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H: Is there a smarter way to integrate?
My textbook uses the following technique to integrate. Let us take the following as an example, evaluate the integral of:
$$f(x) = 6x(x^2 + 1)^5$$
Notice that $[x^2+1]' = 2x$, so $6x\space dx = 3 \space d (x^2+1)$
$$6x(x^2+1)^5 \space dx = \dfrac{6}{2} (x^2+1)^5 \space d(x^2+1)... |
H: Why define vector spaces over fields instead of a PID?
In my few years of studying abstract algebra I've always seen vector spaces over fields, rather than other weaker structures. What are the differences of having a vector space (or whatever the analogous structure is called[module?]) defined over a principal ide... |
H: What is the factor group $\mathbb{Z}/5\mathbb Z$?
I am trying to understand the concept of factor group. The definition of factor group I know is the following: Let $G$ be a group and $H$ be a subgroup of $G$. Then the group of cosets denoted by $G/H$ is called the factor group of $G$ by $H$. Now I am looking at an... |
H: Why is $\langle \operatorname{grad} f, X\rangle_g$ independent of the metric on a Riemannian manifold?
Let $(M,g)$ be a Riemannian manifold and let $f \in C^{\infty}(M)$. Let $X$ be a smooth vector field on $M$. In smooth local coordinates $(x^i)$ on $M$, we can write $g = g_{ij} dx^i \otimes dx^j$ as well as $X = ... |
H: longest path two nodes in common
Let $G$ be a connected graph. Prove that two longest paths in $G$ have atleast one node in common. Note that two longest paths do not neccessarily have the same length.
I began by defining two paths namely $P_1 = <v_1,...,v_k>$ and $P_2 =<u_1,...,u_m>$ where $m\neq k$ because of the... |
H: Finding global max./min.
my task is to figure out the critical points of $f(x,y)=e^y(x^4-x^2+y)$, $\ $$\mathbb{R}^2 \rightarrow \mathbb{R}$, and show which of them is a maximum or minimum. As far as I got, I've shown that the critical points are:
1.: $(0,-1)$ which is neither max. nor min. (char. pol. of Hessian is... |
H: How to solve the irrational system of equations?
Solve the system of equations $$\begin{cases} \sqrt{x+2y+3}+\sqrt{9 x+10y+11}=10,&\\[10pt] \sqrt{12 x+13y+14}+\sqrt{28 x+29y+30}=20. \end{cases} $$
AI: Hint: Try $t=x+y+1$ to get
\begin{cases} \sqrt{t+y+2}+\sqrt{9t +y+2}=10,&\\[10pt] \sqrt{12t+y+2}+\sqrt{28t+y+2}=20.... |
H: What do we call the negation of logical equivalence?
The statement that '$x$, $y$ and $z$ are equivalent' just means
all of $x$, $y$ and $z$ are false, or
all of $x$, $y$ and $z$ are true.
Now suppose its not the case that $x$, $y$ and $z$ are equivalent. That is, suppose:
at least one of $x$, $y$ and $z$ is fal... |
H: What makes the Var(x)+Var(y)=var(x+y) property important?
What makes the Var(x)+Var(y)=var(x+y) property important?
It was taught in my statistics class
AI: In the 18th century Abraham de Moivre considered this problem (his account of which you can find in his book The Doctrine of Chances: If you toss a fair coin $... |
H: Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$.
Calculate the limit: $\lim_{n\to+\infty}\sum_{k=1}^n\frac{\sin{\frac{k\pi}n}}{n}$ Using definite integral between the interval $[0,1]$.
It seems to me like a Riemann integral defin... |
H: What am I doing wrong; evaluating integrals?
$$f(x) = \dfrac{2x}{2-3x^2}$$
$$- \dfrac{1}{3} d (2-3x^2) = 2x$$
$$ \dfrac{2x}{2-3x^2} dx = -\dfrac{1}{3} d (2-3x^2) \cdot \dfrac{1}{2-3x^2} = - \dfrac{1}{3} du \cdot u^{-1} = -\dfrac{1}{3} u^{-1} \cdot du$$
And here I'm stuck, since you can't divide by 0.. What am I doi... |
H: How do I find the limit $\lim_{x \to 1} \frac{1}{1 - x} - \frac{3}{1 - x^3}$
How do I calculate the following limit? $$\lim_{x \to 1} \frac{1}{1 - x} - \frac{3}{1 - x^3}$$
I already transformed this into
$$\lim_{x \to 1} \frac{(1-x)(1-x)(x+2)}{(1-x)(1-x^3)} = \frac{(1-x)(x+2)}{(1-x^3)}$$
but this still has zeroes u... |
H: Kinematics stone thrown upwards past a point, show the following.
