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I solved the following exercise:
Find an integer $n$ such that $U(n)$ contains a subgroup isomorphic to $\mathbb Z_5 \oplus \mathbb Z_5$.
Here $U(n)$ is the group of units modulo $n$.
To solve it I used that $U(st) = U(s) \oplus U(t)$ when $s,t$ are coprime and $U(p^n) = \mathbb Z_{p^n - p^{n-1}}$. I tried the first f... |
I need to proove that for any natural number $n>30$:
$$\pi(4n-3)<n.$$
In this inequality, $\pi(x):\mathbb{N}\to \mathbb{N}$ is the defined as follows:
$$\pi(x):=Card(\lbrace p \ | \ p\leq x\ \ s.t \ \ p \ \ is\ \ prime\rbrace ).$$
Plot:
ListPlot[Table[i - PrimePi[4 i - 3], {i, 31, 200}]]
[]1
Thank you.
|
Well, i have a method to find square roots to any number.
for Eg To find $\sqrt{58}$ or any number. We have to find which perfect square is place before it. 49 is placed before 58.
What we have to do is enter the value in this expression.$\sqrt{a}+\frac{1}{2\sqrt{a}}$ here a is 49. and we get $\space 7+\frac{1}{14}=7.... |
Suppose we have a branching process $\{X_n\}$, where $X_0=1$. How would I go about calculating the covariance $\operatorname{Cov}(X_j,X_i)$ for $i\leq j$?
Not sure how to start, so hint would nice to get me on the right track. I was given a hint that using the concept of conditional probabilities might be helpful here ... |
I would like to ask the following:
Are there "many" sets, say in the interval $[0,1]$, with zero Lebesgue measure but with Hausdorff dimension $1$?
The motivation for this question is the dichotomy between measure and category. There are certainly dense sets with zero Lebesgue measure. But a dense set need not have p... |
Suppose $f(x)$ is a monotone increasing function defined for all $x\in \mathbb{R}$. Show that for any $x_0\in \mathbb{R}$, the one sided limits $$f^+(x_0)=\lim_{x\to x_0^+}f(x) \text{ and } f^-(x_0)=\lim_{x\to x_0^-}f(x)$$ exist and that $f^+(x_0)\geq f^-(x_0)$.
I know that the limit definition is:
$\forall \epsilon>0... |
Let $W_1$ and $W_2$ be subspaces of vector space $V$ satisfying $W_1\cap W_2=\{0\}$ ,how to show $S_1\subset W_1$ and $S_2\subset W_2$ are linearly independent $\implies$ $S_1\cup S_2$ is linearly independent ?
Here is my proof:
let $v\in $ $S_1\cup S_2$, $s_1\in S_1,\ s_2\in S_2$.
Express $s_1=a_1x_1+...+a_ix_i$ for ... |
My Work:
If you take the cartesian product of any set with two arbitrary elements $a$ and $b$, and the resulting set is $\{\{x_1\},\{x_1,x_2\}\}$, then the only possible values for $a$ and $b$ are $x_1$ and $x_2$ by definition of the cartesian product.
This answer seems overly trivial and I think I'm doing something wr... |
If $a\equiv b \pmod{p_i}$ for $i=1,2,\cdots,k$ then prove $a\equiv b \pmod{ p_1p_2\cdots p_k}$
$a\equiv b \pmod{p_1}$ implies $a-b=p_1x_1$
Similarly, $a-b=p_2x_2,\ \cdots,\ a-b=p_kx_k$
So,
$$(a-b)^k = p_1p_2...p_kx$$ where $x=x_1x_2...x_k$
I'm stuck here, please help!
|
The following is the question I'm having a bit of troubles starting:
Musicnotes.com sells sheet music in the following genres: rock jazz, new age, and country. An experiment consists of recording the preferred genre for the next customer, and the number of songs purchased (assume 5 is the max). Consider the following e... |
Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible.
