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H: What is the Cardinality of the euclidean space $\mathbf{R}^N$?
We know that the set of real numbers is not countable by Cantor's proposition and hence higher dimensional Euclidean space is not countable too. However I couldn't find any result about the cardinality of $\mathbf{R}^N$? Is it $\aleph_{n}$ or it has the... |
H: Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$
Problem: Prove that $x^4+x^3+x^2+x+1$ divides $x^{4n}+x^{3n}+x^{2n}+x^n+1$ for all positive $n$ that are not multiples of $5$.
I'd like to get some pointers about how to solve this. No full solutions, just a nudge in the right direction. I've been working... |
H: A square integrable martingale has orthogonal increments
I am really stuck with the following exercise:
$(\Omega,\mathcal{F},(\mathcal{F}_n)_{n\ge1},P)$ a filtered probability space. Let $(X_n )_{n\ge1}$ be a sequence of square-integrable random variables. Define for every $n\ge1$,
$S_n := X_1 +\dots + X_n$ . Su... |
H: How to determine a $\Theta$-class of a Function
I have 6 functions that I have to determine which of 4 given $\Theta$-classes or neither of them.
Example of a function I have been given:
\begin{align*}
\textit{$f_1$}(n) =&(17\textit{n}+1) \\
\end{align*}
The $\Theta$-classes I have been given:
\begin{align*}
\Theta... |
H: Roots Calculation Question
How does one calculate the roots: $$ \sqrt {57-40\sqrt{2}} - \sqrt {57+40\sqrt{2}}$$ manually?
Also, how can one determine which of the pair is bigger:
$ \sqrt {3} +\sqrt {5} $ vs. $ \sqrt {2} +\sqrt {6} $
also, by hand?
AI: $a^2+2ab+b^2 = (a+b)^2$, $a+2 \sqrt{ab} +b=(\sqrt{a}+\sqrt{... |
H: Prove that $ \left(1-\frac{1}{n}\right)^n > \frac{1}{6} $ for $n\geq 2$
Prove that $ \left(1-\frac{1}{n}\right)^n > \frac{1}{6} $ for $n \in \mathbb{N}$, $ n\ge 2$
Indeed, the affirmation is true even if $n$ is not a natural ($ n\geq 2 $ ) and we can prove it using calculus. But, this is part of a question about li... |
H: Which one is BIG
Let we have two event :
Event A: $ab^3$
Event B: $a^3b$
The range of a and b are given as: $-1 < a <0 < |a|< b< 1$
It seems to me a big range to test that which event is big. How to do this quickly without putting lots of numbers to testify?
AI: A = $ab(b^2)$
B = $ab(a^2)$
Because $-1<a<0<b<1$, $... |
H: Distribution of distinct balls in identical boxes
how can I derive a formula for the number of distributions of $n$ different balls in $k$ identical boxes. Where $\mathbf{empty\ box}$ is allowed.
I know this is equivalent to finding the number of ways to partition a set of $n$ labelled (distinct) objects into $k\ \... |
H: cubic equations which have exactly one real root
Question is to check :
For any real number $c$, the polynomial $x^3+x+c$ has exactly one real root .
the way in which i have proceeded is :
let $a$ be one real root for $x^3+x+c$ i.e., we have $a^3+a+c=0$
i have seen that $(x-a)(x^2+ax+(a^2+1))=x^3+x-a^3-a$
But, $a^3... |
H: How show that if Matrix A is not square, it cannot have an inverse.
How to show that if How A is not square, it cannot have an inverse.
Why is the the case and how can I prove it?
AI: Non-square matrices $m \times n$ matrices for which $m ≠ n$ do not have an inverse. However, in some cases such a matrix may have a... |
H: A question about the dihedral group $D_n$
Let the dihedral group $D_n$ be given by elements $a$ of order $n$ and $b$ of order $2$, where $ba = a^{-1}b.$
(a) Show that $a^{-m}= a^{n-m}$ for all integers $m.$
Definition: Let $n$ greater than or equal to $3$ be an integer. The group of rigid motions of a regular n-gon... |
H: Determine the least natural number $k$ such that $a(k)>1$
Let $a(n)$ be a sequence with $a(0)=1/2$ and $a(n+1)=a(n)+(a(n)^2)/2013$, $n$ natural number.
Determine the least natural number $k$ such that $a(k)>1$.
This problem is from Poland proposed to Romanian Masters of Mathematics.
