\\
&= 1 \cdot \frac{2-\... |
H: Looking for a 'quick-and-dirty' resource for proof techniques
I have limited time to spend on the resource (3-6 hours) and never done any proofs and I need to be able to apply deductive, inductive and other proof techniques to some relatively easy propositions (basic number theory, trees etc.).
Please recommend a r... |
H: Do absolute values of differentiable functions with roots always exhibit cusps?
I am interested in the following question because I am hoping to be surprised by an exotic counterexample.
Conjecture: Let $x=a$ be a root of a differentiable function $f$ such that $f(x)>0$ for $k<x<a$ and $f(x)<0$ for $a<x<q$, where ... |
H: Prove that $|\sin z|\geq1$ on the boundary of a square with corners $\pi(m + 1/2)(\pm 1\pm i)$ for $m=0,1,2,3,\ldots$
Prove that $|\sin z|\geq1$ on the boundary of a square with corners $\pi(m + \frac{1}{2})(\pm 1\pm i)$ for $m=0,1,2,3,\ldots$
My approach was to write
$$|\sin z| = |\sin x\cos iy +\cos x\sin iy|$$ ... |
H: How does this fraction simplify?
how does $$1+\cos2\theta = (2-\sqrt2)/2$$ give you $-\sqrt2/2$?
i get that you subtract 1 from the left side, but how does doing so on the right give you $-\sqrt2/2$?
AI: $1+\cos 2\theta=\frac{2-\sqrt 2}{2}=\frac{2}{2}-\frac{\sqrt 2}{2}=1-\frac{\sqrt 2}{(\sqrt 2)^2}=1-\frac{1}{\sqrt... |
H: Question about biased estimator
Assume $X$ is a random variable from a population with normal distribution. Using the likelihood function I get the expression below:
$$\hat{\sigma}_X^2 = \sum_{i=1}^n \frac{(X_i-\mu)^2} n$$
for variance.
I want prove that, $\operatorname{E}[\hat{\sigma}_X^2]=\frac{n-1} n \sigma_X^2... |
H: A weak form of the axiom of choice
I am trying to understand the relationship between two forms of the axiom of choice:
If $T=\{X_0,X_1\cdots \}$ is a family of non-empty mutually disjoint finite sets, then $\cup T$ contains at least one subset having exactly one element in common with each element of $T$.
Let $X\... |
H: Example of a non-smooth $Y\times_X Z$ over $Spec(K)$?
Let $K$ be a field and $X\to Spec(K)$ and $Y\to Spec(K)$ and $Z\to Spec(K)$ three smooth morphisms of schemes.
I have no intuition of relative smoothness. What is an example of such schemes and (not necessary smooth) morphisms $Y\to X$ and $Z\to X$ such that $Y... |
H: predicate logic syntax when universe of discourse is natural numbers/everything
From my course text:
If the universe of discourse is composed of the natural numbers then
we can write:
∀X (odd(X) ⇒ (∃Y (X = 2 x Y)))
If the universe of discourse is everything then we can write:
∀X ∃Y(odd(X) ⇒ (X = 2 x Y))
I know th... |
H: difference between functors and predicates
In my lecture notes for Discrete Structures, the professor introduced a definition on functors in the "Syntax of Predicate Logic" section.
Definition of functors: Let us consider a collection of symbols called
functors (each functor is associated to a natural number n,... |
H: Finding the probability that the population becomes extinct for the first time in the third generation.
I am trying to solve the following problem and am wondering firstly if my solution is correct and alternatively if there is a shorter way to compute this.
In a branching process the number offspring per individua... |
H: what is solution of these improper integrals?
$$\int_{0}^{\infty} x^{n}\sinh x dx$$
$$\int_{0}^{\infty} x^{n}\cosh x dx$$
$$\int_{0}^{\infty} x^{n}\tanh x dx$$
what is solution of these improper integrals?
AI: All three of these hyperbolic functions are nonnegative when $x \geq 0.$ Thus,
$$\int_{0}^{\infty}x^n \sin... |
H: problem related to circles
My problem is:
$AB$ is a diameter of a circle and $BM$ is the tangent at $B$. If the tangent at $C$ on the circle meets $BM$ at $X$ and if $AC$ produced meets $BM$ at $Y$, prove $BX=XY$.