I know I should be able to do this, but I have tried for 3 hours and can't do it. I know its simple but it's driving me mad.
A particle is projected vertically upwards with speed $ u_{0}$ and passes through a point
that is a distance $ h $ above the ... |
H: Totient-like function
I have number written as factors for instance: n = 2 * 3 * 3 * 5. What I have to do is find how many numbers between <1, n) are co-prime to n, which means GCD = 1. It can simply be done using Euler's Totient. But what if GCD = 2 or more? Is there any totient-like function?
UPDATE:
I seeking ho... |
H: Prove that the degrees lie in a range
Let $G=(V,E)$ be a graph with $|V|=n$ and $|E|=m$ prove that
$$
\min_{u\in V} \{d(u)\}\leq 2\frac{m}{n}\leq \max_{v\in V} \{ d(v)\}
$$
now my first intuition is to assume that $\min\limits_{u\in V} \{d(u)\} =0$ holds because it is not a connected graph. And my second assumptio... |
H: Does pointwise equicontinuous and uniformly equicontinuous implies compactness?
If every sequence of pointwise equicontinuous functions $M \rightarrow \mathbb{R}$ is uniformly equicontinuous, does this imply that $M$ is compact?
AI: Consider the subspace $M=\mathbb{Z}$ of the reals, an infinite discrete space, cert... |
H: help me with this regarding hypothesis using chi square distribution
The rope used in a lift produced by a certain manufacturer is known to have a mean tensile breaking strength of 1700 kg and standard deviation 10.5kg. A new component is added to the material which will, it is claimed, decrease the standard deviat... |
H: About completeness of $l^{\infty}$ with respect to sup norm
Let $l^{\infty}$ be the space of all bounded sequences of real numbers $(x_n)_{n =1}^{\infty}$ with the sup norm. I have to show that $l^{\infty}$ is complete with respect to this norm.
Proof: In the proof below I am confused with the sequence $x^n = (x_1^... |
H: Prove that if $ h \circ f = g $ then $ h $ is an $A$-algebra homomorphism.
Let $f:A\rightarrow B,\ g:A\rightarrow C$ be ring homomorphisms. An $A$-algebra homomorphism $h:B\rightarrow C$ is a ring homomorphism which is also an $A$-module homomorphism.
Please prove that if $h \circ f = g$ then $h$ is an $A$-algebra... |
H: What is $\;\int xe^{-x^2} \,dx\;?$
What is $$\int xe^{-x^2} dx\quad?$$
I used substitution to rewrite it as $$\int -\dfrac{1}{2}e^u\, du$$ but this is too hard for me to evaluate. When I used wolfram alpha for $\int e^{-x^2} dx$ I got a weird answer involving a so called error function and pi and such (I'm guessing... |
H: Example of a group which is abelian and has finite (except the $e$) and infinite order elements.
Exercise 7: Show that the elements of finite order in an abelian
group $G$ form a subgroup of $G$
I just solved this exercise but I can't find example of a group which is abelian and has finite (except the $e$) and ... |
H: Total probabilities of being admitted to any university
Let's provide an hypothetical situation in which a student applies to 10 different universities whose number of applicants, admissions and admission rate you can see in the table below.
------------------------------------------------------
| Name | Appl... |
H: Calculate the limit using definite integrals: $\lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2}$
Calculate the limit using definite integrals: $\lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2}$
Well, I started like this:
$\lim_{n\to\infty}2n\sum_{k=1}^n\frac1{(n+2k)^2}=\lim_{n\to\infty}2n[\frac1{(n+2)^2}+...+\frac1{(... |
H: Trigonometric function integration: $\int_0^{\pi/2}\frac{dx}{(a^2\cos^2x+b^2 \sin^2x)^2}$
How to integrate $$\int_0^{\pi/2}\dfrac{dx}{(a^2\cos^2x+b^2 \sin^2x)^2}$$
What's the approach to it?
