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H: We had $m$ (odd number)....
What we will get if we divide $2^{\varphi(m)-1}$ at $m$?
(The answer should be at $m$...)
Thank you!
(Mabye it's something that connect to Euler theorem or Fermat Small Theorem).
AI: You know that
$$2^{\phi(m)} \equiv 1 \pmod{m}$$
Thus
$$2^{\phi(m)-1} \equiv 2^{-1} \pmod{m}$$
Now,
$$2^{... |
H: Show stretch of numbers are composite, n! + 2, n! + 3 etc
Let $n\in\mathbb{N}$ with $n\geq2$. Consider the numbers $n!+2,n!+3,...,n!+ n$. Show that none of them is prime. Deduce that for each positive integer $N$ there is a stretch of $N$ consecutive composite numbers.
What I don't understand is what happens when n... |
H: Grammar derivation
Given these grammar productions:
$$\begin{align*}
&S\to A1B\\
&A\to 0A\mid\lambda\\
&B\to 0B\mid 1B\mid\lambda
\end{align*}$$
And given string $w = 01101$
If I wanted to make a) leftmost derivation and b) rightmost derivation, I thought it would work like this:
a) $S \rightarrow A1B \rightarrow ... |
H: Topological Group $G$ totally disconnected $\Rightarrow$ $G$ hausdorff?
On Wikipedia, I read that a topological group is necessarily Hausdorff if it is totally disconnected. Is that true?
I read it on this page:
http://en.wikipedia.org/wiki/Totally_disconnected_group
If not, does anyone has an example for a totall... |
H: Is $S^0$ a manifold?
Consider a singleton space $\{x\}$, it is a manifold and it is locally euclidean as there is a homeomorphism to $\mathbb{R}^0$. However, consider $S^0=\{-1,1\}$ with the discrete topology, there does not exist any homeomorphism to any open neighbourhood of $\mathbb{R}^k$ for $k \ge 0$, so it ca... |
H: Total k combinations with at least one element from each set.
Given $n$ sets the total amount of ways k of the elements can be combined is given by $$C(|S_1|+|S_2|+...+|S_n|,k)$$
Now suppose we wanted to find the total combinations possible when at least one element from each set must be included. This imposes the ... |
H: Runge-Kutta-Fehlberg Method Problem
Suppose all infected individuals remained in the population to spread the disease. A more realistic proposal is to introduce a third variable $z(t)$ to represent the number of individuals who are removed from the affected population at a given time $t$ by isolation, recovery and ... |
H: Discrete math sets help?
How would I do this question?
suppose U = {1,2...,9}, A= all multiples of 2, B = all multiples of 3, and C = {3,4,5,6,7}. Find C-(B - A).
I really don't know how I would approach this question so can anybody help with this?
AI: The multiples of $3$ in $U$ are $3,6$, and $9$, so $B=\{3,6,9\}... |
H: Show the ring $R = \mathbb{Z}/2\mathbb{Z}[x]/(x^4+x+1)$ is a field
Show that the ring $R = \mathbb{Z}/2\mathbb{Z}[x]/(x^4+x+1)$ is a field. Find the multiplicative inverse of the element [x^2+1] in that field.
From what I know,
$$\mathbb{Z}/2\mathbb{Z}[x]/(x^4+x+1) = \{ a_0 + a_1x+a_2x^2+a_3x^3 + x^4 : a _i = \{0... |
H: matrices equation
Let A, B, J 4x4 matrcies, such that:
$\eqalign{
& {A^t}JA = J \cr
& {B^t}JB = J \cr} $
Prove that:
${(AB)^t}J(AB) = J$
Any help will be appreciated.
AI: Hint:
$(AB)^t = B^tA^t$
$(AB)^tJ(AB)=B^t(A^tJA)B$ |
H: Singularities in (Elementary) Real Algebraic Geometry
I've taken an introductory course in algebraic geometry, and am currently studying Sumio Watanabe's book, "Algebraic Geometry and Statistical Learning Theory". In this book Watanabe gives a, possibly antiquated, definition of real algebraic singularities:
A po... |
H: Why don't I get $e$ when I solve $\lim_{n\to \infty}(1 + \frac{1}{n})^n$?
If I were given $\lim_{n\to \infty}(1 + \frac{1}{n})^n$, and asked to solve, I would do so as follows:
$$\lim_{n\to \infty}(1 + \frac{1}{n})^n$$
$$=(1 + \frac{1}{\infty})^\infty$$
$$=(1 + 0)^\infty$$
$$=1^\infty$$
$$=1$$
I'm aware that this l... |
H: N tosses of a coin ,what is the probability that number of heads are even?
We have a coin with probability of head is p and probability for tail is (1-p).
