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H: Identification of $\mathbb F_2[X]/(X^4+X+1)$
As mentioned in the title I would like to show that we can identify $\mathbb F_2[X]/(X^4+X+1)$ with the set $K$ of polynomials: $p_0+p_1 a+p_2a^2+p_3 a^3$ in a variable $a$ that we assume satisfies $a^4+a+1=0$.
If you have any ideas than can help me with that would be g... |
H: Evaluate $ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$
Evaluate $$ 1 + \frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{4^2}+\frac{1}{7}\frac{1}{4^3}+\dots$$
All i could do was to see that $$\frac{1}{3}=\frac{1}{2.1+1},\frac{1}{5}=\frac{1}{2.2+1},\frac{1}{7}=\frac{1}{2.3+1},\dots$$... |
H: Prove sequence in a complete metric space converges if the series of distances converges.
Assume that X is complete, and let $(p_n)$ be a sequence in X. Assume that $\sum\limits_{n=1}^\infty d(p_n, p_{n+1})$ converges. Prove that $(p_n)$ converges.
(X,d) is a metric space.
Don't quite know how to start. Any help/hi... |
H: What is the cardinality of $\omega^\alpha\cdot n+1$?
How could I determine the cardinality of $\omega^\alpha\cdot n+1$? where $\alpha$ is an ordinal.
I think that, if $\alpha$ is infinite then $|\omega^\alpha\cdot n+1|>\aleph_0$, Is that right?
AI: If $\alpha < \omega_1$ then $\omega^{\alpha} < \omega_1$, so certa... |
H: Is there a function that gives the same result for a number and its reciprocal?
Is there a (non-piecewise, non-trivial) function where $f(x) = f(\frac{1}{x})$?
Why?
It would be nice to compare ratios without worrying about the ordering of numerator and denominator. For example, I might want to know whether the "mag... |
H: Proving composition of functions
I am trying to prove the following theorems:
Let A, B, and C be nonempty sets and let $f : A \rightarrow B$ and $g : B \rightarrow C$.
If $g \circ f : A \rightarrow C$ is an injection, then $f : A \rightarrow B$ is an injection.
If $g \circ f : A \rightarrow C$ is a surjection, th... |
H: Exact sequence of $R$-modules
Let $0\longrightarrow N\overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}L\longrightarrow0$
be a short exact sequence of $R$-modules. Prove that this chain splits iff
$f(N)$ is direct summand of $M$.
Thanks a lot.
AI: Suppose $\;M=f(N)\oplus K\;$ , and define $\;F:M\to N\;... |
H: Existence of a function from $[0,1]$ to an arbitrary measurable set
For any measurable $E\subset \mathbb{R}$ with measure 1, does there exist a continuous function
$$T:[0,1]\to E$$
Such that $\mu\circ T=\mu$, where $\mu$ is Lebesgue measure on the real line?
AI: Hint: A continuous image of $[0,1]$ is compact and c... |
H: Let $R$ be an integral domain, $M$ is free $R$-module with finite basis
Let $R$ be an integral domain, $M$ is free $R$-module with finite basis. Prove that two finite bases of $M$ have the same cardinality.
Help me some hints.
AI: If $R$ commutative ring then there is morphism $f:R\rightarrow k$ on to some field ... |
H: What does it mean for an ultrafilter to have a limit?
I got this question from the construction of the Stone-Čech compactification using ultrafilters given in Wikipedia. There they say that if $F$ is an ultrafilter in a compact Hausdorff space $K$ then it has a unique limit.
I can see this in the case $F$ is princi... |
H: Significance of rank of frechet derivative in definition of manifold?
In studying manifolds, the stipulation that the derivative be full rank is confusing to me on an intuitive level. Can anyone please explain how I should think about this intuitively? What does it mean for the derivative to be full rank? What woul... |
H: Is $\sum_{n=1}^{\infty}\frac{2^nn!}{(n+1)!}$ absolutely convergent?
I'm very uncomfortable with factorials just because I haven't done many of them. But my basic understanding is if I start with (for example) $(n+1)!$ then this is equivalent to $(n+1)*(n)$ and if it were $(n-1)!$ then this is equivalent to $(n-1)*... |
H: Need help with $\int_0^\infty e^{-x}\ln\ln\left(e^x+\sqrt{e^{2x}-1}\right)\,dx$
I need help with this integral:
$$\int_0^\infty e^{-x}\ln\ln\left(e^x+\sqrt{e^{2x}-1}\right)\,dx\approx0.20597312051214...$$
Is it possible to evaluated it in a closed form?
