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H: Definition of notation $\mathbb Z_n$
What does the notation $\mathbb Z_n$ mean, where $n$ is also an integer. I have only seen $n$ being a positive integer up to now. some examples are $\mathbb Z_2$ or $\mathbb Z_3$
This is the context: How to prove $x^{2}+x=1$ has a solution in $\mathbb{Z}_{p}$ if and only if $p=5... |
H: A question on Pixley-Roy topology
Let $R$ be Real line and let $F[R]$ be $\{x\subset R:\text{is finite}\}$ with Pixley-Roy topology.
Definition of Pixley-Roy topology is this: Basic neighborhoods of $F\in F[R]$ are the sets
$$[F,V]=\{H\in F[R]; F\subseteq H\subseteq V\}$$
for open sets $V\supseteq F$, see e.g. here... |
H: The probability of an account being chosen
Current: $140$
1-30 days past due: $80$
31-60 days past due: $40$
61-90 days past due: $25$
Sent for collection: $15$
What is the probability that $2$ current accounts and $2$ 1-30 days past due accounts will be chosen?
Using multiplication rule:
$$\frac{140}{300}\frac{139... |
H: Trigonometry - Addition Theorem Finding Another Trig function
Using the expansion of
$\sin(A + B)$, prove that $\tan 75^\circ = 2 + \sqrt 3$
AI: Remember $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}=\frac{\sin(\theta)}{\sin(90-\theta)}$ so now we've expressed tan purely in terms of sin. Next you need to think of... |
H: Simple question: Finding the number of arrangements
$3$ math books, $5$ English books, $4$ science books and a dictionary are to be placed on a student's shelf so that the books of each subject remain together.
In how many different ways can the books be arranged?
$$3!\space5!\space4!\space4!$$
$3!$ ways to arrang... |
H: Find peak output value using transfer function
I have a filter with Transfer function $H(z)=(1-0.5z^{-1})(1+0.5z^{-1})$ designed for a sampling rate of 800 samples/s.
How to find peak output if a sine of 200Hz and amplitude 4 is applied as input?
AI: Your input signal is
$$x(n)=A\sin n\theta_0$$
with $A=4$ and $\th... |
H: Recursive function into non-recursive
I have to express $a_n$ in terms of $n$. How do I convert this recursive function into a non-recursive one? Is there any methodology to follow in order to do the same with any recursively defined function? Thanks.
$$a_n = \begin{cases} 0 & \text{if }n=1\\ a_{n-1}+n-1 & \text{if... |
H: Expansion of $z^3 \log ( (z-a)/(z-b))$ in $\infty$
I need some hints to evaluate the expansion of $z^3 \log((z-a)/(z-b))$ in $\infty$. I thougt that evaluating $\log(\frac 1 z -a)$ in $z = 0$ may be helpful. How can I proceed ?
AI: Let $u=\frac{1}{z}$ then
$$z^3 \log ( (z-a)/(z-b))=\frac{1}{u^3}\left(\log(1-au)-\lo... |
H: Book recommendations for studying mathematical areas based on set theory
I am at the end of my studies with set theory, and I would like to
continue in fundamental fashion, and study for example calculus based
on set theory. So, I am talking about not calculus the way it is studied
in college, but calculus studied ... |
H: Each finite extension of a field, has a finite number of intermediate extensions.
Prove that every finite extension $K$ of a field $F$, has a finite number of intermediate extensions.
EDIT: All fields here are of characteristic $0$, otherwise we would need to require the extension to be separable.
AI: Each finite e... |
H: For each normal extension of a field, whose Galois group is commutative, each intermediate extension is also normal.
Let $K$ be a normal extension of the field $F$, and let the Galois group $G(K,F)$ be an Abelian group. Prove that each intermediate extension $E$ is also a normal extension.
EDIT: All fields here are... |
H: How does $Ae^{4ix}+Be^{-4ix}=A\cos(4x)+B\sin(4x)$?
$e^{ix}=\cos(x)+i\sin(x)$
$Ae^{4ix}=A(\cos(4x)+i\sin(4x))$
$Be^{-4ix}=B(-\cos(4x)-i\sin(4x))$
What am I doing wrong?
I am trying to find the complimentary function of $\frac{d^2y}{dx^2} +16y=8cos(4x)$
C.F: $x^2+16=0$
$x=4i$ and $x=-4i$
$$Ae^{4ix}+Be^{-4ix}$$
$\ph... |
H: Prove that $x+e^{2x}=1$ have only one solution
I`m trying to prove that this equation have only one solution.
$$x+e^{2x}=1$$
so what I did is to set $\ln$ on this equation and get:
$$\ln(x)+2x=0$$
I need some hint how to continue from here.
Thanks!
