text stringlengths 83 79.5k |
|---|
H: Integral of derivatives and conjugate
For function $f,g$ in the Schwartz class, I want to show that $$\int_{\mathbb{R}}\left(\frac{d}{dx}f(x)+xf(x)\right)\overline{g(x)}dx=\int_{\mathbb{R}}f(x)\overline{\left(-\frac{d}{dx}g(x)+xg(x)\right)}dx$$
But I don't see how to start, since there isn't any easy way to manipul... |
H: Mean value theorem and differentiability
Let $a,b\in \mathbb{R}$, $a<b$ and let $f$ be differentiable real-valued function on an open subset of $\mathbb{R}$ that contains $[a,b].$ Show that if $\lambda$ is any real number between $f'(a)$ and $f'(b)$ then there exists a number $c\in(a,b)$ such that $\lambda = f'(c)$... |
H: Finding the Nth number in a generated list
I am generating numbers as follows:
Let the first digit range from 1 to 2 inclusive.
Let the second digit range from 1 to 3 inclusive. Let the last digit range from 1 to 2 inclusive.
I am then generating all the possible numbers by cycling the last digit, then the middle, ... |
H: I need a proof of a theorem about generated group
how to use definition 2.7 to prove theorem 2.8?
AI: Let $H$ be the collection of all finite products $a_1^{n_1} a_2^{n_2} \ldots a_t^{n_t} (a_i \in X, n_i \in \mathbb{Z})$. Check that :
a) $H$ is a subgroup of $G$ (Use the subgroup test : if $x, y\in H$, conclude ... |
H: What would be the simplified form of this expression?
I'm working on a Homework problem involving Convergence of Random variables and I've arrived at an expression which looks like follows:
$$ M_{X_n}(ju)= \prod_{k=1}^{n}0.5\frac{1-e^{\frac{ju}{2^{k-1}}}}{1-e^\frac{ju}{2^k}}$$
,where $M_X(ju)$ denotes the characte... |
H: What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.
Motivated by:
Tiling an orthogonal polygon with squares,
How to prove that the minimum square partition of a 3X2 rectangle has 3 squares,
Minimum square partitions for 4x3 and 5x4 rectangles,
Wh... |
H: Convolution convergent in $L^\infty$
Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_\epsilon(x)=\dfrac{1}{\epsilon}K\left(\dfrac{x}{\epsilon}\right)$$ Is it always true that $\lim_{\epsilon\rightarrow 0}\|f\ast K_\epsilon-f\|_\infty=0$?
I think it's always ... |
H: Show that $\lim_{h\to 0}[1/(f(a+h)-f(a))-1/hf'(a)]=-f''(a)/2f'(a)^2$.
Given that $f'(x)\ne0$ show that $\lim_{h\to 0}[1/(f(a+h)-f(a))-1/hf'(a)]=-f''(a)/2f'(a)^2$.
By wrriting $f'(a)$ into its limit definitions, LHS seems to be $0$, so how to do this problem?
Thanks.
AI: Let
$$
L(h)=\frac{1}{f(a+h)-f(a)}-\frac{1}{... |
H: How prove this $d(\alpha,W)$ is Gram determinant?
if $W$ is subspace of $V$.and $V$ is $n$-dimensional Euclidean space. and $\alpha\in V$, Define the distance from $\alpha$ to $W$
$$d(\alpha,W)=|\alpha-\alpha'|$$
where $\alpha'$ is $\alpha$ orthogonal projection on the subspace $W$.
such $\alpha_{1},\alpha_{... |
H: Two questions on torsion set
Let $R$ be a ring and $T$ be the set of torsion elements of $R$.
Prove that the only torsion element of $R/T$ is $\{0\}$. I can show that $T$ is an ideal of $R$. Or in fact, $T$ is a normal subgroup of $R$ if we treat $R$ simply as an additive group. If I take this latter view, then I ... |
H: Coordinates of point that is on the end of perpendicular line
Lines AB and CD are perpendicular. Points A, B, D can have any random coordinates and we know there value. Line CD can have any random length and we know its value. How can we calculate coordinates of point C ?
AI: Suppose $A(a_1,a_2),B(b_1,b_2)$ and $D(... |
H: If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$?
I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers being isomorphic to the even integers,... |
H: Minimize distance from a point to a parabola
Find the point on the parabola $$y^2=3x$$ that is closest to the point $(1,7)$.
