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H: Question involving an invertible complex matrix.
Let $A$ be an $n \times n$ invertible matrix with complex entries and call $A = R + iJ$, where $R$ is the real part of $A$ and $J$ is the imaginary part of $A$. Show that there exist a $\lambda_0 \in \mathbb{R}$ s.t $R+ \lambda_0J$ is invertible. Futhermore, conclude... |
H: Show that if $|z| < 1$, then $\displaystyle\frac{z}{1+z} + \frac{2z^{2}}{1+z^{2}} + \frac{4z^{4}}{1+z^{4}} + \frac{8z^{8}}{1+z^{8}} + \ldots$
Show that if $|z| < 1$, then
\begin{align*}
\frac{z}{1+z} + \frac{2z^{2}}{1+z^{2}} + \frac{4z^{4}}{1+z^{4}} + \frac{8z^{8}}{1+z^{8}} + \ldots
\end{align*}
converges
MY ATTEMP... |
H: Does the following series converge or diverge: $\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$?
Which of the following series converge, and which diverge?
$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{2}+1}$
$\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$
$\displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^{2}+n}}$
MY A... |
H: Finding minimal sufficient statistic and maximum likelihood estimator
Question:
Let $Y_1, \dots, Y_n$ be a random sample from a distribution with density function
\begin{align}
f_Y(y ; \alpha, \theta) =
\begin{cases}
\alpha e^{- \alpha ( x - \theta)} & x > \theta,\\
0 & x \leq \theta,
\end{cases}
\end{align}
where ... |
H: Given cups that are $\frac12$, $\frac13$, $\frac14$, $\frac15$, $\frac18$, $\frac19$, $\frac1{10}$ full, can we pour to get a cup $\frac16$ full?
There are seven cups, $C_1$, $C_2$, $\ldots$, $C_7$ and they have the same capacity $V$.
Initial:
Water of $C_1$ occupies $\frac{1}{2}V$
Water of $C_2$ occupies $\frac{... |
H: If $W=\{x \in R^4|x_3=x_1+x_2,x_4=x_1-x_2\}$ show that $W$ is or is not a subspace
If $W=\{x \in \mathbb R^4|x_3=x_1+x_2,x_4=x_1-x_2\}$ show that $W$ is or is not a subspace
I would imagine that vector $x = (a,b,c,d)$ and to show that something is a subspace it has to be closed under addition and scalar multiplicat... |
H: What is the basis for subspace: $W=\{x \in R^4|x_3=x_1+x_2,x_4=x_1-x_2\}$
What is the basis for subspace: $W=\{x \in \mathbb{R}^4|x_3=x_1+x_2,x_4=x_1-x_2\}$?
I previously posted a similar question regarding showing whether this is a subspace but now I wish to find what the basis is.
I know if we have a linear combi... |
H: Abstract algebra olympiad problem for university student
I am looking for abstract algebra problem book for preparring olympiad for university student, any recommendations?
AI: A book is "Putnam and Beyond"
https://link.springer.com/book/10.1007/978-0-387-68445-1 |
H: Couldn't understand a step in solving Homogenous Linear Recurrence relations
I was reading a Wiki on Brilliant.org regarding linear recurrence relations. They have mentioned that, "note that if two geometric series satisfy a recurrence, the sum of them also satisfies the recurrence".
And I am stuck there! How do c... |
H: Smooth approximation identities
Let $G$ be a Lie group. Let $\mathcal U$ be a neighbourhood base at $1$ of G. Does there exist $\{\psi_U:U\in\mathcal U\}$ with the following properties support of $\psi_U$ is compact and contained in $U,$ $\psi_U\geq 0,$ $\psi_U(x^{-1})=\psi_U(x)$, $\int_G\psi_U(x)dx=1$ and $\psi_U$... |
H: Field extension: a conundrum
In the following, I will have a conclusion that is definitely wrong but I don't know why. Your answer and explanation will be greatly appreciated.
Let $Q$ be the field of rational numbers. We know the equation $x^3-2=0$ has three solutions $\sqrt[3]{2}$, $\sqrt[3]{2} \omega$, and $\sqrt... |
H: Prove that the set of all positive integers less than $n$ and relatively prime to n form a group under multiplication modulo n
I came across the problem
Prove that the set of all positive integers less than $n$ and relatively prime to n form a group under multiplication modulo n.
Proving the associativity of mult... |
H: Proving that $f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2}$ is convex
I'm trying to prove:
$f$ is a continuous real function defined in $(a, b)$ such that $$f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2}$$ for all $x, y \in (a, b) \implies f$ is convex.