AI: Hint: Ignore what you did. Consider the function $f(x)=x+e^{2x}... |
H: How to solve the equation $\sum_{k=0}^{n}x^kC_{n}^{k}\cos{k\theta}=0$
Find all real numbers $x$, such that
$$\sum_{k=0}^{n}x^kC_{n}^{k}\cos{ka}=0$$
My idea: we can find this value
$$\sum_{k=0}^{n}x^kC_{n}^{k}\cos{ka}.$$
use Elur $e^{ikx}=\cos{kx}+i\sin{kx}\Longrightarrow 2\cos{2kx}=e^{ikx}+e^{-ikx}$,
$$2x^kC_{n}^{k... |
H: Can I exchange limit and differentiation for a sequence of smooth functions?
Let $(f_n)_{n\in \mathbb N}$ be a sequence of smooth functions converging to some $f$.
Under what circumstances can I exchange limit and derivative?, i.e.
$$\lim_{n\rightarrow \infty} \frac{\partial f_n(x)}{\partial x} = \frac{\partial f(x... |
H: Why $\mbox{Ker }T^{*}\oplus\overline{\mbox{Im }{T}}=X$?
Can you explain me or indicate where can I find a proof, please why:
$$\mbox{Ker }T^{*}\oplus\overline{\mbox{Im }{T}}=X \mbox{ ? }$$
$X$ is a complex Hilbert space.
thanks :)
AI: I'm assuming that $T : X \to X$ is a continuous linear operator.
Let $x \in \bi... |
H: Understanding an example of M. L. Wage, W. G. Fleissner, and G. M. Reed
In this paper of M. L. Wage, W. G. Fleissner, and G. M. Reed, the authors claimed that having a zeroset diagonal does not guarantee submetrizable by showing Example 2. However, the example is very obscure.
The construction of Example 2 is do... |
H: taylor series and uniform convergence
Maybe is a silly question but I am confused...so I hope someone can help me.
Is the convergence of the Taylor series uniform?
To be more specific. We know for example that
$\displaystyle{ e^x = \sum_{n=0}^{\infty} \frac{x^n}{ n!} \quad}$ , $\displaystyle{ \sin x = \sum_{n=0}^{... |
H: Evaluate $ \int^{ \pi/2}_{- \pi/2} \frac {1}{ 1+e^{\sin x} }dx $
Evaluate $ \int^{\pi/2}_{-\pi/2} \frac {1}{ 1+e^{\sin x} }dx $
Solution:
I think odd, even functions are of no use here.
Also we get nothing by taking $e^{\sin x} $ common in denominator.
Also rationalizing the denominator is of no use.
Really I ha... |
H: Prove that $\langle x,y\rangle=x A y^*$ is an inner product
In $\mathbb{C}^2$, I want to show that $\langle x,y \rangle =xAy^*$ is an inner product space. I almost done with other properties of inner product but left only one property: $\langle x,x \rangle >0$ if $x \neq 0$.
To prove this,
$$\begin{align*}
\langle ... |
H: An counterexample of Hahn-Banach theorem in a topological vector space
Problem : Give an example of a TVS $\mathcal{X}$ that is not locally convex and a subspace $\mathcal{Y}$ of $\mathcal{X}$ such that there is a continuous linear functional $f$ on $\mathcal{Y}$ with no continuous extension to $\mathcal{X}$
I thi... |
H: Which set theories without the power set axiom are used occasionally?
To get a set theory without the power set axiom, I could just take an existing set theory like ZF or ZFC, and remove the power set axiom. However, perhaps I would have to be careful how to formulate the other axioms then, or have to add some sent... |
H: Representing a nonzero bilinear alternating form on a two-dimensional space by $\bigl(\begin{smallmatrix} 0&1\cr-1&0\end{smallmatrix}\bigr)$.
So I am having a little bit struggle with some question.
I have a bilenear form $B:V\times V\to \Bbb F$.
$B$ is not the zero form.
$B$ is alternating meaning it is also skew-... |
H: compute pca with this useful trick
A is matrix (m rows, n cols), each row is an object, and each cols is a feature (a dimension). Typically, I compute the pca based on the covariance matrix, that is A'A, A' is the transposed matrix of A.
Today I read a book which presents a useful trick to compute pca, that is if n... |
H: Calculating interest rate of car financing
I want a new car which costs $\$26.000$.
But there's an offer to finance the car:
Immediate prepayment: $25\%$ of the original price
The amount left is financed with a loan: Duration: $5$ years, installment of $\$400$ at the end of every month.