Can you give me some hints? I do... |
H: Prove that $\frac{d}{dx}\int_0^xf(x,y)dy = f(x,x)+\int_0^x\frac{\partial}{\partial x}f(x,y)dy$
Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$. Assume that $$\frac{d}{dx}\int_a^bf(x,y)dy=\int_a^b\frac{\partial}{\partial x}f(x,y)dy.$$
Use the above property and the chain rule to prove that
$$\frac{d}{dx}\int_0^xf(x,y)dy ... |
H: Finding the eigenvalues of a matrix
I was trying to prove the following theorem and got stuck at one point :
Theorem :
Let $A\in M_n(F)$, then the scalar $\lambda$ is an eigenvalue of $A$ if and only if
$$
det(A-\lambda*I_n) = 0.
$$
Proof:
I started off like this ,
A scalar $\lambda$ is an eigenvalue of $A$ if and ... |
H: What are the probability that A is neither symmetric nor skew-symmetric?
A is a 3 × 3 matrix with entries from the set {–1, 0, 1}. Then the probability that A is neither symmetric nor skew-symmetric is:
My thoughts: There can be nine members on a $3*3$ matrix and there are three possibilities for each member. There... |
H: Independence between two random variables and a function of the two random variables
Say we have two random variables, $X$ and $Z$ that are independent. Then let $W=a+bXZ$ be the random variable that is a function of both $X$ and $Z$, $a$ and $b$ are just scalar constants. Then are $X$ and $W$ independent? How abou... |
H: What are the eigenvalues of this symmetric matrix?
Let
$$A=\begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{pmatrix}.$$
I'm trying to find the eigenvalues of $A$, but when I calculate the characteristic polynomial, I get
$$p(\lambda)=-\lambda^3+6\lambda^2+15\lambda+2,$$
and I don't know how to solve $p(\la... |
H: How to find the $S_\kappa$ elements in the product $ \sum_{\kappa=1}^{K} S_\kappa Q_\kappa(r, \theta,\phi) = 1$?
So I have one K-dimensional complex column vector $\textbf{S}$ and a set of K complex functions $Q_\kappa(r,\theta,\phi) \forall \kappa \in \{1,\dots,K\}$ defined in spherical coordinates.
I know all th... |
H: Unique smallest and largest topology ideas
What does it mean to be a smallest topology of a set $X$. I would guess that it would be a topology of $X$ which has least number of elements and similarly for largest topology it would have to be largest number of elements? Am I correct? This looks fairly straightforward ... |
H: Mathematical Analysis: Riemann Integration
This is a question from my Mid-term test which was held last week:
Define $f:[0,1] \to \mathbb{R}$ by $f(x) = x,$ if $x$ is irrational and $f(x) =0$, otherwise
Is $f \in R[0,1]?$
My Proof:
Consider any Partition $P = \{\ x_{0} = 0, ..., x_{n} =1\}$ of $[0,1]$. Then, $L(f,P... |
H: Finding A,B,C s.t $f'(a)+O(h^2)=\frac{Af(a)+Bf(a+2h)+Cf(a+3h)}{h}$
Find constants A,B,C s.t for differtiable three times function f, $f'(a)+O(h^2)=\frac{Af(a)+Bf(a+2h)+Cf(a+3h)}{h}$
I know that $f'(a)+O(h^2)=\frac{f(a+h)-f(a-h)}{2h}$ so I need to solve $$0.5f(a+h)-0.5f(a-h)=Af(a)+B(a+2h)+Cf(a+3h)$$but I don't kn... |
H: Clarification regarding the monotone convergence theorem
I have a simple question about the monontone convergence theorem. Let $f_n : \mathbb{R} \to [0,\infty]$ be a sequence of nonnegative, extended real valued, measureable functions on $\mathbb{R}$. Suppose the sequence is monotone increasing i.e. $0 \leq f_1 \l... |
H: A small bit on partial differentials and general solutions
Consider the equation:
$$\frac{\partial u}{\partial t} - \frac{\partial u}{\partial x} = 0$$
I want to find a general solution.
So I do the following:
$\alpha = ax + bt$, where $\frac{\partial \alpha}{\partial x} = a$, and $\frac{\partial \alpha}{\partia... |
H: How to evaluate the summation $S_b$
This question is from my notebook, not hw or else, only exercise to understand better. I tried by myself. However, since my trail are too trivial, I dont need to write here. i am confused a bit. I want to learn how to solve such question types. I guess I need to use residue. I am... |
H: What is the probability C was the one with the wrong answer?
Students A, B and C each independently answer a question on a test. The probability of getting the correct answer is 0.9 for A, 0.7 for B and 0.4 for C. If two of them get the correct answer, what is the probability C was the one with the wrong answer?