AI: A point on that parabola can be parametrized as $(\frac{y^2}{3},y)$. Now find the distance between $(\frac{y^2}{3},y)$ and $(1,7)$ and minimize it. |
H: Strange mistakes when calculate limits
I have difficulties with calculating the following limits. W|A gives the correct answers for both of them:
$$
\lim_{x \to +\infty} \sqrt{x} \cdot \left(\sqrt{x+\sqrt x} + \sqrt{x - \sqrt x} - 2\sqrt x\right) = \lim_{x \to +\infty} \sqrt{x^2+x\sqrt x} + \sqrt{x^2-x\sqrt x} - 2x... |
H: infinite intersection of 2 uncountable sets
Is it true that if the intersection of 2 uncountable sets is infinite, then the intersection is definitely uncountable?
How do I start disproving/ proving this statement?
AI: False: Construct 2 sets: $A=\mathbb{N}\cup(0,1), B=\mathbb{N}\cup(2,3)$. |
H: Application of Sylow's theorem
Let $p>q$ be primes.
$ (1): \exists $ non-abelian group of order $pq$ $\Longleftrightarrow$ $p \equiv 1 (mod \ q)$
$(2):$ Any $2$ non-abelian groups of order $pq$ are isomorphic to each other.
Proof of claim $(1):$
Suppose $\exists$non-abelian $G$ of order $pq.$ Let $P$ be the $p-$... |
H: How to simplify this summation equation?
If I have
$$
\sum_{n=1}^K nx^{n-1} = \frac{d}{dx}\frac{1-x^{K+1}}{1-x}.
$$
How to calculate this formula based on above equation ?
$$a = \sum_{n=2}^{K}n\left(\frac{\gamma}{2}\right)^{n-1}\rho^n$$
AI: Hint: Factor out $\rho$ and put it outside the summation. $$a = \sum_{n=2}... |
H: Isomorphism of R-modules
Does somebody has an example where the left $R$-modules $R^m$ and $R^n$ are isomorphic for all positive integers $m$, and $n$?
AI: Let $\mathrm{CFM}_\mathbb{N}(R)$ denote the ring of "column finite matrices", where $R$ is some ring.. Then, one can show that $\mathrm{CFM}_\mathbb{N}(R)\to\ma... |
H: How find this continued fraction
Question:
let $x$ such this Continued fraction
$$x=[0;1,3,5,7,9,11,13,\cdots]$$
How find the vaule of $x$.
(can see:http://en.wikipedia.org/wiki/Continued_fraction)
My try: I know this
$$e=[2;1,2,1,1,4,1,1,6,1,1,8,\cdots] $$and
$$\pi=[3;7,15,1,292,1,1,1,2,1,3,1,\cdots]$$
But for my... |
H: Find an value of real number M such that $\frac{2+\sin(n+1)}{2+\sin(n)}\leq M<3$
This sequence is certainly limited
\begin{equation}
a_n=\frac{2+\sin(n+1)}{2+\sin(n)}
\end{equation}
For example, if $\sin(n+1)=1$ and $\sin(n)=-1$, we have: $a_n\leq 3$. But I believe that there is $0<M<3$ such that $a_n\leq M$.
I wou... |
H: Question about $f :\mathbb{R}\rightarrow \mathbb{R}$ defined as $f(x)=|x|^{\frac{3}{2}}$ (TIFR GS $2010$)
Question is :
I am not sure how to check for differentiability,
only thing i know is how to see for differentiability of $f(x)=|x|$
$\lim_ {x\rightarrow 0} \frac{f(x)}{x}=\lim_ {x\rightarrow 0} \frac{|x|}{x}$... |
H: calculus the following functions could you find the minimum of?
In finding the point on the line y=x closest to the point (0,1) which of the following functions could you find the minimum of?
a. $2x^2+2x+2$
b. $x^2-2x+2$
c. $2x^2-2x+1$
d. $2x^2-x+2$
e. $x^2-1$
Anyone can explain?
AI: If the $d$ is the distance, $$d... |
H: How to solve this system of differential equations?
$\begin{cases}
\dot{p}=-p+2x\\
\dot{x}=\frac{1}{2}p + x\\
\end{cases}$
I don't know how I can express $p$ only in terms of $p$'s or $x$ only in terms of $x$'s. Can anyone please help?
AI: $x=(1/2)(p'+p)$, $x'=(1/2)(p''+p')$, $(1/2)(p''+p')=(1/2)p+(1/2)(p'+p)$.
(e... |
H: I need some help with the derivative of this function.
Hey guys i was wondering , what is the derivative function of this function.
f(x) = $\sqrt{x} - e^{-x}$
Any advise will be greated.
