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H: Chain rule method doesn't result with same answer as u-sub. Why?
$\int \ln \left(2x\right)\,\mathrm{d}x$ is the integral in question.
I know how to solve it with the chain rule. $\frac{1}{2x}\times 2x = \frac{1}{x}$
But, because I know $u$-sub method, I wish to use it upon this integral.
$\int \ln \left(2x\right)... |
H: what is $X / \cong $ ?? where $\cong $ is given by
what is $X / \cong $ ?? Suppose $X = \mathbb{R}^2 $. and we define
$$ (x_1,y_1) \cong (x_2, y_2) \iff x_1 + y_1 =x_2 + y_2 $$
With this equivalence relation, we get that the partition is the trivial one since every point is congruent to itself and hence $X$ is pa... |
H: Why is there Inequality in Fatou's Lemma?
I'm studying measure theory for the first time, and I just came across Fatou's Lemma.
Why isn't it true that for any sequence of functions $\left\{ f_n \right\}$ in $L^+$ we always have that $$\int \displaystyle \liminf_{n\rightarrow \infty} f_n d\mu =\liminf_{n\rightarrow... |
H: Pigeonhole principle problem involving inequality 0 < |$\sqrt{x} - \sqrt{y}$| < 1
21 integers are selected from {1, 2, 3, ..., 400}. Prove that two of them, say x and y, satisfy 0 < |$\sqrt{x} - \sqrt{y}$| < 1.
I am confident you have to use and apply the Pigeon Hole Principle. From what I gathered, there are 400 n... |
H: Pigeonhole Principle / Number Theory
Let $S$ be a subset of $A=\{1,2,3,...,1000\}$. Find the largest number of elements in $S$ such that for any $a, b \in S$ with $a>b$, $a-b$ does not divide $a+b$.
I've tried numerous approaches, even brute force (listing), but to no avail. Any help would be greatly appreciated :... |
H: Why aren't multi valued functions invertible?
I recently learnt that functions are invertible if and only if they are bijective. But why aren't multi-valued surjective 'functions' invertible?
AI: Noone would kill you if you wrote $f(x) = x^2$ and then $f^{-1}(4) = \{2, -2\}$ (You kind-of define your inverse as a se... |
H: Proving a map is an automorphism
Let $G$ be a group; for $g\in G$ define $T_g:G\to G$ by $xT_g = g^{-1}xg$ for all $x\in G.$ Prove that $T_g$ is an automorphism of $G$.
Since $T_g$ is onto then for $y\in G$ let $x = gxg^{-1}$ then $xT_g = g^{-1}(x)g = g^{-1}(gyg^{-1})g = y$. Also, one to one is easy to see but ho... |
H: This operator is invertible
I'm thinking if the self-adjoint operators are invertible. I'm really stuck, I don't know even how to begin,I need a hint or or a counter-example if it's not true.
Thanks
AI: The constant zero operator is self-adjoint, but clearly not invertible.
EDIT:
More generally, for an operator on ... |
H: Find $ \int_{0}^{3} \left[- \frac {x^2}{4} - 8\right] \,dx$
Consider the following definite integral
$$A = \int_{0}^{3} \left[- \frac {x^2}{4} - 8\right] \,dx$$
Calculate the Riemann sum that approximates the value of A as a closed-form formula in n (i.e. remove the Sigma using the necessary formulas). Use right-h... |
H: Couples around a table
Find the number of ways in which $n$ men and $n$ women can sit around a round table such that every man can pair off with a woman sitting next to him (to form $n$ couples).
Anyone can help with this problem? Thanks :)
EDIT: Ignore rotations of the table. For example, when $n=1$ there should b... |
H: Infinite sets with cardinality less than the natural numbers
Are there any infinite sets that have a lower cardinality than the natural numbers? Is there a proof of this?
AI: No there are none. If $A$ has cardinality of at most the natural numbers, we may assume that it is a subset of the natural numbers.
One can ... |
H: Number of Nonnegative integers = number of integers
I just came across the idea that says the number of non-negative integers is equal to the number of integers. How is this possible? Isn't it true that non-negative integers are a subset of all integers? Then, it should follow that the number of all integers is big... |
H: Questions on covering space
Following is a paragraph of a paper I am reading:
But I cannot understand this image, maybe it is because I have no idea about coverings. Could anyone explain it to me?
Particularly, you could just explain to me: what does $X_{|U_\alpha}$ mean?
