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H: Integer Part of sequence convergence
I was trying to solve the following exercise.
If $(a_n) \in \mathbb{R}$ and $(a_n)\rightarrow {1}/{2}$ show that $[a_n] \rightarrow 0$ , where $[~]$ the integer part.
I was trying to solve it using the ε-definition. $\forall ε>0 ~ \exists n_0 \in \mathbb{N} : |a_n - 1/2|< ε ~,~... |
H: Direct product of Sylow subgroups
Proposition II.7.5 of Hungerford's Algebra goes
Proposition 7.5. A finite group is nilpotent if and only if it is the direct product of its Sylow subgroups.
Let $G = (\mathbb{Z}_6, +)$. $G$ is abelian, so it is nilpotent and thus by the proposition is the direct product of its Sy... |
H: Sum of the Series , Calculus Homework
I put this into WolfRam and got e^(3/5), but I am trying to figure out how to arrive to that answer?
AI: We know, $$e^x=\sum_{0\le n<\infty}\frac{x^n}{n!}$$
Here $$\frac{3^n}{5^n\cdot n!}=\frac{\left(\frac35\right)^n}{n!}$$ |
H: Looking into mappings
I'm interested in looking at mappings of functions.
For example, how would I come up with a function $f:x\in[0,\infty)|\longmapsto\ (-\infty, 0]$ where $f(x)=x^2.$ Basically I want to glue every point to the right of this function to the left.
AI: HINT: Construct a function mapping positive el... |
H: Is there a possibility for two different primes to have any of its powers to be the same?
Say there are two primes $P_1$ and $P_2$ where $P_1 \neq P_2$. Is there a possibility for some $m$, $n$ ($m \neq 0, n \neq 0$) such that $P_1^m = P_2^n$.
AI: No. Because then $P_1$ divides both sides. And if a prime divides ... |
H: Example of two series with certain properties?
Find 2 series $\sum a_k$ and $\sum b_k$ such that $\sum b_k$ converges conditionally, $\dfrac{a_k}{b_k} \rightarrow 1$ as $k \rightarrow \infty$, and $\sum a_k$ diverges. Can someone give me a hint with this? Thanks.
AI: Let$$b_n=\dfrac{(-1)^n \sqrt{2}}{\sqrt{n}}, \;\;... |
H: Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$.
Given $A$ and $B$ positive-definite matrices and $Q$ unitary matrix, prove that if $A = BQ$, then $A=B$.
$Q$ is unitary, so $QQ^*=I$
If $A$ and $B$ are positive-definite, than $A=A^*$ and $B=B^*$.
$A^*=(BQ)^*=Q^... |
H: Prove if $\{t_n\}_{n\in\mathbb{N}}\to t$ and $t_n\geq 0\forall n\in\mathbb{N}$, then $t\geq 0$
I've been working on some sequence practice problems in Steven Lay's Introduction to Analysis With an Introduction to Proof for my introductory real analysis course, as we are starting our unit on sequences next week. I e... |
H: The order of equalities
First note that I am not a mathematician. I do use it for my studies, but I am not reading anything remotely complicated in regards to maths. That said, here is my question:
Today I found myself wanting to write a probability, first in terms of a fraction and then in terms of a decimal value... |
H: prove that $T''$ is not injective (difficult computation)
Let $T:c_0 \to c_0$ defined by $T(\{s_j\}_j)=\{s_{j+1}-s_j\}_j$. Prove that $T''$ is not injective.
I tried even knowing that $(c_0)' \sim l^1$,in fact if $F\in (c_0)'$ then there exist $s=(s_j)\in l^1$ such that $F=F_s$ and $F_s(t_j)=\sum s_jt_j$ , and simi... |
H: Why is the Uniform Boundedness Theorem not true for all normed vector spaces?
A tentative statement of the Uniform Boundedness Theorem for any normed vector space would be
Let $(T_n)$ be a sequence of bounded linear operators $T_n:X\to Y$ such that $(\|T_n x\|)$ is bounded for every $x\in X$. Then the sequence of... |
H: Proof of natural log identities
I need to prove a few of the following identities from a real analysis perspective- this means I do not have access the $\ln e^2 = 2$ type definition of the log function. I am developing the log function from the definition $log x = \int_1^x \frac1t \mathrm dt$ for $0 < x$.
I need to... |
H: Calculating 6 decimal digits of $3^{\sqrt2}$ using a calculator.
How can we calculate $3^{\sqrt2}$ to 6 decimal digits, using only a
semi-basic calculator (Which has the square root too) and a pen and
paper?