AI: In general, $$\frac{d}{dx}(x^n) = nx^{n-1}$$ and $$\frac{d}{dx}(e^{ax}) = ae^{ax}.$$ Observing that $$f(x) = x^{\frac{1}{2}}... |
H: Prove that there are rational polynomials $p,q$ such that $p(x)(x^4+2x^2+1)+ q(x)(x^4-3x^2-4) = x^2+1$
Prove that there are polynomials $p(x), q(x)$ in $\mathbb{Q}[X]$ such that:
$$p(x)(x^4+2x^2+1)+ q(x)(x^4-3x^2-4) = x^2+1$$
Is it still true if we replace $x^2+1$ with $x+5 $?
So:
I know how to prove this... |
H: Multiplicative function, $\mathbb{Z}[a+b\frac{1+\sqrt{-19}}{2}]$
Could you tell me how to prove that this function is multiplicative?
$R = \mathbb{Z}[a+b\frac{1+\sqrt{-19}}{2}]$
$f: R \ni a+b\frac{1+\sqrt{-19}}{2} \rightarrow a^2 + ab + 5b^2 \in \mathbb{Z}$
Here's my attempt to solve it:
$f(1) = 1$
I multiply $(a+b... |
H: Is my proof for $\lim_{n\to\infty}\frac3{n+1}$ wrong?
To calculate $\lim_{n\to\infty}\frac3{n+1}$ I do:
$$\lim_{n\to\infty}\frac3{n+1}=\lim_{n\to\infty}\frac3\infty=0$$
but my teacher is not convinced so how can I do??
AI: O.K. A proof without $\epsilon$ and $\delta$ can it be as follow.
Assuming that $\lim_{n\to\... |
H: combinations in a circuit
in a circuit there are 3 switches, of which 1 & 2 are in series., 3 is in parralel with 1&2.
all three switches can either be open or closed. how many ways of opening/closing all three different switches are there where there is electricity flowing through the circuit?
i would like to ... |
H: why for every $ f\in C(\sigma(x))$ we have $ \Phi (f(x))= f(\Phi(x))$?
In a book about $ C^* $-algebra, in the section of continuous functional calculus says that:
Suppose $ x $ is a normal element of $ C^*$-algebra $ A $, then the continuous functional calculus has this property that If $ \Phi: A \to B $ is a ... |
H: Complex number ( equation) Find the number of solutions of $z^3+\overline{z}=0$ ...
Problem : Find the number of solutions of $z^3+\overline{z}=0$
Solution : $z^3 =-\overline{z} \Rightarrow |z|^3 = |(-\overline{z})|$
$\Rightarrow |z|^3 = |z| \Rightarrow |z|^3-|z| =0$
$\Rightarrow |z|(|z|^2-1)=0 \Rightarrow |z| =0 ... |
H: Finding the Expected value
You are in a room; seeing n doors in front of you in beginning. You can choose any door you like. The probability for choosing a door is equal for all doors.
If you choose the ath door, it can either take you back to the same position where you begun in ti minutes, or can take you out of ... |
H: Need a counter example for series convergence
I need some advice for constructing a counter example for $\sum\limits_{i=1}^\infty a_i$ converge but $\sum\limits_{i=1}^\infty \frac{a_i}{i}$ diverges.
AI: There is no counterexample.
Take $t_n=\frac{1}{n}$ then $t_n$ is decreasing & positive with limit $0$. It is giv... |
H: Laplace's Equation for a Radial Function (cylindrical co-ord)
I'm working on a question which has lead me to the Laplace equation in cylindrical coordinates. I've looked it up and found that, for the radial component, this is equivalent to
$$\nabla^2\ f = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\parti... |
H: $|a_n - a_m| \le \sum_{k=m}^{n-1} |a_{k+1} - a_k|$
How to show, that every for sequence $(a_n)_{n\ge_1}$ with $n \gt m$ following holds ?
$$|a_n - a_m| \le \sum_{k=m}^{n-1} |a_{k+1} - a_k|$$
AI: Let $d=(n-m)$. We have $n-(d-1)=m+1 $ and
\begin{align}
|a_n-a_m|
=& |a_n\color{red}{-a_{n-1}+a_{n-1}}-a_m|\\
=& |(a_n-a_... |
H: On irreducible polynomial
Prove that the polynomial $f(X)=X^5-9X^3+15X+6$ is irreducible over $\mathbb{Q}\left(\sqrt{2},\sqrt{3}\right)$
Apply Eisenstein's Irreducibility Criterion with $p=3$ we see that $f$ is irreducible over $\mathbb{Q}$. Can conclude that $f$ is irreducible over $\mathbb{Q}\left(\sqrt{2},\sqrt{... |
H: Calculate the area between $f(t) = \cos(t)$ and the $t$ axis as $t$ varies from $0$ to $\pi/4$
The problem below is part of a multiple section question that I'm hoping to solve.