AI: $X|_{U_\alpha}$ should mean "the preim... |
H: Intermediate value theorem for mean
Suppose the function $f:\mathbb{R}\rightarrow\mathbb{R}$ is continuous. For a natural number $k$, let $x_1,x_2,...,x_k$ be points in $[a,b]$. Prove there is a point $z$ in $[a,b]$ at which $$f(z)=(f(x_1)+f(x_2)+...+f(x_k))/k$$.
So I'm thinking about applying the intermediate val... |
H: How to integrate $\int_{0}^{a}x^{n-1}e^{-x}dx$
We know that $$\int_{0}^{\infty }x^{n-1}e^{-x}dx = \Gamma (n)$$
But how do we integrate this?
$$\int_{0}^{a}x^{n-1}e^{-x}dx$$
AI: This integral can be viewed as a recurrance
$$\begin{align*}
I_n &= \int_0^a x^{n-1}e^{-x}dx\\
&= -\int_0^a x^{n-1}de^{-x}\\
&= -\left[x^{n... |
H: $a^{1/n}$ - How do you explain it cannot be $0$ for $a > 0$?
How do you explain formally, $a^{1/n}$ cannot be equal to $0$ for every $a > 0$?
Thanks
AI: If $b=a^{1/n}$, we have $b^n=a$. Now, can $0^n=a>0$? |
H: How to determine the coefficient of binomial
Suppose I have $\left(x-2y+3z^{-1}\right)^4$
How to determine coefficient binom of $xyz^{-2}$?
I've tried using trinominal expansion like this:
$\displaystyle \frac{4!}{1!1!1!} (1)^1 (-2)^1 (3)^2$
AI: You can do this by elementary combinatorics, think about how products ... |
H: Minimize the area of a triangle
Let $A \neq B$ be fixed points outside a fixed circle with centre
$C$. The point $D$ can be chosen freely on the circle. The goal is to minimise
the area of triangle $ABD$. Degenerate triangles (triangles that are merely line
segments) are excluded. In which configurations of $A, B, C$ ... |
H: Matchmaker's problem
This is a past exam exercise I'm unable to solve.
$B$ is finite set.
$h:B\rightarrow \mathcal{P}(G)$ where $\mathcal{P}(G)$ is the powerset of $G$ with the following properties:
for every $x\in B$, $h(x)$ is a finite subset of $G$
$X\subseteq B\rightarrow |X|\leq|\cup \{h(x):x\in X\}|$
I wou... |
H: Is $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} \cup \left\{ 0\right\}$ closed set?
Is it true that $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} \cup \left\{ 0\right\}$ is closed set? I suppose that yes, but I have no idea how can I prove it.
AI: Hint : It would be a bit easy if you can realize :
$\big(\{ \... |
H: Continuity of translation for p=infinite
For $f\in L^p$, $1\leq p<\infty$, the property of continuity of translation holds:
$$\lim_{|x|\to 0}\|f_x-f\|_p=0,$$
where
$$f_x(y):=f(x+y).$$
When $p=\infty$, is continuity of translation still true? If not, can you give a counterexample?
AI: No. Take $f = \chi_{[0,1]}$, th... |
H: Laplace Transform using t-shift (second shift)
$$f(t) = tu(t-π)$$
I know I have to get t in terms of $$(t-π)$$ and to do that I have done
$$ t = a(t-π) + b$$
$$ t = at-aπ + b$$
$$ t = (a-π)t + b$$
$$ (a-π) = 1$$ and $$b = 0$$
Then I think I have done the write thing and being to t-shift with
$$ f(t) = (a-π)(t-π)u... |
H: Number of orbits of a subgroup of the symmetric group $S_9$
Let $H=\langle (3\quad 4\quad 5),(1\quad 2\quad3)(7\quad8\quad9)\rangle \le S_9$ be a subgroup in $S_9$. Find number of orbits and their order.
First I noticed $\mathrm{orb}(6)=\{6\}$. I also think that $\mathrm{orb}(3)=\{1,2,3,4,5\}$ but I can't explai... |
H: Fourier transform of $\exp(-t^2)$ using contour integration.
I am calculating the Fourier transform of $\exp(-t^2)$ using contour integration.
I am left with the integral $\int_{-\infty}^\infty \exp(i\omega t)\exp(-t^2)$.
Usually I would now use the residue theorem, but I cannot find the singularities.
Can someone ... |
H: Show that the topological space ( X, $\tau$ ) is not metrizable
For the topological space ( X, $\tau$ ), with X = {0, 1} and $\tau$ = { $\emptyset$ , {0}, {0,1} } , prove that ( X, $\tau$ ) is not metrizable.