I asked this question from my teacher and he gave me a hint: "Compute the binary digits of $\sqrt2$." ... |
H: Find the Bounding Rectangle of Rotated Rectangle
I have rectangle with co-ordinates(x1,y1) and (x2,y2) and I have to rotate the rectangle an amount of θ about it centre using Rotation Matrix
| cosθ sinθ |
| -sinθ cosθ |
I need to find the co-ordinates of bounding rectangle after rotation.
Before rotation
0,0... |
H: transpose of the exponential operator
Let $X$ be a Banach space and $T:X\to X$ be a continuous and linear operator. What is the transpose operator of $e^T?$
I would like to prove that $e^{T'}=(e^T)'$. At least that equality make perfect sense because they are defined in the same domain. But if $f\in X'$
$$ (e^T)'f=... |
H: Why is normalization of inequalities possible?
I have seen, in many proofs for inequalities, the author does something called normalization. I believe this is only possible for homogeneous inequalities. I saw this in a proof of Nesbitt's inequality:
$$\frac{a}{b+c} + \frac{b}{a +c} + \frac{c}{a+b} \geq \frac32$$
Th... |
H: How do I draw a diagram for a function space?
If one considers a single function, then one can just draw its diagram as a Cartesian product. So it's relatively easy to contribute one's intuition to an argument.
However, when it is a function space, I completely lose my intuition. And I am sure this way of studying ... |
H: Books about manifolds?
I would like to learn about manifolds. Please can someone recommend me a good book to learn about manifolds?
AI: I like
Introduction to Smooth Manifolds by John M. Lee.
The wikipedia article on manifolds is also quite nice and contains a number of references. |
H: Isomorphism of quotient ring of polynomial ring
Are $F_3[x]/(x^2-2)$ and $F_3[x]/(x^2-2x-1)$ isomorphic?
I know that $x^2-2$ and $x^2-2x-1$ are irreducible but how to determine if they are isomorphic or not?
Here, $F_3$ means finite field of order 3.
AI: One of the most important facts about finite fields is that a... |
H: 1-F(x) as F(x) goes to 1
I stumbled into the following statement and I am not sure how to prove it.
The statement is :
Given a distribution function $F(x)$ we can represent the survival function $1-F(x)$ as $-logF(x)$ for $F(x)\rightarrow 1$.
More explicitly $1-F(x)\sim{-logF(x)}$ as $F(x)\rightarrow 1.$
Any refer... |
H: How to really understand the tensor algebra?
If $V$ is a vector space over $F$, then we define $T^r_0(V)=V^{\otimes r}$, then we define the algebra of contravariant tensors to be
$$T(V)=\bigoplus_{r=0}^\infty T^r_0(V)$$
together with the tensor product. But I'm really confused with that. The set $T(V)$ by construct... |
H: if $(ab)^{3}=e\Rightarrow(ba)^{3}=e$, this is true?
for $a,b\in G$ (G is a group)
I have to prove that $if (ab)^{3}=e\Rightarrow(ba)^{3}=e$ or to give an example that this is false...
I belive it false and I try to find an exaple - please help and tell if I'm right (and I'd like to get an exaple) or I wrong...
Than... |
H: What's wrong with this argument that $[0,1]$ is countable?
Every real number in $[0,1]$ has a decimal expansion $0.d_1d_2d_3...$, so construct an infinite tree rooted at 0 where each node has branches leading to $\{0,1,2,3,4,5,6,7,8,9\} $, and let each path through the tree represent the successive decimal digits o... |
H: Is $ord(a)=1$ equivalent to $a=e$?
I write at my notebook:
$ord(a)=?? \Leftrightarrow a=e$
and I forgor to write the number after $ord(a)$, I guess that it was "1", I'm right?
Thank you!!
AI: yes. By definition of order you can "feel" that. |
H: Show that $\{ 1, 1-x , 1-2x + {1 \over 2} x^2\}$ a orthogonal system
I have to show that the following set:
$$A = \left\{ 1, 1-x , 1-2x + {1 \over 2} x^2\right\}$$
is orthogonal system in relative to the inner product
$$\langle f, g\rangle = \int ^\infty_0 f(x)g(x)e^{-x}dx$$.
as far as I know, in order that $A$ w... |
H: Let $\,f$ be a real differentiable function defined on $\,[a,b]$,where the derivative is an increasing function
I am stuck on the following problem:
Let $\,f$ be a real differentiable function defined on $\,[a,b]$,where the derivative is an increasing function and $x_0 \in [a,b]$. Then which of the following state... |
H: Solution of the differential equation $\ddot{x}(t)=\alpha\dot{x}(t)x(t)+\beta x(t)^3$
I have to solve the following nonlinear differential equation:
$$\ddot{x}(t)=\alpha\dot{x}(t)x(t)+\beta x(t)^3$$
with initial conditions:
$x(0)=x_0$ and $\dot{x}(0)=x_1$
Is it possible to solve it without the use of numerical tech... |
H: is the number algebraic?