And this was how I approached, In the figure, we join $B$ to $C$ to have angle $ACB$ equ... |
H: Problem with powers of matrices.
Suppose that the matrix $A$ is as follows:
$$A=\begin{bmatrix}
-3&4\\
2&3\end{bmatrix}$$
We need to prove that $A^{2n + 1}=A$.
The way I tackled this problem is as follows:
If $A^{2n + 1} =A$, then $A^{2n}$ must be the same as the identity matrix $I$.
Thus $(A^2)^n$ must ... |
H: what does "SU2 is homeomorphic to the unit 3-sphere in R4 " mean
Michael Artin's book on algebra
here is a section starting from " $SU_2$ is homeomorphic to the unit 3-sphere in $R^4$ " and then compare the $SU_2$ to the unit 3-sphere $x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1 $ I am a bit confused
this unit-3 sphere ... |
H: Open problems in Algebraic Topology, Geometric Topology and related fields
I've been reading about the Arf invariant and came across the following conjecture (see here):
Each framed bordism class contains a manifold which admits a (possibly different) framing with zero Arf invariant.
Is this conjecture still open? ... |
H: Hom$(G,A)$ $\cong$Hom $(G/G',A)$
Let $G$ is a group and $\varphi: G\longrightarrow A$ be any group homomorphism wherein $A$ is abelian. Then $$\text{Hom}(G,A)\cong \text{Hom} \left(\frac{G}{G'},A\right)$$
What we have here is, for all homomorphism $\varphi: G\longrightarrow A$:
$G'\trianglelefteq \ker(\varphi)\t... |
H: How do I apply The Birthday Problem to Betrayal At The House On The Hill?
I just got a new board game called Betrayal At The House On The Hill. In the game you can trigger a 'haunt' determined by looking up on a table in which Room an Omen Card is found (after some dice rolling). Let us assume that in each game the... |
H: How to motivate inner product spaces
What is the most motivating way to introduce general inner product spaces? I am looking for examples which have a real impact. For Euclidean spaces we relate the dot product to the angle between the vectors which most people find tangible. How can we extend this idea to the inne... |
H: A prime in a separable extension splits completly iff it does so in the galois closure.
I just did the following exercise out of Neukirch's Algebraic Number Theory:
"A prime ideal p of K is totally split in the separable extension L|K iff it is totally split in the Galois closure N|K of L|K"
Now, I managed to solve... |
H: Find the intersect points between a rectangle and a segment
I want to find the (maximum 2) intersect points between a segment (a line that is of finite length defined by a starting and an ending points) and a rectangle. Both of them are "floating" in 2-D space without being bounded to the center (0,0) or to any of ... |
H: Subspaces $\mathcal{C_k}\subset(\mathbb{Z}/2\mathbb{Z})^3$ with $\dim(\mathcal{C_k})=2$
Enumerate all two-dimensional subspaces of the space $(\mathbb Z/2\mathbb{Z})^3$.
Obviously we have $|(\mathbb Z/2\mathbb{Z})^3|=2^3=8$ with
$$(\mathbb Z/2\mathbb{Z})^3=\{v_0,\ldots,v_7\}=\left\{
\begin{pmatrix}0\\0\\0\end{pma... |
H: Characterization of the Closure of a Set
Say I have a metric space $(X, d)$ and a set $A\in X$. I want to prove that if $a \in \overline{A}$ then there is a convergent sequence $\{x_n\}$ that converges to $a$. Could I have any help?
AI: If $a \in \overline{A}$ (the closure of A), then there are two possibilities:... |
H: proof of a lemma on stopping times
Hi I got some problem with the proof of this lemma which is left as excercise
Let $(F_t)$ be a filtration and S,T be stopping times, then show $
\{S=T\},\{S≤T\}，\{S<T\} \in F_S\bigcap F_T$.
Could any one show to how to do it ? thanks in advance.