AI: A useful identity here is ${\rm arccosh\,} z = \ln( z + \s... |
H: For any real numbers $a,b,c$ show that $\displaystyle \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$
For any real numbers $a,b,c$ show that:
$$ \min\{(a-b)^2,(b-c)^2,(c-a)^2\} \leq \frac{a^2+b^2+c^2}{2}$$
OK. So, here is my attempt to solve the problem:
We can assume, Without Loss Of Generality, that ... |
H: Hartshorne Theorem III.5.2 (finite generation of cohomology for coherent sheaves on projective schemes over a noetherian ring)
Hartshorne, Algebraic Geometry, Theorem III.5.2, reads (in part)
Theorem 5.2
Let $X$ be a projective scheme over a Noetherian ring, and let $\mathcal{O}_X(1)$ be a very ample invertible s... |
H: Binary relations: transitivity and symmetry
I've been looking at some examples for transitivity and symmetry.
Suppose $A=\{0,1,2 \} $ and the relation $R=\{ (0,0),(1,1),(2,2),(1,2),(2,1) \}$
Well for starters this is clearly reflixe since $\forall x \in A ,xRx$.
As for symmetry, we define it as : $\forall x,y \in... |
H: Drawing Planar Graphs
Is it possible to draw a planar graph on 11 vertices in which each face (country) has 3 neighbours?
And is there some method after to draw it to confirm that it is in fact indeed planar?
I drew some figures, but then I'm not entirely sure if it is actually planar...
Thanks.
AI: First, the sum ... |
H: Are there a link between series convergence and countability of sets?
Could you please help me understand this question:
Suppose $E \subset [0,1]$ and
for each sequence $(a_n)$ , $a_n \in E$ and there are no duplicate members at $a_n$ , the series $\sum_{n=1}^\infty a_n$ converges.
Prove $E$ is countable set.
I tri... |
H: Boundedness of a real sequence.
Let $\{a_n\}$ be a sequence in $\mathbb{R}$ such that $\sum |a_n||x_n| < \infty$ whenever $\sum |x_n| < \infty$. Prove that $\{a_n\}$ is bounded.
My Attempt :
We have to show $\exists$ $B \in \mathbb{R}^+$ s.t. $|a_n| < B$.
$\sum |x_n| < \infty$ gives $\lim |x_n| = 0$ i.e. for any $\... |
H: Group action with finite stabilizer.
Let $G$ be a group generated by $\{g_1,g_2,\ldots , g_n\}$. Let $X$ be a space with a $G$-action on it, i.e. $G$ is acting on $X$. Suppose for each $x\in X$, the set $\{g_i;g_i(x)=x\}$ is trivial. Does it imply that the stabilizer of $x$, i.e. $\{g\in G;g(x)=x\}$ is finite? In o... |
H: Convergence of events in a probability space with respect to $L^2$
Define for events $X, Y$ that $d(X,Y) = P((X-Y) \cup (Y-X)) $ = $ P(X \bigtriangleup Y) $, show that $d(X_n,X) \rightarrow 0$ if and only if $\chi_{X_n}$ converges in $L^2$ to $\chi_X$ (these are indicator functions which take on values of 1 when i... |
H: Question having multiple answers?
i am studying for a test and i seem to of stumbled across a question which i found the answer to, but there seem to be others answers aswell. I am confused and would appreciate any help.
The question: The slope of a line is double of the slope of another line. If Tan. of the angle ... |
H: solving without a calculator: 2-4*x*arctan(x) = 0.
I was doing some review problems and one of them is find the points of inflection of $\arctan (x)$. I found the second derivative, but was not able solve the problem. The part of the problem I could not solve is in the title.
$2 -4x\arctan (x) = 0$
I get $\fr... |
H: How to prove that minimum of two exponential random variables is another exponential random variable?
How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y)
AI: Note that you must assume that $X$ and $Y$ are independent,
otherwise the result is... |
H: Proving Path-Connectedness
I am working on a homework question that gives $A \subseteq X$ and three points $x, y, z \in A$. Suppose also that there are continuous functions $f, g : [0,1] \to X$ with $f([0,1]) \subseteq A$, $f(0)=x$, $f(1)=y$ and $g([0,1]) \subseteq A$, $g(0)=y$, $g(1)=z$. I am asked to show that t... |
H: Vertices of a median through R?
i am studying for a test and seem to have hit yet another obstacle. Hint/Answers are highly appreciated.
The question: The vertices of a Triangle PQR are P (2,1), Q(-2,3), R(4,5). Find equation of the median through the vertex R.