I was also given:
A real function $f$ defined i... |
H: Let $Y$ be a dense subspace of a space $(X, d)$. Prove that if every Cauchy sequence $(y_n)⊂Y$ is convergent on $X$, so $X$ is complete.
Let $Y$ be a dense subspace of a space $(X, d)$. Prove that if every Cauchy sequence $(y_n)⊂Y$ is convergent on $X$, so $X$ is complete.
Hello,it is difficult for me to use the pr... |
H: Does having non-trivial solutions means trivial solution is also included?
If a system of linear equations have non trivial solutions according to Cramer's Rule (i.e. infinitely many solutions) then it means that zero is also one of it's solutions (since it has INFINITELY many solutions). Now zero is a trivial solu... |
H: Generalized Euclid Lemma
Could anyone check my proof? Thanks in advance.
Statement: If $p$ is a prime and $p$ divides $a_{1}a_{2}...a_{n}$ prove that p divides $a_{i}$ for some $i$.
Proof: Suppose $p$ is a prime that divides $a_{1}a_{2}...a_{n}$ but not $a_{j}$ for all $j\ne i$. Let the set of those $j$ be $J$. Th... |
H: What base does the author take when taking the log of both sides?
I am learning exponential distribution in ThinkStats2 by Allen Downey..
It says that "if you plot the complementary CDF of a dataset that you think is
exponential, you expect to see a function like:
$$
y\approx e^{-\lambda x}
$$
Then, taking the log ... |
H: Find when $\frac{x^5-1}{x-1}$ is a perfect square?
$\textbf{Question:}$Find when $f(x)=\frac{x^5-1}{x-1}$ is a perfect square? where $x \in \mathbb N/ \{1\}$.
I tried upto certain number and somewhat convinced that $3$ is the only solution.But I failed to prove that.Here's what I got so far:
If some prime $p \mid ... |
H: showing that an function $f$ is constant
I am trying to solve the following problem.
$f(z)=u(x,y)+iv(x,y)$ is an analytic function in $D$ ($D$ is connected and open).
If $u, v$ fulfill the relation $G(u(x,y), v(x,y))= 0 $ in $D$ for some function ($G:\mathbb{R^2}\to\mathbb{R}$) with the property
$(\frac{\partial G}... |
H: Does midpoint-convexity at a point imply midpoint-convexity at a larger point?
Let $f:(-\infty,0] \to \mathbb [0,\infty)$ be a $C^1$ strictly decreasing function satisfying $f(0)=0$.
Given $c \in (-\infty,0]$, we say that $f$ is midpoint-convex at the point $c$ if
$$
f((x+y)/2) \le (f(x) + f(y))/2,
$$
whenever $(x+... |
H: What is the largest number of squares that can be cut by the sides of the triangle in this picture?
I came across a question in the Mathematics Olympiad exercise booklet for primary school students in Australia. The questions is presented in the following picture. I can't quite understand what the question is about... |
H: Sign of zeros of $\lambda^2+2\lambda+1-a+\frac{ar}{\delta}=0$ without explicit calculation
I am interested in the zeros of this polynomial in $\lambda$:
$$\lambda^2+2\lambda+1-a+\frac{ar}{\delta}=0$$
where $0<a\leq1$, $r<0$ and $\delta>0$.
How to determine the sign of their real parts without explicit calculation?
... |
H: Prove that for every integer $n$, $n^3$mod$6$=$n$mod$6$
I will use induction to prove this.
Firstly for $n=1$,
$1^3\text{mod}6=1\text{mod}6$
Now we assume that this holds for some $n=k$ and prove that if it holds for $n=k$ it should also hold for $n=k+1$.
$(k+1)^3\text{mod}6=(k^3+3k^2+3k+1)\text{mod}6$
As for $n=k$... |
H: Find the equation of the plane given a line and a point
I’m trying to solve a problem where i have to find the plane equation that contains a given straight line and a given point.
In this photo you can see the equation of the straight Line and the given point $P = (1,-2,3)$.
The answer of the problem is the last L... |
H: Rudin Theorem 1:11: understanding why $L \subset S$
Rudin's theorem 1.11 states:
Suppose $S$ is an ordered set with the least-upper-bound property, $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then $\alpha = \sup L$ exists in $S$, and $\alpha = \inf B$. ... |
H: $\phi_{i}(P)=P\left(x_{i}\right)$, $\psi_{i}(P)=P^{\prime}\left(x_{i}\right)$ for distinct $x_{1}, \ldots, x_{n}$ is basis of$({R}_{2 n-1}[x])^{*}$
I know that 2n distinct evaluation functionals $\varphi_{i}(P)=P\left(x_{i}\right)$ for 2n $x_{i}$ is a basis for $(\operatorname{R}_{2 n-1}[x])^{*}$, but I have no ide... |
H: If the function $Q(x,y)=ax^2+2bxy+cy^2$, restricted to the unit circle, attains its max at $(1,0)$, then $b=0$.