If $Det(A)=k$,and
$Det(A+kadj(A))=0$,
then find the value of $Det(A-kadj(A))$
My attempt:
$Det(A+kadj(A))=0$
$Det(A)Det(A+kadj(A))=0$
$Det(A^2+kAadj(A))=0$
$Det(A^2+k^2I)=0$
Using same steps,I get $Det(A-kadj(A))=\frac{1}{k}Det(A^2-k^2I)$ ... |
I am trying to prove that it is always possible to label the vertices of the $k$-th barycentric subdivision of a $(d −1)$-dimensional simplex with labels
$1, 2, . . . , d$ such that each simplex present obtains all $d$ labels in its
vertices.
For example, in the $2$ dimensional case, the first barycentric subdivision ... |
Prove that the space $\Bbb R_K$ is not regular.
where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$.
[Hint: Look at $0$ and $\{\dfrac{1}{n}:n\in \Bbb Z_+\}$]
My try:
1.Let us consider the set $A=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$ which... |
Assume the psychic is actually just randomly guessing on each flip.
The attempt:
let E be the event in question
number of outcomes per flip = 2
chance of correctly guessing the correct outcome = $ \frac{1}{2}$
notice the key word "at least": indicates that we should be doing a summation
You can get exactly 5 correct,... |
So I've been reading Hatcher and I am unsure what they mean when they say things like $H^\ast(\mathbb{R}P^n;\mathbb{Z}_2)\cong \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$ where $|\alpha|=1$. It is this last part I don't understand. What do they mean by $|\alpha|=1$. Aren't the variables in the coefficient ring merely place ho... |
If $a^n\equiv a \pmod n$ for all integers $a$, does this imply that $n$ is prime?
I believe this is the converse of Fermat's little theorem.
|
The fourth, seventh and sixteenth terms of an arithmetic sequence also form consecutive terms of a geometric sequence. Find the common ratio of the geometric sequence
|
I have an homework question that goes like:
$\cot(\theta) + \tan(\theta) = 2.5 $ is valid on some angles $\theta$ at section $0 < \theta < \pi/2$.
Find the value of $\sin(2\theta)$. (There is no need to find the value of $\theta$ in order to get the answer).
(a) $0.4 \quad$ (b) $0.5 \quad$ (c) $0.6 \quad$ (d) $0.7 \qua... |
I am dealing with the function $f(x)=\begin{cases}
\frac{1}{n} & \text{if }\frac{1}{n+1}<x<\frac{1}{n},\:n\in\mathbb{N},\\
0 & \text{ otherwise.}
\end{cases}$
I want to show it is Riemann integrable without using the fact that every bounded function containing at most countably many discontinuities is Riemann integrabl... |
Suppose $X$ is a topological space with subspaces $X \supset U \supset A$ such that $U$ deformation retracts onto $A$. We know that $H^*(X,U) \cong H^*(X,A)$--one way to see this is to take the long exact sequence of the triple $(X,U,A)$.
I was trying to get an intuition for why the map $(X,A) \to (X,U)$ induces an iso... |
Suppose we define width $w(P)$ of point set $P$ in a plane to be the ratio of the maximum distance to the minimum distance between the points in $P$.
(Assume unique coordinates so that $w(p)$ is finite.)
Can we lower bound $w(p)$ by some function $f(n)$ where $n = |~P~|$?
|
Consider the map $\phi:S^3 \rightarrow S^3$ given by $\phi(x_1,x_2,x_3,x_4)=(x_1,-x_3,-x_2,x_4)$.
Compute the degree of $\phi$. Does $\phi$ preserve orientation?
First I want to point everything that I made:
Some introduction: a regular value of a smooth map $\phi:M \rightarrow N$ is a point $a \in N$ such that for e... |
Find the general solution of the given differential equation:
$$ (x^2-4)\left(\frac{dy}{dx}\right) +4y = (x+2)^2 $$
I found the general solution of the D.E and I got the following correct solution:
$$ y = (x+C)\left(\frac{x+2}{x-2}\right) $$
I know that this is the correct solution and the next part of the question ... |
$\mu_1$ and $\mu_2$ are means of two groups, while $\mu$ is the overall mean.
