text stringlengths 83 79.5k |
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H: Easiest way to find a vector in a span
So $V=\text{span}\{v_1, v_2\}$ where $v_1 = (1, 1, -1)$ and $v_2 = (1, -1, 2)$
I have been given four different vectors and must find which vectors are in $V$. What is the most efficient way to do this? The vectors I have been given are as follows:
$x_1 = (1,0,-1)$
$x_2 = (0,... |
H: Sylow's Theorem Application. Prove $G$ is Abelian.
Assume that $|G|=5^27^2$. Determine the possibilities for $n_5, n_7$ and determine what can be concluded in each case about the $5$-Sylow subgroups and the $7$-Sylow subgroups and prove that $G$ is Abelian.
Proof: Assume that |G|=$5^27^2$. By Sylow's Theorem,
The... |
H: How to work out a double summation without manually finding each answer?
I have the double summation
$$
\sum_{p = 1}^{2} \sum_{q = p}^{3} {(p-1)q}
$$
I know how to work this out if I was to go through and add up every term. My question is, how would I do this in a shorter way, so if the 2 and the 3 were 200 and 300... |
H: Why is the order of $X_{2n}$ is at most $6$ where $X_{2n}=\langle x,y\mid x^{n}=y^{2}=1,xy=yx^{2}\rangle$?
I am reading the book Abstract Algebra by Dummit and Foote.
I am at the begining of the book, and I got to a section about generators
and relations.
The book gives the definition
$$
X_{2n}:=\langle x,y\mid x^... |
H: Finding derivative at a point in a set
If I have a few values for f(x), i.e. {(0,1), (2, 3), (5, 6)}, is there a way to calculate the derivative at, say f(6), without interpolation?
AI: No. You cannot. Given any finite set of coordinates, there is a continuous, nowhere-differentiable function with those points on i... |
H: Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?
Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?
I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.
Thank you.
AI: When $m$ and $n$ are intege... |
H: basic combinatorics(permutations) question
How many ways are there to seat six different boys and six different girls along one side of a long table with $12$ seats? How many are ways if boys and girls alternate sits?
MY try:
For first question, we start with boys, We know there are $P(12,6) = \frac{12!}{6!}$ ways... |
H: Simple algorithm to get the square of an integer using only addition?
This problem was mentioned in passing in a reading and it piqued my curiosity.
I'm not sure where to start. Any pointers? (perhaps square root was meant?)
AI: An algorithm to do this for integers would be (in pseudo-code):
input a
b = abs a
s ... |
H: Representation of an adjoint operator
Let $T \in L(H)$ be a positive self adjoint operator on a Hilbert space $H$ such that $\operatorname{rank} T = 1$. I already know that there exist $y, z \in H$ such that $Tx = \langle x , y \rangle z$ for all $x \in H$. Now I need to show that there exists a $z_1 \in H$ such th... |
H: Cramers rule and inverse in complex numbers.
Ok so my teacher defined the complex number system by saing it is RxR and defined $(a,b)*(c,d)=ac-bd,ad+bc)$
the multiplicative identity is (1,0) and he asked us to find the multiplicative inverse of (a,b) denoted $(a,b)^{-1}$. He told me to use Cramer. My approach was t... |
H: Prove that $f'(a) = \frac{1}{2\pi}\int_0^{2\pi}e^{-i\theta}f(a+e^{i\theta})d\theta$
I know this is to be derived from Gauss' Mean Value Theorem, but I can't get the $e^{-i\theta}$. Where am I going wrong?
$f'(a) = \lim_{h \to 0}\frac{f(a+h) - f(a)}{h} = \lim_{h\to 0}\frac{1}{2pi}\int_0^{2\pi} \frac{f(a+e^{i\theta}+... |
H: Uniform convergence and cauchy sequence
if I have a sequence $(x_n) \subset (C[0,1],||.||_{\text{max}})$ such that
$sup_{s \neq t} \frac{|(x_n(t)-x_m(t)) - (x_n(s)-x_m(s))|}{|s - t |},s,t \in [0,1] $ is convergent to zero. And we have that $||x_n - x ||_{max} \rightarrow 0$.
Can we somehow show then that $sup_{s \n... |
H: Trouble finding an infinite series for these iterations...