Is the number $\alpha=1+\sqrt{2}+\sqrt{3}$ algebraic?
My first attempt was to try a polynomial for which $p(\alpha)=0$ for some $p(x)=a_{0}+a_{1}b_{1}+\cdots +b_{n-1}x^{n-1}$ i. e $x=1+\sqrt{2}+\sqrt{3}$ and then square it many times to get rid of the irrationals. This procedure was futile.... |
H: Necessary and sufficient conditions for the embeddability of a semigroup in a group
According to wikipedia,
The first set of necessary and sufficient conditions for the embeddability of a semigroup in a group were given in (Malcev 1939).[5] Though theoretically important, the conditions are countably infinite in n... |
H: Show that $1<\sqrt{1+x^3}<1+x^3$ for $x>0$
The problem:
Show that $1<\sqrt{1+x^3}<1+x^3$ for $x>0$
How do I solve this using the Fundamental Theorem of Calculus
AI: As other people have pointed out, you certainly don't need the Fundamental Theorem of Calculus to prove these inequalities, but if you do want to use... |
H: Continuous function: $ Z=${$x \in \mathbb{R} : f(x) = 0$} is closed
Consider $f: \mathbb{R} \to \mathbb{R}$ a continuous function. Show that the set $Z=${$x \in \mathbb{R}: f(x) = 0$} is closed
My attempt:
My idea is to show that if $a \in \bar{Z}$ (the set of adherent points), then $f(a) = 0$. Considering only th... |
H: Ring-Homomorphism from $\mathbb{Z}_{2}$ to $\mathbb{Z}_{2n}$
Let $n$ be a positive integer. Then the problem is to show that there is a ring-homomorphism from $\mathbb{Z}_{2}$ to $\mathbb{Z}_{2n}$ if and only if $n$ is odd.
My effort : let $\phi$ be such a ring homomorphism (apart from the zero-map). then if $\phi(... |
H: select the min of a set of sums
I want to the selection of the minimum element in a set of sums; I have a cluster with n elements. In this set, we want to select whats called the clustroid. The clustroid is defined to be the element whos sum of the distance to every other element in the cluster is the minimum. The ... |
H: Is $\int\frac{\cos^5x\sin^3x}{1+\cos2x}dx = \frac{\sin^4x}{8}-\frac{\sin^6x}{12} +C$ or $\frac{\cos^6x}{12}-\frac{\cos^4x}{8} +C$?
$\int\dfrac{\cos ^5x\sin ^3x}{1+\cos 2x}dx = \dfrac{\sin ^4x}{8}-\dfrac{\sin ^6x}{12} +C$ or $\dfrac{\cos ^6x}{12}-\dfrac{\sin ^4x}{8} +C$?
$\int\dfrac{\cos ^5x\sin ^3x}{1+\cos 2x}dx$ =... |
H: Sum of real roots of the equation $x^2 + 5|x| + 6 = 0$?
Sum of real roots of the equation $x^2 + 5|x| +6 = 0$
AI: Hint: It $r$ is a solution then so is $-r$
Note: After seeing Ryan's answer, I realized that the solution set is empty. Thus, Ryan's answer is the correct answer.
Now if we are looking for solutions i... |
H: Calculating derivative
In order to solve a mathematical problem I have to calculate the following derivative:
$\frac{\delta}{\delta k}\frac{11 + \sum_{i = 0}^{k-1}i}{k}$
Does anyone know this derivative?
AI: $$f(k):=\frac{11+\sum_{i=0}^{k-1}i}h=\frac{11+\frac{(k-1)k}2}k=\frac{22+k^2-k}{2k}=\frac k2-\frac12+\frac{11... |
H: If $\lim_{x \to p} f(t)=L$ how do I prove that $\lim_{x \to p} \frac{1}{f(t)}=1/L?$
If $\lim_{x \to p} f(t)=L$ how do I prove that $\lim_{x \to p} \frac{1}{f(t)}=1/L?$
My attempt:
$|\frac{1}{f(x)}-\frac{1}{L}|=\frac{|f(x)-L|}{|L||f(x)|}$
So the only thing stopping me from going on with the proof is the $|f(x)|$ ter... |
H: Help with the chain rule $h(t)=f(t, X(t))$
Assume we have the function $h(t)=f(t, X(t)): \mathbb{R}\rightarrow \mathbb{R}$. How to I calculate $h'$?