Being a high school student , I don't know things like counter integration.(Atleast not taught in India in high school educa... |
H: Convergence of coordinates to zero
Consider a normed finite-dimensional vector space $V$ with some norm $|| .||$
Say a sequence of vectors in this vector space $v_m \rightarrow 0$ where $0$ is the zero vector.
Let $\{b_1,b_2,\ldots,b_n\}$ is a basis for $V$.
Therefore $v_m = \sum_{i=1}^{n} c_i ^m b_i$ for some scal... |
H: Equation with Logarithm in Exponent
How to I solve the following exercise with a logarithm? I've forgotten the "trick" for doing so:
$x^{log_{10} x} =10^4$
AI: To start, take $\log_{10}$ of both sides to get:
$$\log_{10}(x)\log_{10}(x) = \log_{10}(10)^{4} = 4$$
Then just solve $$(\log_{10}(x))^{2} = 4$$
$$\implies ... |
H: Given two sets, finding two non trivial homomorphisms that are not isomorphisms
Is it possible to have two non trivial homomorphisms that are not isomorphisms for given two Groups?
I am specially interested in additive/remainder Group of Integers and multiplicative (arithmetic multiplication as group operator) gro... |
H: Curve defined by a vector
https://i.stack.imgur.com/tD4Bn.png
I'm studying line integrals with a curve as a vector, but I couldn't understand the 'dr' part.
First of all: the curve isn't really a curve, it's like some points where a vector points from the origin. So this 'curve' does not really exists, rigth?
A... |
H: Dynamical limitation and minimization
I come across the following question: Under what conditions (on the series of functions $f_n$ or perhaps the domain of minimization) the following holds
$$
\lim_{n\to\infty}\min_{x_1,\ldots,x_n}f_n(x_1,\ldots,x_n) = \min_{\left\{x_i\right\}_i}\lim_{n\to\infty}f_n(x_1,\ldots,x_n... |
H: Automorphism group of $\mathbb{Z}_p\times \mathbb{Z}_p$
How to determine the automorphism group of $\mathbb{Z}_p\times \mathbb{Z}_p$ where $p$ is a prime? Or more specifically, how to determine the element of order $2$ in this group?
I got stuck here, since I only know that if two finite groups $H$ and $K$, where... |
H: Maximization of minimum difference
Suppose we have a function of the form: $(x_1 - x_2) + (x_3 - x_4) + (x_5 - x_6)$ and we have maximized this summation using linprog (using some constraints which are not important for this matter). This provides us with a value for the different x variables.
The problem I now wan... |
H: Classifying map
Let $\xi=(E,p,B)$ a principal $G$-bundle and $\eta=(P,\pi,Q)$ a real vector bundle such that $\operatorname{rank}(\eta)=n$. We can consider a classificant space $BG$. What is the classifying map $f:X \rightarrow BG$? Why the name "classifying"? How can build classifying map for $\eta$?
AI: Given any... |
H: How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension
I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these spaces are ... |
H: Tell me the ideal selling price to get back a specified number
If I am buying something at xxx,
what is the price to sell it if I want a profit of $2.50 after minus-ing 0.63% (broker fee) from the selling price?
I need to make this into an excel formula, but this is the first step... if you know the excel formula ... |
H: How can this property of definite integrals be true?
In this question, people are saying that the definite integral of $f(x)$ from $0$ to $a$ is equal to the integral of $f(a-x)$ from $0$ to $a$. How can that be true? Simple examples don't work.
AI: Put $z=a-x$ then $dz=-dx$. When $x=0, z=a$ and when $x=a,z=0$. Now... |
H: Linear Approximation
I have an exercise, giving this question.
Find the linear approximation $Y$ to $f(x)$ near $x=a$.
$$
f(x) = x + x^4,\quad a=0
$$
I can see in my result list that it says $Y=x$, however, after multiple tries I can only convince myself that it gives $Y=0$.
Can anyone explain why it gives $1$... |
H: What is the relation between vectors in physics and algebra?
Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. represented like a column matrix). It was really never explain... |
H: Two Algorithm A and B solve the same problem
Two algorithms $A$ and $B$ solve the same problem. $A$ solves a problem of size $n$ with $n^2~2^n$ operations. $B$ solves it with $n!$ operations. As $n$ grows, which algorithm uses fewer operations?
