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H: Understanding Conditional Expectation
I just want to make sure I'm understanding conditional expectation correctly:
Let $X_1,X_2,X_3$ denote three independent coin flips with probability of heads $\frac{1}{4}$ and probability of tails $\frac{3}{4}$, and $X_i=2$ if heads and $X_i=0$ if tails.
Then I'm looking to det... |
H: If $f:\mathbb{R} \to \mathbb{Z}$, why can't we have that $g \circ f$?
In the lecture notes for a course I'm taking, it is stated that:
Unlike multiplication however, we can't reverse order. First of all,
in general it doesn't even make sense to reverse composition. For
example, if we have $f:\mathbb{R} \to \ma... |
H: Solving the 'easy' differential equation $(1 - \phi^2)\phi'' + \phi(\phi')^2 =0$.
I need to solve the following:
$$(1 - \phi^2)\phi'' + \phi(\phi')^2 =0.$$
Is there any standard method I can use?
AI: Just a lot of pattern matching and manipulation. Rewrite the equation as
$$\frac{\phi''}{\phi'} = -\frac{\phi \, \p... |
H: Ring homomorphisms $\mathbb{R} \to \mathbb{R}$.
I got this question in a homework:
Determine all ring homomorphisms from $\mathbb{R} \to \mathbb{R}$. Also prove that the only ring automorphism of $\mathbb{R}$ is the identity.
I know that $\mathbb{R}$ is a field, so the only ideals are $\mathbb{R}$ and $\{0\}$. T... |
H: Linear Transformation - orthogonal projection and orthogonal symmetry compositions
Let $\vec{a}$ a nonzero vector of $V^2$
I know that orthogonal projection of $\vec{u}$ onto the line generated by $\vec{a} \text{ is}$
$P_{\vec{a}}(\vec{u}) = \frac{(\vec{u}\cdot\vec{a})}{\vec{a}\cdot\vec{a}}\vec{a}$
And the orthogon... |
H: How to find the solution coset x in a equation involving cosets?
Is it possible to kindly tell me the steps necessary to find $\overline{x}$ for the following equation?
$$
\overline{3}\overline{x} = \overline{2} \text{ in } \mathbb{Z}_5 \text{ where } \mathbb{Z}_5 \text{ is quotient ring } \mathbb{Z} / \langle 5 \r... |
H: Differentiable functions without an antiderivative
Specifically, why is there no antiderivative, or any possible method of integrating (except numerically) say $\;e^{\csc(x)}$?
(I don't have my computer handy right now so I cant format the formula, sorry about that!)
AI: It's kind of similar to as saying: Why can... |
H: prove that , there is no element $a , b$ of the group $G$ which satisfy this property
let $G=(x) \times (y) $ where $(t)$ is the group generated by $t$ , $|x|= 8 , |y|=4$
let $H=(x^2y , y^2 )$ be isomorphic to $Z_4 \times Z_2 $
prove that , there are no elements a,b of G such that $G=(a) \times (b)$ and
$H = (a^... |
H: Prove $T$ has at most two distinct eigenvalues
The question is from Axler's Linear Algebra text. The $\mathcal{L}(V)$ stands for the space of linear operators on the vector space $V$.
Suppose that V is a complex vector space with dim $V=n$ and $T \in \mathcal{L}(V)$ is such that
$$\text{null} \ T^{n-2} \neq \... |
H: property of equality
The property of equality says:
"The equals sign in an equation is like a scale: both sides, left and right, must be the same in order for the scale to stay in balance and the equation to be true."
So for example in the following equation, I want to isolate the x variable. So I cross-multiply bo... |
H: Solve Modular Equation $5x \equiv 6 \bmod 4$
Here is an modular equation
$$5x \equiv 6 \bmod 4$$
And I can solve it, $x = 2$.
But what if each side of the above equation times 8, which looks like this
$$40x \equiv 48 \bmod 4$$
Apparently now, $x = 0$. Why is that? Am I not solving the modular equation in a right wa... |
H: Eigenvalues of a block matrix
For $X=\left(\begin{array}{cc} A & B\\ C & 0\end{array}\right)$, how are eigenvalues of $X$ related to the eigenvalues of $A$?
AI: Not much can be said. However, if $A$ is square and $X$ is Hermitian (hence $A$ is Hermitian and $C=B^\ast$) and $\lambda_1(M)\le\lambda_2(M)\le\lambda_3(M... |
H: About series representations
Can $\displaystyle \frac{1}{1+\frac{z}{n}}$ have the following series representation?