I thought of letting $g:t \rightarrow(t,X(t))$ and then $h' = g'(t)f'(g(t)) = (1, X_{t})f'(t,X(t))$ but it is not a scalar... so where is the problem?
AI: The problem... |
H: Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$
Is it true that if $X$ is connected, then for every nonempty proper subset $A$ of $X$, we have $\mathbf{Bd} \ne \emptyset$? Does the converse hold?
I start by trying to understand what $\mathbf... |
H: How to get the Galois group for $\mathbb{R}/\mathbb{Q}$
For the real number field $\mathbb{R}$ and the rational number field $\mathbb{Q}$, how to get the Galois group ${\rm{Gal}}(\mathbb{R/Q})$?
AI: Let $f\in Aut_{\mathbb{Q}}(\mathbb{R})$.
Hints:
1) $\forall a,b\in\mathbb{R} [a\leq b \rightarrow f(a)\leq f(b) ]$. T... |
H: Two graphs with the same number of edges and vertices but not isomorphic?
Can anyone give me an example of two graphs that have the same number of edges and vertices but is not isomorphic?
AI: HINT: Any tree with $n$ vertices has exactly $n-1$ edges. Find two non-isomorphic trees with the same number of vertices, a... |
H: Applying prices to augmented matrices
The question is as follows (translated):
A company wants to rent 20 buses. These 2 buses are to hold 1000 people. They can choose between 3 types, 30, 40 and 60 man buses. How many of each kind can the company rent to satisfy the constraints? Solve with Gauss-Jordan.
So I sol... |
H: Galois extensions of Local fields
Let $ L / K $ be a Galois extension of local fields.
My question: why $L / K$ is necessarily of finite degree ??
thanks.
AI: Assuming the inclusion $K\hookrightarrow L$ is continuous, so that $L$ is a topological vector space over $K$, this is true, though the condition that $L/K$... |
H: Analytical solution of nonlinear ordinary differential equation
I have following first order nonlinear ordinary differential and i was wondering if you can suggest some method by which either i can get an exact solution or approaximate and converging perturbative solution.
$$
\frac{dx}{dt} = 2Wx + 2xy - 4x^{3}
$$
$... |
H: tautologies and contradictions with $r$
I'm really struggling to understand tautologies and contradictions.
I've been able to do $(p \rightarrow q) \leftrightarrow (\lnot q \rightarrow \lnot p)$ and I understand why it is a tautology, however I don't understand when they involve the character r. For example,
$$p \... |
H: Simplifying square root with fraction
I'm not sure about this equality $$4(-3+\sqrt {15})/4)^2 = (9-6 \sqrt{15} +15)/4$$
Hope some one can enlighten me. I will be facing more of such fractions, please guide me on how to solve/simplify in easy method.
Thanks :)
AI: I’ll even finish the simplification:
$$\begin{alig... |
H: Are critical points fixed?
Let $M$ be a smooth manifold (compact, connected, without boundary and oriented if you wish) with a smooth action of $S^1$. Let $f:M\rightarrow\mathbb{R}$ be an invariant function $f$. I know how to prove that a fixed point of the action is a critical point of $f$. What I don't know is if... |
H: Monoids as categories; does this construction have a name?
We can view a monoid $M$ as a category with a single object. However, there is another way to make $M$ into a category. Take the elements of $M$ as objects, and define $\mathrm{Hom}(x,y)$ to be set of all triples $(x,y,a)$ such that $ax=y$. Define compositi... |
H: $\mu(A_i)>0$ for countably many $i$
Let $\mu$ be a $\sigma$-finite measure on a $\sigma$-Algebra $\mathcal A$ and $A_i\in\mathcal A$ ($i\in I$) subsets with $A_i\cap A_j=\emptyset$ for $i\neq j$.
Then $\mu(A_i)>0$ for at most countably many $i\in I$.
I know $\sigma$-finite means that there exists a sequence $(A_n)\... |
H: Limit of subtracting fractions from 1
Suppose you have the sequence of fractions $\left\{\frac{1}{a} : a \in \mathbb{N}\right\}$ ($\frac{1}{2},\frac{1}{3}$ and so on).