I don't know how to solve this problem. At first i tho... |
H: Let $N$ be a submodule of $R$-module $M$, $M/N$ is free $R$-module. Prove that $N$ is direct summand of $M$.
Let $N$ be a submodule of $R$-module $M$, $M/N$ is free $R$-module.
Prove that $N$ is direct summand of $M$.
Thanks for any insight.
AI: Hint: The following exact sequence splits (as $M/N $ is free), so ... |
H: $(\sum_{n=1}^{100} n!)^7$ is of the form $7k+5$
Could any one just check for me that $(\sum_{n=1}^{100} n!)^7$ is of the form $7k+5$ or not? I got it that but they asked me to show it is of the form $7k+4$
Thank you!
AI: Observe that $\displaystyle r!\equiv0\pmod7 $ for $r\ge7$
$\displaystyle\implies \sum_{n=1}^{10... |
H: Does there exist a sequence such that $\lim\limits_{n\to\infty}\frac{n(a_{n+1}-a_{n})+1}{a_{n}}=0$?
Question:
Does there exist a positive sequence $\{a_{n}\}$ such
$$\lim_{n\to\infty}\dfrac{n(a_{n+1}-a_{n})+1}{a_{n}}=0?$$
If it exists, can you make an example? if not, why not?
My try: we consider this sequence
$$a... |
H: Limit $\lim_{n\to \infty} \frac{1}{2n} \log{2n\choose n}$
$\lim_{n\to \infty} \frac{1}{2n} \log{2n\choose n}$
I could not approach it beyond these simple steps,
$\lim_{n\to \infty} \frac{1}{2n} \log(\frac{2n!}{(n!)^2})$
$=\lim_{n\to \infty} \frac{1}{2n} [\log(2n)+\cdots +\log(n+1)-\log(n)-\cdots-\log1]$
$=\lim_... |
H: What is the name of this factor-algebra?
In the polynomial algebra $k[x_1,x_2,\ldots, x_n]$ consider an ideal $I$ generated by the polynomials of the form $x_i^k-x_i$, $i=1 \ldots n$ and $k=2,3,\ldots.$
Consider the quotient algebra $A=k[x_1,x_2,\ldots, x_n]/I$.
What is the name of the algebra $A$? Any reference?
... |
H: A polynomial is zero if it zero on infinite subsets
Let $f(t_1, ... , t_n)$ be a polynomial over a field $F$. Suppose there exist infinite subsets $X_1, ... , X_n$ of $F$ such that $f(x_1, ... , x_n) = 0$ for all $(x_1, ... , x_n) ∈ X_1× \cdots ×X_n$. Prove that $f$ is the zero polynomial.
Not sure how to start o... |
H: Spivak Calculus chapter 1, problem 1v
this is my first question here. I am self-studying Spivak's Calculus Fourth Edition and am stuck on Chapter 1 Problem 1v.
The question is to prove that: $x^n - y^n = (x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$
So I expand the right side out to:
$x(x^{n-1}+x^{n-2}y+...+xy^{n-2... |
H: Ideals in a real/complex number field?
Considering a real or complex number field (with traditional addition and multiplication) I see no ideals besides $\mathbb{R}$ and $\{ 0\}$ or $\mathbb{C}$ and $\{ 0 + 0i\}$.
Quick web search gave no satisfactory results. Yet I believe this couldn't be considered trivial, coul... |
H: The trapezoid rule and the integral: $\displaystyle\int_1^4(x-1)(x-4)\,\mathrm dx$
I've tried this many times, and the result I get is always $-4$, I don't know why..
I first get $h$, $ (4-1)/6 $ , then I put $h/2(x)$, $x = 2(-2) + 2(-2)$, which is equal to $1/2(-8) = -4$, what am I doing wrong?
Thanks in advance!... |
H: There exist $n \in \mathbb{N}$ such that $\mathrm{Im} f^{n} + \mathrm{Ker} f^{n} = M$ if $M$ satisfies DCC
Let $M$ be an $R$-module, and let $f$ be an $R$-automorphism on $M$. Prove that if $M$ satisfies the descending chain condition, then there exist $n \in \mathbb{N}$ such that $\operatorname{Im} f^{n} + \operat... |
H: Expectation of a squared random variable and of its absolute value
Is it true that if $\mathbb{E}[X^2]<\infty$ then $\mathbb{E}[|X|]<\infty$? If so, why?
AI: The most direct way to see this, without referring to a single theorem, is to consider the set $E=\{|X| \leq 1\}$. Then if $1_E$ is the indicator function of ... |
H: How to calculate percentage for two different values?