I feel confused about the following proof. Why did the author introduce $\epsilon$ in parametrizing the unit circle? To trace out the circle, one can simply use the closed interval $[0,2\pi]$. Besides, w... |
H: Find all polynomials satisfying $p(x)p(-x)=p(x^2)$
Find all polynomials $p(x)\in\mathbb{C}[x]$ satisfying $p(x)p(-x)=p(x^2)$.
We can see that if $x_0$ is a root of $p$, then so is ${x_0}^2$.
If $0<|x_0|<1$ (or $|x_0|>1$), then we have $|x_0|^2<|x_0|$ (or $|x_0|^2>|x_0|$).
So
repeating this process will give an in... |
H: Is there a formula for $\sum^{n}_{k=1}\binom{ n-k }{h }k$?
I'm trying to show $$\sum ^{400}_{m=1}\binom{ 400-m }{ 3 }m \sim 8.5 \times 10^9\,,$$ but can't seem to find a binomial coefficient identiy.
AI: Hint: Apply the Hockey Stick lemma twice.
$$\sum ^{400}_{m=1} \left[ { 400 -m \choose 3 } \sum_{i=1}^{m} 1 \r... |
H: Show that if 12 integers are chosen there are always two whose sum or difference is divisible by 20.
Also, prove that this is sharp, i.e., one can pick 11 integers so that the sum or the difference of any two of the chosen integers will never be divisible by 20.
I'm trying to solve this problem using the pigeon hol... |
H: Verify and understand Proof of Path connected implies connected
I was “reading” Pugh’s Mathematical Analysis Chapter 2. There, he defines a metric space to be path connected if there exists a continuous function $f : [a,b] \mapsto M$ for all points $(p,q)$ such that $f(a) = p, f(b) = q$.
He then proceeds to prove t... |
H: A Smarter way to solve this system of linear equations?
I am a high school student and when practicing for the SAT I stumbled across this question:
$$
\begin{eqnarray}
−0.2x + by &=& 7.2\\
5.6x − 0.8y &=& 4
\end{eqnarray}
$$
Consider the system of equations above. For what value of $b$ will
the system have exactl... |
H: Classification of groups of order 66
We have that $|G|=66=2 \cdot 3 \cdot 11$, so we have 2,3,11-Sylow. The number of 11-Sylow $n_{11}$ is such that $n_{11} \equiv 1 \ \ (11)$ and $n_{11} \mid 6$, so we have that $n_{11}=1$, and this means that the only 11-Sylow is normal in $G$. So we can say that $\mathbb{Z}_{11... |
H: Does there exist a continuous and differentiable EVEN function whose slope at zero isn't zero?
as I understand, there should not be a case, where the slope at x=0 is nonzero, if the function is even and continuously differentiable at all points including x=0.
AI: Well, just compute the derivative: If $f$ is even an... |
H: Question about Projection operators
Let $P\in L(X)$ be a projection operator, where $X$ is a complex non-trivial banach space, that is $P^2=P$ and $Q\in L(X)$ such that $Q^2=0$, then in my Functional analysis exam it was asked to see that we will have $(P+Q)^2=P+Q$, however I cannot seem to prove this , we wil hav... |
H: Proving that the set of all finite subsets of a countable set is coutable
I am trying to prove the following proposition:
Proposition: Let $S$ be a countable set. Then the set of all finite subsets of $S$ is also countable.
