text stringlengths 83 79.5k |
|---|
H: clarification on accumulation points
Let $A=\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4},\ldots\}$. Then the point $0$ is an accumulation point of $A$. The limit point $0$ does not belong to $A$. Also, $A$ does not contain any other limit points.
Why doesn't $0$ belong to $A$, is it because there are an infinite ... |
H: Conditional expectation (mixed with an iterated expectation) $E[E(X\mid Y)\mid Y]=E(X\mid Y)$
Conditional expectation:
I want to prove $E[E(X\mid Y)|Y]=E(X\mid Y)$
I attempted the following. Is it correct? $$\begin{align*}E[E(X\mid Y)|Y=y]&=\int_{-\infty}^\infty E(X\mid Y=y)f_{X\mid Y}(x\mid y)~dx\\ &=E(X\mid Y=y)\... |
H: If $X$ is well-ordered set, how to prove that $\mathcal{P}(X)$ can be linearly ordered?
I'm having troubles solving the following exercise, proposed in T. Jech, 'Set theory' Exercises 5.4.
Let $X$ is well-ordered set then $\mathcal{P}(X)$ can be linear ordered.
[Let $X<Y$ if the least element of $X\triangle Y$ be... |
H: Applying Stirling's formula in testing for convergence of a sum
I trying to figure which $\beta \in \mathbb{R}$ make the series $\sum_{n=1}^{\infty}\frac{(\beta n)^n}{n!}$ converge. I have tried two tests: ratio test, and approximation by Stirling's formula. I must be making a mistake with at least one of them, bec... |
H: A certain ice cream store has 31 flavors of ice cream available. In how many ways can we...
A certain ice cream store has 31 flavors of ice cream available. In how many ways can we order a dozen ice cream cones if chocolate, one of the 31 flavors, may be ordered no more than 6 times?
$\dbinom{31 + 12 - 1}{12}$ woul... |
H: Formal notation when using the axiom of specification
The axiom of specification states formally that for every property $\varphi$ holds $\forall X\exists Y\forall x(x\in Y\longleftrightarrow x\in X\wedge\varphi(x))$. Since from the axiom of extensionality such a set $Y$ is unique we define $Y:=\{x\in X:\varphi(x)\... |
H: Finding the orthogonal complement of a particular set
Let $\ell^2$ denote the vector space of all square summable sequences with the inner product defined as $\langle x,y\rangle = \sum\limits_{i=1}^{\infty} x_i \bar y_i$, and $\ell_0$ denote the space of sequences that have finitely many non-zero terms. Given $y = ... |
H: Infinite Simple Group as Union of Solvable Groups
Let $A_n$ denote the alternating group on first $n$ natural numbers, and $A_{\mathbb{N}}$ be the union $\cup_{n\geq 1} \,\, A_n$. (In other words, $A_{\mathbb{N}}$ is the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$ which move only finitely many points, a... |
H: Confusion regarding the proof to "every Lindelöf metric space is second countable".
The proof given in my book for "every Lindelöf metric space is second countable" is:
Let there exist an open covering $\{B(x,\epsilon)\}, \forall x\in X$, where $X$ is a Lindelöf metric space. There consequently exists a countable s... |
H: How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
I would like to investigate the convergence of
$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$
Or more precisely, let $$\begin{align}
a_1 & = \sqrt 1\\
a_2 & = \sqrt{1+\sqrt2}\\
a_3 & = \sqrt{1+\sqrt{2+\sqrt 3}}\\
a_4 & = \sqrt{1+\sqrt{2+\sqrt... |
H: How do we know that $|i!| = \sqrt{\pi \operatorname{csch} \pi}$?
(Source: Wolfram Alpha)
Or, to write it out in full,
$$|i!| = \sqrt{\frac{2\pi e^\pi}{e^{2\pi} - 1}}$$
How is this identity derived? Also, knowing this, could we find the exact values for the real and imaginary parts of $i!$?
AI: Recall the functional... |
H: Integrating Factorials
I feel like I'm doing something wrong here: $$\frac{d^n}{dx^n}(x^n)=n!$$
$$ 5!=\frac{d^5}{dx^5}(x^5)$$
$$ \int{5! dx}=\int{\frac{d^5}{dx^5}(x^5)}dx=x\frac{d^4}{dx^4}(x^4)=x*4!$$
Please explain what I'm doing wrong.
AI: $$ \int{5! dx}=\int{\frac{d^5}{dx^5}(x^5)}dx=\frac{d^4}{dx^4}(x^5)=x*5!$$
... |
H: How can I prove that an iterated transformation describes all odd integers?