We toss the coin N times, what is the probability that the number of tosses that show head is even?
What I've tried is to sum over all even k's (k= 0,2,4,...) an... |
H: Why discriminant method gives wrong answer when searching parabola that touches x axis?
Define a so that parabola y = a*x^2 - 5*a*x + 5*a+5 touches x axis.
Touching x axis means that discriminant is 0 (parabola has one double zero on x axis - that is a touch).
Or it can also mean that the vertex is on x axis, which... |
H: Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$?
This a step in my notes which I can't seem to understand clearly. Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$? I... |
H: Where does the function $f(z)=z\bar z+z/\bar z$ satisfy the Cauchy-Riemann equations?
Where does the function $f:\mathbb{C}\setminus\{0\}\to\mathbb{C}, f(z)=z\bar z+z/\bar z$ (where $\bar z$ is the complex conjugate of $z$) satisfy the Cauchy-Riemann differential equations?
I tried to write $f(z)$ as: $$f(z)=x^2+y... |
H: Find the remainder of $7^{2002}$ divided by 101.
This is what I have so far:
Since 101 is a prime and does not divide 7, we can apply Fermat's Little Theorem to see that $$7^{100} \equiv 1 \ (mod \ 101)$$
We can then reduce $7^{2002}$ to $7^{2} (7^{100})^{20} \equiv 7^{2}(1)^{20} \ (mod \ 101)$ which is where I'm s... |
H: normal distribution expected value
In this derivation:
http://www.sonoma.edu/users/w/wilsonst/Papers/Normal/default.html
$$f(x) = \sqrt{\frac{k}{2\pi}}e^{-\frac{k(x-\mu)^2}{2}}$$
they let
$x-\mu = v$
$dx = dv$
and conclude that
$$E(v) = \sqrt{\frac{k}{2\pi}}\int_{-\infty}^\infty ve^{-\frac{kv^2}{2}}dv$$
Why? Should... |
H: When is $\Bbb Q(\sqrt p)\subseteq\Bbb Q(\sqrt[3]q)$, with $p,q$ prime?
Given primes $p, q$, when do we have
$$\Bbb Q(\sqrt p)\subseteq\Bbb Q(\sqrt[3]q)$$
?
At first I tried a linear algebra approach, since in linear algebra, determining when one subspace is inside another is easy, just leading to a system of linear... |
H: Question on Rudin's Proof of the Residue Theorem
The Theorem in question is Theorem 10.42.: If $f$ is meromorphic in $U$, $A$ is the set of poles of $f$ and $\Gamma$ is a cycle in $U-A$ so that $Ind _{\Gamma}=0$ in $U^c$ then
\begin{equation}\frac 1{2\pi i}\int_{\Gamma}f=\sum_{a\in A}Ind_{\Gamma}(a)Res_f(a)
\end{eq... |
H: Prove that $\frac{n^2+(-1)^nn+2}{7n^2+3}$ converges to $\frac{1}{7}$
I want to show that $\frac{n^2+(-1)^nn+2}{7n^2+3}$ converges to $\frac{1}{7}$ using the definition of convergence.
Skratch work:
I need $\mid\frac{n^2+(-1)^nn+2}{7n^2+3}-\frac{1}{7}\mid<\epsilon$.
So I take $\frac{n^2+(-1)^nn+2}{7n^2+3}-\frac{1}{7... |
H: A dense subset of a Hilbert space
I am curious about the following problem:
Consider the Hilbert space (a weighted $L^2(\mathbb{R})$ space):
$$\mathscr{H}=\bigg\{f: \mathbb{R}\to\mathbb{R}\text{ Lebesgue measurable}\,\bigg|\,\int_\mathbb{R} \big|\,f(x)\,\big|^2\exp\Big(-\frac{x^2}{2}\Big)\,\mathrm{d} x<\infty\bigg\... |
H: Differential Equation Question $(\frac{2x^3}{y}-4x^2\cdot e^{4y})dy-(2x^2lny-2x\cdot e^{4y}) dx$
I am trying to find the solution for the following equation
$$(\frac{2x^3}{y}-4x^2\cdot e^{4y})dy-(2x^2lny-2x\cdot e^{4y}) dx$$
what I tried to do is, set
$P(x,y) = (2x^2lny-2x\cdot e^{4y})$
$Q(x,y)=(\frac{2x^3}{y}-4x^... |
H: Find T10(x): the Taylor polynomial of degree 10
I keep thinking that all you do is plug in 0 into the problem for c0 and then solve. And then you plug in 2 for c2...and so on.
I feel dumb...
Thanks!