Calculate the area between $f(t) = \cos(t)$ and the $t$ axis as $t$ varies from $0$ to $\pi/4$.
AI: $$\text{Area}=\int_0^{\pi/4} \cos(t) ... |
H: $f(x)=\sin x^3$ for $x\in \mathbb{R}$ is not uniformly continuous
Question is to prove that :
$f(x)=\sin x^3$ for $x\in \mathbb{R}$ is not uniformly continuous.
What would my first observation in checking uniform continuity is to check if its derivative is bounded.
In this case its derivative $f'(x)=3x^2\sin x^3$ w... |
H: Find the integral of $\tan^4(x) \sec(x)$
I want to know if there is a shorter way to find integral of $\tan^4(x) \sec(x)$
without using reduction formula of $\sec(x)$
because it's really takes a long time.
Thanks all
AI: Write it as
$$
\int\frac{\sin^4x}{\cos^5x}\,dx=
\int\frac{\sin^4{x}}{\cos^6x}\cos x\,dx=
\int\... |
H: Combination and Permutation- examples
I am not sure how to answer this question.
I have a word TOKYO.
a) how many different can I arrange the letters in a row?
b) in how many of these will O's be together?
c) how many will the two O's not together?
My thoughts:
a) I figure out that this is a permutation because ord... |
H: Help With Eigenvectors and Dynamical Systems
I have the following system of differential equations:
$ \frac{d}{dt} \left[ \begin{array}{c} A(t)\\ N(t)\\\end{array} \right] =
\left[ \begin{array}{c c} -(a+b) & 0\\ a & -(a+b)\\ \end{array} \right]
\left[ \begin{array}{c} A(t)\\ N(t)\\\end{array} \right]
$
I believe ... |
H: Extend a function as odd/even periodic function
Let $f$ be the function $f(x) = x^2 + 2 $, where $ 0<x<1 $.
Extend the function $f(x)$
(1) As an odd periodic function with period $2$
(2) As an even periodic function with period $2$
(3) As a periodic function with period $1$
I know what exactly are odd and e... |
H: Simple Modular Arithmetic
We know that, for example,
$2x \equiv 3 \mod 4$
has no solutions since
$2\mid 2x, 2\mid 4,$ however 2 does not divide 3.
So my question is, how does one get from
$2x \equiv 3 \mod 5$
to
$x \equiv 4 \mod 5$?
In other words, could someone please explain the steps involved going from
$2x = 3... |
H: Suppose $f$ is entire and for all $z \in \mathbb{C}$, $f(z)=f(\frac{1}{z})$. Prove that $f$ is constant.
Suppose $f$ is entire and for all $z \in \mathbb{C}$, $f(z)=f(\frac{1}{z})$. Prove that $f$ is constant.
I want to prove $f$ is bounded. Then by using Liouville theorem, $f$ is constant. But I can't proceed. Can... |
H: Any angle divides a plane into two regions
I am studying Euclidian geometry and I noticed that any angle divides a plane into two regions: an inside and an outside. Is there a need for a proof of this (something along the lines of Jordan theorem), or is it just "obvious"?
Browsing the internet, I came across a foll... |
H: Did I do this assignment right? Antisymmetric relation.
I need to prove or disprove, that R is antisymmetric.
This is my set:
$$
R=\{(1,1),(1,2), (1,4), (2,1), (2,2), (3,2), (3,3), (4,4)\}
$$
I proved that it is not antisymmetric in the following manner:
$$
\text{Definition of antisymmetric relation}\\
(x,y)\in R \... |
H: Steps to find gradient of this?
I don't understand how to get from .54 to .55, any help?
Edit:
Wolfram alpha gives the answer as this http://www.wolframalpha.com/input/?i=grad+Ucos%28theta%29%28r%2Ba%5E2%2Fr%29
I can't seem to convert their answer to the answer needed even though it is similar.
AI: Hint: Remember... |
H: Convolution is uniformly continuous and bounded
Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function.
The definition of the convolution is $(f\ast K)(x)=\int_\mathbb{R}f(x-y)K(y)dy$.
There i... |
H: transformation of DFT matrix
$\mathbf{F}$ is a unitary DFT matrix where the $(m,n)$-th entry of $\mathbf{F}$ is given by $\frac{1}{\sqrt{M}}e^{-\imath2\pi(m-1)(n-1)/M}$. Note that $\imath=\sqrt{-1}$.