I feel like this can be done with some basic algebra I've forgotten, but I'm not sure.
Here is some more context:
$$
f = \frac{\sqrt{\dfrac{\sum_{j}{(\mu_j - \mu)^{2}}}{J}}}{\sigma_{y}}
$$
which with two groups (J=2) reduces to:
$$
f = \frac{... |
An $n\times n$ matrix consistent of all ones, will have two eigenvalues: $0$ and $n$. The eigenvector associated with $n$ will be $(1,1,...,1)$, but are there then infinite solutions for the eigenvector associated with $0$? Because, in order for the eigenvalue to be zero, all rows of the eigenvector must add to zero, t... |
Let $C$ be an $n\times n$ matrix, $X$ is $n \times k$, $\epsilon$ is $n \times 1$
This is taken from a simply proof of strict exogeneity in an Econometrics textbook by Hayashi. The explanation he gives is that "Since C is nonsingular, $X$ and $C\cdot X = \tilde{X}$ contain the same information".
The explanation does n... |
A linear group action of a group $G$ on a vector space $V$ is the same thing as a homomorphism from G to the general linear group $GL(V)$.
attempt: Suppose a linear group action of a group $G$ on a vector space $V$ is given. For each $g \in G$ we obtain a map $V → V$, denoted by $\phi(g)$.
Define $\phi(g)(v) = g \cdot... |
Find the radius of convergence and interval of convergence of the series:
$\sum_{n=1}^{\infty}n^n x^{n^4}$
I'm really lost as to how to approach this problem. The other power-series problems were all very straight-forward compared to this one.
I attempted it using both ratio test and root test. I managed to get furthe... |
I always face trouble with these type of integrals.
I need to find $$\int{e^x \frac{x(\cos x -\sin x)-\sin x}{x^2}}dx$$
My problem would be solved if can express $f(x)$ like $g(x)+g'(x)$ but identifying $g(x)$ by trial and error method is sometimes tedious.Is there any easier approach?
|
I am trying to show that if $a^n\equiv a\pmod n$ for all integers $a$ then $n$ is square free.
I have an idea to start with the contradiction that suppose $n=p^2m$ for some prime $p$, then n does not divide $a^{p^2m}-a$ for some integer $a$. Any hints would be appreciated thanks.
|
Prove or disprove that $\forall n \in N$ $, \, n! \in \mathcal{O}(2^n)$
My attempt:
$f(n) = n!$
$g(n) = 2^n$
First I checked if I needed to prove or disprove this statement, and to do so I computed the $\lim_{n \to \infty}{\frac{f(n)}{g(n)}}$ = $\lim_{n \to \infty}{\frac{n!}{2^n}}$ which equals $\infty$ so this state... |
I faced two similar integrals today.
They are
$$\int\frac{1}{x^2(x^{2009}+1)^{ \frac{2008}{2009}}}dx$$
and $$\int\frac{1}{x^2(x^{2009}+1)^{ \frac{1}{2009}}}dx$$
No trigonometric substitution is working here.I tried almost all.What to do?
Moreover binomial expansion was perhaps possible but that would be really very ug... |
How would I write $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ as a sum of geometric series?
|
How many times on average (expected value) must we flip $6$ fair coins before we obtain $3$ heads and $3$ tails?
I know I need $∑ xp(x)$. I just don't know how to apply it.
|
This is my first assignment on these, so I would greatly appreciate your help.
Translate the following into predicate calculus. For each answer,
also state the assumed universe of discourse.
a) "Anyone who was an ancient Roman citizen and tried to kill Caesar was not loyal to Caesar."