$\displaystyle \sum_{k=0}^{\infty}\frac{(-z)^k}{n^k}$
AI: In general for any $w\in \mathbb C$ with $|w|<1$ it holds that $\frac{1}{1-w}=\sum_{k=0}^\infty w^k$ (this is the sum of a geometric series). T... |
H: Nonsingularity of a block matrix
Let $X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$
and:
If $X$ is non-singular, is $A$ non-singular when $B$ is full column rank and $C$ is full row rank?
AI: Counterexample:
$$\begin{pmatrix}0&1\\1&0\end{pmatrix}$$ |
H: condition number of product of matrices
Let $\Phi$ be a $n \times m$ matrix and $C$ be a $n \times n$ diagonal matrix. Let $A = \Phi^{T}C\Phi$ (an $m \times m$ matrix). I am wondering if there is a theorem that relates the condition number of $A$, $\kappa(A)$ to the condition number (or singular values) of $\Phi$ a... |
H: complex exponential equation
I am trying to solve the following exponential equation: $z^{1+i} = 4$ where the argument of $z$ is between $-\pi$ and $\pi$. Here is what I have gotten so far: If $z = a + bi$ then the magnitude of $z$ is $2$ and $arctan(\frac{b}{a}) = -2$, therefore the two solutions of $z$ are in the... |
H: Poisson process (simple question)
Imagine you have two events starting at the same time. The duration time for each event is exponential, with different parameters. Knowing that one of the events is finished (we don't know which) at instant t, how does one get the distribution of the time between t and the moment t... |
H: Can a sequence of a function with a single variable be thought about as a function with two variables?
Long title, but first off is it logically ok to think of $\{f_n(x)\}$ as $f(n,x)$ where $n$ is restricted to a natural number?
Second, would this at all be useful? Thus far in my study of sequences of functions yo... |
H: Question in Hungerford's book
I'm trying to solve this question in Hungerford's Algebra
I didn't use the corollary:
And I used this map: $g:S^{-1}R_1\to S^{-1}R_2$, $g(r/s)=f(r)/f(s)$. I'm wondering how to prove using the corollary, is it hard?
Second, the map I have chosen indeed extends $R$? I identified the el... |
H: Solve $2\tan (x)\cos (x)=\tan (x)$, algebraically where $0 \leq x < 2\pi$
please help me correct if anything is wrong.. or even if i'am right
Solve $\quad 2\tan x\cos x=\tan x,$ algebraically where $0≤x<2π$
$$2\tan(x) \cos (x) - \tan (x) = 0$$
$$\tan (x)(2\cos (x) - 1) = 0$$
$$\text{So, either}\;\;\tan (x) = 0 \... |
H: What am I missing here?
That's an idiot question, but I'm missing something here. If $x'= Ax$ and $A$ is linear operator in $\mathbb{R}^n$, then $x'_i = \sum_j a_{ij} x_j$ such that $[A]_{ij} =a_{ij} = \frac{\partial x'_i}{\partial x_j}$, therefore $\frac{\partial}{\partial x_i'} = \sum_j \frac{\partial x_j}{\parti... |
H: Different kinds of systems
I got interested in learning more about Logic, recently.The first thing i noticed is that this topic is a lot bigger than i expected. As i'm trying to make a sense of it all ( seeing the big picture ) before delving into it, i'm having a lot of questions and i thought of asking help to pe... |
H: Trigonometry: Applications
A helicopter is flying due west over level ground at an attitude of 222 m, and at a constant speed. From point A, which is due west of the helicopter, two measurements of the angle between the ground and the helicopter are taken. The first angle measurement is 6 degrees and the second mea... |
H: Trigonometry: Find the smallest Angles
Calculate the measure of the smallest angle in the triangle formed by the points A (-2, -3), B(2, 5) and C(4, 1).
AI: Hint: Use the distance formula to get the lengths of the three sides and then apply the Law of Cosines. |
H: Is a real number the limit of a Cauchy sequence, the sequence itself, a shrinking closed interval of rational numbers, or what?
I've been studying a collection of analysis books (one of them Bishop's Constructive version) and contemplating the reals. Correct me if I'm wrong, but I feel that I have seen the Cauchy s... |
H: plot a point which is in form of lat and long in a pixel map
I need to plot a point which is in form of lat and long
this point is equivalent to a screen coordinate on pixels on my map.
I want to make a relation in these two point so if you want to plot a lat,long
I can know what is this position on map.
here a g... |
H: Is there a contradiction is this exercise?