AI: By symmetry, you just have to ... |
H: Separation of mathematics and metamathematics
I recall reading that it's important to separate mathematics and metamathematics. What exactly does this mean, and why is it so?
I understand that this question may make no sense without more context, but I've been curious about this for a while and can't seem to find w... |
H: When does an abelian group have a composition series?
There is an exercise in the book "An Introduction to the group theory by J.J. Rose" which can also be found as a proposition in "Abstract algebra by T. Hungerford":
Every finite group has a composition series $^*$.
Now I am doing the exercise $5.9$ of the firs... |
H: Why is the Markov property implied by the existence of a transition matrix?
If $\left(X_n\right)_{n\in\mathbb{N}_0}$ is an $E$-valued stochastic process with distributions $\left(P_x\space:\space x\in E\right)$ satisfying
$$\mathrm{P}_x\left(X_0=x\right)=1$$
and stochastic kernel $\kappa\left(x,B\right)=\mathrm{P}_... |
H: Discussion: Differing definitions for the rank of a set
I've just identified that the definition we used for the rank of a set in my set-theory class (1.) is different than the one I commonly find on the web (2.).
$\text{rank}(A)=\min\{\alpha\mid A\in{V}_\alpha\}$
$\text{rank}(A)=\min\{\alpha\mid A\subseteq{V}_\a... |
H: Why is $1 - 2\cos^2(\frac{\pi n} {4})$ the same as $ -\cos(\frac{\pi n} {2}) $?
Why is $$1 - 2\cos^2(\frac{\pi n} {4})$$ the same as $$ -\cos(\frac{\pi n} {2}) $$
I think the 1 in here, I can just neglect when I look at borders from $[-\infty , \infty]$.
But what's going on with the cosine in here?
AI: If you k... |
H: Homology of submanifold
Let $M$ be a manifold space, Let $H^1(M,\mathbb R)=\{0\}$, then is it possible to get a submanifold $S$ of $M$. such that $H^1(S, \mathbb R)\neq \{0\}$.
If $M$ is simply connected then we can remove a suitable point and then remaining open subset will be sub manifold and which will ha... |
H: Equivalent measures
let $\mu,\sigma$ be finite measures on the complex unit circle $\mathbb{T}$. Would it be correct to say that $\mu\sim\sigma$ implies that $L^2(\mu,\mathbb{T})=L^2(\sigma,\mathbb{T})$ as topological spaces?
AI: I would not say that $L^2(\mu)=L^2(\sigma)$ as topological spaces, meaning that they a... |
H: Prove one standard deviation lies on inflection points
Is my conjecture correct that one standard deviation lies on the inflection points of the normal distribution curve (of the probability density function)? How can it be proved using the standard deviation formula?
AI: We will find the inflection points of the d... |
H: What is $\frac{p\mathbb Z}{p^\alpha\mathbb Z}$?
May I ask someone elaborate how to treat the quotient group $\frac{p\mathbb Z}{p^\alpha\mathbb Z}$ when $p$ is prime? Any answer is appreciated.
AI: Consider the composition of homomorphisms of additive groups
$$
\Bbb Z\longrightarrow p\Bbb Z\longrightarrow p\Bbb Z/p^... |
H: Find a plane perpendicular to a plane passing by point
In $\mathbb R^4$ I have: $$\pi: \begin{cases} x+y-z+q+1=0 \\ 2x+3y+z-3q=0\end{cases}$$
I have to find $\pi' \bot$ $ \pi $ and passing by $P=(0,1,0,1)$. How can I do that? Thanks a lot!
AI: If this is homework, it would be nice you add the tag "homework". :-)
... |
H: What is $\mathbb Z[[t]]$? What are the double brackets?
What does $\mathbb{Z}[[t]]$ mean? Why are there double square brackets?
I can't search through Google, because I can't search Latex.