My attempt:
Ex(R(x) & k(x)) -... |
Graph
I need to show this graph is not planar. I've attempted to find $K_5$ and $K_{3,3}$ as a subgraphs but haven't been successful yet. It's possible but unlikely this graph is planar but I haven't been able to draw it as a plane graph.
|
Would $\displaystyle\int_0^1 t^2x(t)\,dt$ be a linear functional?
For each $x$ in $P$ the function $y$ is defined by $\displaystyle\int_0^1 t^2x(t)\,dt$. I have to show that $y(ax+bz) = ay(x)+by(z)$. I am confused as to whether or not the $t^2$ term messes it up or not.
|
This one looks easy but I still could'nt figure it out.
$$\int ( \frac{1}{\sin x} - \sin x - \sin x \log(\sin x) ) dx$$
I tried substituting $\log(\sin x)=z$ but that's not working.Any suggestions?
|
There is Poisson random variable $X$
$$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$
And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is less than 1).
How can I find the distribution of $Z$?
And what is expectation value?
Expectation is linear, so is expectation value $\lceil\beta... |
In Jech's axiom of choice he proves following lemma (lemma 4.5(b) in his book):
There is an injective mapping from $M$ to $\mathrm{Ord}\times \operatorname{fin}(A)$, where $A$ is a set of atoms and $M$ is the ordered Mostowski model.
I have a doubt to his proof. Its proof considers the orbit
$$\operatorname{orb}(x) =... |
Monotone Convergence Theorem for general measure:
Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1, f_2, ...$ be a pointwise non-decreasing sequence of $[0, \infty]$-valued $\Sigma-$measurable functions, i.e. for every $k\ge 1$ and every $x$ in $X$, $$0 \le f_k(x) \le f_{k+1}(x).$$ Next, set the pointwise limit of th... |
Let $Y=\{f\in L^2(-\pi,\pi):f(t-\pi)=f(t) \text{for almost all $t\in(0,\pi)$} \}$
Show that there exists $g\in Y$ such that $$\|h-g\|_2=\inf \{\|h-f\|_2:f\in Y\}$$
where $h(t)=t$. Compute $\|h-g\|_2$ and specify $g$
I'm struggling with the second part. For the first half of the question, it is enough to show that $Y$... |
I'm working on the following exercise. Why does the solution omit applying induction on $n$? That is, assume $P(n)$ and then use that assumption to prove $P(n + 1)$.
|
I am trying to either find an example of such a set, or prove that no such set exists. I know of examples of dense sets with measure $1/2$ on specific intervals, such as $[0,1]$, but I haven't been able to find any set that satisfies this more general property.
|
Recall the evaluation homomorphism of a ring. For example, if $\mathbb R[x]$ is the ring of polynomials with real coefficients then we can evaluate a polynomial $p(x)=a_0+a_1x+\cdots+a_nx^n$ with respect to $c$ by letting $p(c)=a_0+a_1c+\cdots+a_nc^n$. Note that this operation is a ring homomorphism. Now it is a theore... |
Let $k$ be an algebraically closed field and $m\leq n$. Suppose $\pi:\mathbb{A}^n\to \mathbb{A}^m$ is map which sends $(a_1,\ldots,a_n)\to (a_1,\ldots,a_m)$. If $V$ is an affine algebraic set, then is it true that $\dim(V)\geq \dim(\overline{\pi(V)})$.
My attempt: If $V$ is affine variety, then $\pi|_V$ is a domin... |
If $a|c$ and $b|c$ and $\gcd(a,b) = 1$, prove that $ab|c$.
|
If true, prove else provide a counter example. This is a homework question and I cant figure it out. Please help.
|
Does the degree of a circle function $f:S^1 \to S^1$ simply mean how many times the mapping of $f$ wraps around the $S^1$?
|
Every sequence that was termed as a doob's martingale, I was able to deduce that it was also a martingale. So here are few of my questions:
1) Is it correct to say that every doob martingale is also a martingale?