The following exercise was a resolution to this problem
Let $\displaystyle\frac{2x+5}{(x-3)(x-7)}=\frac{A}{(x-7)}+\frac{B}{(x-3)}\space \forall \space x \in \mathbb{R}$. Find the values for $A$ and $B$
The propose resolution was:
In order to isolate $A$ on the right sid... |
H: Expressing Vectors In Terms of Other Vectors
My professor asked us a few questions in class and asked us to think about them. He's going to reveal the solutions on Thursday, but I want to understand it before he talks about it on Thursday in greater detail. If anyone could explain any of the things below, it would ... |
H: Is there a (real) number which gives a rational number both when multiplied by $\pi$ and when multiplied by $e$?
Besides $0$ of course. What about addition and exponentiation? I would think there's no such number, but I'm not sure if I could prove it.
AI: It is open whether there is such number. The reason is that ... |
H: Relating properties of $H$ to properties of $G/H$
$G$ is a group and $H$ is a normal subgroup of $G$. Prove that $G/H$ is cyclic iff there is an element $a \in G$ with the following property: for every $x \in G$, there is some integer $n$ such that $xa^n \in H$.
Can anyone help, since I have no idea how to work on ... |
H: A numerical aptitude problem
I have got this problem that says : Given digits $2,2,3,3,3,4,4,4,4$ how many distinct $4$ digit numbers greater than $3000$ can be formed.
The options are: $50,51,52,54.$
Is there any way I can logically solve the problem instead of manually counting ?
Also, Almost same problem has ... |
H: How find the $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}(-1)^{m+n}\frac{1}{n(m+2n)}$
find the value
$$\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}(-1)^{m+n}\dfrac{1}{n(m+2n)}$$
I think this is good problem,Thank you everyone
I find
$$\int_{0}^{1}\dfrac{\ln{(1+x^2)}}{1+x}dx=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}(-1)^{m+n}\df... |
H: How to prove $\sum_1^k W_i$ is the smallest subspace of $V$ that contains $W_1\bigcup\cdots\bigcup W_k$
Definition: Let $W_1, \cdots , W_k$ be a subspace of a vector space $V$. The sum is
$\sum_1^k W_i = {x_1+ \cdots + x_k | x\in W_i }$
Proposition:
$\sum_1^k W_i$ is a subspace
$W_i \subseteq \sum_1^k W_i\ \foral... |
H: "Goldbach's other conjecture" and Project Euler - writing 1 as a sum of a prime and twice a square
From Problem 46 of Project Euler :
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
$$9 = 7 + 2 \cdot 1^2$$
$$15 = 7 + 2 \cdot 2^2$$
$$2... |
H: Group of inner automorphisms of a group $G$
Let $G$ be a group. By an automorphism of $G$ we mean an isomorphism $f: G\to G$
By an inner automorphism of $G$ we mean any function $\Phi_a$ of the following form:
For every $x\in G$, $\Phi_a(x)=a x a^{-1}$.
Prove that every inner automorphism of $G$ is an automorphism ... |
H: the angle between $Ax$ and $x = [0, 1]^T$ is
If the trace and the determinant of an orthogonal $2 \times 2$ matrix A are 1, then
the angle between $Ax$ and $x = [0, 1]^T$ is
(i) $25^{\circ}$
(ii) $30^{\circ}$
(iii) $45^{\circ}$
(iv) $60^{\circ}$
I just see the characteristic polynomial is $x^2-x+1=0$, could any one... |
H: Proving that the medians of a triangle are concurrent
I was wondering how to prove Euclid's theorem: The medians of a triangle are concurrent.
My work so far:
First of all my interpretation of the theorem is that if a line segment is drawn from each of the 3 side's medians to the vertex opposite to it, they interse... |
H: Help with understanding a derivation in a book
I'm reading a book on time series analysis and I came across with a derivation I cannot understand. The following picture is a part from my book and I have highlighted with red color the part I'm confused with. I have used other colors to highlight some other points in... |
H: Let S, T be two subspaces of $\mathbb{R}^{24}$ such that $\dim(S) = 19 $ and $\dim(T ) = 17.$
Let S, T be two subspaces of $\mathbb{R}^{24}$ such that $\dim(S) = 19 $ and $\dim(T ) = 17.$
Then
(i) the smallest possible value of $\dim(S ∩ T )$ is 2
(ii) the smallest possible value of $\dim(S + T )$ is 19
(iii) the l... |
H: need to find the rank of $T$
$B=\begin{pmatrix}2&-2\\-2&4\end{pmatrix}$ , $T:M_2(\mathbb{R})\to M_2(\mathbb{R})$ defined by $T(A)=BA$, we need to find the rank of $T$, I have calculated by hand and got rank is $4$, could any one tell me if there is any process with out calculating that?