AI: Hint: The general form of a Taylor series about $0$ is
$$f(x) = \sum_{n= 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$... |
H: Continuity- Image of a function
I have the following topological space:
$\tau=$ {$U\subseteq R: 1\notin U$} U {$R$}
and the following application:
$f: (R, \tau)\to (R, \tau)$
I have to see that if f(1)=1, then f is continuous.
Is $f^{-1}(R)=R$??
AI: The preimage of the codomain is always the entire domain, so $f^... |
H: Smallest values of y for values of x
What are the largest and smallest values of $y=x^3 - 12x + 1$ for values of $x$ in the range $-3$ to $+5$?
So I find the question a bit odd as it is stated, so I'm wondering if anyone can clear up what is meant? What I just did is:
$$y' = 3x^2 - 12$$
$$ x^2 = 4, x = \pm 2$$
$$... |
H: Proof this function is constant
I have the following topological space:
$\tau= \{U\subseteq R: 1\notin U\} \cup \{R\}$
and the following application:
$f: (R, \tau)\to (R, \tau)$
I have already proved that if $f(1)=1$, then $f$ is continuous.
Now, I have to see that if $f$ is continuous and $f(1)=y$ where $y$ is ... |
H: Perfect Fourth Power - Pigeon Hole Principle
Let $a_1, a_2, ..., a_n$ be positive integers all of whose prime divisors are $\le$ 13.
Show that if $n \ge 193$ then there exists four of these integers whose product is a perfect fourth power.
I tried getting many pairs of numbers which multiply to a square but did not... |
H: Find all the values of x such that the given series would converge.
$\sum_{n=1}^{\infty} (5^n (x-9)^n) / (n+9) $
So this is a homework question and we have unlimited tries to check our answer, however, the answer I got as well as my friend who has been helping me with my Calc hw got is considered wrong.
The resul... |
H: Question about one of the first problems in Spivak's Calculus
It's about Chapter I, Problem 21 from Spivak's Calculus:
Prove that if:
$|x - x_0| < \frac{\epsilon}{2}$ and $|y - y_0| < \frac{\epsilon}{2}$
then
$|(x + y) - (x_0 + y_0)| < \epsilon$
$|(x - y) - (x_0 - y_0)| < \epsilon$
I tried somehow making $2\times$ ... |
H: Expected Value of Y = MIN(X, 100) where X~Geometric
Let X = Geometric$(\theta)$ and Y = Min(X, 100). Compute E(Y)
My thoughts are:
Y = g(X) = Min(X, 100).
E(Y) = E(g(X))
= $\sum_{x=0}^{\infty} g(X)P(X=x)$
$$= \sum_{x=0}^{\infty} Min(X, 100) P(X=x)$$
$$= \sum_{x=0}^{100} x P(X=x) + \sum_{x=101}^{\infty} 100 P(X=x)$$... |
H: Calculate the confidence interval for a single variance given two variances (using pooled variance)
I am having trouble trying to interpret why I'm given some value for a homework exercise. It goes like this:
Suppose we're studying the time it takes for a certain industrial process to complete. A recent study, whi... |
H: experimental sequence of number
I'm doing a small numerical experiment. I got, from the first simulations, the following sequence of numbers. I'm trying to imagine a mathematical law behind this sequence. It could be a geometric progression?
1.333333333
1.75
2.491107286
3.835656425
6.219456109
10.44686695
17.998158... |
H: Mathematical induction... divisible by 4
Hello I need to proof that the expression $(9^{n}+3)$ is divisible by $4$.
It is true if I calculate it for $n=1$
for $n + 1$ I got stuck in here:
$9 \cdot 9^n + 3$
I don't know how to continue. Can anyone help me please?
AI: Suppose $9^n + 3$ is a multiple of $4$. Then
$$ ... |
H: Quotient space of S1 is homeomorphic to S1
$S^1=\{z\in\mathbb{C}\mid |z|=1\}$, let $w\sim z$ iff $w=z\vee w=-z$ (identifying antipodal points). Prove $S^1/\sim$ is homeomorphic to $S^1$. Which function should be used to construct a homeomorphism? I am not good at analysis. Thanks!
AI: $f:S^1\to S^1,\ e^{it}\mapsto ... |
H: Derivative of $-2e^{-x^2}x$
$$f(x) = -2e^{-x^2}x$$
Find $f'(x)$:
I have making a small mistake some where in my calculation and I cannot find it.
The answer is stated as:
$$e^{-x^2}(4x^2-2)$$
However I got something different from my work:
$$f'(x)=-2[(e^{-x^2})'(x)+(e^{-x^2})(x)']$$
$$=-2[(e^{-x^2})(-2x)+e^{-x^2}]... |
H: Can we assume independence of random variables?