So I need to calculate the r... |
H: why start the taylor series of $\cos^{2} x$ at $k=1$ and not just $k=0$ as I do not understand the problem with $2^{-1}$
Im using $\cos^2 x=\frac{1}{2}(1+\cos(2x))$ and $\cos x = (-1)^k \frac{(2x)^{2k}}{(2k)!}$ to find the sum for the Taylor series of $\cos^2 x$. I thought I was getting it. When I find the answer $... |
H: Definition of period of a decimal representation of a number
I need to define the period of a decimal representation of a number!!
Thanks in advance!!
AI: Period of a decimal representation of a number may be defined in terms of cyclic numbers:
See https://en.wikipedia.org/wiki/Cyclic_number#Relation_to_repeating_d... |
H: If $T\colon V\to V$ is linear then$\text{ Im}(T) = \ker(T)$ implies $T^2 = 0$
I'm trying to show that if $V$ is finite dimensional and $T\colon V\to V$ is linear then$\text{ Im}(T) = \ker(T)$ implies $T^2 = 0$.
I've tried taking a $v$ in the kernel and then since it's in the kernel we know its in the image so there... |
H: How to count unlabeled balls and labeled boxes case?
How to count unlabeled balls and labeled boxes case
each box can have more than one balls or some box may have no balls
use example , 10 labeled boxes, 8 unlabeled balls
welcome to use polynomial counting
better have closed formula or generating function
would li... |
H: Calculate the sum $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n\times2^{2n+1}}$
I started with
$arctg(x) = \sum\limits_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}$
Then I differentiated to get rid of the denominator. Then divide with $x$ to get $x^{2n-1}$. Then integrate to get $2n$ in the denominator. Then multiply wit... |
H: Any formula to find the maximum $n$th power of $x$ contained in a number?
Is there any formula to find the maximum $n$th power of $x$ contained in a number?
Say I need to find $n$ where $x$ is $2$ and the number is $25$.
So the answer must be $4$.
The problem statement is to find $n$ where $x^n \le m$ and $x^{n+1} ... |
H: Let $A \in M_{nxn} (\mathbb R)$ such that $A^8 + A^2 = I$ prove that $A$ is diagonalizable
Let $A \in M_{nxn} (\mathbb R)$ such that $A^8 + A^2 = I$. prove that $A$ is diagonalizable.
What I did so far:
We know that A is a root of the polynomial $f(x) = x^8 + x^2 -1$. Now we need to figure out what $m_A(x)$ (the... |
H: Calculus - Indefinite integration Find $\int \sqrt{\cot x} +\sqrt{\tan x}\,dx$
Problem : Find $\int \sqrt{\cot x} +\sqrt{ \tan x}\,dx$
My Working :
Let $I_1 = \int \sqrt{\cot x}\,dx$ and $I_2 = \int \sqrt{\tan x}\,dx$
By using integration by parts:
Therefore , $I_1 = \sqrt{\cot x}.\int1 dx - \int\{(d\sqrt{\cot x}... |
H: Comparing fields with same degree
Two part question: Are the fields $\mathbb{Q} (\sqrt[3]{2}, i \sqrt{3})$ and $\mathbb{Q} (\sqrt[3]{2}, i, \sqrt{3})$ identical in algebraic structure? I have in notes that they both have degree of 6 over $\mathbb{Q}$.
How do I show explicitly that $\mathbb{Q} ( i \sqrt{3})$ is onl... |
H: Eigenvectors of real normal endomorphism
A normal endomorphism that has a matrix with only real entries over a complex vector space, has pairwise always pairwise eigenvalues, meaning that we have an eigenvalue and its complex conjugate. now i was wondering whether this statement is also true for eigenvectors. there... |
H: Harmonic number divided by n
How do I prove that $\dfrac{H_n}{n}$ (where $H_n$ is a harmonic number) converges to $0$, as $n \to \infty$?
AI: Let $1\leqslant k\leqslant n$. Using the upper bounds $\frac1i\leqslant1$ for every $i\leqslant k$ and $\frac1i\leqslant\frac1k$ for every $k\lt i\leqslant n$ yields $H_n\leq... |
H: Find $ \int {dt\over 2t+1}$
a simple question, but I'm stuck anyway:
How to integrate this:
$$ \int {dt\over 2t+1} = ? $$
Is it simply: $\ln|2t +1| $
or do I need Chain rule like: $$\ln|2t +1| \cdot \frac{d}{dt}(2t + 1) $$
AI: You've got the right idea about the "form" of the integral, but recall, we need to accou... |
H: Integration of $\int(2-x/2)^2dx$
Got an exam tomorrow and my head is no longer working. Could someone walk through the integration of this function
$$\int\left(2-\frac x2\right)^2dx$$
I understand integration by parts and stuff like that.