This is a question where I want to find "a best" way (or even different ways) to prove my assumption - just to widen my understanding of similar problems and how to approach them. It's a question of proof-strategy. (This is also related to ... |
H: An Attempted Proof of Cantor's Theorem
OK, I have read two different proofs of the following theorem both of which I can't quite wrap my mind around. So, I tried to write a proof that makes sense to me, and hopefully to others with the same difficulty. Please let me know if this is an accurate proof an how I might ... |
H: Finding a power series representation of the function $f(x)=\frac{2}{3-x}$
I feel like I'm on the right track, but I don't know if I need to do something else to finish it off...
\begin{align*}
f(3x)&=\frac{2}{3-3x}\\
&=\frac{2}{3(1-x)}\\
&=\frac{2}{3}\cdot\frac{1}{1-x}\\
&=\frac{2}{3}\cdot\sum_{n=0}^{\inft... |
H: Proving $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$
The question is
Prove that for any $n\in\mathbb N$, $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$
I don't have any ideas how to solv... |
H: Proving convergence of a series
I need to show that the series of general term $$\tanh \frac{1}{n}+ \ln \frac{n^2-n}{n^2+1}$$
converges.
I was thinking to use an equivalence as $n \rightarrow \infty$
We know that $ \tanh \frac{1}{n}= \frac{1}{n} - \frac{1}{6n^3}+ o(\frac{1}{n^4})\sim \frac{1}{n}$
and $\ln\frac{n^... |
H: Proving that the limit $ \lim\limits_{n\rightarrow \infty} (n!)^{\frac{1}{n}}$ diverges to infinity
I came across this limit in some context:
$$ \lim\limits_{n\rightarrow \infty} (n!)^{\frac{1}{n}}$$
I could only say that $n! > n$ implies the limit is greater than or equal to $1$. However, the result seems to be in... |
H: Pattern recognition next shape
Is there any logic for finding the next shape in the blank?
I think it's an hard one.
AI: As the comments pointed out, these types of problems can be really ambiguous. Here's my guess. Note that out of the four options, the main thing that is different is the number of white shapes. S... |
H: Can modulo be used in consecutive multiplications or divisions?
I used to participate in programming competitions and at times I see that the solution should be the remainder when divided with some big prime number (usually that would be 1000000007). In one of the problems, I need to divide a very big number by ano... |
H: Find coefficient of $x^{100}$ in the power series expansion of $\frac{1}{(1-x)(1-2x)(1-3x)}$
I'm trying to find to coefficient of $x^{100}$ of
$$\sum_{n=0}^{∞}a_n x^{n}\ =\frac{1}{(1-x)(1-2x)(1-3x)}.$$
I used the sum:
$$\frac{1}{1-x}\ = 1+x+x^2+\ldots.$$
So : $$\frac{1}{(1-x)(1-2x)(1-3x)}= \left(1+x+x^2+\ldots\rig... |
H: What is the group $\langle U, * \rangle$ where $U$ is the set of roots of unity and * is normal multiplication?
I'm having trouble understanding my textbook in this regard. Everything else seems to make sense, such as the group $\langle \mathbb{Z}_{1000}, + \rangle$ to list an example.
In the text, it is stated th... |
H: Describing a Galois group.
From Robert Ash (Basic Abstract Algebra)
Suppose that $E=F(\alpha)$ is a finite Galois extension of $F$, where $\alpha$ is a root of the irreducible polynomial $f \in F[x]$. Assume that the roots of $f$ are $\alpha_1 = \alpha, \alpha_2, ... , \alpha_n$. Describe, as best you can from the... |
H: Finding angle of intersecting lines
![
$\triangle ABD$ and $\triangle ACE$ are equilateral triangles. Can it be proved that $\triangle ADC$ and $\triangle ABE$ are congruent. Or if given they are congruent what is the value of $\angle BOD$?
Another similar problem with square replacing the equilateral triangles ha... |
H: What went wrong? Calculate mass given the density function
Calculate the mass:
$$D = \{1 \leq x^2 + y^2 \leq 4 , y \leq 0\},\quad p(x,y) = y^2.$$
So I said:
$M = \iint_{D} {y^2 dxdy} = [\text{polar coordinates}] = \int_{\pi}^{2\pi}d\theta {\int_{1}^{2} {r^3sin^2\theta dr}}$.