Let $\mathbf{A}$ be a matrix where the the $(m,n)$-th entry of $\mathbf{A}$ is given by $\frac{1}{\sqrt{M}}(-1)^{m-1... |
H: choosing odd number of balls from $n$ balls
How many ways can an odd number of balls be chosen from $n$ balls?
I tried enumerating, but it's really too tedious. :/ Any help is appreciated.
AI: Hint: #ways of choosing even balls = #ways of choosing odd balls.
and total number of ways of choosing balls = $2^n$ (Why?) |
H: Set problem where $f=A\times B$
The problem states:
If there is function $f:A\to B$ and
$$\left(f=A \times B \right) \iff \left( A= \varnothing \text{ or } |B| = 1\right)$$
(where $A \times B$ is the cartesian product and $|B|$ is the cardinality of $B$)
So I must demonstrate that equivalence.
I've tried to slove ... |
H: Simple Question about the norm.
Suppose we have a Banach space $V$ with a norm $\|\cdot\|:V\to \mathbb{R}$. Is the following true for all linearly independent vectors $x,y\in V$?:
$$\|x+y\|\geq \|y\|,~~~~\|x+y\|\geq \|x\|$$
AI: No, take $V=\mathbb{R}^2$, $x=(1,0)$ and $y = (-1,0.1)$. |
H: Prove $\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$
Prove that $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2}\geq (a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$$ for $a,b,c>0$
Any hints/solutions?
AI: Expanding the terms, we wa... |
H: A plane is a surface.
I am trying to show that a plane is a surface. I posted what I did. I find $σ$ But I cannot find $σ^{-1}$. Also as I said, I need to verify $σ$ is 1-1 continuous and continuous inverse. How can I do all? Please help me to end my proof? Thanks.
AI: You are asking, how to invert the map $(x, y) ... |
H: Proof that the derivative of a linear function is $0$.
We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers.
The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$
Using the theorems of limits we can replace the above limit equation by:$$f'(x)=\frac{\li... |
H: Image of koebe map
Let $f(z)=\frac {z}{(1-z)^2}$ how can I find the image of unit disk under this transformation? Also is it possible to find an explicit expression for the inverse of this transformation?
AI: Note that $$\frac{z}{(1-z)^2} = \frac{1}{4} \left( \frac{1+z}{1-z} \right)^2 - \frac{1}{4}$$ and that $$z \... |
H: How to master integration and differentiation?
We have learnt in school about differentiation and integration, however I find my knowledge fairly poor. I mean I have problems with taking the derivative/integral even simple functions. So I would like to get some links to articles and guides where I could learn the d... |
H: Why is every continuously differentiable function with a uniform bounded derivative lipschitz continuous
I only know how to prove this for functions on a convex set by using the mean value theorem, but is this also true for this general case when nothing is said about the domain of the function besides the fact tha... |
H: Convergent series: am I doing it right?
I have
$$
\sum_{k=1}^\infty \frac{5k-2}{(3{k}^{2}-2)\sqrt[3]{k+6}}
$$
I get (i did not write all solution as it is quite hard to me to put this in LaTeX by myself)
$$
\lim_{k\rightarrow \infty} \frac{{a}_{k+1}}{{a}_{k}}=...=\lim_{k\rightarrow \infty}\frac{15{k}^{3\frac{1}{3}}... |
H: What is the steps of solution for limit of division of trigonometric functions?
Given the $\lim_{x\to \pi} \dfrac{\sin(3x)}{\sin(5x)}$. The answer is $\dfrac 3 5$ but what are the solution steps?
AI: Use L'Hopital's rule (L.H.):
$$\lim_{x\to \pi} \dfrac{\sin(3x)}{\sin(5x)} \quad \overset{L.H.}{=} \quad \lim_{x \to ... |
H: Integrating Real Function in the Complex Plane
Question: Evaluate the integral
$$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+a^2)}=Im\left ( \frac{e^{ix}}{x(x^2+a^2)} \right)$$
Attempt: To evaluate this integral, I use the residue theorem (noting that we have poles $z=0$ and $z=ia$ within our contour):
$$2\pi i... |
H: recurrence relation
It was some time ago I studied recurrence relations and I came across this one that I cannot solve:
$a_{n+3}=-3a_{n+2}+4a_{n}$ with $a_{0}=2$ and $a_{1}=-5$
Ansatz: $a_{n}=r^{0}$ then I get
$r^{3}+3r^{2}-4=0$ with roots $r=1$ and $r=-2$ of double multiplicity
so the solution is
$a_{n}=A+B(-2)^... |
H: how to prove $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $N(0,\frac{a^3}{3})$.