AI: I believe Stirling is your friend. |
H: Prove $\omega^{\omega_1}=\omega_1$ and $2^{\omega_1}=\omega_1$
Prove that $\omega^{\omega_1}=\omega_1$ and $2^{\omega_1}=\omega_1$.
I also found an exercise asking to compute the ordinal number $\omega_1^{\omega}$, but I do not even understand what I am supposed to do, any help?
AI: Ordinal exponentiation is not ca... |
H: How do you respond to "I was always bad at math"?
Here in the U.S., it is my experience that over 75% of adults I meet socially will volunteer that phrase or a variation upon learning that I am a mathematician. I find this frustrating, since almost nobody would brag about being bad at history or English or really... |
H: How should I go about this approaching-infinite limit problem?
I'm doing some exercises of limits approaching infinite, most are simple polynomials where only the highest degree term will matter in the end but for this one I couldn't find a solution (not correct at least).
$$\lim_{x\to-\infty}\frac{x^2+x+1}{(x+1)^... |
H: Let $ \mathbb{N}$, $a \in \mathbb{N} \to a+1 \in \mathbb{N}$
I need to prove the following:
"Let $ \mathbb{N}$, if $a \in \mathbb{N} \to a+1 \in \mathbb{N}$"
thanks in advance!!
AI: There is actually something to prove here. $a+1$ is defined in the Peano axioms as $a+S(0)=S(a+0)=S(a)$. Hence, if $a\in\mathbb{N... |
H: Linear Regression - Proof Sum Adds to Zero
In linear regression, why is $\sum(X_{i} - \mu_{x})$ = $0$? I understand that for ($\sum$ $Y_{i}$ minus the fitted value of Y) = $\sum$ $e_{i}$ this is true but why is this other fact true?
AI: Just break up the sum into two sums, and substitute the definition of
$\mu_x$.... |
H: Understanding the differential $dx$ when doing $u$-substitution
I just finished taking my first year of calculus in college and I passed with an A. I don't think, however, that I ever really understood the entire $\frac{dy}{dx}$ notation (so I just focused on using $u'$), and now that I'm going to be starting calc... |
H: Group equals union of three subgroups
Suppose $G$ is finite and $G=H\cup K\cup L$ for proper subgroups $H,K,L$. Show that $|G:H|=|G:K|=|G:L|=2$.
What I did: so if some of $H,K,L$ is contained in another, then we have $G$ being a union of two proper subgroups, which is impossible due to another result. So none of $H... |
H: Absolute convergence of a real series
I need to show that the following series:
$\sum_{i=1}^n (-1)^n\dfrac{x^2+n}{n^2} $
Is uniformly convergent on any bounded interval, but not absolutely convergent for any real $x$. My first thought was to use the Weierstrass M-Test, however this is pointless as if the above seri... |
H: Is the set theory (ZF) a structure?
According to the definition, generally speaking, a structure $\langle A;R;F,C\rangle$ is such that $A$ is a non-empty set, $R$ is the set of relations, $F$ is the set of functions, and $C$ is a set of constants. For example $\langle\mathbb{R};; +,\cdot, ^{-1};0,1\rangle$ would be... |
H: Closure and limit of a sequence
Let $E$ be a subset of a metric space $(S,d)$. I'm trying to show that an element is in $\overline{E}$ if and only if it is the limit of some sequence of points in $E$.
Suppose there is a sequence $(p_n) \subseteq E$. Then because $E \subseteq \overline{E}$, I know that $(p_n) \subse... |
H: Prove that $e^a e^b = e^{a+b}$
I've read the argument in Rudin, but I think I need a little clarification
\begin{align}
e^a e^b &= \sum_{k=0}^{\infty} \frac{a^k}{k!} \sum_{m=0}^{\infty} \frac{b^m}{m!}\\ &= \sum_{n=0}^{\infty} \frac{ n!}{n!} \sum_{k=0}^{n} \frac{a^k}{k!} \frac{b^{n-k}}{(n-k)!} \\
&= \sum_{n=0}^{\in... |
H: Question regarding the proof of a topological claim
The lecturer in the Topology course I'm taking defined the following:
Given a topological space $X$
we say that:
$X$ is weakly locally compact if for all $x\in X$
there exists a compact nbhd.
$X$ is strongly locally compact if every nbhd of $x$
contains a compa... |
H: Proving that commensurability is transitive
We have that two groups $\Gamma$ and $\Gamma'$ are commensurable if there exist finite index subgroups $G \leq \Gamma$ and $G' \leq \Gamma'$ such that $G \cong G'$. We denote this $\Gamma \approx \Gamma'$.