Now you start with $1$ and subtract every item of the sequence as long as the result is larger than $0$. You would start with subtracting $\frac{1}{... |
H: Number of elements in a group and its subgroups (GS 2013)
Every countable group has only countably many distinct subgroups.
The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have considered some counterexample only like $(\mathbb{Z}, +)$, $(\mathb... |
H: Urn problem - black and white balls
I have a problem with the following exercise:
We have an urn with one black and one white ball. At time 1 you take one of the balls from the urn randomly. Then you take this ball and replace it by two balls of the same colour. For example you take one white ball, then you replace... |
H: Infinite line is closed in $\mathbb{R}^n$
I have been reading the book "Elements of the functional analisys", by Kolmogorov and Fomin. At the chapter of Normed Linear Spaces, page 73 to be precise, the author makes the following definitions:
A linear mainfold $L$ in a normed linear space $\mathbb{R}$ is any set of... |
H: Generalizations of the quadratic formula
The quadratic formula can be used to find the roots of any quadratic polynomial of the form $ax^2 + bx + c$:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The derivation is simple enough and uses a technique called completing the square.
Is there a formula to solve cubic equati... |
H: Expected Value Word Problem
I have another problem, that is so:
The probability that a roulette wheel stops on a red number is 18/37. For each bet on “red” you are returned twice your bet (including your bet) if the wheel stops on a red number, and lose your money if it does not.
If you bet $1 on each of the 10 con... |
H: Proof by induction Involving Factorials
My "factorial" abilities are a slightly rusty and although I know of a few simplifications such as: $(n+1)\,n! = (n+1)!$, I'm stuck
I have to prove by induction that:
$$\sum_{i=1}^n\frac{i-1}{i!} = \frac{n!-1}{n!}$$
I get so far as:
$$\frac{k!-1}{k!} + \frac{(k+1)-1}{(k+1)!} ... |
H: Transitive closure
Given $M=\{n\in\Bbb Z: 0\le n\le 30\}$ find the transitive closure of the relation $R\subset M\times M$ defined by $R=\{(n,m): m=3n+1\}\cup\{(8,16)\}$
So, I know that a transitive closure is the least possible subset that $R$ takes to be transitive.
But, what can I do to know the pairs that are m... |
H: $(a+b)^p = a^p+b^p$ if $p$ prime and $a,b \in \mathbb{F}_p$
Can someone please explain why
\begin{align}
(a+b)^p = a^p+b^p
\end{align}
if $p$ is prime number and $a,b \in \mathbb{F}_p$
I tried to proof it that way
\begin{align}
(a+b)^p = \sum_{j=0}^{p}{p \choose j}a^{p-j}b^j = a^p+b^p + \sum_{j=1}^{p-1}{p \choose j... |
H: On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?)
The Stone-Weierstrass Theorem tells us that we can approximate any continuous $f:\mathbb{R}^n\to\mathbb{R}$ arbitrary well on a compact subset of $\mathbb{R}^n$ by some polynomial. Suppose that $f$ is continuous... |
H: Show that $\sqrt{2+\sqrt{2+\sqrt{2...}}}$ converges to 2
Consider the sequence defined by
$a_1 = \sqrt{2}$, $a_2 = \sqrt{2 + \sqrt{2}}$, so that in general, $a_n = \sqrt{2 + a_{n - 1}}$ for $n > 1$.
I know 2 is an upper bound of this sequence (I proved this by induction). Is there a way to show that this sequence c... |
H: Simple question about parametric equations of a plane in 3D
I'm quite rusty in Linear Algebra.
If you have a plane in 3D with the equation $z=2$, what does $x$ and $y$ equal? Does $x=t$ and $y=t$?
Because if I graph that in Wolfram Alpha, I don't get a horizontal plane in 3D at $z=2$: http://www.wolframalpha.com/in... |
H: Equivalence relation class $\bar{0}$
In the set $\mathbb{Z}$ we define the following relation:
$$a\Re b \iff a\equiv \bmod2\text{ and }a\equiv \bmod3$$
1)Prove that $\Re$ is an equivalence relation. (Done)
2) Describe the equivalence class $\bar{0}$. How many different equivalence classes exist?
My thought on $\ba... |
H: Independence of random variables measure theory
I wish to show that for two random variables $X$ and $Y$, the condition $P(X\leq x, Y\leq y ) = P(X\leq x)P(Y\leq y)$ implies that X and Y are independent.