AI... |
H: Derivation of the quadratic equation
So everyone knows that when $ax^2+bx+c=0$,$$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$
But why does it equal this? I learned this in maths not 2 weeks ago and it makes no sense to me
AI: $ax^2+bx+c=0 (a\neq0)$
$x^2+\frac{b}{a}x+\frac{c}{a}=0$
$x^2+\frac{b}{a}x=-\frac{c}{a}$
$x^2+\fr... |
H: What is the Mirror/PingPong clamp mode algorithm?
I do programming as a hobby, and in a dynamic system various numerical values inevitably change. Those values can be greater than or less than the expected range, in which case they need to be "wrapped" according to a "mode". I am aware of three of these wrapping mo... |
H: How find this inequality$\sqrt{\left(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\right)\left(\frac{y}{x}+\frac{z}{y}+\frac{x}{z}\right)}+1$
let $x,y,z>0$,show that
$$\sqrt{\left(\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}\right)\left(\dfrac{y}{x}+\dfrac{z}{y}+\dfrac{x}{z}\right)}+1\ge 2\sqrt[3]{\dfrac{(x^2+yz)(y^2+xz)(z^2+xy)}{... |
H: Catagorical Definition of Coproduct and Abelian Groups
I have the definition of a coproduct which is as follows:
A coproduct of $\{A_\alpha\}$ in $\mathcal{G}$, where $\mathcal{G}$ is a category and $\{A_\alpha\}$ a collection of objects, is an object $Q$ of $\mathcal{G}$ with $\pi_\alpha:A_\alpha\rightarrow Q$ s.t... |
H: Who has a higher chance of drawing an easy question?
From a pool of N questions n are easy. Two people draw a question without returning it. Who has a higher chance of drawing an easy question?
AI: Assume $N\geq 2$
$P($First person draw an easy question$)=\frac{n}{N}$
$P($First person draw a hard question$)=1-\frac... |
H: Determine $p$ for one to have $\displaystyle\frac{1}{5}+\frac{1}{6}=\frac{1+1}{5+6}=\frac{2}{11}$, in $\mathbb{Z_p}$.
Determine $p$ for one to have $\displaystyle\frac{1}{5}+\frac{1}{6}=\frac{1+1}{5+6}=\frac{2}{11}$, in $\mathbb{Z_p}$.
I know that $p$ must, $13, 43, 61, 101,103$.
AI: Try multiplying both sides by $... |
H: Finding the limit of a sequence? $\lim_{n\to\infty}\frac{3^{n-2}}{4^{m+2}}$
How would the limit of the following sequence be found.
$\displaystyle \frac{3^{n-2}}{4^{m+2}},\;$ as $\;n\rightarrow \infty$.
Would you use the squeeze theorem to find the limit of the sequence like this
$\displaystyle \frac{-3^m}{4^m}<\f... |
H: Show that this is not a countable cover for the graph of a continuous function.
Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Consider any rational number $q$. For any $\epsilon >0$, there is a $\delta > 0$ such that $|q - y| < \delta \implies |f(q) - f(y)| < \epsilon$.
I was using this to try and ma... |
H: Limit of sequences: $ \lim (1 + \frac{r}{n} )^n = e^r $
Consider the sequence $x_n = \left (1+\frac{r}{n} \right )^n $ for $r \in \mathbb{Q} $. I need to prove that $ \lim x_n = e^r $
My attempt of proof (for r>0) is to find a subsequence of $x_n$ that converges to $e^r$, but it will be enough if $x_n$ is a Cauch... |
H: Prove that either$ P(A∩B)=0$ or else $P(A'∩B')=0$
Let $E$ and $F$ be independent with $E = A∪B$, and $F = A∩B$. Prove that either
$P(A∩B)=0$ or else $P(A'∩B')=0$.
AI: We are told that $E$ and $F$ are independent. By the definition of independence, we have
$$\Pr(E\cap F)=\Pr(E)\Pr(F).\tag{1}$$
But $E\cap F)=(A\cup ... |
H: Proving that $\frac{6\tan\phi}{\tan^2\phi-9}=\tan A$
In a triange $ABC$, points $D$ and $E$ are taken on side $BC$ such that $BD=DE=EC$. If $\angle ADE=\angle AED=\phi$, how can we prove that $$\frac{6\tan\phi}{\tan^2\phi-9}=\tan A$$
AI: The key here is recognizing the identity we wish to prove. Note that
$$\tan{A... |
H: Is conjugate of holomorphic function holomorphic?