I am trying to prove that this gives a transitive relation, but I ... |
H: Does half-life mean something can never completely decay?
Caffeine has a half-life of approximately six hours. I understand this to mean that every six hours, the amount of caffeine in the body is half of what it was six hours prior. Does that mean that caffeine never completely leaves the body? It just keeps reduc... |
H: Implicit differentiation: $x^2y - 2x^3 - y^3 + 1 = 0$
Hi I'm stuck on one problem in my study guide. It's the only one with an = 0 at the end of it.
Differentiate:
$x^2y - 2x^3 - y^3 + 1 = 0$
AI: Implicit differentiation:
$$2xy\,dx+x^2\,dy-6x^2\,dx-3y^2\,dy=0\implies (x^2-3y^2)dy=(6x^2-2xy)dx\implies$$
$$y'=\frac{d... |
H: Telescoping sum of powers
$$
\begin{array}{rclll}
n^3-(n-1)^3 &= &3n^2 &-3n &+1\\
(n-1)^3-(n-2)^3 &= &3(n-1)^2 &-3(n-1) &+1\\
(n-2)^3-(n-3)^3 &= &3(n-2)^2 &-3(n-2) &+1\\
\vdots &=& &\vdots & \\
3^3-2^3 &= &3(3^2) &-3(3) &+1\\
2^3-1^3 &= &3(2^2) &-3(2) &+1\\
1^3-0^3 &= &3(1^2) &-3(1) &+1\\
\underline{\hphant... |
H: Piecewise continuous differentiable
if I have a piecewise continuously differentiable function. How do I see that on each open interval, where the derivative is continuous, there is a continous extension on the larger closed interval?
AI: This is true when $f$ satisfies the condition: the lateral limits exist. And ... |
H: Evaluating a summation of inverse squares over odd indices
$$ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$
I want to evaluate this sum when $n$ takes only odd values.
AI: Note that
$$\sum_{n \text{ is even}} \dfrac1{n^2} = \sum_{k=1}^{\infty} \dfrac1{(2k)^2} = \dfrac14 \sum_{k=1}^{\infty} \dfrac1{k^2} = \df... |
H: Does the 80-20 Rule apply to the Buffon process?
Whether the needle crosses a line depends greatly on the angle the needle makes with the line, 90 degrees being of course the most favorable for a line-crossing. Does the 80-20 Rule apply to these angles? That is, since 20% of 90 degrees is 18 degrees, are about 80% ... |
H: An inflection point where the second derivative doesn't exist?
A point $x=c$ is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point.
And a list of possible inflection points will be those points where the second derivative is zero or doesn't exist.
B... |
H: Does the Law of Factor Sparsity apply to factorization?
Does the 80-20 Rule (also known as the Law of Factor Sparsity, the Law of the Vital Few, and the Pareto Principle) apply to factorization? That is, if x is a large positive integer, then are about 20% of the primes not exceeding x sufficient for factorizing ab... |
H: A simple yet hard task for (theoretically) Poisson distribution
Sorry if I don't use the words properly, I haven't learnt these things in English, only some of the words.
Anyway, I'm practicing to one of my exams and sadly this task seemed more challanging for me than it should be. Some kind of explain would help a... |
H: Is pi lying on the ground, and on TV? - and on the sun?
Consider the leaves from a bunch of trees in a terraced plaza in the Autumn. It may well happen that the tiles of the terrace are squares whose length easily exceeds the length of the stem of the leaves (assuming leaves all of the same kind). Cannot this be re... |
H: How to show $\mathbb{Z}[w]/(2,w) \simeq \mathbb{Z}_2$?
Let $\mathbb{Z}[w]=\mathbb{Z}[\frac{1+\sqrt{-15}}{2}]$ be the quadratic integers.