My approach:
If $S$ is countable that means that $S$ is finite or that $S \sim\mathbb{N}$... |
H: $f_{i}(P)=P^{(i)}\left(x_{i}\right)$ for arbitrary scalars $x_{0}, \ldots, x_{n}$ is a basis for $\left(\mathbb{K}_{n}[X]\right)^{*}$
Fix a field $\mathbb{K}$ and a nonnegative integer $n$. Let $\mathbb{K}_{n}[X]$ be the $\mathbb{K}$-vector space of all polynomials in $X$ over $\mathbb{K}$ that have degree $\leq n$... |
H: If $a,b,c$ are sides of a triangle, then find range of $\frac{ab+bc+ac}{a^2+b^2+c^2}$
$$\frac{ab+bc+ac}{a^2+b^2+c^2}$$
$$=\frac{\frac 12 ((a+b+c)^2-(a^2+b^2+c^2))}{a^2+b^2+c^2}$$
$$=\frac 12 \left(\frac{(a+b+c)^2}{a^2+b^2+c^2}-1\right)$$
For max value, $a=b=c$
Max =$1$
How do I find the minimum value
AI: In $\Delta... |
H: Probability of Two Pairs ( Cards game )
Question: Calculate the probability of getting a two pair hand ( e.g., two 8’s, two Queens, and a Knight )
My answer: The probability of getting a two pair hand is :
$$ \frac{13\cdot\binom{4}{2}\cdot12\cdot\binom{4}{2}\cdot11\cdot\binom{4}{1}}{\binom{52}{5}} = \frac{396}{4165... |
H: Distributive law for ideals
Let $A,B,C\triangleleft R$ be Ideals. prove that:
$$B\cap(A+C)\subseteq C+(A\cap B)$$
$$\Updownarrow$$
$$B\cap(A+C)=(B\cap A)+(B\cap C)$$
I managed to proved the upper part from the lower, but I am struggling to prove the second direction.
AI: One direction is straighforward as
$$(B\cap ... |
H: $p(x)$ be a fifth degree polynomial with integer coeffients that has an integral root $\alpha$. If $p(2)=13$ and $p(10)=5$
$p(x)$ be a fifth degree polynomial with integer coeffients that has an integral root $\alpha$. If $p(2)=13$ and $p(10)=5$ then find the value of $\alpha$
I am looking for various other appro... |
H: Is it true that $H(Y|X_1,\dots,X_n)=H(X_1,\dots,X_n,Y)-H(X_1,\dots,X_n)$?
Suppose $X_1,\dots,X_n,Y$ are random variables. Is it true that the conditional entropy can be expressed as the difference between the joint entropy of all variables and the joint entropy of only $X$ variables? That is:
$$H(Y|X_1,\dots,X_n)=H... |
H: Combinatorics: number of fruit baskets of size $n$
I improved a little but that expression about even number of strawberries prevented me to solve this.
How many fruit baskets are there, which should include $n$ fruits with up to 3 bananas, an even number of strawberries, and any number of pineapples and grapes?
AI... |
H: convergence or divergence of infinite rational series
Finding whether the series $$\sum^{\infty}_{k=0}\frac{5k^2+7}{8k^2+2}$$ is converges or diverges.
What i Try:
I am Trying to solve it using ratio test
Let $\displaystyle a_{k}=\frac{5k^2+7}{8k^2+2}$. Then $\displaystyle a_{k+1}=\frac{5(k+1)^3+7}{8(k+1)^2+2}$.
... |
H: Comparability with zero of an ordered semigroup
Is it correct that any ordered semigroup $S$ can be embedded into an ordered semigroup with zero $S_0$ in which every element is comparable with $0$, in a way that the order of $S$ is a subset of the order of $S_0$?
If not, is it always possible to re-order an ordered... |
H: for which a is this integral bounded
I am trying to prove that for a > -1 the following integral :
$$\int_{0}^{\infty} x^a*\lambda * \exp(-\lambda x)dx < \infty$$
with $\lambda$ > 0
Is there a criteria that I can use to do so ?
Thank you very much.
AI: You have to check boundedness at both ends of the interval.
H... |
H: Remainder when divided by $7$
What would be the remainder when
$12^1 + 12^2 + 12^3 +\cdots + 12^{100}$ is divided by $7$ ?
I tried cyclic approach (pattern method), but I couldn't solve this particular question.
AI: In the comments, you recognized that $12^1+12^2+12^3+\cdots+12^{100}$
$\equiv \underbrace{5+4+6+2+3... |
H: What is the correct name for relation like things such as $\in$ and =
= and $\in$ look like relations and you can sometimes treat them like they are. However the domain and codomain of both is the class of all sets. Is there a term for this type of thing?
AI: Ultimately, use the shortest term you don't think will ... |
H: Copula Theory : CDF from Marginals
I have given $(X,Y)$ to be a two-dimensional random vector with Exp(1)-marginals and a Copula
$C(u,v) = uv + (1-u)(1-v)uv$
I need to determine the density of $(X,Y)$, if it exists.
I would assume that it is the product of the density of the components. However, in the question it ... |
H: Why are $\sin,\cos,\tan$ continuous
I'm done with two courses in Analysis, but just can't seem to work out how I'll show the base trigonometric functions to be continuous.
Any references or indications for a simple, preferably elementary proof ?