I have the following problem I want to solve: Let $\delta,\varepsilon>0$ and $X_n$ a sequence of non-negative random variables such that $P(X_n\geq \delta)\geq \varepsilon$, show that with probability one $\sum_{n=1}^{\infty}X_n = \infty$.
I solved this problem very ... |
H: DFA Transition Function Inductive Proof
Show for any state $q$, string $x$, and input symbol $a$, $\hat\delta(q, ax) = \hat\delta(\delta(q, a), x)$, where $\hat\delta$ is the transitive closure of $\delta$, which is the transition function for some DFA.
I think the best way to proceed is by induction and that the f... |
H: Show $\pi\alpha\pi^{-1}$ and $\pi\beta\pi^{-1}$ are disjoint given $\alpha$ and $\beta$ are disjoint for an arbitrary permutation $\pi$
Let $\alpha$ and $\beta$ be disjoint cycles of the same length $s$. Show that for any permutations $\pi$ we have that
$$\pi \alpha \pi^{-1}\quad \text{and }\quad \pi \beta\pi... |
H: Can even-dimensional complex skew-symmetric matrices be made block diagonal by orthogonal matrices?
Suppose an even dimensional matrix Q is complex and skew-symmetric. Can Q be written as $Q= O^T \Sigma O$, where O is orthogonal (i.e. $O^T O = O O^T=1$) and $\Sigma$ is block diagonal with $2 \times 2$ blocks?
AI: W... |
H: if $\mu(A)>0$, and $f
here is my doubt:
Let $(X,\mathcal{T}, \mu)$ be a measure space, $f,g:X\to \overline{\mathbb{R}}$ measurable functions (indeed with finite integral), and $A \in \mathcal{T}$ such that $\mu(A)>0$ and
$$
f(x)<g(x)\quad \forall x \in A
$$
then $\int_A f\ d\mu<\int_A g\ d\mu$.
Any hint in or... |
H: If $p$ is prime, then $n\mid\varphi(p^n-1)$
How can I prove that the value of $\varphi(p^n-1)$ (where $p$ is prime and $n$ is some positive integer) is some multiple of $n$? The purpose of this is to prove that $n$ divides $\varphi(p^n-1)$.
AI: Consider the group of units $\Bbb Z/D\Bbb Z^\times$ where $D=p^n-1$, $p... |
H: If $r, s, t \in R$, then $r \gcd (s, t)$ is associate to $\gcd(rs, rt)$.
I seem to be stumped on this question. For the setting, let $R$ be an integral domain and let $r, s, t \in R$. The question asks
Show that $r \gcd(s, t)$ is associate to $\gcd (rs, rt)$
To start, let $d$ be some $\gcd$ of $s$ and $t$, and ... |
H: Quotient ring of a matrix ring
Let $F$ be a field. Consider the set
$$R = \left\{ \begin{bmatrix}
a & b \\
0 & c
\end{bmatrix} : a, b, c \in F \right\}$$
Think of a non-trivial two-sided ideal of $R$ and describe in a concrete way the quotient ring $R/I$. (Your description should be explicit enough to make th... |
H: Is it possible to simplify the following combination?
$$C(n, r-1) = \frac{(n)!}{(n - (r - 1))! (r - 1)!} $$
Could I simplify any further or break up the $(n - (r - 1))!$ part? I'm having a hard time following a problem in the book and if I could understand this it would really help out.
AI: Note that $$\frac{n!}{(n... |
H: Prove that $\lim_{n \to \infty} \int_0^2 e^{ x^2 / n}\,{\rm d}x$ exists and evaluate it.
I need to show that this limit exists and then evaluate it. It is from a section on uniform convergence of sequences. I know that if $f_n \rightarrow f$ uniformly and each $f_n$ is integrable, then I can bring the limit inside ... |
H: Why is $L^{1}(G)$ unital if and only if $G$ is discrete?
I've seen it stated in several sources and lecture notes for Abstract Harmonic Analysis that for a locally compact group $G$, $L^{1}(G)$ is unital if and only if $G$ is discrete.
What about the locally compact group $\mathbb{T} = \{\lambda\in\mathbb{C}: |\la... |
H: write formula to predict nth term of sequence $1, 1\cdot3, 1\cdot3\cdot5, 1\cdot3\cdot5\cdot(2n-1)$
How can I write a formula for a sequence with the following behavior:
{$1, 1\cdot3, 1\cdot3\cdot5, 1\cdot3\cdot5\cdot7, 1\cdot3\cdot5\cdot7\cdot9$}
1st term is $1$
2nd term is $1 \cdot 3 = 3$
3rd term is $1 \cdot 3 \... |
H: If $G=H\rtimes K$ where $H$ is cyclic, and $K$ is abelian, why is $G$ abelian?