I have two data one is like$=75$ and second data is dislike$=127$, both values are counted separately. But i need to show percentage of this.
For example $40\%$ likes and $60\%$ dislikes for the above something like.
I know to calculate percentage for single val... |
H: Suppose $a_n+b_n$ converges. Does $a_n*b_n$ converges also?
$a_n, b_n$ - sequences
Suppose $a_n+b_n$ converges. Does $a_n b_n$ converge also?
I tried thinking if I can learn something about $a_n$ and $b_n$ by the assumption $a_n b_n$ converges.
I also tried to develop this equation $|a_n b_n - L| < \epsilon$ assumi... |
H: Suppose $\mu(X)<\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q<\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r
Suppose $\mu(X)<\infty$. Show that if $\ f\in\ L^q(X,\mu)$ for some $1\le q<\infty$ then $\ f\in\ L^r(X,\mu)$ for $1\le r<q$
In the solution the assistant wrote this:
If $1\le r<q$ then $q/r\g... |
H: Characterizations of convexity relying only on gradient
Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is once (and only once) continuously differentiable.
Are there any characterizations of convexity that rely only on the gradient $\nabla f$?
In the one-dimensional case this would be that $f'$ is non-decreasing.
A... |
H: Integrability of functions
Briefly justify the following facts:
a) $|x|$ is integrable on $[-1,2]$
b) $x^{\frac{1}{4}}$ is integrable on $[0,9] $.
c) The function $h(x)=\begin{cases}
x^2& x\in[0,1] \\
2x+3 &x\in(1,2]
\end{cases} \quad$
is integrable on $[0,3]$.
d) if $f, g$ are integrable on ... |
H: Diophantine Equation.
How many solutions are there in $\mathbb{N} \times \mathbb{N}$ to the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{1995}?$
How would you solve this? I have tried but am not sure how I should proceed with this.
AI: HINT:
$$\frac1x+\frac1y=\frac1m\iff (m-x)(m-y)=m^2$$ |
H: Normalizing data to a given value.
I would like compare the data by normalizing it to a given value.
for example:
[20,30,40,50,60,70]
How do I normalize the given set of elements to its first value, that is, 20.
Please also let me know if did not understand it correctly.
I understand that I need to take the mean an... |
H: Evaluating $\lim\limits_{x\to \infty} \sqrt{x^2-3x+5} - \sqrt{x^2+2x+1}$
I cannot solve the following limit with radicals:
$$\lim_{x\to \infty} \sqrt{x^2-3x+5} - \sqrt{x^2+2x+1}. $$
AI: Multiply by $$\dfrac{\sqrt{x^2 - 3x + 5} + \sqrt{x^2 + 2x + 1}}{\sqrt{x^2 - 3x + 5} + \sqrt{x^2 + 2x + 1}}$$
This gives us:
$$\lim... |
H: How find this $a_{n}$ such $\lim_{n\to+\infty}\left(\dfrac{a_{n+1}+1}{a_{n}}\right)^n=e$
if postive sequence $\{a_{n}\}$ such
$$\lim_{n\to+\infty}\left(\dfrac{a_{n+1}+1}{a_{n}}\right)^n=e$$
find a example the $a_{n}$ such this condition?
This problem is from analysis book : show that
$$\lim_{n\to\infty}\left(\dfr... |
H: Let $G$ be a finite group and $\phi:G \to K$ be a surjective homomorphism and $n \in \mathbb{N}. $ If $K$ has an element of order $n,$ so does $G.$
Let $G$ be a finite group$\ ,\phi:G \to K$ be a surjective homomorphism and $n \in \mathbb{N}. $ If $K$ has an element of order $n,$ so does $G.$
May I know if my proof... |
H: Ideals generated by a set in a ring without multiplicative unity
It is well known that in a ring $R$ with $1\neq 0$, the ideal generated by $S$ is $$I(S)=\{a_1 s_1 b_1+a_2 s_2 b_2+\dots+a_n s_n b_n: n\in\mathbb{N}, a_i, b_i\in R, s_i\in S\}.$$
Is there a similar expression for the ideal generated by a set in a ring... |
H: Definition of the probability distribution of a random variable?
I have a vocabulary problem. I understand that a "probability distribution" is a function from a sigma algebra to the reals. But then what is a "probability distribution" in "the probability distribution of a random variable?"
AI: A random variable is... |
H: write the trasition map $\phi$ between $\sigma_1$ and $\sigma_2$. Verify $\det( J(\phi))$ and find $T_p(S)$.