Is it possible to do it relying only on $\epsilon$-$\delta$ arguments?... |
H: Interesting problem on determinant (and pinch of number theory)
The digits A, B and C are such that the three digit numbers A88 , 6B8
and 86C are divisible by 72, what is $$\begin{vmatrix}
A & 6 & 8\\
8 & B & 6 \\ 8 & 8 & C \end{vmatrix} \pmod {72}$$
I can (and did) find the individual digits B and C , A... |
H: Constructing the root diagram for $B_2$
I'm trying to self-teach some Lie theory, and in particular I'm trying to construct the root diagram for $B_2$. I've found 8 roots, labelled $\pi_1,\pi_2,-\pi_1,-\pi_2,\pi_1+\pi_2,-(\pi_1+\pi_2),\pi_1+2\pi_2$ and $-(\pi_1+2\pi_2)$.
To draw the root diagram in an $(x,y)$ plane... |
H: Determine if $\int_1^{\infty}\frac{dx}{x^p+x^q}$ converges ...
Determine if
$\int_1^{\infty}\frac{dx}{x^p+x^q}$
converges if $\min(p, q) < 1$ and $\max(p, q) > 1$, where $\min (p, q)$ is the minor of the numbers $p$ and $q$, and $\max (p,q)$ is the major of the numbers $p$ and $q$.
I have doubts of how to arrang... |
H: Prove that $\dim(U_1 \cap U_2 \cap ... \cap U_k) \geq n-k$ and find a case where the equality doesn't hold
Let $V$ be a finite dimensional vector space of dimension $n$. Let $1 \leq k \leq n$ and consider $U_1,...,U_k$ distinct subspaces of $V$, all of dimension $n-1$
a) Prove that $\dim(U_1 \cap U_2 \cap ... \c... |
H: Fourier Transform of char. function of $d$-dimensional unit cube
I want to find the Fourier transform of the unit cube.
So far, I have
$$f(\xi) = \frac{1}{(2\pi)^\frac d 2}\int_{\mathbb{R}^d}\chi_{[-1,1]^d}e^{-i\langle x,\xi\rangle}dx = \frac{1}{(2\pi)^\frac d 2}\int_{[-1,1]^d}e^{-i\langle x,\xi\rangle}dx$$
Now I d... |
H: Proof for the general formula for $a^n+b^n$.
Based on the following observations. That is
$$a+b = (a+b)^1 \\ a^2+b^2 = (a+b)^2-2ab \\ a^3+b^3 = (a+b)^3-3ab(a+b) \\ a^4+b^4= (a+b)^4-4ab(a+b)^2+2(ab)^2\\ a^5+b^5
= (a+b)^5 -5ab(a+b)^3+5(ab)^2(a+b)\\\vdots$$
I came to make the following conjecture as general for... |
H: When the classes of two finitely generated modules are equal in the Grothendieck group
Let $R$ be a Commutative Noetherian ring. Let $G_0(R)$ denote the Grothendieck group of the abelian category of finitely generated $R$-modules i.e. it is the abelian group generated by the isomorphism classes of finitely generate... |
H: A set problem (inspired by geometry)
Here's my problem:
say we have four sets of letters (abcdef) (abde) (abc) (ad). We can only add or subtract those sets in a way that (abc) + (ad) = (aabcd), (abcdef) - (abde) = (cf), but (abc) - (ad) is not allowed. Is it possible to get (b) only with these rules?
(inspired by a... |
H: Questions about the euclidean topology
My general topology textbook just gave the definition of euclidean topology on $\mathbb{R}$ but unfortunately did not provide any examples and I was hopping that someone here could help me with some questions I have. The definition they gave is the following:
A subset $S$ of ... |
H: Is it possible to tell that one of the coin was biased if coins are changed mid experiment?
Suppose I have two coins one is baised i.e its probality of head $\ne 0.5$ and another is a fair one with head and tails as equally likely outcome.
Now I begin tossing the baised coin first and note down the results for very... |
H: Can someone explain the limit $\lim _{n \rightarrow \infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)$?
$$\begin{aligned}
&\text { Find the following limit: } \lim _{n \rightarrow \infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}... |
H: Why do polynomials for higher degrees have large oscillations near the edge of the interval?
In regards to Polynomial Interpolation, especially Lagrange Interpolation, I noticed that near the edges of the interval there are huge oscillations. My question is: Why do polynomials of higher degrees have big oscillation... |
H: Showing that the Diophantine equation $m(m-1)(m-2)(m-3) = 24(n^2 + 9)$ has no solutions
Consider the Diophantine equation $$m(m-1)(m-2)(m-3) = 24(n^2 + 9)\,.$$ Prove that there are no integer solutions.
One way to show this has no integer solutions is by considering modulo $7$ (easy to verify with it).