I know intuitively it can't be but don't know how to prove it. The only idea I have is that {0, 1} cannot ... |
H: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n\in H$ for all $x\in G$?
Remarks. The answer is positive whenever $H$ is normal, e.g., for $n=2$. In general, by using the normal core of $H$, one can find an $m\ge 1$ such that $x^... |
H: Central Limit Theorem: asymptotic distribution
I keep reading in the documents about the asymptotic distribution of the CLT. I am learning things by myself so try to figure it out on my own and don't have a teacher to ask the question to.
So it's my understanding that the variance of the sample mean is:
$Var(\bar X... |
H: Elliptic Curve: Multiplying points over a finite field
Let $E$ be an elliptic curve over a finite field $\mathbb{F}_q$ where $q$ is prime. Let $P$ be a point on $E$.
Consider the point $Q=(q+1)P=P+\cdots+P$, which is $P$ added to itself $q+1$ times. Due to the fact that we are in the finite field $\mathbb{F}_q$, I... |
H: A linear operator $T: V \rightarrow V$ commuting with all linear operators is a scalar multiple of the identity.
Let $\mathbb{K}$ a field, $V$ a vector space over $\mathbb{K}$. If $T:V\to V$ commutes with all other linear operators $V \to V$, then there exists $\lambda \in \mathbb{K}$ such that $T= \lambda I$, wher... |
H: Arbitrary intersection of uncountable subfields of $\mathbb{C}$
What is about arbitrary intersection of uncountable nested subfields of $\mathbb{C}$?
Does it have to be uncountable too or it can be countable?
Edit: {\it Nested} subfields of $\mathbb{C}$ means that we are considering an infinite chain of field $F_1,... |
H: Extensions of continuous bounded functions
If $u:U \rightarrow \mathbb{R}^{n}$ is bounded and continuous can $u$ always be extended such that $u \in C(\bar{U})$? and is $u$ uniformly continuous?
AI: No, this is not always possible, let $n = 1$ and $U = (0,1) \subseteq \mathbb R$, define $u(x) = \sin(x^{-1})$. Then ... |
H: Upper bound of an integral on a circle segment
Say you have the integral:
$$\int_S f(z)dz = \int_S \frac{1}{x^{12}+1}dz$$
Where $S$ is a circle segment that runs from $R$ to $Re^{i\pi/6}$. An example question suggests it is possible to estimate this integral with:
$$\left| \int_S f(z)dz \right| \le \frac{\pi R}{6}... |
H: Algebra Homomorphism
This is a follow-up to a question I asked here yesterday. It's coming from a (non-examinable) exercise sheet and I really can't get my heard around how this question is posed and to be approached.
For clarity I'll re-type relevant parts of the initial post.
Suppose we have some field $K$ and n... |
H: Laplace Transform using t-shift
$$f(t)=\begin{cases}cos(πt), & 1\leq t < 4 \\ 0, &elsewhere \end{cases}$$
Okay, I attempted to write it in terms of step functions and I got
$$ f(t) = cos(πt)u(t-1)-cos(πt)u(t-4)$$
But Now I'm not sure how to get the $$cos(πt)$$ for both parts in terms of (t-1) and (t-4) so then I c... |
H: Why is $\mathbf{Rel} \cong \mathbf{Rel}^{\mathbb{op}}$?
I haven't yet fully grasped Category Theory so I am doing the exercises in Awodey's book for the first chapter, and exercise 2a) is confusing me very much.
The question is to prove or disprove that $\mathbf{Rel} \cong \mathbf{Rel}^{\mathbf{op}}$. I get that th... |
H: Finite group of two generators
My question is simple :
Any finite group of two generators is cyclic, semidirect sum, or direct sum ?
AI: No. The group $A_5$ is generated by $(123)$ and $(12345)$. It isn't cyclic and cannot be represented as a nontrivial semidirect product, let alone a direct product. |
H: What do $x\in[0,1]^n$ and $x\in\left\{ 0,1\right\}^n$ mean?
$x\in[0,1]^n$
$x\in\{0,1\}^n$
Thank you in advance.
AI: If $x=(x_1,\ldots,x_n)\in [0,1]^n$, then $0\leq x_i\leq 1$ for all $i$. If instead $x=(x_1,\ldots,x_n)\in \{0,1\}^n$, then $x_i=0$ or $x_i=1$ for all $i$.
For example, if $n=2$, then $[0,1]^2$ is th... |
H: Calculate the length of AC
The diameter $AB$ of the circle is $10\,\text{cm}$. The length of $BC$ is $6\,\text{cm}$. Calculate the length of $AC$.