AI: Hint: $\displaystyle \int u'(x)(u(x))^2dx=\dfrac{(u(x))^3}{3}+C$ |
H: Not Equivalent Interpretation
Can someone provide an interpretation to show that the following are not equivalent:
$$\forall x \in D, P(x) \vee Q(x)\;\;\text{vs.}\;\;(\forall x \in D, P(x)) \vee (\forall x \in D, Q(x))$$
They seem equivalent.
AI: Let $D=\Bbb R$ and $P(x)$ mean $x\ge 0$. Can you find a suitable prop... |
H: find the limit $\lim \limits_{x \to {\pi/2}} \frac {\sin x -(\sin x)^{\sin x}} {1-\sin x+\log (\sin x)}$
$\displaystyle \lim\limits_{x \to {\pi/2}} \frac {\sin x -(\sin x)^{\sin x}} {1-\sin x+\log (\sin x)}$
Solution :
We can solve this question by L' Hospital rule
But it will be a bit tedious
Is there any other ... |
H: A markov chain inequality in Billingsley that should be an equation?
In the section on Markov chains in Billingsley's Probability and Measure (3e) we have the following inequality on page 120 in the proof of Theorem 8.3,
$$
\begin{align*}
p_{ji}^{(m)} &=
P_j([X_m=i] \cap [X_n = j \text{ i.o.}])\\
&\le \sum_{n>m} P... |
H: Is my understanding of the solution for $a$ in $\sum^\infty_{n=0} e^{na} = 2$ correct?
This was a homework problem that I got and I was required to find the value of $a$ in $$\sum^\infty_{n=0} e^{na} = 2$$ So I did the following. I removed the $\sum$ and along with the indexer $n$ and was left with $e^a = 2$ where ... |
H: cardinality of sets
Prove if $ |A| < |B| $ and $ |B| \leq |C|$
then $ |A| < |C|$
I know that $|A| < |B|$ means there is a one to one mapping of A onto a SUBSET of B but no one to one mapping from A to B.
I also know that $|B| \leq |C|$ means there is a one to one mapping of B onto C.
I am going to try to prove thi... |
H: Given $T(A) = A^t$ in $M_{n\times n}(\mathbb R)$. Find the polynomials and find if it's diagonalizable
Given the vector space $M_{n\times n} (\mathbb R)$ and a transformation $T(A) = A^t$ (transpose):
Find $m_T$, $P_T$ (the minimum polynomial and the characteristic polynomial respectively.)
Find if $T$ is diagon... |
H: Weak convergence in $L^2$ and CDF
Assume that for sequence $X_n \in L^2(\Omega,F,P)$ which converges in distribution to CDF $F_X$ ($F_n(t)\rightarrow F_X(t)$ for every point of continuity of $F_X$), we have also that $X_n$ converges weakly to $Z$ in $L^2(\Omega,F,P)$ $(\mathbb{E}X_nY \rightarrow \mathbb{E}ZY$ for e... |
H: integrating by parts $ \int (x^2+2x)\cos(x)\,dx$
I seems to be stumped in this integral by parts problem. I have
$$ \int (x^2+2x) \cos(x)\,dx $$
step 1- pick my $u , dv, du, v$
$$u=x^2+2x$$
$$du=(2x+2) \,dx$$
$$dv = \cos(x)$$
$$v= \sin(x) $$
step 2- apply my formula $uv - \int v \,du$
$$(x^2+2x)(\sin(x))-\int(\si... |
H: percentage problem for salary
In $\;2002,\, 2003, \, \text{and}\; 2004\;$ the total income of Jerry was $\$36,400$.
His income increased by $15\%$ each year. What was his income in $2004$?
Any hints or solution will be welcome.
Thanks in advance.
AI: $
\begin{align} I_0 & : \quad \text{income earned in}\;2002.... |
H: Every principal ideal domain satisfies ACCP.
Every principal ideal domain $D$ satisfies ACCP (ascending chain condition on principal ideals)
Proof. Let $(a_1) ⊆ (a_2) ⊆ (a_3) ⊆ · · ·$ be a chain of principal ideals in $D$. It can be easily verified that $I = \displaystyle{∪_{i∈N} (a_i)}$ is an ideal of $D$. Since $... |
H: How many ways are there to represent the number $N$?
I was given a task that doesn't require any special knowledge of math, but got stuck with it. Here it is:
How many ways are there to represent the number $N$ in the
following way: $$ N = a_3 \cdot 10^3 + a_2 \cdot 10^2 +
a_1\cdot10+a_0 \ \ \ (1)$$ $$ a \i... |
H: Limit in a sequence
I know that $\lim\sqrt[n]{a}=1$(where $a > 0$ is a real number).
I know also that $\lim{\frac{1}{n}}=0$.
But, can you explain me why $\lim\sqrt[n]{2 + \frac{1}{n}}= 1$ ?