If $f(z)$ is holomorphic, does it follow that $g(z)=\overline{f(z)}$ is holomorphic?
I'm looking at $$\lim_{z\rightarrow a}\dfrac{g(z)-g(a)}{z-a} = \lim_{z\rightarrow a}\dfrac{\overline{f(z)-f(a)}}{z-a}$$
Can we pull the limit out to get $\overline{\lim_{z\rightarro... |
H: Order Topology on $\mathbb{Z_+}$
Why is the order topology on $\mathbb{Z_+}$ a discrete one? I understand that the discrete topology will have all subsets of $\mathbb{Z_+}$ which means that all subsets of $\mathbb{Z_+}$ are open which is not necessarily true.
AI: Why do you say not every subset is open? Every subse... |
H: Prove : If $I = (p(x))$ is a prime ideal in $F[x]$ then $p(x)$ is irreducible.
I have to show :
If $I = (p(x))$ is a prime ideal in $F[x]$, where F is a field, then $p(x)$ is irreducible.
In the book I use, there is the proof of the converse which uses Euclid's Lemma.
I tried to suppose $I=(p(x))$ is an ideal, the... |
H: Is the following sentence a tautology: $(p\Rightarrow q)\vee(r \Rightarrow p)\vee(r\Rightarrow s)\vee(r\Rightarrow q)$?
If both $p$ and $q$ are false then ($p\Rightarrow q$) is true.
If either $p$ or $q$ is true then one of ($r\Rightarrow p$) or ($r\Rightarrow q$) is true.
If both $p$ and $q$ are true then all are... |
H: Show that $ x_n = \left(1 + \frac{r}{n} \right)^n $ has an upper bound
I asked this question but maybe my doubt was not enough clear. So I will ask something more specific: Show the sequence $x_n = \left(1 + \frac{r}{n}\right)^n$ for $ r \in \mathbb{Q}, r>0$ has an upper bound.
I tried to show it as we do for $a_n ... |
H: How was this derivative simplified further?
Here is the steps on how the derivative gets simplified:
We are given the point (0,0) for it to be evaluated at.
\begin{align*}\tan(3x+y) &= 3x \\
(3+y^{\prime}) \sec^2(3x+y) &= 3 \\
y^{\prime} &= \frac{3-3\sec^2(3x+y)}{\sec^2(3x+y)} \\
&= \frac{-3\tan^2(3x+y)}{\tan^2(3x+... |
H: Cauchy's integral formula problem
Let $D$ be an open disk in $\mathbb{C}$ and let $\overline{D}$ be the closure of $D$. Suppose $f:\overline{D}\to\mathbb{C}$ is continuous on $\overline{D}$ and analytic on $D$, prove that
$$f(w)=\frac{1}{2\pi i}\displaystyle\int_{\partial{D}}\frac{f(z)}{z-w}\mathrm{d}z$$
for any $w... |
H: Constructing regular grammar
I'm trying to make a regular grammar for this language:
$$ L = \{ a^ncb^m(cc)^p : n\ge 1, m\le 1, p\ge 0\} $$
Where the alphabet is $ \Sigma = \{a,b,c\}$
It seemed like right-linear. This may be disastrously wrong, but here's how I started:
$S \to aA$
$A \to aA | cB $
$B \to bC | C $
$... |
H: Computing complex integral with absolute value
What is the value of $\int_{|z|=1}|z-1||dz|$?
By definition, if $z$ is parametrized by $z=e^{i\theta}$, the integral is $$\int_0^{2\pi}|e^{i\theta}-1|\cdot|ie^{i\theta}|d\theta = \int_0^{2\pi}|e^{2i\theta}-e^{i\theta}|d\theta$$ It has an absolute value, so I don't know... |
H: Calculating Variance to the power of a variable
Let X be standard uniform random variable. That is, X has the density $f_x(x) = 1$ for 0 < x < 1 and 0 elsewhere. Suppose that we toss a fair coin (independently of the value of X) and set $Y = X$ if the coin shows heads, and $Y = 1$ if the coin shows tails.
Calcula... |
H: Volume of a solid involving integration by parts.
Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y=e^x$ and the line $x=\ln 5$, rotated about the line $x= \ln 5$
Here is my work:
$$V= 2\pi \int _{a}^{b}(\text{shell radius})(\text {she... |
H: Hard Integral $\frac{1}{(1+x^2+y^2+z^2)^2}$
Prove that $\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2}\, dx \, dy \, dz = \pi^2$
I tried substitution, trigonometric substitution, and partial fraction decomposition, but I can't... |
H: Limit as $x$ approaches $0$ for a function.