This is a curious problem I've been stuck on.
Suppose $G=H\rtimes K$, where $H$ is cyclic of order $n$, $K$ abelian with $\gcd(|K|,\varphi(n))=1$, $\varphi$ being the totient function. Why is $G$ actually abelian?
I let $h$ be a gener... |
H: Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$
For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of $\frac{\sin x}x=\prod_{n=1}^\infty(1-... |
H: Power series representation
I'm trying to find the series representation of $ f(x)=\int_{0}^{x} \frac{e^{t}}{1+t}dt $. I have found it using the Maclaurin series, differentiating multiple times and finding a pattern. But I think there must be an easier way, using the power series of elementary functions. I know tha... |
H: If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.
An algebraic integer is a complex number that is a root of a monic polynomial with coefficients in $\mathbb{Z}$.
Let $\alpha$ and $\beta$ be algebraic integers. Then any solution to $x^2+\a... |
H: How can I further simplify $(a \le b) \lor (b \le a)$ to prove that it is a tautology?
Over $\mathbb{Z}$, $aRb \iff a \le b \lor a = 3b$. Determine if it is total.
I think it is:
Have arbitrary elements $a,b \in \mathbb{Z}$.
We have to prove that $aRb \lor bRa$, which can be written as:
$$(a \le b \lor a = 3b) \l... |
H: Combinations question confusing
There are 8 men and 7 women from which a group of 4 with at least 2 men must be selected. Find how many possible groups there are.
There are 2 ways which both make sense to me but only one method is right.
Correct method:
(No. of groups with 2 men, 2 women) + (No. of groups with 3 me... |
H: Probability density function of a summation of continuous random variables
Let $Z_{i} = \tau + X_{i}$, where $X_{i}$ is a exponential random variable ($X_i \sim \varepsilon(\lambda)$), $0<\tau, \lambda < \infty$
Assume $X_i$ are independent random variables.
Suppose $T_{n} = Z_{1} + Z_{2} + \cdots + Z_{n}$, find t... |
H: Extended Proof of the Theorem that a bounded analytic function is constant.
I am having trouble feeling convinced by my proof and more importantly - feeling confident in my working out. The question reads
(a) Let $f$ be an entire function such that there exist real constants $M$ and $N$ such that $|f(z)|<M|z|+N$ ... |
H: Show that this set of polynomials is ideal in F[x]
In $\mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $J$ be the set of elements of polynomials that have coefficients that add to zero (so $a_0 + a_1 + ... + a_n = 0$). Show that $J$ is an ideal of $\mathbb{F}[x]$. I know that the proof of this statement is meant ... |
H: Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$?
Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$? I am going through the solution of a problem I'm working on and this seems to be what they are saying. If its true, why so? I see that... |
H: How to prove that $P[X_1=X^{(i)}]=\frac{1}{n}$?
Let $X_1$, $X_2$ be two samples draw from a continuous distribution, then I think there is no reason to say that $X_1\leq X_2$ or $X_1\geq X_2$, so we may have
$$P[X_1\leq X_2]=P[X_1>X_2]=\frac{1}{2}$$
more generally, let $X^{(1)},\ldots,X^{(n)}$ denote the $X_1,\ldo... |
H: If $f(x) = \cos x$, explain, without taking the derivative, how you would find the $f^{(99)}(x)$?
My theory:
derivative of $\cos x = - \sin x $
derivative of $-\sin x = -\cos x $
derivative of $-\cos x = \sin x.$
cycle occurs three times but then what do you do??
Is there a good way to solve this?
AI: The cycle o... |
H: Proving that the closed unit square in the plane is compact.
My thoughts on proving this statement is as follows:
Suppose $G_a$ is an open cover of $Q= [0,1] \times [0,1]$. For each $x$ in $[0,1]$, there is some ball around $x$ with radius $r_x$ such that it covers $[x-r_x, x+r_x] \times [0,1]$. Since $[0,1]$ is cl... |
H: Is this a valid way to show a sequence does not split?
Suppose you have
$$
0\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}\to 0
$$
viewed as additive groups. Let $i$ be the inclusion of $\mathbb{Z}$ into $\mathbb{Q}$. I want to say this is not a split sequence, since there is no retract $f:\mathbb{Q}\to\mathbb... |
H: Checkers on a Chessboard
Given 2k pieces on a k by k chessboard, prove that there is always a sequence of pieces $K_1, K_2 \ldots K_{2n}$ such that $K_1$ and $K_2$ are in the same row, $K_2$ and $K_3$ are in the same column, $K_3$ and $K_4$ are in the same row ... $K_{2n-1}$ and $K_{2n}$ are in the same row, and $K... |
H: Orthonormal Matrices-Intuition
Why is it, geometrically, that the row space is ALSO orthonormal? What exactly does the transpose LOOK like? Normally the row and column space are two separate things, but in the case of an orthonormal matrix, you have that the row space is actually the inverse of the column space, so... |
H: Equivalence Relations Proof dealing 3 dividing x + y
Consider the relation $S$ on the Natural Numbers defined by $\quad x\,S\,y\quad$ if $3$ divides $\quad x + y.\quad$ Prove $S$ is not an equivalence relation.