I am approaching this problem from a measure theoretic perspective. So in particular I can write that $P(X\leq ... |
H: Prove $\lim\{s_n\}=+\infty\iff\lim\left(\frac{1}{\{s_n\}}\right)=0$
Prove the following: Given that $\{s_n\}$is a sequence of positive numbers. Then $$\lim s_n=+\infty\iff\lim\left(\frac{1}{s_n}\right)=0$$
My attempt at proving this:
For $\lim s_n=+\infty\implies \lim\left(\dfrac{1}{s_n}\right)=0$, suppose $\lim ... |
H: Dimension of a vector space-regarding
I want to find the dimension of the vector space $V=\{u\in \mathbb{R}^3:Mu^{t}=u^{t}\}$, where $M=\begin{pmatrix} 1&0&0\\ 0&\cos \theta& -\sin \theta\\ 0& \sin \theta& \cos \theta\end{pmatrix}$, $0<\theta<\dfrac{\pi}{2}$.
I feel no vector $u$ except $(0,0,0)$ will satisfy this.... |
H: cardinality of $S_{\mathbb{N}}$
When I am proving something, I got a doubt. what is the cardinality of $S_{\mathbb{N}}$, the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$?
I hope it is countable, because that will make my life easier.
Thanks in Advance.
AI: It is not countable: it is $2^{\aleph_0}=\mathfr... |
H: How to convert parametric form to a single algebraic equation?
I'm pretty sure this is impossible to do but here is my attempt.
Parametric form:
$$x=1+t\\y=2+2t\\z=3+3t$$
Attempt:
$$(x,y,z)=(1+t,2+2t,3+3t)$$
That didn't really get me anywhere, so here I tried to put it into symmetric form:
$$x-1=\frac{y-2}2=\frac{z... |
H: Combinatorial Proof Of A Number Theory Theorem--Confusion
I came across a combinatorial proof of the Fermat's Little Theorem which states that
If $p$ is a prime number then the number ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$.
Let me write down the proof.
PROOF
We have pearls of $a$ colours .... |
H: Taking "Absolute Value Operator" as a common factor?
If I have an equation like this and Im trying to solve for X
|x| + 4|x| = 40
Can I take the absolute Value (Modulus) as a common factor?
|x + 4x| = 40
and the proceed to solve for X?
AI: In general $|a|+|b|\ne|a+b|$
See here, for the general relation
We can wri... |
H: Why does $e$ seem to be an intuitive number?
I often find two numbers roughly "in the same ballpark" if they are within a factor of about $e$ of each other. For example, if I know computers generally cost upward of $\$1000$, then $\$2700$ would probably be the most I would a priori feel is a reasonable upper limit ... |
H: Normal Subgroups and their Qualities
Hope all you are healthy and in peace
Suppose N is a normal subgroup of G. If every subgroup of N is normal also in G, could we deduce that centralizer of N in N is N itself ?
AI: Short answer: no. Suppose $G$ is the Quaternion group and $N = G$. Then $N$ is normal in $G$. Every... |
H: Convergence of a function and interval topology
Let $(X,\leq)$ be a well-ordered set that contains exactly one element $x$ such that $y<x$ for uncountably many values of $y\in X$. Let $f(x)=1$ and $f(y)=0$ for all other values of $y\in X$. For the interval topology $\tau$ on $X$, show that for every sequence $u_n\t... |
H: Meaninig of a symbol at the Circle Group
I have $U_{20}$ (at the meainig of the Circle Group),
What is the meaning of $W_{20}^{8}$??
What is the 20 and what is the 8?
Thank you!
AI: Assuming you mean $W = \omega$, then I suspect $\;\omega^8_{20}\;$ denotes the $8$th root of unity in $U_{20}= \left\{\omega^k_{20} = ... |
H: Rate of change of radius and volume
They put a gas bubble in someone's eye. The volume of a gas bubble changes from $0.4$ $cc$ to $1.6$ $cc$ in $74$ hours. Assuming that the rate of change of the radius is constant, find
(a) The rate at which the radius changes;
(b) The rate at which the volume of the bubble is ... |
H: Balls and bins with 2 balls and 2 bins
I'm trying to understand a proof in a paper I'm reading. It relies on a balls and bins problem.