I have the following iteration, which first I want to get a general form for "$p$" terms, and then I want to put it into a sum.
$$E_1 = \frac{e}{a^2} - \frac{e}{(2a)^2} + \frac{e}{(3a)^2} - ... \frac{e}{(pa)^2}$$
$$E_2 = \frac{e}{a^2} - \frac{e}{(a)^2} + \... |
H: Reduction of Order: $t^{2}y''+3ty'+y=0$, $\quad t>0$; $\quad y_{1}(t)=t^{-1}$
I am working an exercise from Elementary Differential Equations and Boundary Value Problems Ninth Edition by Boyce and Diprima, and I think there is mistake\typo. On page 173 Section 3.4 exercise 25.
The book is correct I dropped the mi... |
H: Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
Let $X$ and $Y$ be independent random variables. Prove that $f(X)$ and $g(Y)$ are independent for any choice of measurable functions $f$ and $g$.
This sounds very obvious, but I have no idea how to approach it.
EDIT: Two random variables ... |
H: Showing one set is a subset of another
Let's say you have sets $A,\, B,$ and $C.$
How would you show that $[(A-B) - C]\subseteq (A-C)$ using a venn diagram or logical translations?
How can this even be done when you don't know the members of $A,\, B,$ or $C$?
AI: You'd need to use logical translations, implicitly (... |
H: Axiom of countable choice
I apologize in advance for my neophytic question. Let $(A_n)$ be a countable family of disjoint sets. Why is it not possible define a sequence $(x_n)$ with $x_n\in A_n$ using recursion? It seems clear to me that this should be possible, but I am clearly making a ridiculous mistake.
Thanks ... |
H: Index relation between two primitive roots
Let n be a positive integer, and x an integer such that gcd(n, x)=1. Suppose g and h are primitive roots mod n. Show that:
$ind_{h}(x) = ind_{h}(g) \cdot ind_{g}(x) (mod {\phi}(n))$
I've been staring at it for about an hour now. I know I can rewrite $ind_{h}(x)$ as $h^{ind... |
H: Hint for real analysis question
Suppose that $f$ is one-to-one and continuous on [$a,b$]. Prove that $f$ is either strictly increasing or strictly decreasing on [$a,b$].
AI: Hint Use the intermediate-value theorem to get a contradiction if you assume
$$\exists x,y \in [a,b] : f'(x) < 0 \wedge f'(y) > 0$$
Use Weiers... |
H: Determining consistency for every $b$ in $\mathbb{R}^m$
I have the matrix
$$\left[\begin{array}{cc|c}
1 & 0 & -3 \\[0.55ex]
-1 & 0 & 3
\end{array}\right]$$
Basically I can see that adding row 1 to the other row will result in an all zero row. What is wrong with this? According to my notes I just... |
H: A question about similar matrices: $Id$ and $-Id$
Currently, I'm trying to understand the idea of matrix similarity. As a toy example, I am thinking about $Id$ and $-Id$. Now, I do not think that these matrices are similar, and here is my proposed reasoning:
Assume that there exists an invertible matrix $P$ such ... |
H: Why can't I make this claim about my confidence interval?
Say out of a sample of 200 penguins, I find that 192 of them like chocolate. From this sample data, I create the $95\%$ confidence interval for the proportion of penguins that like chocolate:
$$
\left(\frac{192}{200} - 1.96 \times \sqrt{\frac{(\frac{192}{200... |
H: Calculate the rate of emission
A source usually emits particles at a rate of 60 per minute. But how many particles would
be emitted in one hour if (a) the usual rate of emission was increased by a factor of 60?
(b) the usual rate of emission was increased by 25%? (c) the usual rate of emission was
decreased by 7... |
H: Using linear algebra to solve algebra
I feel like in my class all we do is memorize rules, lists of rules, terminology and more lists of rules. I have absolutely no clue how to apply any of these concepts. I have the matrix (which is representing a linear set of equations I think):
$$\left[\begin{array}{ccc|c}
... |
H: Question about the matrix representation of the differentiation map on the subspace generated by $\{1, t, e^{t}, e^{2t}\}$
As mentioned in a previous post (I think), I've been trying to learn some linear algebra, and so I've begun to post little questions whose answers I'm sure are obvious to most here; this is jus... |
H: A linear transformation satisfying $P^2 = P$
Prove that a linear transformation $P \colon V \to V$ of a finite dimensional vector space satisfies $P^2 = P$ if and only if there exists a basis with respect to which $P$ can be written as a block matrix
$$P = \begin{bmatrix} I & 0 \\ 0 & 0 \end{bmatrix}.$$
Hence dete... |
H: A metric space such that all closed balls are compact is complete.