I am curi... |
H: Bounding $\|Ax\|_2$ in terms of $\|A\|_1$ and $\|x\|_2$
Given a matrix $A$ of size $m\times n$ and a vector $x$ of size $n\times 1$, how can be bouund the $l_2$ norm $\|Ax\|_2$ in terms of $\|A\|_1$ and $\|x\|_2$, where $\|A\|_1$ is the sum of the absolute values of all entries of the matrix $A$.
AI: One useful and... |
H: A doubt about Theorem 14 in textbook Algebra by Saunders MacLane and Garrett Birkhoff
I'm reading the proof of Theorem 14 in textbook Algebra by Saunders MacLane and Garrett Birkhoff.
Any permutation $\sigma \neq 1$ on a finite set $X$ is a composite $\gamma_{1} \cdot \cdot \gamma_{k}$ of disioint cyclic permutati... |
H: How do I Find Preparatory Exposure?
I always prefer to be prepared ahead of time, but I am not
sure of how to be for a career in mathematics. How do I immerse myself in a formal, mathematical environment
without necessarily enrolling in a university, and at the
same time take a general survey of the profession?
AI:... |
H: Effect of squaring while finding roots of unity
Consider $$b=\frac{1}{b}\rightarrow b^2=1$$
Clearly $b=\pm1$ But if we square the above equation on both sides and then solve
$$(b=\frac{1}{b})^2\rightarrow b^2=\frac{1}{b^2}\rightarrow b^4=1
$$
And we know fourth root of unity are $1,i,-1,-i$ why am i getting e... |
H: Suppose $R$ is $(3, 5)$ and $S$ is $(8, -3)$. Find each point on the line through $R$ and $S$ that is three times as far from $R$ as it is from $S$.
Suppose $R$ is $(3, 5)$ and $S$ is $(8, -3)$. Find each point on the line through $R$ and $S$ that is three times as far from $R$ as it is from $S$.
I'm confused rega... |
H: Problem with Summation of series
Question: What is the value of $$\frac{1}{3^2+1}+\frac{1}{4^2+2}+\frac{1}{5^2+3} ...$$ up to infinite terms?
Answer: $\frac{13}{36}$
My Approach:
I first find out the general term ($T_n$)$${T_n}=\frac{1}{(n+2)^2+n}=\frac{1}{n^2+5n+4}=\frac{1}{(n+4)(n+1)}=\frac{1}{3}\left(\frac{1}{n+... |
H: Solve $\int_{|z|=5} \frac{z^2}{(z-3i)(z-3i)}$
Solve$$\int_{|z|=5} \frac{z^2}{(z-3i)(z-3i)}.$$
So I am currently working on the unit about the Cauchy Integral Formula, where$$\int\frac{f(z)}{z-z_0}=2\pi i f(z_0).$$
The zero $z=3i$ lies within the circle $|z|=5$ so $z_0=3i$ and we can rewrite the integral so$$\int_... |
H: Rudin 2.2 Solution Explanation
The question is to prove that the set of all algebraic numbers is countable. He gives the hint that for every positive integer N there are only finitely many equations with n + $\vert a_0 \vert$ + $\vert a_1 \vert$ + ... + $\vert a_n \vert$ = N. I looked at the solution given at https... |
H: Question about parametric equations
This is a question from MIT 18.01 single variable calculus on parametric equations:
I have the answers, but I don't quite understand it, especially the equation circled in pink. What does it mean? Moreover, I don't get how it equates to $\theta = \frac{\pi}{2} - \frac{\pi}{6}t$.... |
H: Vector sequence in $l^2$ is Cauchy
I am trying to prove that the sequence of vectors in $l^2$ $ \{ v^{(n)} \} _{n \in \mathbb{N} }$ which is defined as $ \underline v^{(n)}:= \sum_{k=1}^{n} e^{-k \alpha} \underline e^{(k)}$ , $\alpha > 0$
where $\{ \underline e^{(n)} \} _{n \in \mathbb{N} }$ denotes the canonical... |
H: Spivak's Calculus on Manifolds theorem 2-9 why is continuous differentiable needed
In the third last line, it is said that because each $g_i$ is continuously differentiable at $a$ then the constructed function $g$ is also differentiable at $a$.
I do not see why "continuously" differentiable is need cuz I think a f... |
H: Convergence in probability implies mean squared convergence
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{F}$ measurable random variables. Let $X$ be another $\mathcal{F}$ measurable random variable. I have $X_n \rightarrow X $ in probabili... |
H: Degree of minimal polynomial over $\mathbb{Q}$
Can someone please help with this question.