I want to show that $\mathbb{Z}[w]/(2,w) \simeq \mathbb{Z}_2$. It seems very clear, but how can I show the isomorphism rigorously by isomorphism? I found somewhat similar using 3r... |
H: How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$
How can I prove the following conjectured identity?
$$\mathcal{S}=\sum_{n=1}^\infty\frac{(4\,n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^... |
H: Proof of matrix inequality involving trace and max-operator
Let $C\in\mathbb R^{N\times N}$ be a real positive-semidefinite matrix, $I\in\mathbb R^{N\times N}$ the identity matrix and $W,V\in\mathbb R^{N\times N}$ two real-valued random matrices. Is it true that
$$tr\left[(C-I)WCV^T\right]\leq max\left\{tr\left[(C-... |
H: A basic proof on Morse Function
The questions is to show if $f_t$ is a homotopic family of functions on $R^k$, show that if $f_0$ is Morse in some neighborhood of a compact set $K$, then so is every $f_t$ for sufficiently small t.
I get the previous question leads to the proof, but couldn't reach the question:
det$... |
H: What happens to the infinite monkey theorem when there are an infinite number of keys on the typewriter?
What happens to the infinite monkey theorem when there are an infinite number of keys on the typewriter? So what is the probability of a finite string of keys like the works of Shakespeare being typed up. Thanks... |
H: What set theory axioms do I need to believe in uncountable ordinals?
Math people:
The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof. It is not intuitively clear to me that uncountable ordinals or cardinals should exist. What axioms of set theory are needed to... |
H: Unique largest normal pi-subgroup
Let $\pi$ be a set of prime numbers. A finite group is said to be a $\pi$-group if every prime that divides its order lies in $\pi$. If $G$ is finite, show that $G$ has a unique largest normal $\pi$-subgroup (which may be trivial and may be all of $G$).
What I did: suppose $|G|=p_1... |
H: Recurrence Relations with single roots
I have the following recurrence: $a_{n+3}=3a_{n+2}-3a_{n+1}+a_n$
with initial values $a_1 = 1, a_2 = 4, a_3 = 9$
I have found the characteristic equation to be $x^3-3x^2+3x-1$ and the root to be 1.
My text book is not helpful in how I should go about solving this when I have a... |
H: Good calculus exercises/problems?
I can't enroll in a university this year, so I'm studying calculus at home, but the only exercises about calculus that I find are the easy ones. Do you know a great page where I can find not only calculus exercises, but problems as well?
I want to find about:
hard limits,
derivat... |
H: Sum of Logarithm Arguments
This is a very simple question I suspect but I just cannot seem to nail it...
I have values for $X,Y,Z $, where $X =\log (x)$, $Y = \log (y)$ and $Z = \log (z)$ and I need to calculate $x + y + z$, well actually $\log(x + y + z)$ would suffice. Is there a clever way of doing this other t... |
H: Redundance in statement of second morphism theorem
The standard statement of the Second Morphism Theorem found in my textbook and Wikipedia is as follows:
Let $N$ be a normal subgroup of $G$ and $H$ be any subgroup of $G$.
$HN = \{hn | h \in H, n \in N\}$ is a subgroup of $G$
$H \cap N$ is a normal subgroup of $H... |
H: Probability AND/OR
Suppose we have a bag of $10$ balls, and each ball is a unique colour.
If we randomly select $3$ balls from this bag, without replacement, I want to find out the chances of correctly guessing the colour of all $3$ balls (if we have chosen $3$ colours beforehand).
I have come down to two possible ... |
H: Is the compactness theorem (from mathematical logic) equivalent to the Axiom of Choice?
Or more importantly, is it independent of the axiom of choice. The compactness theorem states the given a set of sentences $T$ in a first order Language $L, T$ has a model iff every finite subset of $T$ has a model. So for any n... |
H: distribute m pennies to n people, what is the expectation of coins one would obtain
Assume there are $m$ pennies and $n$ people. We want to distribute the pennies to the people by uniformly picking a vector $(x_1,...,x_n)$ from the set of all vectors satisfying $x_1+...+x_n=m$, where $x_i$ is the number of coins gi... |
H: Big-O notation, prove the following: $\sum\limits_{k=3}^n(k^2 - 2k)$ is $O(n^3)$.
Use the definition of Big-O notation to prove the following:
$\sum\limits_{k=3}^n(k^2 - 2k)$ is $O(n^3)$.
Can someone please give me some hints on how to expand $\sum\limits_{k=3}^n(k^2 - 2k)$?