I know an equivalence relation is one that is reflexive, symmetric, and transitive. I believe that S doe... |
H: Checking if proof is correct
I would like to check if my proof of this proposition has been correctly done. I would also like help on proving part (c). Thanks in advance.
Proposition: Let $H$ be a subgroup of a group $G$, and let $N$ be the normalizer of $H$. Prove that:
(a) $H$ is a normal subgroup of $N$
(b) $H... |
H: Find a basis for $p \in P_2(\mathbb R) $ with $p(7) = 0$?
Find a basis for the subset: $$ S = \{\;p \in P_2(\mathbb R)\;\; |\;\; p(7) = 0\; \} $$
I'm not sure how to approach this question.
$$
p(7) = a_0 + 7a_1 + 49a_2 = 0
$$
$$
a_0 = -7a_1 - 49a_2
$$
$$
a_1(-7 + x) + a_2(-49 + x^2) = 0
$$
Is this even in the right... |
H: What does it mean for a function $u : D \to \mathbb{C}$ to be harmonic, $D \subset \mathbb{R^2}$?
On page 167 of David Ullrich's "Complex Made Simple", he defines $u : D \to \mathbb{C}$ to be harmonic, $D \subset \mathbb{R^2}$, to be harmonic in $D$ if it is twice continuously real differentiable and $u_{xx} = u_{y... |
H: Compute Left Eigenvectors
How does one compute the left eigenvectors of a matrix? I cannot seem to quite get the answer.. I don't care what the matrix is. Let's just say I have matrix $A$ and have found the 'right' eigenvectors $e$ and I want to compute the left eigenvectors. Do we $A^Te = c$ Then the left eigenvec... |
H: Find expected value of this discrete distribution
My knowledge on probability topics is a bit rusty, so I was hoping you guys could help me.
Let X be the amount of products a person buys.
The probability that the person buys 1 to 12 is 60%.
From 13 to 20, 35%.
From 21 to 100, 5%.
I need to find the expected number ... |
H: Is the "first nonzero digit" function surjective?
For sets $A= \{x \in \mathbb{R}: 0< x< 1 \}$ and $B=\mathbb{Z_+}$ let $f$ be a function $\space f:A \rightarrow B$ such that $f(x)$ is the position of the first nonzero digit of $x$, ex.g. $\space f(0.2)=1$, $\space f(0.02)=2$. Determine whether $f$ is injective an... |
H: Differentiate $f(x)=\frac{2-x^2}{3x+x^2}$
I am double checking a question I was given on a test that was marked wrong and I am not sure how I got it wrong.
Find $f'(x)$:$$\begin{align*} f(x) &=\dfrac{2-x^2}{3x+x^2}\\
f'(x) &=\dfrac{(2x)(3x+x^2)-(2-x^2)(3+2x)}{(3x+x^2)^2}\\
f'(x) &=\dfrac{3x^2+4x-6}{(3x+x^2)^2} \end... |
H: Prove that a sequence diverges if and only if its subsequence diverges
Prove that $(x_n)_n$ diverges if and only if for every $a\in\mathbb{R}$, there exists an $\epsilon > 0$ and a subsequence $(x_{n_k})_k$ for $(x_n)_n$ such that for all $k\in \mathbb{N}$, $|{x_{n_k} - a}| \ge \epsilon$.
Thank you!
AI: If a seque... |
H: Projected area of a paralelogram over a plane
Let $u=\hat{i} +\hat{j} +\hat{k}$ and $v=\hat { i } - \hat { j } -\hat { k }$ two vectors that are the two coterminal sides of a paralelogram. Compute the projected area by this paralelogram over the plane whose unit normal vector is $n=\hat {i} + 2\dfrac{\hat {j}}{3}... |
H: Braid group B3 pure group
I will denote b1 as the twist of first two strands and b2 as the twist in the last two strands so that I have my two generators. When finding the kernelof B3 to S3 why is it wrong to list b1inverseb1 for the kernel?is it because by definition it is identity of braid group? What is an examp... |
H: Prove regular language closed under min and max
Given some regular language $L$, show that $L$ is closed under the following operations:
$$\begin{align*}
\min(L) &= \{w\mid w\in L,\text{ but no prefix of }w\text{ is in }L \}\\
\max(L) &= \{w\mid w\in L,\text{ but for no }x\text{ other than }\epsilon\text{ is }wx\in... |
H: Prove that $\Bbb R^2 - \{0\}$ is homeomorphic to $S^1 \times \Bbb R$.