Here is what I'm trying to figure out: We want the maximum number of balls in a bin. We have 2 balls and 2 bins. We have a lower bound of 3/2. How do I see that the lower bound is 3... |
H: The set $F_1\subset X_1$ is closed set in $X_1$ if and only if there is an closed set $F$ in $X$ such that $F\cap X_1=F_1$
I need th proving this theorem:
Let $(X, d)$ is metric space and let $(X_1, d_1)$ is its subspace. The set $G_1\subset X_1$ is open set in $X_1$ if and only if there is an open set $G$ in $X$ s... |
H: Proving A Trigonometric Identity- Double Angles
$(\cos(2x)-\sin(2x))(\sin(2x)+\cos(2x)) = \cos(4x)$ I'm trying to prove that the left side equals the right side. I'm just stuck on which double angle formula of cosine to use.
AI: $$(\cos(2x)-\sin(2x))(\sin(2x)+\cos(2x))=(\cos^2(2x)-\sin^2(2x)) = \cos(4x)$$
From
$$\c... |
H: Does a point lie on a line with a parametric equation
Does the point $(0, 5, 5)$ line on the line with the parametric equations:
$x = 3 - t\\y = 2 + t\\z = 2 + 2t$
This is the first time I see one of these, right now I assume it is as simple as solving $t$ and plugging it into the equations as such
if $x = 0$ then ... |
H: Easy Probability Question (Independent Events)
Suppose I have two doors. One of them has a probability of $1/9$ to contain X, the other has a probability of $2/3$ to contain X. Then, supposing I pick randomly one of the two doors, what is the probability that it contains X?
(If one contains X, the other can also co... |
H: Proving that $\sum_p\frac{1}{p+1}$ diverges
How does one prove
$$\sum_{p\in\Bbb P}\frac1{p+1}=\infty.$$
Where $\Bbb P$ denotes the set of prime numbers.
I have attempted forming an inequality by playing around with Euler's work on the reciprocals of primes. Robjohn showed me an inequality in chat that I do not un... |
H: Trigonometry- Double Angle Trigonometric Equations
$$\tan(3x)=((\tan x(3-\tan^2x))/(1-3\tan^2x))$$ I'm just having a hard time seeing where the double angle formula fits in with verifying this equation.
AI: Using a Adicional formula: $$\tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y},$$ we have:
$$\tan(2x+x)=\frac{\t... |
H: The most complete reference for identities and special values for polylogarithm and polygamma functions
I am looking for a book, paper, web site, etc. (or several ones) containing the most complete list of identities and special values for the polylogarithm $\operatorname{Li}_s(z)$ and polygamma $\psi^{(a)}(z)$ fun... |
H: Probability that $a^2 \equiv 1 \pmod{10}$ when $a$ is chosen randomly from a set
Out of the set $\{1,2,...,n\}$ we choose randomly a number $a$. Find the probability $p_n$ that $a^2 \equiv 1 \pmod{10}$, and find $\lim_{n \to \infty} {p_n}$
Ideas anyone?
AI: All squares mod $10$ end with a $0,1,4,5,6,9$. We know ... |
H: Is this equation with two unknowns solvable?
Can this equation be solved? If so how?
I would like to find both $X$ and $Z$ .
$4.33=\dfrac{0.4397-Z}{X-0.4397}$
where $Z$ is known to be in the range of $0.1931$ to $0.2352$
and $X$ is known to be in the range of $0.3549$ to $0.5576$
Ps. I am new here so if I could phr... |
H: Prove that $\{z\in \mathbb{C}, |z|=1\}\cong\mathbb{R}/\mathbb{Z}$
How to prove that $T=\{z\in \mathbb{C}, |z|=1\}$, $T\cong\mathbb{R}/\mathbb{Z}$?
AI: Let $\phi: \mathbb{R} \to \mathbb{C}$ be given by $\phi(t) = e^{2 \pi i t}$.
Suppose $\phi(s)=\phi(t)$, what does that say about the relationship between $s$ and $t$... |
H: How can the derivative of arc length be anything except zero?
If a and b are constants, then the above definite integral (arc length) has to be some constant.
How can the derivative of s be anything except zero (this is contradicted in the blue box)? This method of evaluation is for line integrals by the way.
AI:... |
H: Showing that $\mathcal{M} \preccurlyeq \mathcal{N} \implies \mathcal{M} \equiv \mathcal{N}$.
Suppose that $\mathcal{M} \preccurlyeq \mathcal{N}$. Then by definition we have that $\mathcal{M}$ is a substructure of $\mathcal{N}$ s.t. for any (possibly empty) tuple $\overline{a}$ from $M^n$ and for any $\phi \in L$, ... |
H: When does $\mathbb E_\mathbb P[X]=0$ imply $\mathbb E_\mathbb P[X\mid\mathcal E]= 0$ $\mathbb P$-a.s.
Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $\mathcal E\subseteq\mathcal F$ a sub-$\sigma$-algebra, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. My question is, when does $\mathbb... |
H: open cover and boundedness
Use the open cover characterization of compactness to prove that if $f:[a,b]→X$ is a continuous function and $X$ is a metric space, the $f$ is bounded.
Proof:
To show $f$ is bounded we must show that $f([a,b])$ is a bounded subset of $X$. So if $f([a,b]) \subseteq X$ then $[a,b]$ is compa... |
H: Expectation throwing balls into boxes
I was thinking about this situation:
Suppose there are $n$ boxes. In each box we randomly throw one of the balls numbered $1,2,\ldots,k$, independently of other boxes. Let $X$ be the number of boxes with ball number $1$. What is $E[X]$, and what is $E[\dfrac{1}{X}\mid X>0]$?
I... |
H: Random variables $X$ and $Y$ that are not independent
I am trying to solve a stat question, here is the question:
Give an example of two random variables $X$ and $Y$, each taking values in the set $\{1,2,3\}$ such that $P(X = 1; Y = 1) = P(X = 1)P(Y = 1)$, but $X$ and $Y$ are not independent.
A friend says that the... |
H: Compute $\langle 5 \rangle$ in integers
Compute $\langle 5 \rangle$ in integers.
I thought the answer would have been $$\{5^n| \text{ for $n$ in integers }\}$$
However my teacher has marked it $$\{5*n|\text{ for $n$ in integers }\}$$
What have I done wrong?
AI: The integers do not form a group under multiplicat... |
H: Computing $E[1/X]$ for binomial $X$
Let $X$ be a binomial distributed with parameters $n$ and $p=1/k$. Then we have $E[X]=n/k$. But what is $E[\dfrac{1}{X}\mid X>0]$? Is there a nice closed formula getting the exact value or approximation?
AI: I don't know a closed formula for this, but you could simply calculate i... |
H: eigenvalues of $AB$ are eigenvalues of $\sqrt{B} A \sqrt{B}$
Suppose $A,B$ are symmetric positive definite matrices. An author claims that the spectrum of $AB$ is the spectrum of $\sqrt{B}A\sqrt{B}$. Why?
Certainly they have the same trace by cyclic permutation. And they have the same determinant by multiplicativit... |
H: Line integrals giving different values depending on what it is integrated against?
For example, take a look at the solution to 4:
Considering the interval is always the same (from x=1 to x=8), how can it give different values depending on what you integrate against?
AI: It's probably easiest to explain it startin... |
H: Does a differentiable $f[g(x)]$ imply a differentiable $g(x)$ or the reverse?
Does a differentiable $f[g(x)]$ imply a differentiable $g(x)$?
Does a differentiable $g(x)$ imply a differentiable $f[g(x)]$?
Thanks in advance
AI: Not at all. Take $f$ to be the $0$ function, so that $f\circ g$ is readily differentiable,... |
H: What is the closure of these sets?
I am working through the problems in Topology by Munkres. This comes from Section 17, #17 on page 101.
Consider the lower limit topology on $\mathbb{R}$ and the topology given by the basis $\mathcal{C}$, where $\mathcal{C} = \{[a,b)\text{ such that }a \lt b, a\text{ and }b \text... |
H: Increasing in each point implies increasing.
$\newcommand{\R}{\mathbb R}$ Let $f: \R \to \R$ be a function. We say $f$ is increasing in $p \in \R$ iff $\exists \delta > 0 : x \in (p-\delta,p), y \in (p,p+\delta) \Rightarrow f(x) \leq f(p) \leq f(y)$. We say $f$ is increasing if $f(a) \leq f(b)$ if $a \leq b$.
If... |
H: Proving the interior of a set [Homework]
I have to find the interior of the following set:
$E = [0,5] \cup (5,7)$, and prove it.
I found the union of the set to be $[0,7)$, and thus found the interior to be $(0,7)$.
I'm not sure how to prove this is true using the definition of a closed set i.e. there exists a delt... |
H: How much math does one need to know to do philosophy of math?
I'm looking for advice from mathematicians who also study philosophy of math (PoM). Due to interest I'd like to study PoM as a hobby, but I'm worried if I don't understand math well enough from a pure math perspective I will make errors in reasoning abou... |
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