AI: The easiest way to conclude that it... |
H: Related rates, calculus
Suppose that $k^{2} + h^{3} = 9$. Find $\frac{dh}{dt}$ when $k=1$ and $\frac{dk}{dt} = 3$ ans $= \frac{1}{2}$. I'm differentiating with respect to $t$ but I cannot get the answer if you could show me the steps, or how to approach this question, would be very helpful :)
AI: Treating $k$ and $... |
H: Ratio of corresponding sides of similar triangles, given the areas.
The area of two similar triangles are 72 and 162. what is the ratio of their corresponding sides?
AI: When linear dimensions a scaled by the factor $\lambda$, area is scaled by the factor $\lambda^2$. Here, we have $\lambda^2=\dfrac{162}{72}=\dfrac... |
H: Geometry of the space $C[a, b]$ respect to the norm $\lVert x \rVert_{\infty} = \max_{t\in [a,b]}\lvert x(t)\rvert$.
I have studied that the space $C[a, b]$ of all scalar-valued (real or complex)
continuous functions defined on $[a, b]$ is a Banach space with respect to the norm $\lVert x \rVert_{\infty} = \max_{t\... |
H: Let $f(z)=\frac{z^3}{(z-\pi)^3(z+5)^2}$ and let $C$ be $|z|=3$.Then $\int_C{f(z)dz}=0$ because :
Which of the following options are true?
Let $f(z)=\dfrac{z^3}{(z-\pi)^3(z+5)^2}$ and let $C$ be $|z|=3$.Then $\int_C{f(z)dz}=0$ because :
(A) the residue is $0$ at its only pole within $C$.
(B) the sum of the resid... |
H: Cylindrical Shell method conceptual question
I am self-teaching myself calculus for the summer to get ready for the actual class. Let us say that we have a region bounded above by the curve $y = 2 - x^2$ and below by the curve $y=x^2$ from $x=0$ to $x=1$. Suppose that the region is revolved around the $y-$axis.
Wh... |
H: First Homology Group and Abelianization
On the Wolfram Mathworld article for Commutator Subgroup, it states that the first homology group is the abelianization, $$H_{1}(G) = G \big/ [G,G]$$
which totally blows my mind because I've only seen the commutator subgroup in the context of Lie algebra representation theory... |
H: Pages 6 and 27 are on the same (double) sheet of a newspaper, how many pages are there in the news paper altogether
I was never really good at maths, trying to get back into it.
I have this question:
Pages 6 and 27 are on the same (double) sheet of a newspaper, what are the page numbers on opposite sides of the ... |
H: Probability that a divisor of $10^{99}$ is a multiple of $10^{96}$
What is the probability that a divisor of $10^{99}$ is a multiple of $10^{96}$? How to solve this type of question. I know probability but I'm weak in number theory.
AI: HINT: $10^n=2^n5^n$, so the divisors of $10^n$ are the numbers of the form $2^... |
H: The maximal ideal space of $A$ is contained in the unit ball of $A^\ast$
On page 281 of Rudin's book Functional Analysis, let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$, $K$ be the norm-closed unit ball of $A^*$, then $\Delta\subset K$ (by Theorem 10.7). Can someone give me more detail?... |
H: Formula for series $\frac{\sqrt{a}}{b}+\frac{\sqrt{a+\sqrt{a}}}{b}+\cdots+\frac{\sqrt{a+\sqrt{a+\sqrt{\cdots+\sqrt{a}}}}}{b}$
All variables are positive integers.
For:
$$a_1\qquad\frac{\sqrt{x}}{y}$$
$$a_2\qquad\frac{\sqrt{x\!+\!\sqrt{x}}}{y}$$
$$\cdots$$
$$a_n\qquad\frac{\sqrt{x\!+\!\sqrt{\!x+\!\sqrt{\!\cdots\!+\s... |
H: Conditional Expectation with independent sub-sigma fields
Let X and Y be bounded random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider two independent sub-$\sigma$ fields $\mathcal{G}$ and $\mathcal{H}$ of $\mathcal{F}$. We assume X is $\mathcal{G}$-measurable and Y is $\mathcal{H}$-... |
H: Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies $E(f(X_n)) \to f(0)$.
The following is problem 4 from Section 4.2 of "A Course in Probability Theory" by Kai Lai Chung.