But when I calculated that I got t... |
H: Path space of $S^n$
Suppose that $p,q$ are two non conjugate points on $S^n$ ($p \ne q,-p$). Then there are infinite geodesics $\gamma_0, \gamma_1, \cdots$ from $p$ to $q$. Let $\gamma_0$ denote the short great circle arc from $p$ to $q$; let $\gamma_1$ denote the long great circle arc $p(-q)(-p)q$ and so on. The s... |
H: The sum of square roots of non-perfect squares is never integer
My question looks quite obvious, but I'm looking for a strict proof for this:
Why can't the sum of two square roots of non-perfect squares be an integer?
For example: $\sqrt8+\sqrt{15}$ isn't an integer. Well, I know this looks obvious, but I can't p... |
H: $\mathbb{R}^k$ and $\mathbb{R}^k$ are trivially diffeomorphic.
Is this claim correct?
If so, is it because identity is the diffeomorphism?
AI: Yes, the identity map is a diffeomorphism. The reason is that it is smooth and its inverse (the identity map again!) is smooth. Finally, its bijective for obvious reasons.
T... |
H: Prove that if $f_n\to f$ uniformly and $\int_0^\infty|f_n|\le M$, then $\int_0^\infty|f|<\infty$
Again while preparing to calculus I found another interesting question:
Prove or give counterexample that if $f_n\to f$ uniformally $[0,\infty]$ and $\forall n\in\mathbb {N}\int_0^\infty|f_n|dx\le M$, then $\int_0^\inf... |
H: Proof that Scheme T implies reflexivity
In my modal logic book it's written that, for each frame $F(S,R)$ the accessibility relation $R$ is reflexive IF AND ONLY IF the scheme T:$\square A \implies A$ is valid in $F$.
Even if I can easily prove that reflexivity $\implies$ T, I can't prove that T $\implies$ reflexiv... |
H: Differentiate the following w.r.t. $\tan^{-1} \left(\frac{2x}{1-x^2}\right)$
Differentiate : $$ \tan^{-1} \left(\frac {\sqrt {1+x^2}-1}x\right) \quad w.r.t.\quad \tan^{-1} \left(\frac{2x}{1-x^2}\right) $$
AI: the question have been solved by assuming tan inverse is “arc tan“.
put x=tanθ and simplify
→arc tan... |
H: Convergence of $\int\limits_{-\infty}^{\infty}\frac{\sin{x^2}}{x}dx$
I had to prove that the integral $\int\limits_{-\infty}^{\infty}\frac{\sin{(x^2)}}{x}dx$ converges. I thought splitting it to $$\int_{-\infty}^{-1}\frac{\sin{x^2}}{x}dx+\int_{-1}^{0}\frac{\sin{x^2}}{x}dx+\int_0^1\frac{\sin{x^2}}{x}dx+\int_1^\infty... |
H: How to prove that a circle passing through the center of the circle of inversion invert to a line?
link to the referenced picture: http://www.flickr.com/photos/90803347@N03/9220374271/
In order to prove the Arbelos Theorem, as in the picture above, one need to prove that the semicircle $C$ invert to line $l$, as we... |
H: Determine if the following sequence converge in the quadratic mean
"For integer $n$ let $f_n(x) = \dfrac{1}{\sqrt{1 + nx^2}}$
say if the sequence $f_n$ converges in quadratic mean."
This is what I have concluded so far:
$$\lim\limits_{n\rightarrow \infty}f_{n}(0) = 1$$
For $x \neq 0$
$$\lim\limits_{n\rightarrow \in... |
H: Why difference quotient of convex functions increases in both variables
Let $f: \mathbb R \rightarrow \mathbb R$ be a convex function and
$$
g(x,y)=\frac{f(x)-f(y)}{x-y} \textrm{ for } x\neq y.
$$
I wish to prove that $g$ is increasing function in both variables.
Thanks
AI: Fix $y_2 > y_1 > x$ and define
$$ \phi(... |
H: Mandelbrot set incorrect picture
I'm writing an algorithm to generate the Mandelbrot set in Java. However, the final picture is incorrect. It looks like this
I was wondering if the algorithm was incorrect.
public void Mandlebrot() {
float reMax=2;
float imMax=2;
float reMin=-2;
float imMin=-2;
... |
H: If $f$ has a pole at $z_0$, then $1/f$ has a removable singularity
I tried a few examples and I think that the following in complex analysis holds:
If a function $f$ has a pole at $z_0$, then $1/f$ has a removable singularity at this point. Is this correct?