I am trying to solve the following exercise:
Let $(X,d)$ be a metric space that has the property that for any $x\in X$ and $r>0$, the closed ball
$$\bar{B}(x,r):=\{y\in X:d(x,y)\leq r\}$$
is compact. Show that $X$ is complete.
I think I have a... |
H: Any Set with Associativity, Left Identity, Left Inverse is a Group
Related Link: Right identity and Right inverse implies a group
Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra
I will present my proof (distinct from those in the link) for critique and then ask my question. $G$ is... |
H: Prove a Field is a commutative ring with identity
I was just introduced to rings and fields recently and am in need with help for the following proof.
Prove: A field $F$ is a commutative ring with identity and with at least two elements, such that for all $a \ne 0$ and $b \in F$, the equation $ax=b$ has a unique s... |
H: Integration by parts in $ \int_a^b t^3\sqrt{1+2t^2}dt $
So I have an integral:
$$
\int_a^b t^3\sqrt{1+2t^2} dt
$$
My first instinct here is to integrate by parts.
So I choose:
$$
u= \sqrt{1+2t^2}
$$
$$
v'= t^3
$$
so
$$
u' = \frac{2t}{\sqrt{1+2t^2}}
$$
$$
v = \frac{t^4}{4}
$$
using the form:
$$
\int_a^b f(x)g'(x)dx... |
H: Is the differential equation $y'=x+y$ separable?
I'm at a loss. My guess is no, but I'm new to doing these problems. It can not be solved with cross multiplication but there are other ways of solving these problems I'm sure. Thank you for any help.
AI: No. There is no way to manipulate this ODE so that the method o... |
H: Proving that f'(x) is even if f(x) is odd and differentiable
I've seen some proofs but I don't really get it..I find it hard to understand..
I've done this so far:
\begin{eqnarray} f'(x) &=& \lim_{h \to 0} \frac{f(x) - f(x+h)}{h} \textit{ (since f(x) is odd } f(-x) = -f(x)) \\ &=& \lim_{h \to 0}
\frac{-f(-x) + f(... |
H: How can I show that a map is an inner product?
Question: Prove that the map
$$\operatorname{Mat}_{nxn}(\mathbb{F}) \times \operatorname{Mat}_{nxn}(\mathbb{\mathbb{F}})\to \mathbb{F}, \quad (A,B) \mapsto tr(B^{t}A)$$
is an inner product on $\operatorname{Mat}_{nxn}(\mathbb{F}).$
I know that we have to show that the... |
H: Real Analysis homework hint
a) Let $f: [a,b] \rightarrow [a,b]$ be continuous. Prove that there exists a $c\in[a,b]$ such that $f(c)=c$.
b) Does this result still hold for continuous function $f: [0,\infty )\rightarrow[0,\infty)$? Either prove it or provide counter-example.
AI: For the first part, consider the func... |
H: What is inverse tangent?
I recently started thinking about what inverse tangent is. It is obvious that the definition of tangent is $\frac{\sin x}{\cos x}$, however, what is inverse tangent?
I first thought $\tan^{-1} x = \frac{\sin^{-1} x}{\cos^{-1} x}$, but it didn't seem right when I graph it out.
One interestin... |
H: How to solve the system $t\frac{dx}{dt}=-x+yt$, $t\frac{dy}{dt}=-2x+yt$?
Could you show me how to solve the following simultaneous differential equations? I tried substitution such that $u=xt$, yet I couldn't find the solution.
$$\frac{dx}{dt}t=-x+yt$$
$$\frac{dy}{dt}t=-2x+yt$$
AI: $(tD+1)x=ty$ and $(tD-t)y=-2x$ th... |
H: Does the set of rational numbers between 0 and 2 have the least upper bound property?
Let $A = \{ a \in Q : 0 < a < 2\}$
Does A have the least upper bound property?
Definition: $A$ has the least upper bound property if $\forall$ nonempty $B \subseteq A$, if $B$ has an upper bound in $A$, then it has a least upper b... |
H: What is the characteristic of $\mathbb{Z_4}\times\mathbb{Z_6}$?
I show the characteristic is the $lcm$ of $4$ and $6$.
Suppose we have a ring $\mathbb{Z_a}\times\mathbb{Z_b}$.
Let $(1, 1) \in \mathbb{Z_4}\times\mathbb{Z_6}$ (since it's an identity).
Then for some $k$, we have $k(1, 1) = (k, k) = 0$.
Thus $k = 0 \m... |
H: Question regarding proof of Fatou's lemma
I'm really confused with the step enclosed in red. Can someone please be kind enough to explain to me why does it follow the part in red?