Let $f$ be a bounded uniformly continuous function in $R^1$. Then $X_n\to 0$ in pr. implies $E(f(X_n)) \... |
H: Help with functions
How many functions $f\ne0$, $f:\mathbb{Z}\to\mathbb{C}$
Periodic in an integer $a$ are there so that
$$f(x+y)=f(x)f(y)$$
What I have sofar is that
$$f(a)=f(2a)=f(a)f(a)$$
So that $f(a)=1$,
Also if I multiply $f(x)$ together with itself $a$ times I get that $$f(x)^a=f(x)f(x)f(x)....f(x)=f(ax)=1$... |
H: Diamond diagram for Correspondence Theorem
This paragraph appears in Isaacs' Algebra (chapter on homomorphisms).
We comment briefly on the interpretation of the Correspondence theorem in terms of lattice diagrams, at least in the case where $\phi$ is the canonical homomorphism $G\rightarrow G/N$. If we have a latt... |
H: Find a measurable function such that $f(x)\le \alpha$ for $x\in E_\alpha$
Theorem: Given $\{ E_\alpha \}_{\alpha \in \mathbb{R}}\subset \mathcal{M}$ such that $E_\alpha \subset E_\beta$ for $\alpha < \beta$. We have also that $\bigcup_{\alpha \in \mathbb{R}}E_\alpha=X$ and $\bigcap_{\alpha \in \mathbb{R}}E_\alpha=\... |
H: A product identity involving the gamma function
I have reduced this problem (thanks @Mhenni) to the following (which needs to be proved):
$$\prod_{k=1}^n\frac{\Gamma(3k)\Gamma\left(\frac{k}{2}\right)}{2^k\Gamma\left(\frac{3k}{2}\right)\Gamma(2k)}=\prod_{k=1}^n\frac{2^k(1+k)\Gamma(k)\Gamma\left(\frac{3(1+k)}{2}\righ... |
H: Check proof for a statement of linear independence involving 5 matrices in $M_2(\mathbb{R})$
Let $A,B,C,D,E \in M_2(\mathbb{R})$
I'm asked to prove or disprove that if the set $A = \{EA,EB,EC,ED\}$ is linearly independent so the set $\{A,B,C,D\}$ is linearly independent.
I was having troubles with matrices algebra ... |
H: Nonsingular affine curve which is not unmixed
Let $C$ be any nonsingular curve in $A^3_{\mathbb C}$. Can a point be an irreducible component of $C$? I am not able to find an example of such $C$.
AI: First, note that a point is an irreducible component of a variety if and only if its complement is closed, i.e. if an... |
H: Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$
I'm trying to prove rigorously that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$. Where $f$ is the pdf of the random variable $X$.
I can't find a proof on the wikipedia article, or if it's there then it's disguised enough that I can... |
H: Cardinality of the set of divergent sequences
How to find the cardinality of the set of divergent sequences? Let's name this set $A$.
I know that cardinality of the set of sequences equals $2^{\lvert \mathbb{N}\rvert}$, so $\lvert A\rvert\le 2^{\lvert \mathbb{N}\rvert}$.
How to strictly prove to the other side?
Tha... |
H: Proving that none of these elements 11, 111, 1111, 11111...can be a perfect square
How can i prove that no number in set S
S = {11, 111, 1111, 11111...}
Is a perfect square.
I have absolutely no idea how to tackle this problem i tried rewriting it in powers of 10 but that didn't really get me anywhere...
Thanks in... |
H: Construction of the projective plane over $\mathbb{F}_3$
I have a question about constructing projective plane over $\mathbb{F}_3$. We first establish seven equivalence classes $P= \{ [0,0,1], [0,1,0], [1,0,0], [0,1,1], [1,1,0], [1,0,1], [1,1,1] \}$.
Given a triple $(a_0, a_1, a_2) \in \mathbb{F}^3_3 \setminus (0,... |
H: Exponential Growth, half-life time
An exponential growth function has at time $t = 5$
a) the growth factor (I guess that is just the "$\lambda$") of $0.125$ - what is the half life time?
b) A growth factor of $64$ - what is the doubling time ("Verdopplungsfaktor")?
For a), as far as I know the half life time is $\d... |
H: Ideals of quotient algebras.
Suppose $I$ and $J$ are ideals of a Lie Algebra L.
I know that we have the fact that:
$\frac{I+J}{J} \cong \frac{I}{I\cap J}$
Prove that the ideals of $\frac{L}{I}$ - the quotient algebra of L defined by $x + I$ $x \in
L$ are of the form $\frac{K}{I}$ where K is an ideal of L containi... |
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