AI: If $f$ has a pole at $z_0$ then there is an open $U$ ... |
H: Prove question $(A\setminus B) \cup (B\setminus C) = A\setminus C$ , $ (A\setminus B)\setminus C= A\setminus(B\cup C)$
I want to prove the following statements but for do it I need some hint.
\begin{align}
\tag{1} (A\setminus B) \cup (B\setminus C) &= A\setminus C\\
\tag{2} (A\setminus B)\setminus C&= A\setminus(B\... |
H: Series Question $\sum_{n=1}^{\infty}\frac{4^n}{7^{n+1}}$
I`m trying to check if the following series are convergent.
$$1)\sum_{n=1}^{\infty}\frac{4^n}{7^{n+1}}$$
$$2)\sum_{n=1}^{\infty}(-1)^n\frac{5^n}{4^{n+2}}$$
what I did so far for the first one is:
$$\sum_{n=1}^{\infty}\frac{4^n}{7^{n+1}} = q=\frac{4}{7}<1 \ri... |
H: Homotopy Theory and extensions/liftings.
I found the statement: suppose that in the extension problem we have a map f': A -> E homotopic to f, and f' extends. Then it does not follow that f extends. Similarly, if the map g in the lifting problem is homotopic to a map that lifts, it does not follow that g itself li... |
H: find symmetric line of given two line
I have one question. Suppose that we have two lines given by equations
$$y=2x+3$$
$$y=-2x+11$$
I want to find all equations of lines which these two given lines have same distances from them in plane.As I know symmetric means that the distance between it and the two given ... |
H: For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$?
Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|<1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that
$$\left|\sum_{i=1}^n a_iz_i\right|<\s... |
H: $K$ is a basis for $W$, and $L$ is a basis for $U$. Is $K\cup L$ is a basis for $U + W$?
Question:
$V$ is a vector space over field $F$ , $U,W$ are subspaces of $V$. Is the next statement true or false?
"$K$ is a basis for $W$, and $L$ is a basis for $U$. Therefore $K\cup L$ is a basis for $U + W$".
What I did: (I... |
H: The Effect of a Transpose on a Matrix Inequality
In the solution to an exercise I came across the following: $y^TA_N \geq c_N^T \rightarrow A_N^Ty \geq c_N$. Now I was wondering, is it in general true that an inequality remains valid when 'taking transposes on both sides'? If so, what is the proof for this?
AI: Gen... |
H: Rearranging power series expansion to get parameter on denominator
How can we rearrange
$$T=\dfrac{k V+g}{gk}\bigg(kT-\dfrac{1}{2}k^{2}T^{2}+\dfrac{1}{6}k^{3}T^{3}\bigg),$$
to get
$$T=\dfrac{2V/g}{1+k V/g}+\dfrac{1}{3}k T^{2}$$
?
AI: So, $$kT-\frac12k^2T^2+\frac16k^3T^3=\frac{gTk}{kV+g}=\frac{Tk}{\frac{kV}g+1}$$
$... |
H: Cardinality of the quotient field of power series ring
Let $k$ be a field which is countable and let $x$ be an indeterminate over $k$. I have hard time to prove $$\operatorname{card} k((x)) = \operatorname{card}\mathbb R.$$ Thank you.
AI: When your field is $\mathbb{Z}_{2}$ Define a bijection between $k[[x]]$ and t... |
H: Find the volume of the body bounded by $x^2 + y^2 + z^2 = 4$ and $x^2 + y^2 = 1$
Find the volume of the body bounded by $x^2 + y^2 + z^2 = 4$ and $x^2 + y^2 = 1$.
This is the last subject in our syllabus and I am afraid that my professor had not any time to teach it before the end of the semester, and left us won... |
H: Is this a linear transformation?
Let $T$ be a transformation from $P_2$ to $P_2$ (where $P_2$ is the space of all polynomials with degree less than or equal to $2$)
$$T(f(t)) = f''(t)f(t)$$
I'm tempted to say that this is not a linear transformation because
$$T(f(t) + g(t)) = (f''(t) + g''(t))(f(t) + g(t))$$
Which ... |
H: How to find the limits $\lim\limits_{h\rightarrow 0} \frac{e^{-h}}{-h}$ and $\lim\limits_{h\rightarrow 0} \frac{|\cos h-1|}{h}$?