AI: Suppose $X^+$ is the set of $x$ on which $\varphi(x)>0$ and $X^0$ is the set on which $\varphi(x)=0$. Then
$$\int_E\overline\varp... |
H: How many 3-subsets of $\{1,2,\ldots,10\}$ contain at least two consecutive integers?
Let A = {1, 2,..., 10}. How many three-element subsets of A contain at least two consecutive integers?
I believe there are $\displaystyle \tbinom{10}{3}$ total 3-subsets of A. To find the subsets containing at least two consecuti... |
H: How to calculate a series with binomial terms invovled
I'm studying probability and having trouble in understanding the following calculation
How to get from left to the right on the first line, with the condition that m could only be even numbers? Any hint would be appreciated, thanks~~
AI: We write the same thin... |
H: Show that $T$ is a subring of $R$.
Let $R$ be a multiplicative (commutative) ring with multiplicative identity. For $b \in R$ let $$T = bR = Rb = \{rb : r \in R\}$$ be the subset of $R$ consisting of multiples of $b$. Show that $T$ is a subring of $R$.
All I basically have to do is just say for any $c \in R$ we h... |
H: Extending a linearly independent subset into a basis
I have that $S = \{(1,2,1,0,0)\}$ is a linearly independent subset of $V$, where
$V= \{(x_1,x_2,x_3,x_4,x_5):x_1 - 2x_2 +3x_3 - x_4 + 2x_5 = 0\}$.
I now have to extend $S$ into a basis for $V$. Is there a better way rather than trying different sets?
AI: If you ... |
H: Limit of integral of sequences
Calculate
$$\lim_{n\to\infty}\int_{-\pi/4}^{\pi/4}\frac{n\cos(x)}{n^2x^2+1}\,dx$$
I don't know how to calculate the integral and the sequence is not monotone or dominated by a $L^1$ function, so I'm stucked. Any idea?
AI: How about integration by parts?
\begin{align*}
{\int\frac{n... |
H: Computing the sum of a Catalan sequence-- Random-walk motivated
How would one go about computing the following?:
$$\sum_{n=0}^\infty (.5)^{2n+1} \cdot \frac{{2n}\choose{n}}{n+1}$$
The motivation is that this gives the probability that a random walk on a number line will hit the point $0$ given that we move with pro... |
H: Proof of continuity using Epsilon-Delta
Let
$$f(x) = \left\{ \begin{array}{lr} x^2 \cos\left(\frac{1}{x}\right) &: x \ne 0 \\ 0 &: x = 0 \end{array}\right.$$
Prove that $f(x)$ is continuous for all $x \in \mathbb{R}$.
I can use a theorem which states: let $I, J \subseteq \mathbb{R}$ be intervals. Let $f:I \t... |
H: If $A(A(x)) = x$ for all $x$, then $A$ is one to one and onto
Suppose that $A$ is a mapping from a set $S$ to itself and $A(A(x)) = x$ for all $x \in S.$ Prove that $A$ is one-to-one and onto.
Can someone please break this down, define one to one, and onto? I am new to this terminology.
AI: A function $f$ is cal... |
H: Conditional pdf
$f(x) = 2(1-x)$ for $0 < x < 1$
Given that $X$ exceeds $0.5$ what is the probability that $X$ is less than $0.75$?
How do I go on about thia problem?
I can calculate the probability of exceeding $0.5$ by just taking the integral from $0.5$ to $1$ but I don't know where to go from there
AI: Let $A$ ... |
H: A subset of $SL(2,\mathbb{C})$
Let $\mathcal{H}$ be the real vector space of $2 \times 2$ complex Hermitian matrices. Set
$$\mathcal{K} := \{A \in SL(2,\mathbb{C}) : \forall H \in \mathcal{H}, \space A^{-1}H = HA^* \}.$$
Here $A^*$ denotes the conjugate transpose of $A$. I'm trying to determine which matrices are e... |
H: Derivative of function raised to a power, using the chain rule
How do I find the derivative of the function $f(x)= (2x+1)^2$? I've tried doing this problem and am not fully sure that I am correct. I found the derivative to be $f'(x) = 8x+4$. Is that correct?
AI: Yes, the derivative is correct.