I'm doing a mock exam and I'm not sure how to work out the length of $AC$. Any ideas? Thanks.
AI: The Pythagorean Theorem applies: the right angle is $\angle ACB$, by ... |
H: Nth Term of Fractions
How do I work out the Nth term of these fractions?
$$1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots$$
Would I need to change them all into decimals but that would be quite complicated and then go from there? Yes I can see a pattern, the denominators are squared numbers but what do I... |
H: What causes the change in the expected value of the product of random variables?
The following question is part of a homework exercise on portfolio theory that I have to do.
Suppose that $Y$ is a random variable representing the returns on an investment. Now, let $f$ be a continuous, concave, strictly increasing ... |
H: partitions that contain two singleton blocks one and n.
How many partitions [n] contain at least one of the singleton blocks 1 and n? I am having trouble doing this problem using the Sieve formula. Is it possible? and if so how do i go about it?
AI: I would use inclusion-exclusion on this one. Your answer is equal ... |
H: Prove that $n$ is prime $\iff \forall a\in\mathbb{Z}(\gcd(a,n)=1\lor n\mid a)$
I need to prove that given $n\in\mathbb{N}$ ($n>1$), $n$ is prime $\iff \forall a\in\mathbb{Z}(\gcd(a,n)=1\lor n\mid a)$.
I proved the first part, assuming that $n$ is prime and proving that for every $a$, $\gcd(a,n)=1$ or $n\mid a$, but... |
H: Question on the proof that $C(\Omega)$ is a Frechet space
I am using Rudin's book on Functional Analysis. I am studying the proof that the space $C(\Omega)$ of continuous functions on an open set $\Omega \subseteq \mathbb{C}$ is a Frechet space. I encountered a detail that I can't understand and I hope you guys her... |
H: What is the best way to solve this high school exercise?
Can you share with me how would you best solve this exersise to a high school student?
Show that $f(x)=x^2-6x+2$ , $x\in(-\infty,3]$ is $1-1$ and find its inverse.
AI: Hint: To prove the function is 1-1 use the definition
$$ f(x_1)=f(x_2) \implies x_1 = x_2 ... |
H: Tangent vectors as curves equivalence relation
I do not understand the definition of the equivalence relation that is defined on the curves creating a tangent vector space.
Let $X$ be any manifold, a point $x \in X$, two curves $\alpha:(-a,a) \to X, \beta:(-b,b) \to X$. Then $\alpha$ is equivalent to $\beta$ at $x$... |
H: free subgroups of $SL(2,\mathbb{R})$
In the example section of the wikipedia article on the the Ping Pong lemma, you can see how to construct a free subgroup of $SL(2,\mathbb{R})$ with two generators
$$ a_1 =
\begin{pmatrix}
1 & 2 \\
0 & 1
\end{pmatrix},
\ \ \ \ \ a_2 =
\begin{pmatrix}
1 & 0 \\
2 & 1
\end{pmatri... |
H: What are the chances of winning with a specific card in Spades
In the game of spades, a standard deck is shuffled then all the cards are dealt in a clockwise manner until each of the 4 players has 13 cards. The first play of the game is for each player to throw their lowest club (clubs are ordered from low to high:... |
H: Let $G=\langle x\rangle$ be cyclic of order $n$. Prove that $\langle x^r\rangle⊆\langle x^s\rangle$ iff r is a multiple of $s$ modulo $n$.
Let $G=\langle x\rangle$ be cyclic of order $n.$ Prove that $\langle x^r\rangle\subseteq \langle x^s\rangle \iff r$ is a multiple of $s$ modulo $n.$
I know you have to approach... |
H: Prove, square of quadrilateral is the sum of squares of 4 triangles
Let $A_1$, $B_1$, $C_1$, and $D_1$ - midpoints of the sides $AB$, $BC$, $CD$ and $DA$ convex quadrilateral $AВСD$. Directs $AC_1$, $ВD_1$, $CA_1$ and $DВ_1$ - divide it by $5$ quadrilaterals and $4$ triangles. Prove that the area of the central q... |
H: Denoting the set of initial segments of a binary sequence
The index is an infinite, innumerable binary sequence in $\{0,1\}$. $ I= \{f \mid f: \Bbb {N} \to \{0,1\}\} $
Is there a way to get a set $X_i$ from the infinite index number $10110\ldots$
$ X_i = \{1,10,101,1011,10110,\ldots\}$ where $i \in I$ and $i=10110\... |
H: p.d.f of function of random variable
Suppose $X$ is a continuous random variable with p.d.f:
$$
f_X(x) = \begin{cases}1 & x \in [0, \frac{1}{2}) \\ 1 & x \in [1, \frac{3}{2}) \\ 0 & \text{otherwise} \end{cases}
$$
What is the p.d.f of $Y = {(X - 1)}^2$?