AI: Since $2<2+\frac1n\le 3$, yuo can compare your sequence with $\sqrt[n]2$ and $\sqrt[n]3$. |
H: Integer Linear Programming (ILP): NP-hard vs. NP-complete?
I was thinking about examples where a problem is NP-hard but was not NP-complete and ILP came to mind.
It is obviously NP-hard but is it NP-complete? I.e., is it in NP? Given a certificate (the alleged minimum), we certainly cannot check if it is the minim... |
H: Pullback of a coproduct in an Abelian category is 0?
Let $i_P:P\rightarrow P\oplus Q$ and $i_Q:Q\rightarrow P\oplus Q$ be a coproduct in an abelain category $\mathcal A$, let $t:T\rightarrow P$,$u:T\rightarrow Q$ be arrows such that $i_Pt=i_Qu$, I have to prove $t=u=0$.
The only thing I came up with was to show tha... |
H: Existence a certain subgroup of a group
Let $G$ be a finite group such that $G=P\rtimes Q$ where $P\in {\rm Syl}_p(G)$; $P\cong \Bbb{Z}_p\times \Bbb{Z}_p$
and $Q\in {\rm Syl}_q(G)$; $|Q|=q$ ($p, q$ are primes).
Can we classify groups $G$ which contain a subgroup ... |
H: Representation of matrix A = BC
This is straight from a textbook (Cullen): Show that every $m \times n$ matrix $A$ of rank $r$ has a representation as $A = BC$, where $B$ is an $m \times r$ matrix whose columns are the first $r$ linearly independent columns of $A$ and where $C$ is an $r \times n$ matrix in row-red... |
H: Correlation Coefficient - $\rho(X,Y)$.
If I have two aleatory variables
$$X=\begin{pmatrix}2&3&4&5&6&7&8&9&10&11&12\\ \frac{1}{36}&\frac{2}{36}&\frac{3}{36}&\frac{4}{36}&\frac{5}{36}&\frac{6}{36}&\frac{5}{36}&\frac{4}{36}&\frac{3}{36}&\frac{2}{36}&\frac{1}{36}\end{pmatrix}$$ and $$Y=\begin{pmatrix}2&3&4&5&6&7&8&9&1... |
H: Prove or disprove: if $f$ and $fg$ are continuous then $g$ is continuous.
Prove of provide a counterexample:
Suppose that $f$ and $g$ are defined and finite valued on an open interval $I$ which contains $a$, that $f$ is continuous at $a$, and that $f(a)\neq 0$. Then $g$ is continuous at $a$ if and only if $fg$ is c... |
H: Limit of a Sequence involving cubic root
I succeed in finding the following limit applying binomials and squeeze theorem:
$$\lim(\sqrt{n+1} - \sqrt{n}) = \lim\frac{1}{\sqrt{n+1} + \sqrt{n}} = 0$$ because $0 \leq \frac{1}{\sqrt{n+1} + \sqrt{n}} \leq \frac{1}{\sqrt{n}}$
But I need help because I'm not finding any way... |
H: Isomorphisms between Orderings
If $h$ is an isomorphism between $(P,<)$ and $(Q,\prec)$ then show $h^{-1}$ is an isomorphism between $(Q,\prec)$ and $(P,<)$
DEFINITION: $h$ is an isomorphism between $(P,<)$ and $(Q,\prec)$ then $h(p_{1}) \prec h(p_{2})$ whenever $p_{1}<p_{2}$.
* I have this definition from definiti... |
H: Question about normal subgroup and isomorphism relation
Maybe by using the following theorem:
If $N$ is a normal subgroup of $G$, then the function $\phi: G \to G/N,$ given by $\phi(g)=gN$ yields a surjective homomorphism from $G$ to $G/N$ with $\ker(\phi)=N$.
I want to solve the following exercise:
$G=$ the grou... |
H: Did I derive this correctly?
I derived this $$(2x+1)^2 \sqrt{4x+1}$$
and got $(8x+4)(\sqrt{4x+1})$+$\frac{2}{\sqrt{4x+1}}(2x+1)^2$
Is this correct?
I ask because Wofram Alpha gave me a different answer.
Thanks in advance.
AI: Your answer:
$$\begin{aligned}
&(8x+4)(\sqrt{4x+1})+\frac{2}{\sqrt{4x+1}}(2x+1)^2
\\
=& \... |
H: understanding the proof of $\int_A p'(t)dt=\frac{1}{2}\int_0^1 p'(t)dt$
I need to understand the proof of a theorem from a book called "geometric group theory by Graham and Roller".
The theorem says:
Let $p:[0,1]\rightarrow \mathbb{R}^n$ be $C^1$-path. Then there exists an open subset $A\subset[0,1]$ such that $... |
H: Convexity of a function depending on value of parameters
Check out convexity of a function $J(u)=cu^r$, $J:[a,b]\rightarrow R$,
$0<a<b<\infty$, depending on values of parameters $c,r\in R$.