AI: That is the ring of formal power series in $t$ with integer coefficients, i.e., of $$\sum_{n=0}^\infty a_nt^n,$$ with $a_n\... |
H: Minimal polynomial of $i\frac{\sqrt{3}}{2}+\frac{1}{2}$
I'm looking for the minimal polynomial of $\alpha = i\frac{\sqrt{3}}{2}+\frac{1}{2}$ in $\mathbb{Q}[x]$. A polynomial with a root $\alpha$ is
$$(2(x-\frac{1}{2}))^4-9.$$
A computer algebra system shows me that the polynomial is irreducible. I looking for a wa... |
H: Must a space homeomorphic to $\Bbb R\setminus \Bbb Q$ have a countable complement?
There is a problem from a list suggested practice problems that I am having issues with. It says:
Suppose that $X$ is a subspace of the real line $\mathbb{R}$ which is homeomorphic to the space of irrational numbers. Is the complemen... |
H: Image of a morphism
According to Wikipedia, image of a morphism $\phi:X\rightarrow Y$ in a category is a monomorphism $i:I\rightarrow Y$ satisfying the following conditions:
There is a morphism $\alpha:X\rightarrow I$ such that $i\circ\alpha=\phi$.
If $j:J\rightarrow Y$ is a monomorphism and $\beta:X\rightarrow J$... |
H: Finding a coefficient of a unknown to have a unique solution in a system
I am resolving the following matrix:
1 2 1|1
-1 4 3|2
2 -2 a|3
Where I have to find all the values of $a$ so that the system can have a unique solution. Doing elementary row operations I ended up with:
1 2 1 |1
0 1 2/3|1/3
0 0 2+a|4
Whe... |
H: How to check if a point is inside a rectangle?
There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
AI: Let $P(x,y)$, and rectangle $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3),D(x_4,y_4)$
Calculate the sum of areas of $\triangle APD, \tr... |
H: Problem with basic probability concepts.
I saw this question and calculated 1/16 as the answer.
Q1) A lucky boy has been getting heads every time in three tosses of a coin, what is the probability that he will get heads again in the fourth tossing of the coin?
Then, this question came along and I calculated 1/36 as... |
H: Proving a polynomial irreducible over finite field
Possible Duplicate:
How do I prove that $x^p-x+a$ is irreducible in a field of $p$ elements when $a\neq 0$?
How to prove that $x^p - x - 1$ is irreducible over $F_{p}[x]$.
I thought proving that this polynomial has no roots over $F_p$ but that is not enough to ... |
H: A non-abelian $p$-group $G$
There is some facts about finite non abelian $p$-groups over the site. For example, when $n=3$: Nonabelian groups of order $p^3$. I have found the following problem in my very old works unsolved, claiming:
$G$ is a finite non-abelian $p$-group, $|G|=p^n$. Then $|G'|\neq p^{n-1}$
When $... |
H: Distance Between Two Sets of Points
Consider two sets of $N$ $n$-dimensiononal points each:
$$\mathcal{X}= \lbrace \mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N \rbrace,$$
$$\mathcal{Y}= \lbrace \mathbf{y}_1,\mathbf{y}_2,\dots,\mathbf{y}_N \rbrace,$$
where $\mathbf{x}_i,\mathbf{y}_i\in\mathbb{R}^n$.
Is there a metr... |
H: A question about composition of functions
If $f(x)=g(h(x))$, why is $f^{-1}(x)=h^{-1}(g^{-1}(x))$ ?
AI: Think about dressing your feet. Here are the instructions
Put on socks
Put on shoes
What is the reverse of this operation?
Remove shoes
Remove socks.
You must undo the operations in the reverse order in w... |
H: homology with field coefficients
Let $X$ be a topological space and $F$ be a field. I want to understand why the homology $H_n(X,F)$ is an $F$-vector space : that is how do we define the multiplication of a class in $H_n(X,F)$ by a scalar $\alpha\in F$?
My guess: if we take a class $c\in H_n(X,F)$ as a formal sum ... |
H: When can a birational map of curves be extended?
I'm looking at algebraic curves at the moment (irreducible varieties of dimension 1). I have already had a great answer concerning the closed sets of a curve, so I'd thought I'd try my luck on stack exchange again; though I wouldn't be surprised to find out that this... |
H: Point spread function (PSF) expressed as a convolution and a sum
The Wikipedia article on the Point Spread Function (link) discusses how an imaging system can be conceptually described using linear system theory. A convolution of the PSF with the image in the spatial domain is equivalent to a multiplication in the... |
H: Meaning of Point Evaluation
I read in some general measure theory books and there is always like "define measure $x$ to be the point evaluation at $y$..." but when I look around online and some other books there is no mention on what is "point evaluation". Can anyone explain to me what is point evaluation?