To insert the detai... |
H: basic differential question
I need guidance on this problem. Could someone lead me in a direction of how I should go about doing this question. Is there some sort of proof involved in this question? No need to solve the question just some guidance would be good.
show that $f′(−x)=f′(x)$ for all x
EDIT:
f is a diffe... |
H: Find critical numbers of $g(t)= t(4-t)^{1/2}$ where $t<3$
The question is to find any critical numbers of the function, which is $$g(t)= t(4-t)^{1/2} , \qquad t<3$$
I know that in order to find the critical numbers, I first have to find the derivative and then set that equal to zero. I'm not sure I'm using the c... |
H: Marginal density of bivariate density that is a circle
I have the following density function:
$f(x,y) = \frac{2}{\pi}$ for $x^2 + y^2 \leq 1$ and $y > x$.
I figured out that this represents half of the unit circle (the upper half when cut along the line $y=x$). I would like to find the marginal density of Y. To do ... |
H: Monotone convergence theorem to evaluate improper integral
My book says that the equality $$\int_{(0,1]}x^{-3/4}d\mu=\lim_{t\rightarrow 0^+}\int_{[t,1]}x^{-3/4}d\mu$$ follows from the monotone convergence theorem. Why is it so? I can't see how to apply monotone convergence theorem here.
AI: Hint: For each $k\in\mat... |
H: Find the natural numbers $a$ and $b$ so that $a\cdot b$ has the largest possible value but $a + b = x$ must hold.
Is there a way to find the natural numbers $a$ and $b$ so that $a\cdot b$ has the largest possible value but $a + b = x$ must hold. It's easy small numbers but is there any way, through calculus or othe... |
H: Probability of a 5 card hand from a standard 52 deck containing all 4 suits
The answer to this is $\dfrac{4 {13 \choose 2} {13 \choose 1} {13 \choose 1} {13 \choose 1}}{ {52 \choose 5}}$,
but what I'm trying to figure out is why $\dfrac{{13\choose 1}{13\choose 1}{13\choose 1}{13\choose 1}{48 \choose 1}}{{52\choose... |
H: Riemann sum of $e^x$
I understand the how to sum the area under $e^x$ from, say, $[0,1]$ — but how do you sum from $[-1,1]$?
AI: For the left Riemann sums, evaluate $e^x$ at $x=-1+\frac{2k}{n}$, for $k=0$ to $n-1$.
The same method that you used for $[0,1]$ then works, for we can take the $e^{-1}$ "out."
Added: If ... |
H: How to differentiate this function?
Hey I need help differentiating this specific function. I've attempted for quite some time now and my answer seems off. Can someone please differentiate this so I can compare my answer to yours? Thanks in advance
$${y}=\frac{870}{q}+3500 \cdot \frac{e^{(3 q+4 )/820}}{q}.$$
AI: Th... |
H: Find parametric equations
Find parametric equations for a particle moving two full revolutions clockwise around a circle of radius $2$ centered at $(3,-1)$. In other words give equations for $x(t)$ and $y(t)$, and specify the time interval.
Is $x=3+2cost$
and $\ \ $$y=-1+2sint$
correct?
I only have several exampl... |
H: Derivative: chain rule applied to $\cos(\pi x)$
What is the derivative of the function $f(x)= \cos(\pi x)$?
I found the derivative to be $f^{\prime}(x)= -\pi\sin(\pi x)$. Am I correct? Can you show me how to find the answer step by step?
This is a homework question:
What is $x$ equal to if $-\pi\sin(\pi x)=0$?... |
H: Prove that there exists only 2 solutions for $x^2 \equiv 9 \pmod {p^k}$, ($p$ an odd prime > 3 and $x$ a natural number < $n$)
It appears that the only two solutions are always $3$ and $p^k-3$, I want to prove this, here has been my approach, I think I am close but just missing something, would really appreciate an... |
H: Exponential function word problem
From this word problem I know that I need to first plug in 99 into the average cost and I find my number to be 59.82083
And I know i need to do the derivative of the equation which is, but it isnt simplified, I have it simplified on my paper:
When i plug in 99 into that derived ... |
H: Prove : $\dfrac{a}{ac+1}+ \dfrac{b}{ab+1}+ \dfrac{c}{bc+1} \le \frac 12 (a^2+b^2+c^2)$
$a;b;c\in \mathbb{R}^{+}$ such that $abc=1$
Prove : $\frac{a}{ac+1}+ \frac{b}{ab+1}+ \frac{c}{bc+1} \leq \frac{1}{2}(a^2+b^2+c^2)$
AI: $\frac{a}{ac+1} \leq \frac{a}{2\sqrt{ac}} = \frac{1}{2} a \sqrt{b}$. So it suffices to show th... |
H: calculus question about finding a limit of a sequence of functions.