How to work around to find the limit for these functions :
$$\lim_{h\rightarrow 0} \frac{e^{-h}}{-h}$$
$$\lim_{h\rightarrow 0} \frac{|\cos h-1|}{h}$$
For the second one i think that th... |
H: How to find maxima and minima of a function involving a factorial
i need to find the value of y when the bell curve for the following function reaches its maximum , i can solve the problem easily on a MAS software but i needed to know a more mathematical approach . here's the simplified equation
$$\frac{3^{-y}}{(... |
H: Checking Differentiability for given function
Find if the function $x\mapsto |\sin (x)-1|$ is differentiable at $x=\pi /2$ .
I get stuck at $$\lim_{h \rightarrow 0}{ \left|{\cos h \over h}\right|}$$
AI: HINT:
With Graham Hesketh's solution, the differentiability of $(1-\sin x)$ is established here
As $$\frac{d f(... |
H: Minimum and maximum of $ \sin^2(\sin x) + \cos^2(\cos x) $
I want to find the maximum and minimum value of this expression:
$$ \sin^2(\sin x) + \cos^2(\cos x) $$
AI: Following George's hint,
Because $-1\le \sin x \le 1$, and $\sin x$ is strictly increasing on $-1\le x\le 1$, we see that $\sin (\sin x)$ (and hence $... |
H: Questions regarding Holder's and Minkowski's inequality
I've some questions regarding Holder's and Minkowski's inequality as given in my text:
Does the author consider the case $q=\infty$ in the equality case of lemma 1.1.36?
Shouldn't the author mention $C_1,C_2$ are not all zero in the equality case of lemma 1... |
H: what is the explicit form of this iterativ formular
I am not sure, if there is an explicit form, but if there is, how do I get it?
This is the formula:
$$c_n=\frac{1-n \cdot c_{n-1}}{\lambda}$$
where $\lambda \in \mathbb{R}$ and $n \in \mathbb{N}$
I already tried some forms for c via trail and error, but I couldn't... |
H: Calculator for real and complex parts of a polynomial.
Does anyone know of a nice calculator for calculating the real and complex parts of a complex polynomial. Say, for example, I want to write the polynomial $p(z)=z^3+2z^2+1$ as a function $f(x,y)=(f_1(x,y),f_2(x,y))$, but I do not want to calculate $f_1$ and $f... |
H: I'm looking for several ways to prove that $\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$
I'm looking for several ways to prove that $$\int_{0}^{\infty }\sin(x)x^mdx=\cos\left(\frac{\pi m}{2}\right)\Gamma (m+1)$$
for $-2< Re(m)< 0$
AI: Here is a fake proof: Let $x = \sqrt{t}$. Then
$$ \int_{0}^{... |
H: Does there exist a group of even order which every element is a square?
Does there exist a group of even order which every element is a square? I know in any group of odd order every element is a square. I am not sure the case of even order. Any suggestion?
AI: A simple counting argument, noticing that a group of e... |
H: Immersion of Quaternions
Does there exist an immersion of the Quaternion Group in the Symmetric Groups $S_6$ and $S_7$?
If it does exist, can you give me an explicit description of that immersion?
AI: The Sylow 2-subgroups of $S_6$ (or $S_7$) are isomorphic to $D_8 \times C_2$ (here $D_8$ is the dihedral group of o... |
H: Use weak induction to prove the following statement is true for every positive integer $n$: $2+6+18+\dots+2\cdot 3^{n-1}=3^n-1$
Use weak induction to prove the following statement is true for every positive integer $n$: $$2+6+18+\dots+2\cdot 3^{n-1}=3^n-1$$
Base Step: Prove it is true for $n$.
Inductive Hypothesis:... |
H: How to find the element $x\in \mathbb Z_n$ such that $f(x)=1$, where $f\in \mathrm{Aut}(\mathbb Z_n)$?
Let $0<m<n$ and $(m,n)=1$. Consider the map $f\in \mathrm{Aut}(\mathbb Z_n)$ such that $f(1)=m$. Which element $x\in \mathbb Z_n$ has the property that $f(x)=1$?
AI: Hint 1
Let us denote the class of $x \in \Bbb{... |
H: Question related to first order partial derivatives
If The funtion $f: \Bbb R^2 \to \Bbb R$ has directional derivatives in all directions at each point in $\Bbb R^2$ then the function $f$ has first order partail derivatives at each point in $\Bbb R^2$
How can I prove this? Please do not downvote because of not say... |
H: I need a better explanation of $(\epsilon,\delta)$-definition of limit
I am reading the $\epsilon$-$\delta$ definition of a limit here on Wikipedia.