Unfortunately this question does not make the slightest sense to me. I know how to find limits but this is confusing. Any help will be appreciated...
$$\lim_{x\to 0}\frac{\sin(x^m)}{(\sin x)^p}$$
where $m$ and $p$ are positive integers. (Hint: the answer will depend on
$m... |
H: Solving a second order homogenous ODE with non-real complex roots
I'm working on solving this problem:
$$ y''+ 48y = 0 $$
For a typical homogenous ODE with real roots we let $y=e^{rx}$ and solve for the roots $r_1$ and $r_2$:
$$
y=e^{rx}$$ $$
y'=re^{rx} $$ $$
y''=r^2e^{rx} $$
$$ r^2e^{rx} + 48e^{rx} = 0 $$
$$ e^{rx... |
H: Is this a correct way to express $\left|f(x)\right| \leq \left|x\right|^9$?
If $\left|f(x)\right| \leq \left|x\right|^9$, then, is it correct to say that
$f(x) \leq x^9$ and $f(x) \geq -x^9$ ?
If it is not, could someone explain why? Thank you.
AI: To expand on what Mark Bennet said, your formulation is almost corr... |
H: Question about calculating $2^{32101}\bmod 143$?
I am trying to calculate $2^{32101}\bmod 143$ with using paper and a calculator.
What bothers me is that $32101 = 47 \times 683$ and $143=11 \times 13$, so they are not prime numbers. This means i can't use Fermat's little theorem. I tried also to solve it by square... |
H: How do I prove by induction?
For example if i wanted to prove:
$1^2 + \dots + n^2 = \frac {n(n + 1)(2n + 1)} {6}$
by induction.
I'm not sure where to start.
Thanks.
AI: Prove for the base case, n=1:
$$1^{2} = \frac{1(1+1)(2+1)}{6} = \frac{2\cdot3}{6}=1$$
The "sum" of just $1^2$ is indeed 1. Base case proven.
Now ... |
H: Partial differentiation with respect to multiple variables?
Let's say you have $y(v,t)$ where $v(x(s,t),t) = y(s,t)$.
I know that
$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t} $
However, I see in my textbook that the second partial deriva... |
H: What function space do $u(x)=\sin(x)$ and v$(x)=e^{-2x}$ belong to?
I'm trying to follow notes, a question is presented in the middle of the notes that isn't answered.
$u(x)=0$ when $-1\leq x<0$
$u(x)=x^2$ when $0\leq x \leq 1$
The $u(x)$ function belongs to $C^1[-1,1]$. What space do $u(x) = \sin x$ and $v(x) = e^... |
H: Regular expression for a language
I made a regular expression to match this language but I'm not sure it's right. Perhaps someone can show me where it deviates.
The language:
$L = {a^{n} c b^{m} (cc)^{p} : n \geq 1, m \leq 1, p\geq 0}$
The expression:
$r = a a^{*} c (\lambda + b) (c c)^{*}$
AI: Yes, that’s exactly... |
H: Find the equation of the tangent line to the curve at the given point. $y = 1+2x-x^3$ at $(1,2)$
I have the equation $y = 1+2x-x^3$ and the point $(1,2)$.
When I work it out I come up with the derivative of $2-3x^2$.
When I apply the point I come up with a slope of $-1$ and a tangent line of $y=4-x$.
Can someone wo... |
H: Language made by a regular expression
I created a language from this regular expression but I'm not sure about it, especially where I wanted to use the $w$ to express a sequence of terminals.
The expression:
$r = a a ^{*} (b + bb + bbb) (a + b) (a + b) ^ {*}$
The alphabet:
$\Sigma = \{a,b\}$
The language:
$L = \{ a... |
H: Existence of a certain 2-coloring of a graph
There are n studio apartments in a building. Some of the apartments are connected with each other by direct phone line. Prove that it is possible to assign to each apartment a female or a male in such way that each person has direct connection with at least as many peopl... |
H: How can we directly see that the number of random walks starting and ending at the origin is ${n\choose n/2}^2$?
In an infinite two-dimensional square-shaped grid, we define four directions, north, south, east, west. We thus have $4^n$ random walks of length $n$. If we end where we started, for every north step we ... |
H: How does $\zeta(i\pi)$ converge?
$$\zeta(i\pi) = \sum_{r=1}^{\infty}r^{-i\pi} = \sum_{r=1}^{\infty}e^{-i\pi \ln(r)} = \sum_{r=1}^{\infty}\operatorname{cis}(-\pi\ln(r))$$
Did I mess up somewhere in the steps above? I can't see how the last expression would converge.