Let's plot $g(X) = {(X - 1)}^2$ over the interval $0 \leq x \l... |
H: term for a "squared simplex"
The set of points
$$\{(x_0,...,x_n)|\forall{i}: x_i \in [0,1], \ and \ x_0+..+x_n=1\}$$
is an n-simplex.
What can I call a set of points:
$$\{(x_0,...,x_n,y_0,...,y_n)|\forall{i}: x_i,y_i \in [0,1], \ and \ x_0+...+x_n = 1 \ and \ y_0+...+y_n=1\}$$ ?
for $n=1$, a 1-simplex is a line se... |
H: Solid Angle Calculation - Understanding a formula
I'm currently reading a paper and try to understand this one formula. The problem is: In an n dimensional space. A cone with half-angle $\theta$ is given (the top of the cone is in the origin). We are interested in the solid angle of the cap cut out by this cone of ... |
H: Given a non-abelian group $G$ with $|G|=p^3$ and $p$ prime, how do I show that $|Z(G)|=p$?
Given a non-abelian group $G$ with $|G|=p^3$ and $p$ prime, how do I show that $|Z(G)|=p$? $Z(G)$ is as always center of $G$.
It is easy to see that $|Z(G)|\in\{p, p^2\}$, but how do I exclude $p^2$. Thanks in advance!
AI: Hi... |
H: Showing that function are equal almost everywhere in Sobolev Spaces
Consider the Holder space $C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})$ and the Sobolev Space $W^{1,p}(\mathbb{R}^{n})$. Take $u_{m} \in C_{c}^{\infty}(\mathbb{R}^{n})$ such that Morrey's Inequality we have $||u_{m}||_{C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})} ... |
H: Negating the Definition of a Convergent Sequence to Find the Definition of a Divergent Sequence
My task is to write a precise mathematical statement that "the sequence $(a_n)$ does not converge to a number $\mathscr l$"
So, I have my definition of a convergent sequence:
"$\forall\varepsilon>0$ $\exists N\in\Bbb R$ ... |
H: which one of the following probability mass function can define a probability distribution?
a) $f(x)=(5-x^2)/6$ for $x=0,1,2,3$
b) $f(x)=x/15$ for $x=1,2,3,4,5$
c) $f(x)=1/2^x$ for $x=0,1,2,3,4$
d) $f(x)=1/4$ for $x=2,3,4,5,6$
Can you please suggest me how to solve these questions ?
AI: Hint: The probabil... |
H: Convert output of atan to range -1, 1,-1
I'm trying to convert the output of an arctan function from a range of -PI,0,PI to -1,1,-1 - in fact I've succeeded! But it's not very elegant:
angle = Mathf.Atan2(y, x);
angle = Mathf.Abs(angle);
angle /= Mathf.PI;
angle = angle - 1f;
angle = Mathf.Abs(angle);
angle *... |
H: Show that $\sqrt{n+1}-\sqrt{n}\to0$
Let $\ a_n=\sqrt{n+1}-\sqrt{n}$. I have to show that $\lim_{n\to \infty}a_{n}=0$.
How should I start? Do I have to use any theorem?
AI: Use the fact that
$$\sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}}.$$ |
H: Help with Elementary Vector subspace proof
I am having trouble getting started with a proof of the following statement (I translated it from German, so bear with me):
"$V$ is a vector space and $U$ and $W$ are vector subspaces of $V$. Prove that the following two are equivalent:
a) $U \cap W = \{0\} $
b) Every $v \... |
H: Covering an area equally with layers of non-tesselating polygons
A series of hexagons on an hexagonal lattice means that the every point in the entire area is covered by one polygon only.
A grid of octagons will not tesselate, leaving square holes such that 4/18 of the area is not covered.
Is it possible to stack m... |
H: evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n}$
I am trying to compute the sum $$\sum_{n=1}^\infty \frac{n^2}{3^n}.$$
I would prefer a nice method without differentiation, but if differentiation makes it easier, then that's fine.
Can anyone help me?
Thanks.
AI: Here's a bit of a slick trick. Let's put $$S=\sum_{n=1}... |
H: Approximating the modulus of a Complex Function near a point.
Let $\Omega$ be a domain in $\mathbb{C}$, and let $z_0 \in \Omega$.
Let $f$ be analytic on $\Omega$.