Let $$f_n = n \chi_{[0,\frac{1}{n} ]}$$
I want to find $ \lim f_n$. I think it is $0$ since if we fix $x$ then can find $x > \frac{1}{N} $ By archimedes. So $f_n(x) $ must be $0$ for all $n > N$ is this correct?
AI: It seems like $f_n(x)$ approache... |
H: Norm of a Kernel Operator
So, I was practicing some problems and considered the space $X = C[a,b]$ with the $L_{1}$-norm. I consider the operator
$$Tf(x) = \int_a^b k(x,y)f(y)\,dy$$, where $k(x,y)$ is continuous in both of its variables.
So, I find this operator is bounded:
$$\|T\| \le \operatorname{max}_{a\le x \... |
H: Showing that the Class of Cyclic Groups Aren't Axiomatizable
The class of finite cyclic groups are not axiomatizable, for if we supposed they were by some set of sentences $\Sigma$, then there would exist a model for $\Sigma$ of at least order $n$ for all $n \in \mathbb{N}$ (i.e., the model $Z_{n+1}$ for each $n$).... |
H: How to minimize $\bar{A}.\bar{C}+\bar{A}.B+A.C$ further?
$\bar{A}.\bar{C}+\bar{A}.B+A.\bar{B}.C+B.C$
$=>\bar{A}.\bar{C}+\bar{A}.B+A.\bar{B}.C+\color{Orchid }{(A+\bar{A})
}.B.C$
$=>\bar{A}.\bar{C}+\color{blue}{\bar{A}.B}+\color{green}{A}.\bar{B}.\color{green}{C}+\color{green}{A}.B.\color{green}{C}+\color{blue}{\bar{... |
H: Conditional Joint PDF given a value
Given $f(x,y) = \frac{6-x-y}{8}$ and $0<x <2$ and $2<y<4$
What is $P(2<Y<3|X = 1) $
How do i approach this problem?
I got the marginal PDF for x $\frac{21}{16} - \frac{3}{8}$
AI: In analogy with regular probabilities, where
$$\mathbb{P}(A|B)=\frac{\mathbb{P}(AB)}{\mathbb{P}(B)},... |
H: Find equation of tangent line
Find the equation of the tangent line at parameter values $\theta=\pi/6$ and $\theta =5\pi/4$ to the cycloid given by
$$x(t)=r\theta-r\sin \theta$$
and
$$y(t)= r-r\cos \theta$$
with $\theta\in [0,2pi]$
At which parameter values is the tangent line to the cycloid given above horizont... |
H: identifications: sanity check
A continuous surjective map $p: X \to Y$ is called identification if $O$ is open in $Y$ if and only if $p^{-1}(O)$ is open in $X$. For surjective maps it is true that $f(f^{-1}(O)) = O$ for any set $O$. Can one define identification equivalently as follows?
A continuous surjective map ... |
H: Generating Markov Chains recursively
$X_0:\Omega\rightarrow I$ is a random variable where $I$ is countable. Also $Y_1,Y_2,\dots$ are i.i.d. $\text{Unif}[0,1]$ random variables.
Define a sequence $(X_n)$ inductively as $X_{n+1}=G(X_n,Y_{n+1})$, where $G:I\times[0,1] \rightarrow I$. Show that $(X_n)$ is a Markov cha... |
H: Are universities teaching math too fast?
I love math and I feel happy to learn it. But in universities, I think there's too much to learn. For example, in my university, there are mathematical analysis(I,II,III), linear algebra(I,II), ODE, topology, differential geometry,..., 24 courses altogether. In addition, we ... |
H: Probablity a randomised four digit number does not have two specific consecutive numbers
I am trying to work out the probability a four digit number does not have two consecutive numbers, for example two consecutive 5's, not starting with a 0 is assumed.
Now I could work out how many numbers in this range contain t... |
H: Counting down by halving to 0
Say that you are counting down from 10. You say how long is left after half the amount of time you said how long was left (Like 10, 5, 2.5, 1.25, 0.125, etc.). Because when you halve repeatedly you can never get down to 0, wouldn't you have to say how long is left infinitely many times... |
H: Question about diagonal entries of inverse matrices?
Assume $A$ is a symmetric positive semidefinite matrix with diagonal zero and all other entries are less than one. Also assume $D$ is a diagonal matrix with all entries in diagonal are positive and less than $1$.