No idea where to even begin. There is a hint: this requires construction of an explicit function.
AI: Hint: The plane minus the origin can be written as
$$\Bbb{R}^2 \setminus \{0\} = \{(r \cos{t}, r \sin{t}) : 0 < r < \infty, 0 \le t < 2\pi\}$$
D... |
H: If two matrices both multiplied by the same vector are equal are the matrices equal?
Assume A and B are n x n matrices.
If Av$_k$ = Bv$_k$ then is A = B where v$_k$ is a vector in R$^n$?
AI: No. Counterexample:
$A=\left[ \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right]$, $B=\left[ \begin{array}{cc} 1 & 0 \\ 0 & ... |
H: Ring homomorphism is injective but not surjective
Let $F$ be a field, let $R_1=F[x]$ be the ring of polynomials with coefficients in $F$, and let $R_2$ be the ring of all functions from $F$ to itself, with addition and multiplication defined as the usual operations on functions with values in a ring. The function
$... |
H: Partitions on C = {i,-1,-i,1}
Let $ C = \{i, -1, -i, 1\}$ , where $ i^2 = -1 $. The relation $R$ on $C$ given by $xRy$ iff $xy = \pm 1$ is an equivalence relation on $C$. Give the partition of $C$ associated with $R$
I would really appreciate any help. I am not really sure where to start with this. Is it as simple ... |
H: Connected $T_3$ space
I wanted to prove that a connected $T_3-$space containing atleast two points must be uncountable.
My attempt is as follows:
Let $X$ be a connected $T_3-$ space and let $x,y\in X$. Then there exist disjoint open sets $G,H$ containing $x$ and $y$ respectively. Since $X$ is connected, therefore, ... |
H: Partitioning a set with a relation on it
Let R be a relation on a set A that is reflexive and transitive but not symmetric.
Let R(x) = {y: xRy}. Does the set a = {R(x): x ∈ A} always form a partition of A?
I really don't know where to start with this one. I know that R(x) is the same as x/R except R is not an equi... |
H: If $|f|$ is Hölder continuous, what about $f$?
Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function such that $|f|$ is Hölder continuous with exponent $0<\alpha\leq1$. Does it follow that $f$ is also Hölder continuous with the same exponent?
I thought of this statement a few days ago and had tro... |
H: Produce two distinct topologies of R such that the first is strictly finer than the second but the two are homeomorphic to each other.
Need to prove that they are homeomorphic to each other.
So if T_1 and T_2 are topologies on set X with T_1 contained in T_2, every element of T_1 will be in T_2 and T_2 is the "fin... |
H: Proving Convergent Series made by continuous function $f$, $a_n=f(1/n)$
Let $f$ is continuous on an interval around 0, and let $a_n=f(1/n)$
Prove that if $f''(0)$ exists and $f(0)=f'(0)=0$, then $\displaystyle{\sum_{n=1}^{\infty}{a_n}}$ converges.
This is problem from Spivak calculus 4th, chapter 23 Exercise 6.
If ... |
H: finding all the roots (including complex)of the equation
Find all the roots of $z^4=16(z+2i)^4$.
Can someone help me teach/ guide to solve this equation?
AI: Just take the square root of both sides. Then affix with +/- to get 2 cases for that equation. Then for each case, get the square root again. Then affix the r... |
H: How does $1^\infty=\infty$?
I remember hearing in school long ago that $1^\infty=\infty$.
I was just wondering if anyone could explain this in laymen's terms?
AI: $1^\infty$ is indeterminate, as this Wikipedia entry explains. Or perhaps you remember something like $\lim_{n\to\infty}\sqrt[n]n=1$ ? |
H: Evaluate $\int_0^e{W(x)}\,\mathrm{d}x$
The function $W(x)$ satisfies $W(x)e^{W(x)}=x$ for all $x$. Evaluate $$\int_0^e{W(x)}\,\mathrm{d}x$$
I tried integrating $xe^{-W(x)}$ but can't see how to do it.
AI: Integrate by parts with $u=W(x)$ and $\mathrm{d}v=\mathrm{d}x$
$$\int{W(x)}\,\mathrm{d}x=xW(x) - \int xW'(x)\... |
H: The number of monomials of a given degree
I'm trying to understand why the number of the monomials of degree $d$ in $n+1$ variables is $C_{n+d,n}$. If someone could help me to remember how to solve this, I would be glad.