I know a definition of a convexity:
"Function J,defined on convex interval U, is convex on U if $J(\alpha u + (1-\alpha) v)... |
H: Solving distributional differential equation
How to solve differential equation in $\mathcal D'(R)$:
$$u''+u=\delta'(x),$$ where $\delta$ is Dirac Delta function?
Solution of homogeneous problem is $C_1\cos{x}+C_2\sin{x}$, so using the variation of parameters, I got that the final solution to the problem should be:... |
H: showing that $S=\sum^n_{i=1} \ln(X_i)$ is a complete sufficient statistic
We have a random sample $X_1,X_2,\ldots,X_n$ from a probability with density:
$$
f(x)=\theta x^{-(\theta +1)}
$$
given that $x>1$ and $0$ else.
now the question is:
Show that $S=\sum^n_{i=1} \ln(X_i)$ is a complete sufficient statistic and u... |
H: If modulus of each one of eigenvalues of $B$ is less than $1$, then $B^k\rightarrow 0$
Let $B$ be a $n\times n$ matrix and let $X$ be the set of all eigenvalues of $B$. Prove that if $|m|<1$ then $\lim \limits_{k\rightarrow\infty}B^k=0$, where $m=\max X$.
Thanks.
Actually, there isn't a order involved. Sorry. The c... |
H: extension of a valuation of $K$ to $K(X)$
Let $A$ be a Krull ring and $p$ a prime ideal of height 1. Then $A_p$ is a DVR with corresponding valuation $v$ on the field of fractions $K$ of $A$.
Question: Can we extend this valuation to an additive valuation of $K(X)$?
Remark: Matsumura in his Commutative Ring Theory... |
H: Recurrence relations: How many numbers between 1 and 10,000,000 don't have the string 12 or 21
so the question is (to be solved with recurrence relations: How many numbers between 1 and 10,000,000 don't have the string 12 or 21?
So my solution: $a_n=10a_{n-1}-2a_{n-2}$. The $10a_{n-1}$ represents the number of stri... |
H: Matrix similar to a companion matrix
I am currently intensively reading my linear algebra notes under dim light and was wondering whether it is true, that a an endomorphism whose minimal polynomial has the same degree as the dimension of the vector space is similar to a companion matrix?
AI: The answer is yes and w... |
H: Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
Can you help me to prove that
$$(x+y)^n\leq 2^{n-1}(x^n+y^n)$$
for $n\ge1$ and $x,y\ge0$.
I tried by induction, but I didn't get a result.
AI: Look. I also have tried to do it by induction. It is obvious that it holds for $n=1$ and $n=2$. Assume that it also ho... |
H: Is the cartesian product of groups the product of a normal subgroup and its quotient group?
I'm studying elementary group theory, and just seeing the ways in which groups break apart into simpler groups, specifically, a group can be broken up as the sort of product of any of its normal subgroups with the quotient g... |
H: algorithm to determine complexity of algorithms?
Given a decision problem X, can there exist an algorithm A which, given any algorithm B which solves X in finitely many steps, determines whether B runs in polynomial time? If such an A exists, when is it possible for A to run in polynomial time?
AI: For some decisio... |
H: Looking for bounds of a recursively defined sequence
I'm looking for the tightest upper and lower bounds on the sequence defined recursively by $a_{0}=1$
and $a_{n}={\displaystyle \sum_{k=0}^{n-1}\frac{4}{n^{2}}a_{k}+c\cdot n}$
for $c>0$. It is obvious that $a_{n}\in\Omega\left(n\right)$ and I managed to show t... |
H: Why $O(\epsilon^{-1})\ll O(\epsilon^{-3/2})$
When looking for the approximate roots of $\epsilon^2x^6-\epsilon x^4-x^3+8=0$, since this is a single perturbation problem, we need to track down the three missing roots, so we consider all possible dominant balances between pairs of terms as $\epsilon\to 0$.
Now suppos... |
H: Methods to show polynomials are irreducible
I would like to show that $x^3 + x^2 - 2x - 1$ is an irreducible polynomial over $\mathbb{Q}$. What are my standard lines of attack to solve this problem? Typically I go to Eistenstein, but it does not apply to this polynomial (I believe). I'm familiar also with Gauss' le... |
H: Is constructing a function that DNE a sufficient counterexample to show the function does not diverge to $\infty$?
Prove or disprove: If $f(x)\to 0$ as $x\to a^+$ and $g(x)\geq 1$ for all $x\in \mathbb{R}$, then $g(x)/f(x)\to\infty$ as $x\to a^+$.