Sphere $$S=\{(x,y,z) \mid x^2+y^2+z^2=R^2\}$$
$$
\sigma_1(u,v)=(u,v, \sqrt{1-u^2-v^2}) \\
\sigma_2(u,v)=(\tilde u, \sqrt{1-\tilde{u}^2 -\tilde{v}^2}, \tilde v)
$$
I guess $\{\sigma_1, \sigma_2\}$ forms an at... |
H: Evaluate the following using Simpson's rule
I was asked to evaluate the following using Simpson's rule (by using 2 strips). Below is the function and my answer to it. What am I doing wrong?
$$\begin{align}
\color{red}{\int_1^{1.6}\dfrac{\sin2t}{t}\,\mathrm dt} & \color{blue}{=\int_1^{1.6}\dfrac{\cos2t^2-\sin2t}{t^... |
H: Evaluate $\lim\limits_{n \to\infty} \left(\frac{n-1} {2n+2}\right)^n$
What is the easiest way to evaluate this limit?
$\displaystyle{\lim_{n \to\infty} \left(n-1 \over 2n+2\right)^n}$
$$
\text{Is this possible ?$\,$:}\quad
\lim_{n \to\infty}\left(n/n - 1/n \over 2n/n + 2/n\right)^n
=
\lim_{n \to\infty}\left(1 - 1/n... |
H: References on Ring and Module Theory
Next semester I will take a course about Ring and Module Theory. Can anyone tell me the best texbooks and problems books about it. I only know some famous problems books such as Exercises in Classical Ring Theory, Exercises in Modules and Rings, Exercises in Basic Ring Theory.
A... |
H: Converting CFG to CNF
I have the following problem of converting CFG to CNF:
$$
\begin{aligned}
S \Rightarrow\,& bA \mid aB\\
A \Rightarrow\,& bAA \mid as \mid a\\
B \Rightarrow\,& BB\mid bs\mid b
\end{aligned}
$$
I know that Chomsky normal form only has productions of the type $A\Rightarrow BC$ and $A ... |
H: If $P$ is a prime ideal of $R$, $\sqrt{P^{n}}=P\ \forall n\in\mathbb{N}$?
Let $R$ be a commutative ring. If $P$ is a prime ideal of $R$, $\sqrt{P^{n}}=P\ \forall n\in\mathbb{N}$?
AI: Yes that is true. Note that $P^n\subseteq P$ and taking radicals of both sides gives the $\subseteq$ direction of the equality. So we... |
H: Isomorphism of rings under different operations
i am new to this site. i have been reading through its posts and question and they are really amazing. however, i found a link to a question asked about one year ago and a question i don't know how to tackle came to my mind. this is the question.
Let R be a ring with ... |
H: Measurable with Respect to a Complete Space
Let $f:(X, A,\mu) \to [0,\infty]$ have a Lebesgue integral. Show that $f$ is measurable with respect to the completion of the sigma algebra $A$.
I know that I have to show that the pre-image of every subset of $X$ is contained in the completion of $A$, but I'm having tro... |
H: Finding the volume between two concentrical hemispheres
I have a sphere with the equation $x^2 + y^2 +z^2 = b^2$ with a another sphere $x^2 + y^2 +z^2 = a^2$ located inside of it so that $0<a<b$. I am trying to find the volume between these two spheres above the $xy$ plane, so as I understand they would both be hem... |
H: $\frac{x_1^2}{\cos^2 \left( \arctan \frac{x_2}{x_1} \right)} =x_1^2 +x_2^2 $?
Is the relationship true? I don't seem able to prove it myself. I know that $$\cos(\arctan(x)) =\frac{1}{\sqrt{1+x^2}}$$ but I am at a loss here.
Thank you.
AI: So, $$\cos\left(\arctan \frac{x_2}{x_1}\right)=\frac1{\sqrt{1+\left(\frac{x_2... |
H: Lebesgue measure nested sequences
I am asked to prove the following:
Let $E \subset \mathbb R$ be Lebesgue measurable. Then there is a sequence of open sets $(O_n)$ and a sequence of closed sets $(F_n)$ such that $F_n \subset E \subset O_n$, $F_n \subset F_{n+1}$ and $O_{n+1} \subset O_n$ for all $n$. Furthermore $... |
H: How to prove Trigonometry equation?
how to solve following equation
$$
\tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{1}{9}\right)
=
\cos^{-1}\left(\frac{3}{5}\right)
$$
How to prove the above equation?
AI: $\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcomman... |
H: Determine if the number $ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $ is rational
$ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}} $
I have tried to raising it to the square, but I can't obtain the result.