Prove that degree of minimal polynomial over $\mathbb{Q} $ of $\zeta_{7} $ , a primitive 7th root of unity is not a prime number.
I thought as $\zeta_{7} $ =$1^{1/7} $ so I can write $1^{1/7} $ =x which implies $x^{7} $ =... |
H: Show that convergence in probabiltiy plus domination implies $L_p$ convergence
I want to show that if random variable $X_n $ converges to $X$ in probability (Let $(\Omega, \mathcal{A},P)$ be the probability triple) and $|X_n| < Y \,\,\forall\, n$ then $X_n$ converges to $X$ in $L_p$.
Here's my attempt so far:
Since... |
H: Why do partitions correspond to irreps in $S_n$?
As stated for example in these notes (Link to pdf), top of page 8, irreps of the symmetric group $S_n$ correspond to partitions of $n$. This is justified with the following statement:
Irreps of $S_n$ correspond to partitions of $n$. We've seen that conjugacy classes... |
H: $\{x\in X: f(x)=g(x)\}$ is closed in $X$ if $f$ and $g$ are continuous on $X$.
could you help me demonstrate the following please
Let $f,g:(X,d_X)\rightarrow\mathbb{R}$ be continuous functions. Prove that $\{x\in X: f(x)=g(x)\}$ is closed in $X$.
It is a result that is very interesting to me, but I really do not kn... |
H: Uniform limit of a sequence in $C_{c}^{0}(\mathbb{R}^n)$ is in $C_{c}^{0}(\mathbb{R}^n)$
I am trying to prove the next:
Let $(f_k)$ be a sequence in $C_{c}^{0}(\mathbb{R}^n),$ the space of continuous functions with compact support from $\mathbb{R}^n$ to $\mathbb{R}.$Let $K$ be a compact set in $\mathbb{R}^n$ which ... |
H: Is $f(x)=\sum_{n=1}^{\infty}\frac{x}{1+n^2x^2}$, $x\in[0,1]$ continuous on $[0,1]$
Let $f(x)=\sum_{n=1}^{\infty}\frac{x}{1+n^2x^2}$, $x\in[0,1]$.
Question: Show that $f$ is Lebesgue integrable and determine whether $f$ is continuous on $[0,1]$.
For the first part, I have no problem, I showed that it is Lebesgue int... |
H: The vector $\mid \phi \rangle \langle \phi \mid \psi \rangle$ is the projection of a vector $\mid \psi \rangle$ along the vector $\mid \phi \rangle$?
I am currently studying the textbook Mathematical methods of quantum optics by Ravinder R. Puri. When going over some basic facts associated with bra-ket notation, th... |
H: Minimize the maximum inner product with vectors in a given set
Given a finite set $S$ of non-negative unit vectors in $\mathbb R_+^n$, find a non-negative unit vector $x$ such that the largest inner product of $x$ and a vector $v \in S$ is minimized. That is,
$$
\min_{x\in \mathbb R_+^n,\|x\|_2=1}\max_{v\in S} x^Tv... |
H: A quiz question based on finding sum of series
This question was asked in my analysis quiz and I had no clue in the exam how it could be solved.So, I am asking here .
Find the sum of series $\sum_{n=0}^{\infty} \frac{n^2} {2^{n} } $ .
Unfortunately, I have no idea on how to approach this problem.
AI: We start w... |
H: number of submodules of direct sum of simple modules
Let $M$ be a simple $R$ module. show that the number of submodules of $M \oplus M$ can be infinite.
AI: Consider $M=R=\mathbb{R}$. This is a simple $\mathbb{R}$-module, obviously. Note that for each $\lambda\in\mathbb{R}$ the set
$$M_\lambda:=\{(x,\lambda x): x\i... |
H: implicit differentiation and taking limit on derivative
I have this equation $x^3-xy^2+y^3=0$ and I want to know the value of the derivative at $(0,0)$. Through implicit differentiation I find $y'=\frac{y^2-3x^2}{3y^2-2xy}$. Now for $x=0,y=0$ this fraction becomes an indeterminate form. Upon graphical inspection I ... |
H: linear combination of periodic sequence is also periodic?
Let $x_t$ and $y_t$ real periodic sequences such that the least common multiple of their periods exists. Then, given a constant $a$, $x_t+ay_t$ is also periodic with period, say, $p$.
Does $x_t-ay_t$ also need to have period $p$?
Thanks in advance.
Observat... |
H: Any linear transformation in $\mathbb{C}$ (complex vector space) is a multiplication by $\alpha \in \mathbb{C}$
In the Linear Algebra Done Wrong book, one of the exercises was to show that any linear transformation in $\mathbb{C}$ is a multiplication by $\alpha \in \mathbb{C}$.