Counterexample: Let $f(x)=0$ and $g(x)=1$ for all $x\in\mathbb{R}$. ... |
H: Find the limit using a calculator
We have $u_0 = 6$ and $u_{n+1} = \dfrac{1}{2} u_n + \dfrac{1}{u_n}$. We can use our graphing calculator to make a 'web diagram' (no idea what it is called in English, and I can't find it. It sometimes resembles a spider's web).
When I use my calculator for very high values of n I ... |
H: Fibonacci sequence, prove by induction that $a_{2n} \leq 3^n$
Let ${a_n}$ be the Fibonacci sequence. Prove by induction that $a_{2n} \leq 3^n$ (the Fibonacci sequence is defined as $a_1=1$, $a_2 = 2$, and $a_n = a_{n-1} + a_{n-2}$.)
What I know
$3 \leq a_{2k-2} \leq 3^k $
We need to prove that $a_{2k+2} \leq 3^{k... |
H: Ham sandwich for measuers implies the classical one
Ham sandwich theorem for measures:
Let $\mu_1,\mu_2,\mu_3 $ be finite Borel measures on $\mathbb{R^3}$ such that every hyperplane has measure $0$ for each of $\mu_i$. Then there exists a hyperplane $h$ such that $\mu_i(h^+)= \frac{1}{2}\mu_i(\mathbb R^3)$ where $... |
H: Existence and uniqueness of God
Over lunch, my math professor teasingly gave this argument
God by definition is perfect. Non-existence would be an imperfection, therefore God exists. Non-uniqueness would be an imperfection, therefore God is unique.
I have thought about it, please critique from mathematical/logica... |
H: Divergence and Levi-Civita connection
Let $M$ be a level set of a function in $\mathbb R^3$. Then the mean curvature of $M$ is given by the trace of the second fundamental form which is a divergence term involving the Levi-Civita connection. My question is, why is this the same as the "usual" divergence of the norm... |
H: a sigma-algebra on a countable set is a topology
I am trying to prove the statement in the title, i.e. that
a sigma-algebra $\Sigma$ on a countably infinite set $X$ is a topology on $X$.
I feel like I have an intuition of why this is true, but I can't articulate it. I'd like to request some hints to get me thin... |
H: How to change this geometric progression into a direct one
We have the progression:
$$u_0 = 5$$
$$ u_{n+1} = 3- 0.5u_n \space ( n = 1,2,...)$$
I want to change this into a 'direct' progression, so I tried:
$$ 5 \cdot -0.5^n + 3$$ but this actually fails for $n=0$. How can I edit this so it also holds for $n=0$?
AI:... |
H: Odd part of $n-1$ and primes
Using $n=11$ as an example:
Step 1 : 11 - 1 = 10. Get the odd part of 10, which is 5
Step 2 : 11 - 5 = 6. Get the odd part of 6, which is 3
Step 3 : 11 - 3 = 8. Get the odd part of 8, which is 1
Continuing this operation (with $11-1$) repeats the same steps as above. Ther... |
H: The sum of two periodic functions need not be a periodic function
Let $f(x)=x-[x]$ and $g(x)=\tan x$.
How could we see that $f(x)-g(x)$ is not a periodic function?
This will show that the sum of two periodic functions need not be a periodic function.
I hope the answer has enough details so that I could catch you.... |
H: for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$
How to show that for $z\in\mathbb C-\{0\},~\dfrac{1}{1+nz}\to0.$
I've tried triangle inequality couldn't arrive at any conclusion.
Please help me.
AI: Presumably you mean as $n \to \infty$?
The triangle inequality is your friend. In particular, $|1+nz| \ge n|z|-1$.
If... |
H: number theory: Let $m>n$ for $m,n\in\mathbb{Z}$, prove if $k$ divides $m$ and $k$ divides $n$ then $k$ divides $m\bmod{n}$
Let $m>n$ for $m,n\in\mathbb{Z}$, prove if $k$ divides $m$ and $k$ divides $n$ then $k$ divides $m\bmod{n}$.
How should I approach this question?
I only got $m=qk$ and $n=pk$ if $\frac{m}{n}=... |
H: How to compute $\mathbb{P}(\lambda X>4)$ directly?
Given a random variable $X$ which is exponentially distributed i.e. $X\sim E(\lambda)$. Calculate $\mathbb{P}(X-\frac{1}{\lambda}>\frac{3}{\lambda})$.
My working:
$\mathbb{E}(X)=\frac{1}{\lambda}$, $Var(X)=\frac{1}{\lambda^2}$. Then $\mathbb{P}(X-\frac{1}{\lambda}>... |
H: $\sup_{x\in A}x \sup_{y\in B}y=\sup_{x\in A,y\in B}xy$
Let $A$ and $B$ be two sets of nonnegtive numbers. Prove that
$\sup_{x\in A}x \sup_{y\in B}y=\sup_{x\in A,y\in B}xy$.