AI: Your mistake is thinking that $\zeta(i \pi)$ i... |
H: Prove by induction that $1^3 + \dots + n^3 = (1 + \dots + n)^2$
I'm suppose to prove by induction:
$1^3 + \dots + n^3 = (1 + \dots + n)^2$
This is my attempt; I'm stuck on the problem of factoring dots.
AI: Your attempt looks OK as far as it goes (except for a missing superscript $2$ at one point, but that's not ca... |
H: Is this set of integer sequences countable?
I'm faced with a set of strictly increasing functions $\Bbb N\to \Bbb N$, i.e. positive integer valued sequences. The only thing I know about them is that they are pairwise eventually disjoint, by which I mean that given $U$, $V$ in this set, $U(\Bbb N)\cap V(\Bbb N)$ is ... |
H: Conditional Probability of census burea
The question seems very straight forward , but when i attempted it the answer is wrong.
I just took the probability of all three groups and multiplied by themselves the intersections and addition them.
this is the question.
In early 2001, the United States Census Bureau start... |
H: Can someone explain these matrix operations to me?
So I have a row reduced echelon form matrix, that is $\begin{bmatrix}
I & F\\
0 & 0
\end{bmatrix}$.
The transpose of this matrix is thus $\begin{bmatrix}
I & 0\\
F^T & 0
\end{bmatrix}$.
This is all well and good and makes sense, but why does their product equal t... |
H: Summation and Time Complexity
So I'm studying for my data structures midterm that's this monday, and my professor gave out a sample midterm with the answers, but I'm having a hard time understanding one of the questions.
Here's a screen cap:
Could someone give me a walk through of the math that's happening in each... |
H: Why are the roots of the polynomial $z^N = a^N$ equal to $z_k = a \ e^{j\frac{2 \pi k}{N}}$?
I am trying to understand equation 3.28 from this image in my book.
I get everything that the author is saying, except for when he finds the roots, (zeros), of $z^N = a^N$. Of course, there are going to be $N$ roots. He s... |
H: Invertibility of a square matrix with zero diagonal elements and positive non-diagonal elements
$M$ is square and
$$M(i,j)=0, i=j$$
$$M(i,j)>0, i\ne j$$
Is $M$ full-rank or invertible?
Actually the $M$ I am studying has much stronger properties but I guess the simple conditions above might be enough to make $M$ no... |
H: 8-digit sequences invoving exactly 6 different digits
How many 8-digit sequences are there involving exactly six different digits? How can I approach this problem?
AI: HINT:
Count the ways to choose a set of $6$ digits to use.
Given a set of $6$ digits, either you use one digit $3$ times, or you use two digits t... |
H: Suppose $F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous functions. If $f$ and $g$ are measurable, then
Suppose $F: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous functions. If $f$ and $g$ are measurable, then $h(x) = F( f(x), g(x) ) $ is also measurable.
Proof:
For all $a$, $ \{ x : h... |
H: Conditional Probability injection-molding
I thought about this questions for a while now. I understand the fact that the foreman forgets to shut up the injection is 0.48. then the questions confuses me when they to tell us the probability of the defective molding will be produced 0.04 to 0.19 in early morning. th... |
H: Question about conformal maps.
By definition, a diffeomorphism $\sigma:(M,g)\to (N,h)$ is called conformal if $\sigma^*h=ug$. Another definition I've seen in other contexts is that conformal maps are ones that preserve angles. Now I've considered maps $\sigma$ from $M$ to $N$ that preserve angles (the angle between... |
H: Find the formula for a derivative
Suppose: $$n(x)=\frac{1}{x}+1$$
Find $n'(x)$
The book indicates the answer is: $$\frac{-1}{x^2}$$ However I am not sure how it got that conclusion. Would anyone mind walking me though the problem step by step?
AI: (Hints)
You can write $\frac1x + 1 = x^{-1} + 1$
The derivative of ... |
H: Equation of a tangent line: $f(x)=(x-4)(x^2-5)$, at $(2,2)$
Find an equation of the tangent line to the graph of f at the given point:
AI: What you want to use is the product rule for finding the derivatives. $\frac{d}{dx} uv = u'v + v'u$
So $u = (x-4), v = (x^2-5)$, find u' and v' and substitute into the equation ... |
H: Seeking a combinatorial proof of the identity$1+3+\cdots+(2n-1)=n^2$
I would appreciate if somebody could help me with the following problem
Q: Seeking a combinatorial proof
$$1+3+\cdots+(2n-1)=n^2$$
AI: Consider a bag with balls numbered from $1$ to $n$. Number of ways of choosing $2$ balls with replacement is $n^... |
H: Minimum size of directed path an orientation of a graph with a given independent set size
The problem I am working on is as follows:
Let $G$ be a graph with $n$ vertices and at most $r>=2$ independent vertices (no two are adjacent). Prove that if D is an orientation of G which does not contain a directed cycle the... |
H: How is this matrix called (two diagonals)?