AI: "Poi... |
H: Determining whether or not spaces are separable
I've been going over practice problems, and I ran into this one. I was wondering if anyone could help me out with the following problem.
Let $X$ be a metric space of all bounded sequences $(a_n) \subset \mathbb{R}$ with the metric defined by
$$d( (a_n), (b_n)) = ... |
H: What is the Inverse function of $f(x)=192x-16x^2$?
What is the inverse function of: $$f(x)=192x-16x^{2}$$
I have been finding myself going in a circle in trying to complete this problem which otherwise looks simple, but for some reason I am at a block. Could someone help explaining how to go about this problem?... |
H: Trilateration to know relative distance to a point.
I've been trying to find the distance $d$ in this system where I only can know the distances $a0, a1, b0, b1, c0, c1$ but not the position of $A, B, C, P0, P1$.
I tried 2D trilateration using nonlinear least squares method but I'm not being able to solve it.
Is i... |
H: Regular Tetrahedron
If $ABCD$ is a regular tetrahedron with $AB=l, $what is the measure of the angle between $AB$ and $ACD$ or mathematically $\angle(AB, (ACD))$ and I ask you, please, to explain the method. Thanks.
AI: If the side length is 1, then the height of a face is $\sqrt3\over2$ by Pythagoras. Let $P$ be ... |
H: I have a hard time solving integrals of trig equations. Any advice?
I'm working through a section of integrals of trig functions, and I always have the hardest time knowing off the bat the next step to make. I know the techniques, but knowing which step to take is difficult. There's a tree of possible solutions a... |
H: Is every inductive set also transitive?
I stumbled upon this exercise:
Prove that, if $X$ is an inductive set, then $Y=\{x \in X\colon x \subset X\}$ is inductive.
I easily proved that $\emptyset \in Y$ but then I could not find a way to show that $\forall y \in Y, y \cup \{ y \} \in Y$.
Is every inductive set ... |
H: Stolz region - stuck with the proof
Given two points $z_1,z_2$ such that $ \lvert z_i\rvert<1$, show that for every point $z\ne 1$ in the closed triangle with vertices $z_1,z_2,1$ following holds: $$ \frac{\lvert 1-z\rvert}{1-\lvert z\rvert}\le K,$$ where $K$ is a constant that depends only on $z_1, z_2.$
Determine... |
H: Can true randomness come out of mathematical rules?
For example, prime numbers, they seem very random, and they are defined by a simple set of rules. I can't see how real randomness could exist in the real world, but what about mathematics?
AI: Mathematics doesn't purport to generate randomness from scratch.
Probab... |
H: Integer solutions of $p^2 + xp - 6y = \pm1$
Given a prime $p$, how can we find positive integer solutions $(x,y)$ of the equation:
$$p^2 + xp - 6y = \pm1$$
AI: Certainly if $p=2$ or $p=3$ you cannot find such $x$ and $y$. For the others, note that ny prime $\gt 3$ is of the form $6k\pm 1$.
Suppose that $p=6k+1$. Th... |
H: The direct image of the inverse image of a surjective map
If $f:A\rightarrow B$ is a surjection and $H\subseteq B$, how to show that
$$
f(f^{-1}(H))=H
$$
AI: Let $x\in H$, since $f$ is surjective there is some $a\in A$ such that $f(a)=x$, and this $a\in f^{-1}(H)$ by definition (since $x\in H$). Therefore $f(a)=x\i... |
H: Cardinality of the set of all functions from blank to blank
I'm trying to determine if a bunch of examples constitute countable or uncountable sets. One of which is the set of all functions from the positive integers to the positive integers. Word on the street is that it's uncountable, but I can't really convinc... |
H: Would this space be homeomorphic to the set of irrationals?
I've been reviewing various problems dealing with interesting homeomorphisms, and I came across this one.