2) What is the difference between the two then? When is it better to model a sequence of random variables a... |
Cleaner version: $$\left \| T(x) \right \|=\left \| x \right \|,\forall x\in V\Longleftrightarrow \left \langle T(x),T(y) \right \rangle=\left \langle x,y \right \rangle$$
I know that $\left \langle x,y \right \rangle=\frac{1}{4}\left \|x+y \right \|^2-\frac{1}{4}\left \|x-y \right \|^2$, for converting between inner p... |
If $Y_1, Y_2, \ldots , Y_n$ denote a random sample from an exponential distribution with mean $θ$, then $E(Y_i)=θ$ and $V(Y_i)=θ^2$. Thus, $E(\bar Y)=θ$ and $V(\bar Y)=θ^2/n$, or $σ_Y=θ/\sqrt{n}$. Suggest an unbiased estimator for θ and provide an estimate for the standard error of your estimator.
With these $Y_i$ and ... |
Is there a name for the subset of graph theory dealing with vertices and edges of distinct classes? For example, I could have a graph in which each vertex must be either blue, yellow or red and each vertex must be either dashed or solid.
I'm working on a problem with these characteristics but I have no background in g... |
Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$
That is, every elements of the vector is a series with the other terms in $\vec{c}$. Can I prove $\vec{c}=0$?
I tried solving the equations for $i=0,i=1$ and so on but I get nothing of this method. Maybe i... |
Define a category as follows: the objects are categories and the arrows are embeddings (full and faithful functors). Is this category Cauchy complete or complete?
|
$$\int{(2008 x^{2009}+2009x^{2007})}\cos(2008x) dx$$
I tried to integrate the second term by parts and also the first.None of the terms seems to cancel out.I don't know why by parts is not working.Any other method possible?
Only hints will suffice!
|
I'm having some difficulty understanding the solution to a particular Laguerre expansion.
The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < f|g> = \int_{0}^{\infty}f^*(x)g(x)e^{-x}dx $$
I know the first 4 polynomials $L_{n}(x)$, which states that $$ e^{-x} = a_{0}L_{... |
Let $S$ be the unit circle in $\mathbb{C}$, standard topology. Define the equiv. rel. $\sim$ on $\mathbb{R}$ as $x\sim y\iff x - y\in\mathbb{Z}$. I would like to prove that $\mathbb{R}/\sim$ is homeomorphic to $S$.
Idea: First, $\mathbb{R}/\sim$ is the reals modulo $1$. So define $z\mapsto e^{2\pi i z}$. This is a bije... |
If $P$ is a transition probability matrix of a finite state regular Markov Chain, and $m_{ij}$ is the mean return time, how can I show that $m_{ij} = 1+ \sum_{k \neq j}P_{ik}m_{kj}$? It seems rather trivial but I don't know how to get the $k \neq j$ part. Does anyone have any ideas?
|
Is there any genuine approach to find pair of positive integers $(x,y)$ such that both their sum and product is a perfect square?
One pair is $(5,20)$ but it looks to me that this question can be solved by hit and trial only which is very difficult thing to do.
|
How can I solve this using only 'simple' algebraic tricks and asymptotic equivalences? No l'Hospital.
$$\lim_{x \rightarrow0}
\frac
{\sqrt[3]{1+\arctan{3x}} - \sqrt[3]{1-\arcsin{3x}}}
{\sqrt{1-\arctan{2x}} - \sqrt{1+\arcsin{2x}}}
$$
Rationalizing the numerator and denominator gives
$$
\lim_{x \rightarrow0}
\frac
{A(\ar... |
Is the evolute always a regular curve? (that is, the tangent vector of this curve is nonzero at any time)
|
I have to test convergence of improper integral
$$ \int_{0}^{\infty} \frac{x\log(x)}{(1+x^2)^2}\,\mathrm dx$$
I write as $\log(x) \leq x$ . So $x\log(x) \leq x^2$. So $ \frac{x\log(x)}{(1+x^2)^2} \leq \frac{x^2}{(1+x^2)^2}$ . Now using comparison test with integral $\frac{1}{x^2}$ to get original integral convergent. ... |
I heard somewhere (and just read on a physics forum) that the gradient of a smooth function $f$ on a manifold $M$ can be defined when $M$ is equipped with an affine connection on its tangent bundle, i.e. the Riemannian metric is not strictly necessary.