means $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
AI: This is a normal random variable as a barycenter of normal random variables.... |
H: Using the ratio Test to see if a series converges or diverges
I need to find if $\sum_{n=1}^\infty\dfrac{3^n}{2^n+1}$ converges or diverges. I am trying to use the ratio test, which gives $\dfrac{3(2^n)+1}{2(2^n)+1}$. I am struggling to rearrange this into a form where it is clear what it tends to as n tends to inf... |
H: Does $\pi \ | \ 2 \pi$
Does $\pi$ divide $2 \pi?$
Clearly $\frac{2 \pi}{\pi}=2$ and 2 is an integer, so it would seem to make sense to say that $\pi \ | \ 2 \pi$.
Does it make sense to write, for example, $$\pi \ | \ x \implies \sin(x)=0?$$
AI: This isn't completely formal, as usually divisibility is only defined o... |
H: Irreducible polynomial $f$ as quotient - effect on $\mathbb{Z}_5[x]$
I want to get a better understanding of quotient rings so I have two questions.
Let $f(x) = x^2 + 2$
Let $R = \mathbb{Z}_{5}/(f(x))$
Now as $f$ is irreducible in $\mathbb{Z}_{5}$ we have that $R$ is a field with elements being all polynomials in $... |
H: Evaluate the limit $\lim_{x\rightarrow 0} \frac{\sqrt{1-\sin(5x)}-\sqrt{1+\sin(5x)}}{x^2+x}$
Trying to find
$$\lim_{x\rightarrow 0} \dfrac{\sqrt{1-\sin(5x)}-\sqrt{1+\sin(5x)}}{x^2+x}=\lim_{x\rightarrow 0} \dfrac{(1-\sin(5x))-(1+\sin(5x))}{(x^2+x)(\sqrt{1-\sin(5x)}+\sqrt{1+\sin(5x)})}=\lim_{x\rightarrow 0} \dfrac{... |
H: How to find $S$ such that $S = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)3^n}$?
How to find $S$ such that $$S = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)3^n}$$
I know I have somehow to manipulate a integrals and derivative but I can't see how.
I'd love if anyone can solve me this question with full solution.
Thanks in adv... |
H: The continuity of the expectation of a continuous stochastic procees
Let $X_t$ be a continuous stochastic process on a filtered space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$.
Is $\mathbb E[X_t]$ necessarily a continuous function?
My first answer would be no. For example if $X_t$ admits densities $f(t,x)$, ... |
H: If $|X_{n}| \leq Y$ almost surely, show that $\sup_{n}|X_{n}|\leq Y$ almost surely as well.
Suppose $|X_{n}|\leq Y$ a.s., each $n$, $n=1,2,3,\cdots$. Show that $\sup_{n}|X_{n}|\leq Y$ a.s. also.
This seems pretty intuitive to me, since if $|X_{n}|\leq Y$ a.s., it is bounded above by $Y$, and the sup is the least of... |
H: complex numbers and 2x2 matrices
Is it correct that set ${\mathbb C}$ is isomorphic to the set of following 2x2 matrices:
$$\left(
\begin{array}{cc}
a &-b\\
b &a
\end{array}\right)
$$
$a \in {\mathbb R}$ and $b \in {\mathbb R} $?
In other words: are these two sets identical?
AI: Yes. Provided that $a^2+b^2 \neq 0... |
H: Probability Revision
An urn contains $2$ black balls and $3$ white balls. Two balls are drawn at random from the urn. Find the probability that both balls are black.
My approach towards this problem:-
No. of black balls = 2
No. of white balls = 3
Total no. of balls = 5
2 balls are drawn at random
P(E) = ($2$C2)/$5... |
H: permuting digits of a number
Given a number $N$ with upto 18 digits, We need to find how many numbers smaller than $N$ can be formed using the same digits.
Eg:
for 725
we can form 527,572,257,275 hence answer=4
AI: It's not going to have a nice formula, but you can formulate a recursive algorithm to compute it. T... |
H: Tempered fundamental solutions
According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a tempered fundamental solution, namely a $E\in\mathscr{S}'$ with $P(\partial) E=\delta_... |
H: Probability of baby be right handed
I'm trying to resolv this problem but I think i'm not getting the correct answer. The anwser says that there is 50% probability that the baby is right handed if the test accuracy is 90%, and 69% probability the baby is right handed if the accuracy of the test is 80%.