It says that
$f(x)$ can be made as close as desired to $L$ by making the independent variable $x$ close enough, but not equal, to the value $c$.
So this means that... |
H: What's the difference between $ \mathbb{Z}/4\mathbb{Z}$ and $ 4\mathbb{Z} $?
Can someone please explain the difference between $ \mathbb{Z}/4\mathbb{Z} $ and $ 4\mathbb{Z} $?
From my understanding (please correct where I'm wrong): the group $4\mathbb{Z}$ has only four elements, $\{0,1,2,3\}$, and the group $\mathb... |
H: The rank of a bunch of vectors in $\mathbb{R}^4$
Please help me solve this:
In $\mathbb{R}^4$ how can I calculate the rank of the following vectors: $$a=(3,2,1,0), b=(2,3,4,5), c=(0,1,2,3), d=(1,2,1,2), e=(0,-1,2,1).$$ I know that since $\#\{a,b,c,d,e\}=5$ it's a linearly dependent set in $\mathbb{R}^4$ because $\d... |
H: Show that $\mathbb{R}^n\setminus \{0\}$ is simply connected for $n\geq 3$
Show that $\mathbb{R}^n\setminus \{0\}$ is simply connected for $n\geq 3$.
To my knowledge I have to show two things:
$\mathbb{R}^n\setminus \{0\}$ is path connected for $n\geq 3$.
Every closed curve in $\mathbb{R}^n\setminus \{0\}, n\geq ... |
H: Maximum Likelihood Principle; Local vs. Global Maxima
In the statement for estimating parameters through the Maximum Likelihood Principle (MLE), there is no mention of whether to choose a local maximum or a global maximum. (In my very limited reading so far) From the examples given in various textbooks/lecture note... |
H: Integral dependence of an algebraic element
Let $A$ be a UFD, $K$ its field of fractions, and $L$ an extension of $K$. Then, let $\alpha \in L$ and let $f_\alpha \in K[x]$ be its minimal polynomial over $K$.
Is it true that $\alpha$ is integral over $A$ if and only if $f_\alpha$ has coefficients in $A$?
AI: Yes. B... |
H: Residue/Laurent series of $\frac{z}{1+\sin(z)}$ at $z=-\pi/2$
For some reason, I just can't quite figure out how to easily calculate the Laurent series for the following function:
$$
f(z)=\frac{z}{1+\sin(z)},\quad z_0=-\frac{\pi}{2}
$$
I don't really need the whole series, just the residue. The function has a zer... |
H: Wolfram double solution to $\int{x \cdot \sin^2(x) dx}$
I calculated this integral :
$$\int{x \cdot \sin^2(x) dx}$$
By parts, knowing that $\int{\sin^2(x) dx} = \frac{1}{2} \cdot x - \frac{1}{4} \cdot \sin(2x) +c$. So I can consider $\sin^2(x)$ a derivative of $\frac{1}{2} \cdot x - \frac{1}{4} \cdot \sin(2x)$, a... |
H: Trigonometric manipulation of complex number, how does this step occur?
I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says:
"Let $z$ be a complex number. Suppose we wish to find th... |
H: A question about Banach algebras: showing that $\operatorname{Sp}a \subset D_o \cup D_1$
Maybe this problem be easy for a person that have study in Banach Algebra; please give me a hint.
Let $e=0$ or $1$, and $a$ be an arbitrary element in a Banach algebra $A$. Let
$D_o$ and $D_1$ be the disks in the complex plane ... |
H: What does it mean by a matrix being bounded?
Does it means each entry is bounded?
Also, finally, given the domain is orthogonal group, I am aware that the range of $F$ is identity, which is closed. But how can I show the domain is?
By the way, it comes to my mind that - does this function has inverse: $F(A) = AA^T$... |
H: Is Dirichlet function Riemann integrable?
"Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$.
On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded and continuous almost everywhere, which the Dirichlet function satisfies.
... |
H: $p \mid x^2 +n\cdot y^2$ and $\gcd(x,y)=1 \Longleftrightarrow (-n/p) = 1$
Let $n$ be a nonzero integer, let $p$ be an odd prime not dividing $n$. then $ p \mid x^2 + n\cdot y^2$ and $x,y$ co-prime $ \Longleftrightarrow(-n/p) = 1 $
How can i prove this? by $(-n/p)$ i mean the Legendre symbol.
For $\implies$ i have ... |
H: Cube roots don't sum up to integer
My question looks quite obvious, but I'm looking for a strict proof for this. (At least, I assume it's true what I claim.)