Is this true? If so, how can this be done?
|
Let $G$ be a finite group, $A\subseteq G$ and put $A^{-1}=\{ a^{-1}:a\in A\}$.
Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?
|
A total of $M$ balls are divided between two urns A and B. A ball is chosen uniformly at random.
If it is chosen from urn A then it is placed in urn B with probability $b$ and otherwise it is returned
to urn A. Similarly, if the ball is chosen from urn B then it is placed in urn A with probability
$a$ and otherwise it ... |
Suppose $u_n=\operatorname{Pr}\{X_n=0|X_0=1\}$ What is $\operatorname{Pr}\{X_j=0|X_i=k\}$, where $\{X_n\}$ is a branching process and $k\geq 0$, if we were to write the answer in terms of the aforementioned $u_n$'s?
|
First the definition:
Polynomial $q(x) \in \mathbb{Z}_p[x]$ of degree $n$ is called primitive, iff:
$q(x) \mid x^{p^n-1}-1$
$\forall k : 1 \leq k \leq p^{n}-1$ : $q(x) \nmid x^k - 1$
Now the polynomial from my exam, where I should show that it is primitive:
$p(x)=x^6+x^5+x^2+x+1, \quad p(x) \in \mathbb{Z}_2$
So i... |
I want to evaluate $\displaystyle \lim_{(x,y)\to (0,0)}\frac{\ln(1-x^2-y^2)}{x^2+y^2}$.
Any idea how to prove the answer is -1? I don't see an easy way to simplify this.
|
We have a sphere with the following equation:
$x^2+y^2+z^2=4$
We seek to find the partial derivative, with respect to $x$, of this equation. We think of this equation as a function of three variables
$f(x,y,z)=4$
From this equation, we can conclude the following:
$\frac{df}{dx}=0$
Using the chain rule, we have:
$0=f_x\... |
If there is a function $r(x)$ that is infinitely differentiable, prove that $k(x)$ is also infinitely differentiable if $k'(x) = r(k(x))$ for all $x$.
I am trying to somehow prove using induction. However, after taking many derivatives($1$st, $2$nd, $3$rd order), I have not been able to get to the necessary result.
|
$$Z=\frac{X}{X+Y}$$
$(X,Y)$ are iid r.v.s with $$f(x)=\lambda e^{-\lambda x}$$
We are asked to condition on $Y$ to derive the distribution of $Z$; $F(t)$ and $f_Z(z)$.
I don't know where to start, exactly. Is this a transformation? I don't know what it means to "condition on" a variable.
|
I need some help to solve this integral:
$$\int_{4x}^{\infty} \frac{w^{\frac{m}{2}}e^{-a\sqrt{w+2\sqrt{xw}}}}{\sqrt{w^2-4xw}}\mathrm{d}w.$$
Thank you
|
The theorem says:
Suppose $c_n \geq 0$ for $1,2,3 ...$. $\sum c_n$ converges, $\{s_n\}$ is a sequence of a distinct points in $(a,b)$, and $\alpha (x) = \sum^{\infty}_{n=1} c_n I(x-s_n)$. Let $f$ be continuous on $[a,b]$. Then
$$\int ^b _a f d\alpha = \sum^{\infty}_{n=1} c_n f(s_n).$$
So I was wondering does $\{s_n\}$... |
How can I find a subgroup of order $k$ in $U(n)$ or an element of
order $k$?