$ \sqrt{8+2\sqrt{10+2\sqrt{5}}} - \sqrt{8-2\sqrt{10+2\sqrt{5}}}= k $
$ 2\sq... |
H: If $X_j$ is iid Unif$(-1,1)$ and $Y_n=\frac{\sum X_j}{\sum X_j^2+\sum X_j^3}$, show that $\sqrt{n}Y_n\rightarrow N(0,3)$ in distribution.
If $X_j$ is iid Unif$(-1,1)$ and $\displaystyle Y_n=\frac{\sum X_j}{\sum X_j^2+\sum X_j^3}$, show that $\sqrt{n}Y_n\rightarrow N(0,3)$ in distribution.
I know from central limit ... |
H: Prove by induction that $a-b|a^n-b^n$
Given $a,b,n \in \mathbb N$, prove that $a-b|a^n-b^n$. I think about induction. The assertion is obviously true for $n=1$. If I assume that assertive is true for a given $k \in \mathbb N$, i.e.: $a-b|a^k-b^k$, I should be able to find that $a-b|a^{k+1}-b^{k+1}$, but I can't do ... |
H: How to show the metric space is complete?
the space is the Real line with bounded metric (i.e. $d/(1+d)$, $d$: euclidean).
We thought that since nd the space real line with euclidean metric is complete and the bounded metric is smaller than d, then any cauchy sequence that converges with eucliedan metric will also ... |
H: Convergence of sum and multiplication
Iv'e got a question and I find it difficult to me.
Let $ a_n,b_n$ be sequences. Assume that $a_n+b_n$ converges.
Is $a_n\cdot{b_n}$ essentially converges?
Prove.
Thank you!
AI: No. Consider $a_n = n$, $b_n = -n$. |
H: $A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but not compact?
Is it true that set $A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but is not compact. We consider space $(\mathbb{R}^2, d_C)$ where $$d... |
H: $\max\{\chi(G):G$ embeds on projective plane$\}=6$
My lecture notes in Discrete Mathematics state that
$$
\max\{\chi(G) \; : \; G \text{ embeds on projective plane} \}=6,
$$
but I have no idea where this comes from. $\chi(G)$ is the chromatic number of $G$. How does such an emebedding looks like?
AI: The complete ... |
H: How to prove surjectivity part of Short Five Lemma for short exact sequences.
Suppose we have a homomorphism $\alpha, \beta, \gamma$ of short exact sequences:
$$
\begin{matrix}
0 & \to & A & \xrightarrow{\psi} & B & \xrightarrow{\phi} & C & \to & 0 \\
\ & \ & \downarrow^{\alpha} & \ & \downarrow^{\beta} \ & \ & \... |
H: Permutation and combination of letters
I need help with the following question:
"Given that a computer can only type letters A,B,C,D and E, how many ways can I type in 6 letters such that they must contain at least all of the different letters?
ie. AABCDE ABCDEA ABCDEB"
My professor said that the number of ways is... |
H: set of linear and continuous functions
Let E,F be normed vector spaces
$ \mathcal L(E,F) = \{ f \in Hom(E,F) | f-continuous\} $
Why
1) $Hom(\Bbb K^n,\Bbb K) = \mathcal L (\Bbb K^n,\Bbb K) = \Bbb K^n $
2) $\mathcal L (\Bbb K^n, F) = F^n$
AI: First part of 1) says that any linear map $\Bbb K^n\to \Bbb K$ is continuou... |
H: i.i.d. random variables with continuous distribution function are equal with probability 0
Let $X_1, X_2, ...$ be i.i.d. (real) random variables with continuous distribution function (in particular, we're not assuming absolute continuity).
How can I show that $P[X_n = X_m] = 0$ for $n \ne m$?
It's pretty clear fo... |
H: Proof of convexity of $f(x)=x^2$
I know that a function is convex if the following inequality is true:
$$\lambda f(x_1) + (1-\lambda)f(x_2) \ge f(\lambda x_1 + (1-\lambda)x_2)$$
for $\lambda \in [0, 1]$ and $f(\cdot)$ is defined on positive real numbers.
If $f(x)=x^2$, I can write the following:
$$\lambda x_1^2 + ... |
H: Finding number of solutions.