Why can't the sum of two cube roots of positive non-perfect cubes be an integer?
For example: $\sqrt[3]{100}+\sqrt[3]{4}$ isn't an integer. Well, I know thi... |
H: How do I show that this Limit of 2 variables is zero?
How do I show that :$$\lim_{(x,y)\to(0,0)}xy\frac{x^2-y^2}{x^2+y^2}=0?$$
I'm stumped...
AI: $$
0 \le (x-y)^2 = x^2+y^2 - 2xy\implies|xy| \le \frac{x^2+y^2}{2}
$$ |
H: How can I geometrically (or geographically) group items together?
I'm a programmer, and I'm working on a project that takes a bunch of photos and separates them into groups by their gps coordinates. I have no experience in things like geometric group theory so I'm not even sure if that's the field that would help m... |
H: Regular open sets and semi-regularization.
In a Hausdorff space $(X,\tau)$, we can generate a coarser topology, say $\tau'$, by taking its base to be the family of regular open sets in $(X,\tau)$. (Semi-regularization of $(X,\tau)$)
Given that it's already proven, how to we proceed to prove that the "regular open ... |
H: Evaluating: $\int^{n}_{1}[\ln(x) - \ln(\lfloor x \rfloor)] dx $
I am attempting to evaluate the integral:
$$\int^{n}_{1}\ln(x) - \ln(\lfloor x \rfloor) dx $$
To a form:
$$f(x) + O(g(x))$$
where $g(x) \rightarrow 0$ as $x \rightarrow \infty $
How do I compute that f(x) or atleast some type of series representation f... |
H: Computing the value of logarithmic series: $Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $
Given a series of the type:
$$Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $$
How does one evaluate it?
Something I noticed was:
$$Q(1,n) = \ln(1) + \ln(2) + \ln(3)+ \cdots+\ln(n) = \ln(1\cdot 2\cd... |
H: Why don't we define division by zero as an arbritrary constant such as $j$?
Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is the only number such that $0*j \not= 0$ ,... |
H: How to prove that $n\log n = O(n^2)$?
How to prove that $n\log n = O(n^2)$?
I think I have the idea but need some help following through.
I start by proving that if $f(n) = n \log n$ then $f(n)$ is a member of $O(n\log n)$. To show this I take a large value $k$ and show that $i\geq k$ and $f(i) \leq c_1\cdot i\log(... |
H: Line integral, parabola
I'm brushing up on some multivariable-calc and I'm stuck on the following problem:
Calculate:
$$E = \int_\gamma \frac{-y \cdot dx+x \cdot dy}{x^2+y^2}$$
for $\gamma$ which is the parabola $y=2x^2-1$from $(1,1)$ to $(\frac{1}{2}, -\frac{1}{2})$.
I've done the following:
Let $$x(t)=t \implies ... |
H: Checking my proof related to directional derivatives
Please can somebody check my answer? Tell me and explain me my mistakes and so on if there is. Thank you for helping :)
Question:
Suppose that the function $f:\Bbb R^n \to \Bbb R$ is continuously differentiable.
Let $x$ be a point in $\Bbb R^n$. For $p$ a nonzer... |
H: Show that$ f(x)=x^5-3$ is solvable by radicals over $\mathbb{Q}$.
I was reading about solvability of quintics by radicals, but unfortunately there were no many examples and I am afraid that I do not understand the whole concept. How to show $x^5-3$ is solvable by radicals over $\mathbb{Q}$?
AI: The splitting field ... |
H: Estimation of variance.
Given $X_1, X_2,\dots, X_n$ independent random variables with the same distribution, if we define $S^2_N = \displaystyle\frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar X_n)^2$ show that $S_N^2$ converges almost surely to $\sigma^2$ -variance-
It seems that i have to prove the following: $P\{(\lim {S_... |
H: change of bases - matrix representation of linear maps
I am trying to solve a problem and got stuck. I suppose I made a stupid mistake somewhere, could somebody explain where?
Let $B = \{v_1, v_2, v_3\}$ be a basis of a vector space $V$, and let
$B' = \{v_1, v_1 + v_2, v_1 + v_2 + v_3\}$. Now define a linear map... |
H: Question related to partial differentiablity and directional derivative
$\mathbf {Question:}$
Define a function $f:\Bbb R^2 \to \Bbb R$ by
$f(x,y)=$ $(x/|y|)\sqrt {x^2+y^2}$ if $y\not = 0$
$f(x,y)=0$ if $y=0$
$\mathbf{a)}$ prove that the function $f$ is not continuous at the point $(0,0)$
$\mathbf{answer-a:}$
I n... |
H: is $f(x)$ in big-$O$ of $g(x)$ assuming the following?