Let $z=z_0+re^{i\theta}$ for $r$ small.
Assume that $f(z_0) \neq 0$ and $f'(z_0) \neq 0$.
I want to show $|f(z)| = |f(z_0)|\big(1+\lambda r \cos(\theta+\... |
H: $G=\{(x,y)\in\Bbb R^2 | 1\leq x^2 + y^2\leq16 \text{ and } 0\leq x \text{ and }0\leq y \}$ Calculate $\underset{G}{\int\int}xy^2dxdy $
I'm making some exercises for my analysis exam, and i'm having trouble with this exercise.
$G=\{(x,y)\in\Bbb R^2 | 1\leq x^2 + y^2\leq16 \text{ and } 0\leq x \text{ and }0\leq y ... |
H: How to solve $\lim_{x\to-\infty}x^2e^x$
How to solve the following limit?
\begin{eqnarray}
\\\lim_{x\to-\infty}x^2e^x\\
\end{eqnarray}
According to the website WolframAlpha, L'H rule can be used here but it is $-\infty/1$ instead of $-\infty/-\infty$, $\infty/\infty$ or $0/0$. So how to solve this question?
Than... |
H: What is the remainder when $2^{1990}$ is divided by $1990$?
I actually do not have the basic idea on how to approach these type of questions....so please tell me a generalized method about all this too.
It came in RMO, and the question is:
What is the remainder when $2^{1990}$ is divided by $1990$ ?
AI: Using Ferma... |
H: Is $\lambda(n) + \max\limits_{p\mid n} v_p(n)\leqslant n$?
Given an integer $n = \prod\limits_{n\mid p}p^{v_p(n)}$, is
$$\lambda(n) + \max\limits_{p\mid n} v_p(n)\leqslant n$$
where $\lambda(n)$ is the Carmichael function?
AI: For a single-prime factor, i.e. $n=p^v$, this is true, since
$$\begin{align}
\lambda(p^... |
H: Why am I getting half the correct answer by using Green's Theorem?
I had this homework problem that asked me to use Green's Theorem to solve it, so I did. Unfortunately, my answer was wrong. I looked for an error in my reasoning, but did not find it. I eventually solved by way of the line integral, which is usually... |
H: Prove that $f$ is constant
Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be an entire function. if there exists $\delta> 0$ and $w\in \mathbb{C}$ such that
$$\left | f(z)-w \right | \geq \delta \qquad \forall z\in\mathbb C $$
Prove that $f$ is constant.
AI: Let $g(z) = \frac{1}{f(z)-w}$, then $g$ is entire and $|g(z)| ... |
H: Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous
$$d_\infty = \max|x_i - y_i|$$
$$d_1 = \sum_{i=1}^n |x_i - y_i|$$
The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, which I did using $\epsilon = ... |
H: Bound for analytic function on a disk given values at 0
I am trying to prove the following problem from An Introduction to Complex Function Theory by Palka.
If $f$ is analytic in the unit disk $D(0,1)$, and if $f(0)=f'(0)=\ldots= f^{(k)}(0)=0$, and $|f^{(k)}(z)|\leq 1$ for every $z\in D(0,1)$, then show that $|f(z)... |
H: For sets $A,B,C$, $(A\setminus B)\subset (A\setminus C)\cup (C\setminus B)$
First of all, I am sorry for my bad english, I am from Brazil :-) I have problem with proof for some set theory task.
Here it is:
$A,B,C$ are three sets. Show that:
$$(A\setminus B) \subset (A\setminus C) \cup (C\setminus B)$$
It is clear... |
H: Proving that $\frac{1}{2} \le \frac{1}{2^n+1} + \frac{1}{2^n+2} + ... + \frac{1}{2^n + 2^n}$
I'm having trouble proving that:
$$\frac{1}{2} \le \frac{1}{2^n+1} + \frac{1}{2^n+2} + ... + \frac{1}{2^n + 2^n}$$
Edit: The next step is actually a mistake. I've put up a comment to the accepted answer explaining why it's ... |
H: I am trying to prove that for all $x \ge 1, (x^{\frac{1}{n}}) \to 1$.
$\textbf{Proof:}$
Let $x \ge 1$ be arbitrary and note that $\forall n \in \mathbb{N}$, $x^{\frac{1}{n}} > 1$.
So $ \forall n \in \mathbb{N},\ x^{\frac{1}{n}} - 1 \ge 0 > -1$.