How many solutions does this equation have
$$2 \cos^2\left(\frac12 x \right) \sin^2 x = x^2+x-2$$
where $0 \lt x \le \displaystyle\frac \pi9?$
I observed that $2 \cos^2\left(\frac12x\right)$ can be written as $1+\cos x$. Simplifying $\sin^2 x,$ we get
$$(1+\cos x)^2(1-\cos x)=x^2+x-2$$
... |
H: Intro analysis - contraction mappings
A function $ f: \mathbb{R} \rightarrow \mathbb{R} $ is called a contraction mapping if there exists a positive constant K < 1 such that
$ |f(x) - f(y)| \leq K |x-y| $
d) Suppose $f:\mathbb{R} \rightarrow \mathbb{R} $ is a contraction mpaping and for any $ x_0 \in \mathbb{R} $,... |
H: How to find the derivative of $(2x+5)^3(3x-1)^4$
How to find a derivative of the following function?
$$\ f(x)=(2x+5)^{3} (3x-1)^{4}$$
So I used:
$$(fg)'= f'g + fg'$$ and $$(f(g(x)))'= f'(g(x)) + g'(x)$$
Then I got:
$$ f(x)= 6(2x+5)^{2} + 12(3x-1)^{3}$$
and I don't know how to get the solution from this, which is: $... |
H: Lagrange multipliers - perturbation of constraints
I have been spending some time learning about Lagrange multipliers lately.
Something is puzzling me though. Reading around (also on Wikipedia) I saw multiple time the interpretation that lagrange multipliers represent the rate of change of the optimal value of the ... |
H: Thinning a Poisson Process
Suppose that events are produced according to a Poisson process with an average of lambda events per minute.
Each event has a probability $p$ of being Type A event, independent of other events.
Let the random variable $Y$ represent the number of Type A events that occur in a one-minute pe... |
H: Does Wolfram Alpha fail for $x^x$?
WolframAlpha generates the following graphic for $f_{(x)} = x^x$:
$f_{(x)} = x^x$
Can anyone explain me why this graphic looks like above? I mean why the real part for negative, non-integer, but rational numbers looks like that?for $0^0$?
To clarify my question:
We know that:
$... |
H: Probability mean is less than 5 given that poisson distribution states it is 6
I want to find the probability that mean is less than 5 given that poisson distribution states it is 6
ie find p(x<5|x~po(6))
Here is the actual question:
Two grocers agree that the daily demand for a particular item has Poisson distribu... |
H: find the limit of ${(n + 1)^{{1 \over {\sqrt n }}}}$
I really tried to think of a way to find the limit of this sequence but couldn't think of something useful. I tried to use Bernoulli's inequality to squeeze the sequence, but it didn't come in handy.
$$\mathop {\lim }\limits_{n \to \infty } {(n + 1)^{{1 \over {... |
H: Norm of operator $T_x(f) = f(x)$
Let $X$ be a normed vectorspace and $X'$ be the dual space of $X$. For $x \in X$ we can define $T_x: X' \to \mathbb F$ by $T_x(f) := f(x)$. This is indeed an operator in $X''$. I read that $\| T_x \| = \| x \|$ but I could not figure out how to prove this. Can this be done by means ... |
H: Question about second condition of pumping lemma
I don't think that I fully understand how to use the pumping lemma to prove that a given language is not regular. I'm reading Sipser and according to him the definition of the pumping lemma is:
"If A is a regular language, then there is a number p ( the pumping lengt... |
H: PDE using Fourier Transform
Using the Fourier Transform, solve:
$u_t=u_{xx}+\alpha u$ with $\alpha>0$, for $x \in \mathbb{R}, t>0$
with initial data $u(x,0)=f(x)$, with $f$ continuous in $\mathbb{R}$
Apllying Fourier transform in equation and initial data, we obtain
$\partial_t^2 \hat u(\xi)=-\xi^2\hat u + \alpha ... |
H: Organic firm sales drop
Assume a grocery company grows from having 180 to 210 shops and at the same time experiences a 7% drop in sale. How much would their sale have dropped if they had not opened the extra stores.
I can see that to be status quo they should have grown by (210-180)/180=16.66 %, but I can't figure ... |
H: Existence of inaccessible cardinals implies the consistency of ZFC
I wonder if any can sketch for me in very broad lines the proof of the fact that the existence of inaccessible cardinals implies the consistency of ZFC? I don´t know much about set theory, but I find it extremely interesting that this should be the ... |
H: Invertible matrices
Please help with the next question.
Let A and B be two invertible matrices such that $A+B \neq 0$.
Prove or disprove that A+B is invertible.
Thanks!
AI: What you can conclude with these matrices?
$$A=I_2\ \text{and}\ B=\mathrm{diag}(-1,1)$$ |
H: Symmetric and Transitive closures
Given a relation $R$, is the symmetric closure of the transitive
closure of $R$ equal to the transitive closure of the symmetric
closure of $R$? If yes, prove it. If not, give a counterexample.