I need to write an algorithm for solving this matrix but I wanted to first make a search online and that's why I need its name.
AI: You can rearrange terms, to get the following system
$$
\left [ \begin{array}{ccccccc}
d_1 & a_{2n+1} \\
a_1 & d_{2n+1} \\
& & d_2 & a_{2n} \... |
H: Prove that if $(v_1,\ldots,v_n)$ spans $V$, then so does the list $(v_1-v_2,v_2-v_3,\ldots,v_{n-1}-v_n,v_n).$
Prove that if $(v_1,\ldots,v_n)$ spans $V$, then so does the list $(v_1-v_2,v_2-v_3,\ldots,v_{n-1}-v_n,v_n).$
Proof:
Suppose that $V = \text{span } (v_1,\ldots,v_n).$ Then for any $v \in V$, there exist $a_... |
H: Am I missing something with this question about nested compact sets?
On my homework, I have a question that states:
Let $X$ be a compact space, and suppose $\{ F_n \}$ is a countable collection of nonempty closed subsets of $X$ that are nested. Show that $\cap_n F_n$ is nonempty.
It seems trivial since the finite i... |
H: Characteristic polynomial of a linear operator T
Suppose that $V$ is a finite dimensional vector space with $B_1$ and $B_2$ as its ordered basis and let $T$ be a linear operator on $V$.
Then the matrices $[T]_{B_1}$ and $[T]_{B_2}$ are similar.
Why are they similar?
Also, because they are similar, it would follow t... |
H: Can we ever get an irrational number by dividing two rational numbers?
If we try to divide any two random arbitrarily long rational numbers like
103850.2387209375029375092730958297836958623986868349693868398659825528365...
and
127.123123123...
Is it guaranteed that the result is also a rational number?
AI: The q... |
H: Right-linear grammar from regular expression
I made a right-linear grammar that from this regular expression:
The alphabet is:
$Σ = \{a, b, c\} $
Regular expression:
$r = cc^*(ba)^*bb$
My solution, it seems a little too short like I'm leaving something out. Maybe someone can see where I went wrong on the right-line... |
H: How to show a set is nowhere dense
I want to show that $A=\{(x_1,x_2,x_3,0,0,...,0): x_i\in \mathbb{R}\}$ is nowhere dense in $\mathbb{R}^{n}$ with usual topology.
I know that I have to show $(\overline{A})^{0}=\emptyset$. But I couldn't find $\overline{A}$. Please help.
AI: $\overline{A} = A$ because of $(x_n) \in... |
H: Probability that exactly $n$ trials are required is $\binom{n-1}{k-1}p^{k}(1-p)^{n-k}$
Independent trials that result in a success with probability $p$ are
successively performed until a total of $k$ successes is obtained.
Show that the probability that exactly $n$ trials are required is
$$\binom{n-1}{k-1}p^{... |
H: Doubt in the definition of graded rings
My doubt is quite simple, I didn't understand what is the operation $\cdot$ in this definition:
Thanks
AI: Given a ring $R$ and any two subsets $X\subseteq R$ and $Y\subseteq R$, we can define the subset
$$X\cdot Y=\{x\cdot y\mid x\in X,y\in Y\}$$
where $x\cdot y$ is just th... |
H: Suppose $A$ is a subset of $\mathbb{R}^n$ with $m_{*}(A) = 0$, then show that $m_{*}(A \times R^m) = 0$ in $\mathbb{R}^{n+m}$.
Suppose $A$ is a subset of $\mathbb{R}^n$ with $m_{*}(A) = 0$, then show that $m_{*}(A \times R^m) = 0$ in $\mathbb{R}^{n+m}$.
The result seems intuitive, but I'm not sure how to prove it.
... |
H: Why distribution in PDE is so like the distribution in Probability?
I just learnt distribution in PDE class.
Campared with distribution in probability, they have:
Same name
Similar concept for Delta function
Same converge mode(converge as distribution)
...
AI: Probability distributions are measures, and a measure ... |
H: How to get average of Arun's weight?