Here $U(n)$ is the group of units modulo $n$.
For example, if $n=700$ and $k=6$ I know that since $700=5^2 \cdot 7 \cdot 2^2$ we have $U(700) = U(7) \oplus U(4) \oplus U(5^2)$ and since $U(7) \cong \mathbb Z_6$ a cyclic subgroup of order s... |
I've heard of a number of ways that people have derived a cubic formula (I've even heard of a number of different ways to show the formula itself too). What I want to know is how to derive it without any substitutions (like the wikipedia article on it). I know it will be messy, but at least I'll be able to understand i... |
Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function such that $Re(f)\le |p(z)|$ for some polynomial, can we derive that $f(z)$ is a polynomial.
If $p(z)$ is constant, then this can be shown by considering $e^f$. If we instead consider $|u(z)|\le |p(z)|$, then it can also be shown. But if we do not establish the lowe... |
Suppose $f: \mathbb{R} \rightarrow \mathbb{R} $ is differentiable and $g: \mathbb{R} \rightarrow \mathbb{R} $ is infinitely differentiable, i.e. $ g \in C^{\infty}(\mathbb{R})$, where we know $f'(x) = g(f(x)) $ on $\mathbb{R}$. Show that $ f \in C^{\infty}(\mathbb{R})$.
Thus I have to show that $f$ is infinitely diff... |
I'm having trouble grasping what it means for two points to be conjugate on a Riemannian manifold. Could someone provide a geometric or intuitive explanation for this?
For clarification: given a geodesic $\gamma: [0,a] \to M$, the point $\gamma(t)$ is conjugate to $p=\gamma(0)$ if there exists a Jacobi field $J$, not ... |
Let $A \oplus B \simeq A' \oplus B $. Does it follow that $A\simeq A'$?
Many thanks in advance!
|
Suppose I have a diagonalizable matrix $M$ which has all its eigenvalues between $a$ and $b$. Is it possible to scale $M$ to $M_S$ such that all the eigenvalues of $M_s$ lie in the interval $[-1,1]$?
One method I came across:
Scale such that
$$
M_s=\frac{M-(b+a)/2}{(b-a)/2}.
$$
But, this is not working. Does anyone ... |
I was referring to a publication on BCH codes in GF(64) (http://ipnpr.jpl.nasa.gov/progress_report2/42-38/38O.PDF) where I noticed that the Parity Check Matrix (attached image) that they have derived is at first glance different from a parity check matrix of a standard (63, 56) BCH code.
My question is: How has this Pa... |
1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ cyclically. Then show that six points $A_1, B_1, C_1, A'_1, B'_1, C'_1$ lie on a conic.
2-Let $ABC$ and $A'B'C'$ be two t... |
I have been using a 6dof LSM6DS0 IMU unit (with accelerometer and gyroscope) and am trying to calculate the angle of rotation around all three axes. I have tried many methods but am not getting the results as expected.
Methods tried:
(i) Complementary filter approach - I am able to get the angles using the formula prov... |
The statement of this problem asks to show that if $A$ and $B$ are two distinct convex $n$-gons there is a continous family of convex $n$-gons such that the first in that family is $A$ and the last is $B$. My first intuition is that convexity ought to be utilized at some point in this proof. Perhaps consider establishi... |
Suppose that we have $AB=(AB)^{\ast}$ and $A=A^{\ast}$, does this implies that $B=B^{\ast}$? ($A^{\ast}$ is the Hermitian adjoint of $A$.)
I have a feeling that they might not be equal in general. If $B\ne B^{\ast}$ then what condition(s) do we need to add in order that $B=B^{\ast}$? $A,B\ne 0$.
Edit: According to th... |
(Don't say 'cube' or 'rectangular prism')
I'm looking for a generic name for polyhedra with 6 Faces, 8 Vertices, and 12 Edges where each face could be any quadrilateral shape: rectangle, rhombus, trapezoid, etc.