Well for this, the only reasoning I can come up with is if I have an arbitrary tra... |
H: If a "group" has two identities then is not a group
The story goes like this: A friend and I found this old exercise:
Let $G=\Bbb R-\{-1\}$ and $a*b:=a+b+ab$, is $(G,*)$ a group?
I say that $(G,*)$ is not a group because for any $a\in G$ follows that $0*a=0+a+0a=a+0+a0=a*0$ and also $1*a=1+a+1a=a+1+a1=a*1$, hence... |
H: Integral of $\int \frac{\sin^22x}{\cos2x}\,\mathrm dx.$
I would like to solve the following integral
$$\int \frac{\sin^22x}{\cos2x}\,\mathrm dx.$$
I know that the derivative of $\cos$ its $\sin$ but how its help me?
Any suggestions?
Thanks.
AI: Hint:
$$\int \frac{\sin^2 2x}{\cos 2x}~dx ~ = ~ \int \sec 2x... |
H: On finding a limit by dividing by the highest exponent
Sometimes it’s easy to divide by the highest exponent to find a limit.
$${{{n^3} + 4{n^2}} \over {\root 3 \of n + \root 4 \of {{n^3}} }}$$
So, in the above example (which I just made up; there’s nothing special about it), you should divide by $n^3$. How do y... |
H: Jacobson radical subset of maximal ideal
If we define the jacobson radical $J$ to be the intersection of all maximal right ideals then I am trying to show that if we have a maximal two sided ideal $M$ we must have $J\subseteq M$?
Any ideas
AI: The Jacobson radical of $R$ is the intersection of all right primitive i... |
H: Did I solve this problem correctly? $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n^2}{n^3+4}$
For the following series:
$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n^2}{n^3+4}$
I found $b_n$ to be $b_n=\frac{n^2}{n^3+4} \gt 0$ and then I took the limit of this and found it to be zero.
AI: You can prove that $b_n$ is decreasing by u... |
H: How to show that two groups makes $S_n$
I need to show that:
$S=\left\{(12),(13),...,(1n)\right\}$ generates $S_n$
$S=\left\{(12),(123\cdots n)\right\}$ generates $S_n$
How do I show that each one of them generates $S_n$?
Thank you!
AI: Can you prove that every element of $S_n$ is equal to a product of transposi... |
H: Find roots of a trigonometric equation
I've been struggling on find the roots of this equation for a while.
$$
f(x)=\cos(2x) - 2\sin(x)\cos(x).
$$
I've already ended transforming all $\cos$ relation to $\sin$ ones, but I'm now stuck on:
$$
f(x)=1-2\sin^2(x) - \sin(2x).
$$
What should I do now?
AI: You have
$$f(x)... |
H: On differentiating an integral with respect to a function
Let $f,g:\mathbb{R}^n \rightarrow \mathbb{R}$, and let
$$
Q = \int \! g(\mathbf{x})f(\mathbf{x}) \, \mathrm{d}\mathbf{x}
$$
What is the result of the following differentiation?
$$
\frac{\partial}{\partial g}Q = \frac{\partial}{\partial g} \int \! g(\mathb... |
H: On the definition of the direct sum in vector spaces
We say that if $V_1 , V_2, \ldots, V_n$ are vector subspaces, the sum is direct if and only if the morphism $u$ from $V_1 \times \cdots \times V_n$ to $V_1 + \cdots + V_n$ which maps $(x_1, \ldots, x_n)$ to $x_1 + \cdots + x_n$ is an isomorphism.
Looking at the d... |
H: algorithm question how prove that $(n+a)^b = \Theta(n^b)$
this question doctor in college give us as home work
but I don't know how approve it
AI: Recall that $f \in \Theta(g)$ whenever $k_1g(x) \leq f(x)$ an $f(x) \leq k_2g(x)$ for some constants $k_1$, $k_2$ and all $x$ greater than or equal to some $N$. In this... |
H: Decomposition of polynomials
It is a very simple question but I'm stuck in decomposing this: $x^3+2x^2-2$. I can't find the $x-c$ (Ruffini's rule) form that can enable me to decompose it. Is it possible to decompose?
If I can solve it, I will be able to resolve an entire math problem!. It is an elementary question... |
H: When is $c v^\top y y^\top v \ge ||v||^2$
Given $c\in R$ being some constant, $v, y \in R^n$, I want to find conditions for which the following inequality holds true:
$$c v^\top y y^\top v \ge ||v||^2$$
EDIT:
Note that $y y^\top$ is an $n \times n$ matrix.
AI: The way the product is written, $v^\top y^\top y v$ mu... |
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