This is an aptitude problem I found in a website, question no 9:
In Arun's opinion, his weight is greater than 65 kg but less than 72 kg. His brother does not agree with Arun and he thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother's view is that h... |
H: How prove $p$ have contains infinite prime numbers
give the natural Numbers $a,b,c$,and such $a<b<c$, if exsit prime number $p$, such
$$p+a,p+b,p+c$$ are Composite numbers,
show that:such this contidtion $p$ have contains infinite prime numbers.
My idea: for example $a=2,b=3,c=4$.
(1)then we take $p=23$ is pr... |
H: Prove $\mu(V)=\mu(E) $, when $V = \cap_{k=1}^{\infty} {V_k} $
Prove $\mu(V)=\mu(E)$, when $V = \cap_{k=1}^{\infty} {V_k}$
$E \subset {V_k}$,For each k , $\mu(V_k) < \mu(E) + \epsilon\cdot 2^{-k}$
AI: Since $V=\bigcap_{i=1}^{\infty} V_i\subseteq V_k$ for any positive integer $k$, we have $$\mu(V)\leq \mu(V_k)<\mu(... |
H: Basic examples of monoids?
What are some (simple/elementary) examples of noncommutative monoids with no additional structure? I'm having a hard time thinking of examples of "pure" monoids that aren't monoids simply because they are groups...
I've read this and this and some of this, but would like more examples tha... |
H: Why is n mod 0 undefined?
I tried to find out what $n$ mod $0$ is, for some $n\in \mathbb{Z}$. Apparently it is an undefined operation - why?
AI: I might say it depends on how you define what it means to mod out by a number.
A typical first way of thinking about mods is to say that $a \equiv b \pmod d$ if $a = b + ... |
H: Problems in the Ramanujan Class Invariant $G_n$?
In page 290 of his second notebook , Srinivas Ramanujan defines 2 functions $G_n$ and $g_n$. And then proceeds to give a table of $G_n$.
But looking at the papers of Bruce C. Berndt, Heng Huat Chan, and LiangCheng Zhang -(http://www.math.uiuc.edu/~berndt/articles/t... |
H: What is meant by linearity of a dot product?
I would like to know what is meant by linearity of a dot product.
Thank you
AI: It means that the dot product satisfies two properties:
If $u$, $v$ and $w$ are vectors such that $\cdot$ and $+$ make sense, $$u \cdot (v + w) = u \cdot v + u \cdot w$$ and vice-versa:
$$(u... |
H: Simple proof about XOR, (possibly a duplicate question)
Prove Whether this statement is True or False:
Other than solving by Truth Table
If $A \oplus b = A \oplus C$ then $A=C$
I saw this question online and I've been thinking about it for days now and I'm pretty sure this question may have been asked already b... |
H: Show that $\det(A-\lambda B)$ is a nonconstant polynomial if $B$ is invertible
Let $A$ and $B$ be arbitrary complex square matrices. If $B$ is invertible, show that $$p(\lambda)=\det(A-\lambda B)$$ is a nonconstant polynomial in $\lambda$.
AI: If $B$ is invertible, the leading term is ${\sf det}(B)(-\lambda)^n$ whi... |
H: Extending a $C^1$ function up to the boundary
Let $U \subset \mathbb{R}^N$ be an open bounded set. let $f \colon U \to \mathbb{R}$ be a $C^1$ function.
I know that it is not always possible to extend $f$ continuously up to the boundary of $U$ since it could be unbounded.
Now, it sounds reasonable that this is possi... |
H: Question on using the integral of conditional probability to get unconditional probability
I'm trying to solve this problem in my homework assignment and I get different result from the answer. I know the answer is right, but at the same time I also don't see where I did wrong in my solution. So here's the problem:... |
H: finding $f$ function
For which of the sets $\mathbb{X}:= \mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$, there exist a function $f:\mathcal{P}(\mathbb{X})\rightarrow\mathbb{X}$ that for all $A\in\mathcal{P}(\mathbb{X})\setminus \lbrace\phi\rbrace$, $f(A)\in A?$ I want to find $f$ function? please help me.
... |
H: Operation of Sine and Cosine
Use condition $\displaystyle\sin\theta-\cos\theta=\sqrt{2}$,
Please find the value of $\displaystyle\frac1{\sin^{10}\theta}+\frac1{\cos^{10}\theta}$.
AI: If we set $a=\sin\theta,b=\cos\theta$
we have $a-b=\sqrt2$ and
as $a^2+b^2=1,(a-b)^2=2\iff -2ab=1$
$\displaystyle (a^2-b^2)^2=(a^2+b^... |
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