Therefore I can write $x^{\frac{1}{n}} = 1 + ( x^{\frac{1}{n}} -1 )... |
H: How to find the exact value of an upper bound for an exponential random variable
Suppose the waiting time between people entering a store can be modeled by the exponential random variable $X$ with parameter $\lambda=5$. If you use markov's inequality you can find the $P(X\ge 20)$ is $.25$. How would I find the exa... |
H: Finding the limit of $\frac{e^{x}+x-\cos(2x)}{x^2}$
How would one find the limit for the following problem.
$x\rightarrow\infty$
$\frac{e^{x}+x-\cos(2x)}{x^2}$
I did the hospital rule.
$\frac{e^x+1+2\sin(2x)}{2x}$
But now I am stuck I did this but I feel it diverges.
$e^x+1+2\sin(2x)*\frac{1}{2x}$
AI: $$\lim_{x\to\... |
H: Find the domain of $g(x) = \ln \left( {\frac{x}{{x - 1}}} \right)$
I know the argument of the function has to be greater than 0, so:
$\eqalign{
& \left( {\frac{x}{{x - 1}}} \right) > 0 \cr
& x > 0 \cr} $
however in this case $x \ne 1$, $x \ne 0$ as they result in an answer which is undefined, so I think it's ... |
H: I've seen "hyperbolic rotation" - from this: generalization to multisection rotation: is this possible?
This question is more in recreational mathematics area
By accident I came across the concept of "hyperbolic rotation" where we use a matrix containing $\cosh$ and $\sinh$ instead of the trigonometric $\cos$ and ... |
H: $C[0,1]$ endowed with integral norms
Consider the following norms on $C[0,1]$: for each $p\in [1,+\infty]$ $||x||_p=(\int_0^1|x(t)|^pdt)^{1/p}$ (this $p$-norm is induced from $L^p$ wich contains $C[0,1]$). Are they equivalent? Of course, $||x||_p$ is not equivalent to $||x||_{\infty}$ (it is easy to build a counter... |
H: How could you express the following double integral in term of a single integral?
How could I express the $$\int\int e^{(x^2+y^2)^2} dA$$ in terms of a single integral with respect to r where D is a disk with center (0,0) and radius 1
AI: Change to polar coordinates:
$$x=r\cos t\;,\;\;y=r\sin t\;,\;\;0\le t\le 2\pi... |
H: Find the $P{X < Y}$ from this joint density function $f(x, y) = 6x^2y$
I have a joint density function given as:
$$f(x, y) = 6x^2y \\
0\le x\le 1 \\
0\le y\le 1$$
Now I am asked to find the $P\{X < Y\}$, as well as the $P\{X < 2Y\}$
In order to solve this, I did the following integrals:
\begin{align}
P\{X < Y\} &=... |
H: Does the interval (a, a) contain any real numbers, assuming the interval is on the real line?
The title pretty much describes my question. I apologize if the language is not as mathematically precise as it could be. Many thanks in advance for your thoughts.
AI: Hardly. The interval $(a, a)$ is empty; it doesn't e... |
H: Inverse of Natural Projection?
May this be a silly question, but can I construct an inverse of a natural projection $p$ from a module $M$ to its quotient module $M/A$? Of course more than one element can be assigned for each coset in $M/A$, but if we limit the inverse's range to direct complement of $A$, I think su... |
H: How to find the number of positive integral solutions for the equations $\frac1x+\frac1y=\frac1{n!}$?
I was trying to solve a question over hackerrank and the question link is EQUATION
How to approach for it?
AI: Hint :
$$ \frac{1}{x} + \frac{1}{y} = \frac{1}{n!} \Rightarrow x = \frac{n!y}{y-n!}$$
but $$ x \in \B... |
H: Finding the limit of $(1+2x)^{3\csc(2x)}$
Find the limit $\displaystyle{\lim_{x\to0}(1+2x)^{3\csc(2x)}}$
I did the following $(1+2x)^{3\csc(2x)}=e^{\ln(1+2x)3\csc(2x)}$, took $\ln$ on the lim getting
$3\frac{\ln(1+2x)}{\frac{1}{\csc(2x)}}$
I did hospital rule
and got
$\lim3\cdot2\frac{\frac{1}{2x}}{\frac{-1}{\csc... |
H: Finding the limit of $( 1 + a + a^2 + \ldots+ a^n)/(1 + b + b^2 + \ldots+ b^n)$
I have some problems finding the limit of $$\frac{ 1 + a + a^2 +\cdots + a^n}{1 + b + b^2 + \cdots + b^n}.$$
$0\le a,b \le +∞$
Here is what I got :
Forcefuly factorize $a^n$ and $b^n$ :
$$ \frac{a^n ( 1 + \frac1{a} + \frac1{a^2} +... |
H: If every element of $R$ is irreducible or a unit, then $R$ is a field.