Can we say that the diagonal entries of $[D+A]^{-... |
H: Integral of a disk automorphism
Let $|\alpha|<1$ and $\psi_{\alpha}(z)=(\alpha-z)/(1-\bar\alpha z)$. I want to prove that $$\frac 1 {\pi} \int\int_{\mathbb{D}}|{\psi_{\alpha}}^{'}|dxdy = \frac{1-|\alpha|^2}{|\alpha|^2}\log\frac{1}{1-|\alpha|^2}$$
I calculated ${\psi_\alpha}^{'}(z)=(|\alpha|^2-1)/(1-\bar\alpha z)^2$... |
H: multi-variable chain rule
The question (stewart 14.5.50) reads:
If $u = f(x,y)$, where $x=e^scos(t)$ and $y=e^ssin(t)$, show that $$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = e^{-2s} \left[\frac{\partial^2u}{\partial s^2} + \frac{\partial^2u}{\partial t^2}\right] $$
Moving $e^{-2s}$ over... |
H: Question about unity and composition of morphisms (Category theory)
I am reading Awodey's book of category theory and I have some confusion regarding the definitions. Maybe these are very basic questions but I feel I am missing something. This question has helped me, especially Qiaochu Yuan's answer, and probably d... |
H: Find the probability one of a sample is less than a value
I'm having trouble trying to solve this question, the context is pH acidity in rain:
mean = 3.719
sd = 0.546
You are given that the probability that a rainfall collection has a pH less than 3.2 is 0.17. In a random sample of 10 rainfall collections, find th... |
H: Compact and Connected sets
Let $\{A_n\}_{n=1}^{\infty}$ be a decreasing sequence of nonempty compact connected sets in a Hausdorff space. I want to prove that $\bigcap\limits_{n=1}^{\infty}A_n$ is nonempty, compact and connected.
If I can show that $\bigcap\limits_{n=1}^{\infty}A_n$ is nonempty, then all will be pr... |
H: A semiprime only has $4$ factors
It seems quite trivial, but I can't figure out how to explain that in general a semiprime $pq$ only has $4$ factors (namely $1, p, q, pq$). Can anyone give me a small proof?
AI: It is not true when the semi-prime is the square of a prime. Here is a hint for the case where the primes... |
H: What is a perfect square in mod n
I have been stuck with a question on eliptic curves lately. I need to know whether perfect square mod n is different than a normal perfect square.
And also is 3 a perfect square in mod 13?
AI: Yes, 3 is a perfect square $\bmod 13$ because $4^2 \equiv 16 \equiv 3 \bmod 13.$ All norm... |
H: For $n \in \mathbb{N}$ how many times do I have to do this: $k=\lfloor \frac{n}{2} \rfloor$ till $k=1$?
For $n \in \mathbb{N}$ how many times do I have to do this: $k=\lfloor \frac{n}{2} \rfloor$ till $k=1$?
For example $11 \Rightarrow 5 \Rightarrow 2 \Rightarrow 1$
And a bunch of other examples lead me to believe,... |
H: rephrasing this theorem about identifications
One can prove the following:
Let $X,Y$ are topological spaces, $f: X \to Y$ continuous and surjective and $Y$ with the identification topology induced by $f$. If $B \subseteq Y$ is such that $f^{-1}(B) = A$ is closed then $B$ with the subspace topology has the identifi... |
H: What on earth is (B|A)?
I'm stumped and getting nowhere with this question:
Description:
The Air Pollution and Mortality data of 60 cities were collected in a study. 11 can be considered to have high hydrocarbon pollution potential levels. Suppose that two cities are picked at random from the list. (That is, two ci... |
H: Pigeonhole Principle - numbers between $1$ and $100$
Of the set $A=${$1,2,...,100$}, we will choose $51$ numbers. Prove that, among the $51$ chosen numbers, there are two such that one is multiple of the other
My notes:
1) There are $25$ prime numbers between $1$ and $100$ ;
2) There are $26$ odd and non-prime numb... |
H: Help on abstract algebra proof?
Similar question here Let $R$ be the set of all integers with alternative ring operations defined below. Show that $\Bbb Z$ is isomorphic to $R$. The difference is that in attempting to answer my own problem, I can't.
For any integers $a,b$, define $a\oplus b=a + b - 1$ and $a\odot... |
H: If the interior of $A$ is empty, must $A$ be countable?
Let $X$ be a second metrizable space and $A$ is a subset of $X$. If the interior of $A$ is empty, must $A$ be countable?
Thanks!
AI: This is false in every uncountable separable space; in particular, every uncountable, second countable, metrizable space.
Let $... |
H: Help me prove the following probldem regarding continuity of functions.