The problem ... |
H: Help with intro to number theory question
Let $a$, $b$, $m$ and $n$ be integers with $m > 0$ and $n > 0$. If $(n,m)\mid(a−b)$, then the system
$$\begin{cases}
x\equiv a\pmod m \\
x\equiv b\pmod m
\end{cases}
$$
has a solution. Describe how to find a solution.
Can anyone explain how to solve this for me? I have be... |
H: $\gcd$ of polynomials over a field
I have the polynomials $f,g\neq 0 $ over a field $F$. We know also that $\gcd(f,g)=1$ and
$$
\det \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}\neq 0.
$$
I need to prove that $\gcd(af+bg,cf+dg) = 1 $ for every $a,b,c,d \in F$.
I really do not kno... |
H: Boolean Algebra - Why is the result 1?
Given:
= !(A * (!B + C)) + !(!B * !C)
= !A + (B * !C) + !B + C
Where:
! = NOT
+ = OR
* = AND
I'm having some trouble to why !A + (B * !C) + !B + C simplifies to 1? Can someone shed some light on this please? It would be very much appreciated.
AI: Use the distributive property ... |
H: Algebraic structures with one binary operation
I need your help.I am studying about algebraic structures.I have a question that I can't solve ..
Let $G=\langle A,*\rangle$ be a monoid with the identity $e$. Let $a\in A$.
İs it possible that $a$ has two different left-inverses, $L_1$ and $L_2$?
Thank you very much ... |
H: Cardinalities of Sylow subgroups
I must be missing something very simple here.
Problem: Let $|G|=56$. Let H be a Sylow 7-subgroup of G, and suppose H is not a normal subgroup of G. What are the cardinalities of the Sylow 7-subgroups of G and the Sylow 2-subgroups of G?
My attempt: The cardinalities of the Sylow ... |
H: Elementary number theory problem for homological algebra
Hello to everybody: I'd like to know if the following statement is true or not, since if it's false it will help me solving a problem for exact sequences of modules.
$``$Given $(a,b,m) \in \mathbb{Z}^3$ if $[a,m]= [b,m]$ and $a \equiv b (\textrm{mod} \ m)$ th... |
H: Convolution convergent in $L^p$
Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow 0}\|f\ast K_t-f\|_p=0$.
I choose a compactly supported function $g\in C^\infty$ such that $\... |
H: Prove that a Language is Non-Regular Using Closure Properties
Use the closure properties of regular languages and a language $B$ known to be non-regular to prove that a language $A$ is not regular.
My understanding is that the closure properties only apply when both languages are regular. So, I'm not sure what such... |
H: Is it possible to formulate any line or curve in 2d space ?!
I wonder how we can explain any kind of lines or curves as a formula!
more clear,can we say that "any kind of lines or curves have a formula but we cannot find their formula"?!
AI: It makes no sense just to say if you have a line or curve, can you find a ... |
H: Prove the following combination?
I need a quick proof of the following combination
$$\binom{n+1}1+\binom{n+1}2+\binom{n+1}3+\dots++\binom{n+1}{n+1}=2^{n+1}-1$$
AI: Daniel Fischer has already given you a very nice answer. However, it might be nice to see an alternative more combinatorial approach.
Imagine you are tr... |
H: Two general questions regarding intervals in $\mathbf{R}$
Does an open interval contain infinitely many closed intervals?
Is the interval $(-1,1)$ equal to $[-a,a]$ where $a<1$? Why?
AI: HINT for the second question: If $a<1$, then $a<\frac12(a+1)<1$.
Added: For the first question, let $(a,b)$ be a non-empty close... |
H: Taylor/Maclaurin Series Exam Question.
Show that if x is small compared with unity, then $$f(x)=\frac{(1-x)^\frac{-2}{3}+(1-4x)^\frac{-1}{3}}{(1-3x)^\frac{-1}{3}+(1-4x)^\frac{-1}{4}}=1-\frac{7x^2}{36}.$$
I've expanded all the brackets of f(x) up to the 2nd order and I'v ended up with $$f(x)=\frac{2+2x+\frac{37x^2... |
H: Set logic to propositional logic
How would you convert set logic to propositional logic? In particular, I'm not sure how to handle converting $\subseteq$
For example:
$$A-(\bar{B} \cup \bar{C}) \subseteq B \cap C$$
My attempt at converting to propositional logic:
$$ x \in A \land (x \in B \lor x \in C) \implies x ... |
H: Sum of the series $\sum_{n \ge 0}{\frac{x^{4n+1}}{(4n+1)!}}$
I want to determine the sum of the series $$\sum_{n \ge 0}{\frac{x^{4n+1}}{(4n+1)!}}$$
I know this has to do with the sum
$$\sum_{n \ge 0}{\frac{x^{n}}{(n)!}}=e^x\;\; \forall x\in \mathbb R$$
But i can't see how to start. Thank you for your help!!