This is not for homework, but I seem to be stuck a would like a hint please. The question asks
If every nonzero element of an integral domain $R$ is either a unit or irreducible, then $R$ is a field.
The question looks non-threatening, and I'... |
H: A problem related to Rouche's Theorem.
In Rouche's theorem, If we replace analytic property of functions $f(z)$ and $h(z)$ with meromorphic then this theorem will not be valid anymore.
I want to illustrate this fact by producing some $f(z)$ and $h(z)$ which are meromorphic on some bounded domain D (where D have pi... |
H: Area below and above a curve compensate
Consider the graph below:
Don't worry about the units in the axes, it might look like a physics question, but my question is purely mathematical. Basically, the work done is the area under this force/displacement graph. For $x_0 = 10$, this work done is $100J$, which in our ... |
H: How to prove that $\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n)$
I know that $$\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n),$$ but I cannot find a way how to prove it. I tried induction but it did not work. On wiki they say that I should use differentiation but I do not know how to apply it to binomial coefficient.
Than... |
H: Expectation of throwing $n$ balls into $n$ bins
Suppose we throw $n$ indistinguishable balls in $n$ bins at random. The throws are independent. What is the expected number of empty bins? What is the expected number of bins with one ball.
Using indicator random variables, expectations, some sloppy math and some qu... |
H: Proof about pointwise $\lim\limits_{x\to c} f(x)g(x)$
Let $f$ be bounded and let
$$\lim_{x\to c} g(x) = L$$
Prove:
(a) If $L=0$ then $\lim\limits_{x\to c} f(x) g(x) = 0$
(b) if $0<L<\infty$ then $\lim\limits_{x\to c} f(x) g(x)$ exists.
(c) $L=\infty$ then then $\lim\limits_{x\to c} f(x) g(x) = \infty$.
... |
H: True or False: Continuous Functions (Extreme Value Theorem)
Do my justifications seem appropriate?
a.) Every function $f:[0,1]\rightarrow \mathbb{R}$ has a maximum.
True; if $f:[0,1]\rightarrow \mathbb{R}$, $f$ wil be closed and bounded above, so it will have a maximum.
b.) Every continuous function $f:[a,b]\righ... |
H: an example of a simple ring which is not a division ring
I was thinking about a simple ring and I found a question that every simple ring is a division ring. I think this is not a correct theorem and I want to find an example.
AI: A matrix ring over a field $M_n(F)$ is simple but not a division ring for $n>1$.
I th... |
H: $S_n$ and its subgroups
Show that $A_n$ is unique in $S_n$ with index $2$.
I'm trying to use Quotient Group and Lagrange's Theorem to approach this problem but I'm still clueless. Can anyone tell me how to do this problem? Thanks.
AI: As I comment above, for n=1,2,3,4, this can be done explicitly. Assume n $\geq 5$... |
H: define the reals in a non-archimedean elementary extension of the real field.
Can it be done?
We have the real field $(\Bbb R,+,-,\times,0,1,<)$, of course $(0,1,-,<)$ are definable using the rest.
We take an elementary non-archimedean extension. Can we define the original set of reals in it?
AI: No, you cannot: Yo... |
H: Center is never a maximal proper subgroup
Prove that center is not a maximal proper subgroup of group $G$.
AI: Hint: We can assume that $Z(G)\neq G$. If $x\in G\setminus Z(G)$ what can you say about $C_G(x)$? |
H: Injective function: example of injective function that is not surjective.
Does there exist an injective function that is not surjective? Could I have an example, please?
AI: There are many examples. It just all depends on how your define the range and domain.
For example $\operatorname{f} : \mathbb{R} \to \mathbb{R... |
H: Does $A \Delta N = A \cup N'$ in this proof?
Please have a look at this topic:
$\sigma$- ideal
The answer says:
$A\Delta N=A\cup N'$ with $N'=N\setminus A$.
I do not see why this is correct..
Usually it is $A\Delta N=(A\setminus N)\cup (N\setminus A)$.
Unfortunately the helper does not answer and I really need a re... |
H: $\int_0^{2\pi} \sqrt{1-\cos(x)}\,dx = 4\sqrt{2}$. Why?
According to the textbook, and Wolfram Alpha the above is correct.
Here is the step by step procedure from Wolfram Alpha for evaluating the indefinite integral:
Take the integral: $$\int\sqrt{1-\cos(x)}\,dx$$ For the integrand $\sqrt{1-\cos(x)}$, substitute $u... |
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