Assuming that: $f(n)=O(g(n))$ and $f(n)$ and $g(n)$ are nondecreasing and always bigger than 1
Is the following necessarily true?
$$f(n)\log_2(f(n)^c)=O(g(n)\log_2(g(n)))$$
And also, could you explain why? Thanks.
AI: If you can prove that $\log f(n) = O(\log... |
H: Algebraic proof of De Morgan's law with three sets
Given:
$A'$ $\cup$ $B'$ $\cup$ $C'$ $=$ $(A$ $\cap$ $B$ $\cap$ $C$ $)'$
Problem:
Show how the identity above can be proved using two steps of De Morgan's Law along with some other basic set rules (i.e. an algebraic proof).
I wasn't aware that De Morgan's Law had m... |
H: The preimage of continuous function on a closed set is closed.
My proof is very different from my reference, hence I am wondering is I got this right?
Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of continuous function on closed set is closed.
Let $D$ be a ... |
H: Proving that a Particular Set Is a Vector Space
Let $V$ be the set of all differentiable real-valued functions defined on $\mathbb R$. Show that $V$ is a vector space under addition and scalar multiplication, defined by
$$(f+g)(t) = f(t) + g(t),\quad (cf)(t) = c[ f(t)],$$ where $f, g \in V$, $c \in F$.
Since ad... |
H: $(\cdot,\cdot)$ in Banach spaces?
I have been doing some research on fixed point theorems, and they have brought me around to papers from the 1960s in functional analysis in Banach spaces.
I think that today it is common practice to use either $\langle\cdot,\cdot\rangle$ or $(\cdot,\cdot)$ to mean an inner product ... |
H: Calculate sum of squares of first $n$ odd numbers
Is there an analytical expression for the summation $$1^2+3^2+5^2+\cdots+(2n-1)^2,$$ and how do you derive it?
AI: $$ 1^2+3^2+5^2+\cdots+(2n-1)^2 = \sum_{i=1}^{n}(2i-1)^2 = \sum_{i=1}^{n}(4i^2)- \sum_{i=1}^{n}(4i) + \sum_{1}^{n}1=\dots.$$
Now, you need the identitie... |
H: studying compact $\partial$-$n$-manifolds via closed $n$-manifolds?
What would be counterexamples to the following statement:
It is not true that any $n$-manifold with boundary is a $n$-manifold with finitely many embedded disjoint open disks removed, since that would mean that its boundary is a disjoint union of s... |
H: Is this a characterization of well-orders?
While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I can immediately see it's true for countable orders, but not for uncountable... |
H: Laplace equation with weird boundary condition
So, guys, here's my problem. I have this differential equation
$$
U''_{xx}+U''_{yy}=0
$$
with these boundary conditions:
$$
U'_{y}(x,0)=0
$$
$$
U'_{y}(x,\pi)=0
$$
$$
U(0,y)=0
$$
$$
U(\pi,y)=1+\cos(2y)
$$
Now, I obtain this solution for the first three conditions:
$$
\s... |
H: Prove that if $\langle x,z\rangle = 0$ for all $z,$ then $x=0.$
I just wanted to check if my reasoning in this proof was correct.
The question is as follows.
Let $\beta$ be a basis for a finite dimensional inner product space $V.$
a) Prove that if $\langle x,z \rangle = 0$ for all $z \in \beta,$ then $x=0.$
OK, I... |
H: How to show $G(p)$ is a subgroup of $G$ an abelian group
If $G$ is an abelian group and $p$ is a prime then
$$G(p):=\{g \in G: \mathrm{ord}(g)=p^n, n \geq 0\} \leq G.$$
I think I may be making this problem too difficult. I know $G(p)\neq \varnothing$ as $\mathrm{ord}(e)=p^0$ so $e \in G(p)$.
If $a,b \in G(p)$ ... |
H: Field extension $k(a)$
Well, first I write some definitions:
Let $K|k$ be a field extension. Then $k(a)$ denotes
$$
k(a)=\bigcap\{F:k\subseteq F \subseteq K \,\ a\in F \}
$$
and is the smallest field of the extension $K|k$ such that $a$ belongs to it.
With this, and given $a$ in $K$, I set
$$
\begin{array}{... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.