AI: Rec... |
H: Bounding for convolution convergence
Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow 0}\|f\ast K_t-f\|_p=0$.
I choose a compactly supported function $g\in C^\infty$ such ... |
H: power series help for real this time
I'm given $$f(x)= \frac{1}{x+2} $$
I know i have to make it look like:
$$f(x) = \frac{1}{1-x} $$
but I have no clue how to go about that. Any suggestions? or maybe I could divide by 2 throughout?
AI: Hints:
$$\frac1{2+x}=\frac12\frac1{1+\frac x2}\;,\;\;\left|\frac x2\right|<1\if... |
H: Prove $(3x^2+3) \geq (x+1)^2+1$
$(3x^2+3) \geq (x+1)^2+1$
I tried using a direct proof but I think I got stumped along the way.
$3x^2+3 \geq x^2+2x+2$
$2x^2+1 \geq 2x$
$2(x^2) +1 \geq 2x$
$x^2 + (1/2) \geq x$
How can I make this appear more clear? I don't think this is evident that it is true.
AI: Try completing t... |
H: $\dim(\mbox{im}(f)) = \dim(U)$ and $\dim(\ker(g)) = \dim(V)-\dim(U)$
Let $U, V, W$ be vector-spaces over the same field $K$ with linear transformations $$f:U\to V , g:V\to W $$ so that $$g\circ f$$ is an Isomorphism.
Show that $$\dim(\mbox{im}(f)) = \dim(U)$$ and $$\dim(\ker(g)) = \dim(V)-\dim(U)$$$$$$My idea is to... |
H: Normal spaces in box and uniform topology
Is $\mathbb{R^\omega}$ normal in product topology? In the uniform topology? and In box topology?
My attempt:I know it is normal in uniform topology because it is metrizable. I would gues that it is normal in product topology. But do not know how to go about proving it.
AI: ... |
H: What are the angle brackets in Linear Algebra?
In my linear algebra book, they have angle brackets around two different vectors, so it looks like this: $\langle\mathbf{u_2},\mathbf{v}_1\rangle$. They don't use angle brackets to define vectors, but use regular parenthesis instead.
For the Gram-Schmidt process, they ... |
H: Tiling a $23 \times 23$ square by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles
A $23 \times 23$ square is tiled by $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles. Prove that at least one 1 x 1 tile must be used. Find such a tiling with exactly one $1 \times 1$ tile.
Hint: put a number in each $1 \times 1$... |
H: A Noetherian module annihilated by a power of maximal ideal must has finite length.
Let $M$ be a Noetherian $R$-module and $P^kM=0$ from some maximal ideal $P$ of $R$ and some integer $k$. How to show that $M$ has finite length?
The length of a module is defined to be the maximum length of the chain of submodule:... |
H: Convergence of formal power series substitution
Prove that the substitution of formal power series $F(G(x))=\sum_{k\geq0}f_k \frac{G(x)^k}{n!}$ converges for every $F$ if and only if $G(0)=0$
AI: Let's call the degree (the $x$-adic valuation really) of a formal power series the lowest exponent appearing in a non-ze... |
H: Derivative of $e^{-x} - xe^{-x}$
Find $f'(x)$:
$$f(x) = e^{-x} - xe^{-x}$$
$$f'(x) = -e^{-x}-(x)'(e^{-x})+(x)(e^{-x})'$$
$$f'(x) = -e^{-x}-e^{-x}-xe^{-x}$$
$$f'(x) = -2e^{-x}-xe^{-x}$$
$$f'(x) = -e^{-x}(2+x)$$
However the answer states:
$$f'(x) = e^{-x}(x-2)$$
I am not sure where I made a mistake.
AI: $$f'(x) = -e^... |
H: Number invariant problem: replacing any two numbers $a$ and $b$ with $a - 1$ and $b + 3$
Numbers 1, 2, 3, ..., 2014 are written on a blackboard. Every now and then somebody picks two numbers $a$ and $b$ and replaces them by $a - 1$, $b + 3$. Is it possible that at some point all numbers on the blackboard are even?... |
H: When should I use the *Central Limit Theorem*?
I am facing the following question:
Assume you have invited a $100$ people to a party. The probability that one would decide to come to the party is $0.75$.
What is the probability that more 70 but not more than 80 people will decide to come to your party?
I can tell t... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.