Is the product of the space of irrationals and the space of rationals homeomorphic to the space of irrationals?
I haven't been able to make any prog... |
H: Final step in solving differential equation
I can't figure out the last step of the following question:
Show that the general solution of the differential equation: $dy/dx + y\tan x=1$ is given by: $y\sec x=\ln(\sec x + \tan x) + C$
This is what I've worked out so far:
To make things easier let $u(x)=\exp(\int\! ... |
H: basic t-test statistics
I'm writing an article, and I'm on a deadline so please be kind to me, I'd like to test for statistical significance of the difference before and after an intervention but I'm no real math genious
My sample size is 63 pre and post-intervention interviews, and I have two different mean scores... |
H: Is it possible to extend phi to higher order polynomials?
Below, $x=\phi$ when $n=2$:
$$x^n-\sum_{i=1}^{n}x^{n-i}=0$$
($\phi$ being the golden ratio)
Is there a way to express $x$ in terms of $\phi$ for $n>2$?
AI: When $n=5$, we are talking about the roots of $x^5-x^4-x^3-x^2-x-1=0$. I suspect that, like most polyn... |
H: Exact deconvolution of two matrices using numerical techniques
Suppose that I am given two $n \times m$ matrices $\bf{A}$ and $\bf{C}$, and let $\bf{B}$ be a matrix that is convolved with $\bf{A}$, such that:
$\bf{A} * B = C$
In the above, $*$ is the convolution operator.
I am searching for a numerical procedure (a... |
H: Differential Equation Problems
I was hoping that someone could explain to me the major steps involved in the following derivation, as I am fairly new to differential equations:
http://gyazo.com/da8b0c7ea5502bd3000fdbd6e9424339
AI: There must be some extra information given beyond the quoted lines. Assuming that $$... |
H: Question about the cardinality of a space
I've been having conflicting thoughts about the following problem, and I was wondering if anyone could help me out.
Is is true that the cardinality of every regular separable space does not exceed $2^{\omega}$?
Thank you for your time.
AI: This is false: $\beta\omega$ is... |
H: Looking for a couple of injections
This is an extension of my last question I guess. If I'm not following protocol in that regard I sincerely apologize!
Basically, I'm considering the set of all functions from the positive integers to itself, and the set of all functions from the positive integers to the set {0,1}... |
H: On weak closure again.
I worked in this problem too:
On the weak closure
But after that I could not think of anything that can help me about the $l_p$ case. I mean, $\{ n^{1/p} e_n \}$ has $0$ as a weak accumulation point but no subsequence of this set weakly converges to $0$ (with $p \in [1,\infty)$) ?
I tried t... |
H: Given a group $G$ and $H\leq G$ is there a homomorphism $f: G \rightarrow G$ s.t $f(H) = id$ and $f(x) \neq id$ if $x\notin H$?
The motivation for this question is that I am trying to Read Basic Algebra 2 (despite a somewhat weak background for the book) and if this is true then it is fairly easy to show that moni... |
H: Is $\{3\}$ a subset of $(0, 3)$?
$\{3\}$ is a subset of $(0, 3)$, true or false?
I'm inclined to say true, however my textbook is indicating that it's false because $\{3\}$ is a set and $(0,3)$ is an interval. Perhaps I'm reading that out of context. Am I on the right path?
AI: No $3\not\in (0,3)$. Threfore, $\{3... |
H: Interesting question about the Sorgenfrey line
I was looking over previous comprehensive exams, and I saw this question about the Sorgenfrey line:
Can the Sorgenfrey line be represented as the union of three metrizable subspaces?
I haven't made any progress with this one, and I was wondering if anyone could help ... |
H: finding umvu estimator of parameter $\theta$
Suppose $X_1,\ldots,X_n$ are a random sample of uniform distribution on interval $(0,1)$. If
$$
Z_i = \begin {cases} X_i & \text{w.p. }\theta, \\
-X_i & \text{w.p. }1-\theta, \end {cases} \qquad 0<\theta<1
$$
and $P$ denote the number of pozitive $Z_i$, how can find um... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.