What is this called?
|
Is there an alternative approach to the definition of prime numbers based on the definition of the natural logarithm? These are my thoughts about it, the questions are at the end:
Basically when a natural number $n \gt 1$ is composite at least there are two natural numbers $a,b$ such as: (1) $1 \lt a \le b \lt n$, and... |
Show that the equation of a tangent in a point $P\left(x_0, y_0\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, could be written as: $$\frac{xx_0}{a^2} + \frac{yy_0}{b^2} = 1$$
I've tried implicit differentiation $\to \frac{2x\frac{d}{dx}}{a^2}+\frac{2y\frac{d}{dy}}{b^2} = 0$, but not sure where to go... |
Suppose a theorem proves something about a family of distance measures, with this the triangle inequality assumption. How limiting this assumption is in reality?
What are some real-world examples of distance measures which don't obey the triangle inequality?
Edit:
The triangle inequality is the following requirement ... |
In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the equation
\begin{equation}
u_{z\overline{z}}+\frac{2\rho_u}{\rho}u_zu_{\overline{z}}=0
\end{equation}
where the indices denote ... |
If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$.
While it is possible to show that using the definition of binomial and negative binomial, I am more interested in showing that directly using their p.m.fs.
|
It is exactly the same question as Partial Differential Equation about Rotation question. Sadly, I gain nothing useful from the above post. Or I should say I am not familiar with the terms in the question and I have no idea what approach should I use.
Consider the constant coefficient PDE $$a_{11}u_{xx}+2a_{12}u_{xy}+... |
Let $C$ be a collection of sets, and $\bigcup C \in V_{\omega}$ where $V_{\omega}$ is the collection of hereditarily finite sets. Is it possible to show that $C\in V_{\omega}$?
YES. Because $V_\omega$ models every axiom of set theory except the infinity axiom;
i.e. $\bigcup C \in V_{\omega}\:\Rightarrow\:C \subseteq \m... |
Let $A$ and $B$ be non-trivial groups. We construct their (restricted) wreath product as follows.
Denote by $A^{(B)}$ the set of all function from $B$ to $A$ with finite support, and equip it with pointwise multiplication to make it a group. The wreath product $A\wr B$ is the semidirect product $A^{(B)}\ltimes B$. The... |
EDIT: I'm too tired for math and the answer to my question is explained in a comment below. Should this post be removed? Not sure if it adds much to the community given that it was all just me misreading something, but then I don't know how this website works too well anyways.
It's my first time here, so if I make mist... |
First we find a tangent line of function $y^2=16-2x$ at $T(6,2)$:
$y_t-y_0=f'(x_0)(x-x_0)$ where $x_0=6,y_0=2$
$y^2=16-2x\Rightarrow y=\sqrt{16-2x}$ or $y=-\sqrt{16-2x}$
Derivative of $y=\sqrt{16-2x}$ at $x_0=6$ is
$f'(x_0)=\frac{-1}{\sqrt{16-2x_0}}=-\frac{1}{2}\Rightarrow $ tangent of $f(x)$ at $T(6,2)$ is $y_t=-\fra... |
Let $u:\mathbb{C}\to \mathbb{R}$ be a subharmonic function. Do we have that the limit $\lim_{r\to \infty}\frac{1}{rlog(r)}(\int_0^{2\pi}u(re^{i\theta})d\theta)$ converges to a (possibly infinite) value? I have been looking at this for a few hours and I cannot determine the answer. Could anyone show me a solution or giv... |
I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup of $N_G(Q)$, the normalizer of $Q$ in $G$.
Obviously when trying to find an example, you should look at non-commutative... |
What's the maxima of $$2^{\sin(x)}+2^{\cos(x)}$$ I found max by taking logs and then differentiating and equating to $0$ at $x=45°$ so the answer is $2^{\frac{\sqrt{2}+1}{\sqrt{2}}}$ am I right or I am missing something}
|
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