Thanks for your help.
AI: If $x \in A$, then $x \le \sup A$. Similarly, $y \le \sup B$. Since $x,y \ge 0$, we have $xy \le \sup A \sup B$. It... |
H: Derivative of Trace of Matrix wrt parameters
I have the following function which I need to find the derivative of
$$L=trace(\Sigma K^{-1})$$ where $K$ is a function of $\theta$ and $\Sigma$ is constant.
If I'm correct what I need to do to find $\frac{\partial L}{\partial \theta}$ is $\frac{\partial L}{\partial K}\t... |
H: Changing Summation Index Question
I'm sorry if this seems like a very novice question, but I am still relatively new to the world of discrete math ( still in 9th grade). I've been reviewing some of the concepts I learned in a chapter from Concrete Mathematics (Graham,Knuth,Patashnik) about Sums, and I seem to hav... |
H: Orthogonality and linear independence
[Theorem]
Let $V$ be an inner product space, and let $S$ be an orthogonal subset of $V$ consisting of nonzero vectors. Then $S$ is linearly independent.
Also, orthogonal set and linearly independent set both generate the same subspace.
(Is that right?)
Then
orthogonal $\r... |
H: Contraction Mapping
$$f(x)=\begin{pmatrix}1/4 & 0 & 1/2 \\ 0 & 1/3 & 0\\ -1/2 & 0 & 1/4 \end{pmatrix}\begin{pmatrix}x_1 \\ x_2\\ x_3 \end{pmatrix}, \qquad\forall x=\begin{pmatrix}x_1 \\ x_2\\ x_3 \end{pmatrix} \in \Bbb R^3 $$
I am able to prove that this function is a contraction w/ standard Euclidean metric via us... |
H: How to prove " $¬\forall x P(x)$
I have a step but can't figure out the rest. I have been trying to understand for hours and the slides don't help. I know that since I have "not P" that there is a case where not All(x) has P... but how do I show this logically?
1. $\forall x (P(x) → Q(x))$ Given
2. $¬Q(x)$... |
H: Contraction Map on Compact Normed Space has a Fixed Point
Let $K$ be a compact normed space and $f:K\rightarrow K$ such that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ Prove that $f$ has a fixed point.
AI: Edit: Note that $f$ is continuous. (Why?) Define $$g(x)=\lVert f(x)-x\rVert$$ for al... |
H: Is This A Derivative?
I am in a little over my head. This all began with my reading how each level of pascals triangle adds to $2^n$, where n=row# starting with n=0. I then though, "wouldn't it be clever if the rows added to something else--like say $3^n$ instead?" Or even better generalize it for any constant, $a^... |
H: Finding the probability from a markov chain with transition matrix
Consider the Markov Chain with state space $S=\{v,w,x,y,z\}$, transition matrix below:
$$\left[\begin{array}{cccccccccc}
0 & 0.4 & 0.6 & 0 & 0\\
0 & 0.5 & 0.5 & 0 & 0\\
0 & 0 & 0 & 0.1 & 0.9\\
0 & 0 & 0& 0.2 & 0.8\\
0.7 & 0 & 0.3 & 0 & 0
\end{array}... |
H: Is $f$ a differentiable function?
Hello everyone I have this problem, Can somebody help me with this?
$f:\mathbb{R}^2\rightarrow{}\mathbb{R}$ is defined by:
$$f(x,y) = \left \{ \begin{matrix} \ln\left(\displaystyle\frac{x}{y}\right) & \mbox{if } xy\geq{0}
\\ 0 & \mbox{if }xy<0\end{matrix}\right. $$
Is $f$ a differ... |
H: Characteristic polynomial of a matrix $A$ is $x^7$ and $\operatorname{rank}(A)=4$ and $\operatorname{rank}(A^2)=1$. Classify $A$.
I know how to do this problem the 'long' way. I was wondering if there
was an easier, less computationally cumbersome way to do this.
Here is the question: Let $A$ be a square matrix ove... |
H: Area bounded by a parametric curve
Be $\gamma: \mathbb{R}\rightarrow \mathbb{R}^3$ defined by:
$$\gamma(\theta):= (\cos(\theta), \sin(2\theta), \cos(3\theta))$$
and be $$S:=\lbrace t\gamma (\theta): t\in [0,1], \hspace{0.1cm} \theta\in\mathbb{R}\rbrace$$
Calculate the area of $S$.
I have problems understanding this... |
H: Condition the tensor product of a vector is not singular
I am generating a matrix with a vector through outer product in my code, and resulting matrix is singular. Is there a condition I can use to check the vector and figure out where the singularity comes from?
AI: The outer product of two nonzero vectors always ... |
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