Let $f:(a,b) \rightarrow \mathbb{R}$ be a continuous function. Let $x_1,x_2,x_3,\dots,x_n \in (a,b)$. Prove that there exists a point $c \in (a,b)$ such that
$$f(c) = \dfrac{f(x_1)+f(x_2)+......+f(x_n)}{n}
$$
AI: Since $x_1, x_2, \ldots, x_n \... |
H: If $r\leq p\leq s$, prove that $\|f\|_p\leq \max(\|f\|_r,\|f\|_s)$
Let $X$ be a measure space with measure $\mu$ and $\|f\|_p=\left(\int_X |f|^p\; d\mu\right)^{\frac{1}{p}}$ be the standard $p$-norm for complex-valued $f$. I want to prove the following result, given $0<r\leq s$:
For all $p\in [r,s]$, $\|f\|_p\leq ... |
H: Direct sum of vector spaces and quotients
Suppose $V$ is a finite dimensional vector space, and $W_{1}, W_{2}$ are subspaces of $V$. Assume $V=W_{1}\oplus W_{2}$. Does it necessarily follow that $V/W_{1}$ is isomorphic to $W_{2}$ (as vector spaces)?
I feel like this is a well-known fact, but I have seen enough coun... |
H: cutting a cake without destroying the square toppings
There is a square cake. It contains N toppings - N disjoint axis-aligned squares. The toppings may have different sizes, and they do not necessarily cover the entire cake.
I want to divide the cake into 2 rectangular pieces, by either a horizontal or a vertical ... |
H: Tossing coins - basic probability example
Oh no, another coin tossing problem? Yes. I've read more than a dozen of coin tossing questions here but I didn't find anything helpful.
Let's have an experiment: I have $3$ identical coins and a pot to throw them in.
So I throw all the coins into the pot. What is the proba... |
H: Prove : $\frac{1}{a}+\frac{1}{b}-\frac{1}{c}< \frac{1}{abc}$
$a;b;c>0$ such that $a^2+b^2+c^2=\frac{5}{3}$. Prove :
$\frac{1}{a}+\frac{1}{b}-\frac{1}{c}< \frac{1}{abc}$
AI: $$
c(a+b)\leq \frac{c^2+(a+b)^2}{2}=\frac{a^2+b^2+c^2}{2}+ab=\frac 56+ab
$$
which is the rearrangement of your inequality and even the stronger... |
H: Showing a set is not compact by describing an open cover that doesn't have a finite subcover
I would like to prove that the following set is not compact by stating an open cover for it that has no finite subcover.
$E=\{x\in\mathbb{Q}:0\leq x\leq2\}$
I'm having trouble thinking of one. A hint would be appreciated.... |
H: Finite abelian unramified $p$-extension of a number field
Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur as Galois group but I don't know how the "unramified" cond... |
H: Evaluating a limit for a odd rooted radical
This is not a question which is much of an issue with the limit per se. Algebrically...I am kinda flat at this juncture...
$\frac{2+x}{\sqrt[5]{x^{5} -9}}$
My questions:-
Highest power of x, that I should take inside the radical? I believe it's 5 given the root of the ra... |
H: For what $k$ does $k \sin A + \cos 2A = 2k - 7$ have a solution?
The equation $k \sin A + \cos 2A = 2k - 7$ has a solution, if:
$k >6$
$k>2$
$k<7$
$2\leq k\leq 6$
Although I did figure out the answer to be the last option using a bit of guess and all, but I need an exact way of solving this.
Thanks in advanc... |
H: How to show the standard $n$-simplex is homeomorphic to the $n$-ball
I am trying to show the standard $n$-simplex is homeomorphic to the $n$-ball.
Here, the standard $n$-simplex is given by $$\Delta^n=\left\{(x_0,x_1,\cdots,x_n)\in\mathbb{R}^{n+1}:\sum x_i=1,x_i\geq0\right\}$$ and the $n$-ball is given by
$$B^n=\{x... |
H: How prove this$\sum_{i=0}^{m-1}\binom{n-1+i}{i}x^ny^i+\sum_{j=0}^{n-1}\binom{m-1+j}{j}x^my^j=1$
let $m,n$ be positive numbers,and $x,y>0$ such $x+y=1$,show that
$$\sum_{i=0}^{m-1}\binom{n-1+i}{i}x^ny^i+\sum_{j=0}^{n-1}\binom{m-1+j}{j}x^jy^m=1$$
My try:
$$\sum_{i=0}^{m-1}\binom{n-1+i}{i}x^ny^i=\sum_{i=0}^{m-1}\binom... |
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