Here's the proof in the solutions par... |
H: Determine extrema of a multivariate function defined on a set
Given a function $$f(x,y)=2+2x^2+y^2$$ and the set $$A:=\{(x,y)\in\mathbb R^2 | x^2+4y^2\leq 1\}$$ and $f:A\to\mathbb R$ how do I determine the global and local extrema on this set? Normally I would not find this hard since it would just be determining $... |
H: How to prove that for every integer $k \geq 2$ we have $k^{1/k} \leq e^{1/e}$ without using the first derivative test?
I just stumbled across this cool property, by doing some calculus I could prove it, the function $f(x) = x^{1/x}$ has a local maximum at $x = e$ and the derivative changes sign at that point, but I... |
H: Prove that if $\alpha$ is any cycle of length $n$, and $\beta$ is any transposition, then ${\alpha, \beta}$ generates $S_n$
Question 6 from the above set of exercises has me a bit stumped. I see that you can rotate then relabel $\alpha = (a_1 a_2 \cdots a_n)$ such that $\alpha = (1 2 \cdots n)$ and $\beta = (1 m)$... |
H: How to prove that a sequence is Cauchy
How do I show if $x_n = \frac{n^2}{n^2 -1/2}$ is a Cauchy sequence? (using the definition of Cauchy sequence)
My attempt: A sequence is Cauchy if $ \forall \epsilon>0$ $ \exists N \in \mathbb N$ $\forall m,n \geq N$ :|$x_n -x_m$|$\leq \epsilon$
|$x_n -x_m|=|\frac{n^2}{n^2 -1/2... |
H: Number of permutations for hiring mr.X
Let be $9$ candidates for a job hiring (mr.X included). There are $3$ judges. Every judge defines a priority list ($1st-9th$) for the candidates (1st being the best and 9th the worst). A candidate is being hired only if they exist in the first $3$ spots of each of the three li... |
H: Does $ \sum\left( (n^3+1)^{\frac{1}{3}} -n \right) $ converge or diverge?
The test I know are Cauchy's root test, Cauchy's integral, Raabe's test, logarithmic test and D' Alembert's Ratio Test. I dont know which test I can use to prove that this series converges?
$$\sum \left( (n^3+1)^{\frac{1}{3}} -n \right) $$
AI... |
H: Group of order 28 with normal subgroup of order 4 is abelian
Herstein ch2.11 q19
Prove that if $G$ of order 28 has normal subgroup of order 4, then $G$ is abelian.
My attempt: The 7-sylow subgroup lies in center. So $\circ(Z)=7, 14$ or $28$.
For $\circ(Z)=14$, $G/Z$ is cyclic. But this argument fails for $\circ(... |
H: Why does it make sense to talk about the 'set of complex numbers'?
In my complex analysis course we've discussed quite a few times the idea that $\mathbb{C}$ is really 'the same thing' as $\mathbb{R}^2$ with the added complex multiplication operation. I've also read a number of the popular posts here including this... |
H: Measure theory $\mu(\lim \inf E_n) \leq \lim \inf E_n$
Let $(E_n)_{n \in \mathbb{N}}$ be some measurable sets. Define $\lim \inf E_n$ to be
$$\bigcup_{n \in \mathbb{N}} \bigcap_{m = n}^\infty E_m$$
I want to show that $\mu(\lim \inf E_n) \leq \lim \inf E_n$. I thought we might take an increasing sequence $(F_n)$ wh... |
H: If $ax+by+cz = 0$, show that $span(x,y)=span(y,z)$
If we have $ax+by+cz = 0$ show that $span(x,y)=span(y,z)$
My steps:
$$x = -\frac{b}{a}y-\frac{c}{a}z\Rightarrow x\ \in\ span(y,z)$$
$$z=-\frac{a}{c}x-\frac{b}{c}y\Rightarrow z \in\ span(x,y)$$
But we still have not proved that $span(x,y)=span(y,z)$ and I'm not enti... |
H: Index of cyclic subgroup $\langle h \rangle$ in a subgroup $H$ with finite index in a finitely generated group $G$
Let $\phi: H\to \mathbb{Z}$ be a homomorphism with finite kernel.
Let $h \in H$ such that $\phi(h) \not=0$.
Can someone help me understanding why exactly the cyclic group $\langle h\rangle$ has finite ... |
H: Why isn't conformal mapping more flexible?
I have been spending some time familiarizing myself with the basics of conformal mapping, and found myself somewhat stumped with the limitations of some of the methods I have encountered. Möbius transformations or Schwarz-Christoffel maps, for example, have very strict req... |
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