Thanks.
AI: The stars and bars argument in combinatorics shows that the number of ways to place... |
H: Uniformly continuous bijection from $X$ to the Cantor set
Let $X$ be a metric space, and $C$ be the Cantor set (equipped with the standard topology).
Let $f: X\to C$ be a uniformly continuous function. Assume that $f$ is
a bijection. Does it follow that $f$ is a homeomorphism?
I know that if $X$ is compact, th... |
H: Limit of $\frac{1}{a} + \frac{2}{a^2} + \cdots + \frac{n}{a^n}$
What is the limit of this sequence $\frac{1}{a} + \frac{2}{a^2} + \cdots + \frac{n}{a^n}$?
Where $a$ is a constant and $n \to \infty$.
If answered with proofs, it will be best.
AI: With $S_n = \frac{1}{a} + \frac{2}{a^2} + \frac{3}{a^3} + \cdots \frac... |
H: probability problem related to card shuffling
You have five cards numbered 1,...,5. You shuffle them, so that at the end every permutation of them is equally likely to show up. What is the probability that no card 'i' will end up in the i-th card after shuffling?
So I'm figuring out this problem. I got stuck becau... |
H: Cauchy's Problem Question $x(x^2+1)dy-(3x^2+1)ydx= x(x^2+1)^2dx,\ \ y(1)=2$
I want to solve the following equation
$$x(x^2+1)dy-(3x^2+1)ydx= x(x^2+1)^2dx,\ \ y(1)=2$$
what I chose to do is to order the equation to $()dy+()dx=0$ then to find integrating factor. so what I did :
$$(x^2+x)dy=(x(x^2+1)^2+(3x^2+1)y)dx$$
... |
H: How prove this inequality $f(a)\le f(b)$
Suppose $f(x)$ is continous on $[a,b]$,and for any $x_{0}\in [a,b]$.
the limit
$$\varliminf_{x\to x^{-}_{0}}\dfrac{f(x)-f(x_{0})}{x-x_{0}}\ge 0$$
show that
$$f(a)\le f(b)$$
My try: I found this problem is same as
How prove this $f(a)\le f(b)$
But for my problem,this ... |
H: An inequality in $L^p$-spaces
Let $\{f_k\}_{k=1}^{\infty}$ be a sequence in $L^p(\Omega,\Sigma,\mu)$ for $1\leq p<\infty$. Suppose $0<c=\inf_k \lVert f_k\rVert_p\leq \sup_k \lVert f_k\rVert_p=C<\infty$ and $f_if_j=0$ for $i\neq j$. Let $(a_k)_{k=1}^{\infty}$ be a sequence of real numbers such that $\sum_k \lvert a_... |
H: Let P1 and P2 be path connected. Prove that P1 x P2 is path connected.
So we know that separately they are both path connected.
That in the topological space X $\forall x,y \in X$ $\exists$ $f: I \rightarrow X$ such that $f(0)=x$ and $f(1)=y$.
Given P1 and P2 are path connected.... ok not sure what now....
AI: Le... |
H: Proof via equivalence laws; $(a \lor b) \equiv (b \lor a)$?
Is this a correct progression to prove that $p \rightarrow (q \rightarrow r) \equiv q \rightarrow (p \rightarrow r)$?
$$\begin{align} p \rightarrow (q \rightarrow r) & \equiv p \rightarrow (q \rightarrow r) \\
& \equiv \neg p \lor (q \rightarrow r) \text{ ... |
H: is this conjecture true or false?
I want to know if this conjecture istrue or false
$$\Large e^{\frac{ \ln x}{x}} \notin \mathbb{Z} $$
for every $x \in \mathbb{R} \setminus \{1,-1,0\} $
AI: You should ask this only for $x>0$, as the expression is not well defined otherwise. You can rule out the case $x\in(0,1)$ eas... |
H: How can i get the all digit of a number from 1st to last?
For example:
Given number=1234;
Failing,how can i separate all the digits from 1st to last?like 1,2,3,4
AI: Hint
$\lfloor \log_{10} x \rfloor$ is the number of digits (starting with $0$) and
$\left\lfloor \frac{x}{10^i} \right\rfloor {\rm mod}\ 10$ is the $i... |
H: Does $g(f(x))$ imply $g(x)$?
If $g$ is a function of $f(x)$ does this imply that $g$ is a function of $x$?
If yes, am I allowed to write the chain rule as:
$$\frac{{d[g(f[x])]}}{{dx}} = \frac{{d[g(x)]}}{{d[f(x)]}} \cdot \frac{{d[f(x)]}}{{dx}}
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerb... |
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