AI: $T$ is a linear map of s... |
H: Can anybody validate this WolframAlpha computation?
Can anybody validate this WolframAlpha computation?
http://www.wolframalpha.com/input/?i=GCD%5BDivisorSigma%5B1%2Cx%5D%2C+DivisorSigma%5B1%2Cx%5E2%5D%5D
Thank you!
AI: PolynomalGCD is a simple operation: it takes the GCD of polynomials. It doesn't know about trans... |
H: How can i create a presentation of a group ?
in Dummit and Foote , the notion of presentation is introduced in section 1.2 which talks about dihedrial group of order $2n$.
and after this , it was rare to talks about presentation throw the exercises or the material of the sections . but suddenly in chapter 4 , it a... |
H: Every normal matrix is diagonalizable?
Sorry, I mis-typed the question. The real question is that
Every normal matrix is diagonalizable?
and the answer is False.
Can you give me a counter example?
AI: Let
$$A=\left(\begin{matrix}1&i\\
i&-1\end{matrix}\right)$$
then $A$ is a symmetric complex matrix and its charact... |
H: Proving divisibility of numbers
Let us take a two digit number and add it to its reverse.We have to prove that it is divisible by 11.
Same way,if we subtract the larger number from the other,it is divisible by 9.How can we explain these?
AI: Hint: any two digit number $ab$ can be represented in base 10 as: $$ab=10... |
H: This tower of fields is being ridiculous
Suppose $K\subseteq F\subseteq L$ as fields. Then it is a fact that $[L:K]=[L:F][F:K]$. No other hypotheses are needed (I'm looking at you, Hungerford V.1.2).
Now obviously $[\mathbf{C}:\mathbf{R}]=2$. But consider the fact that the algebraic closure of $\mathbf{R}(t)$ has c... |
H: recurrence relation with induction
The following recurrence relation:
$T(n)=T(n-1)+n=1+\frac{n^2+n}{2}=\theta(n^2)$, so this mean that:
there is $c_1, c_2, n_o > 0 : c_1n^2<=1+\frac{n^2+n}{2}<=c_2n^2$,
the second inequality it's easy, but how to prove by induction the first:
$c_1n^2<=1+\frac{n^2+n}{2}$?
AI: Choose ... |
H: Approximate Nash Equilibrium
I am sort of confused by the notion of approximate Nash equilibrium. I will try to express my confusion in the following exercise.
Problem. Is it true that for every two player game where every player has $n$ available actions and all payoffs $\in [0,1]$, there exists approximate $\epsi... |
H: How does this strange phenomena happen in quotient of groups ?
in my question , here , i learned a strange fact from the comments which was a surprise for me on the answer of landsacpe !
and this surprise it :
if $G$ is a group , $H$ and $K$ are two normal subgroups of $G$ such that $H$ is isomorphic to K then ... |
H: Invariance under transition of integral
I am reading a calculus book, and in section on invariance under transition of integral I get the following passage I don't quite understand:
Suppose we have a function $s(x)$. Then we introduce another function $t(x)$, such that $t(x)=s(x)+c$, i.e. $t(x)$ is $s(x)$ shifted t... |
H: Problem in computing of integral by substitution.
I want to compute an integral like
$\int_0^{+\infty} \ln(1+x)e^{-x}\,\mathrm dx$.
Then denote $\mu = 1-e^{-x}$, so $x=-\ln(1-\mu)$. Substitute this into the integral, we get
$$\int_0^1 \ln(1-\ln(1-\mu))(1-\mu)\,\mathrm d(-\ln(1-\mu)) = \int_0^1 \ln(1-\ln(1-\mu))\,... |
H: Well definition of algebra homomorphism
Suppose we have unitary $\mathbb{K}$-algebras $A,B$ and $A$ is generated by $\{a_1,...,a_n\}$. Let $f: A \rightarrow B$ be an unitary algebra homomorphism satisfying $f(a_i) = b_i$ where $b_i$ are any elements of $B$. When is $f$ well defined?
I know that $f$ does not have... |
H: RObin problem (Laplace equation)
Let
$\Delta u = 0 $,
$ \frac{\partial u}{\partial v}(x) + \alpha u(x) = 0 $
be the Laplace equation with Robin conditions.
How do I prove it has at most one solution.
If I could prove that any two solutions differs only by one constant, I would prove this using the unicity of the t... |
H: Integral related to harmonic functios
It suppose to be a easy task. But I couldn't solve it (I guess I can't learn much analysis).
If $ u $ is harmonic, in the middle of my problem, I need to prove that the integral
$ \int _ {\partial \Omega } \displaystyle\frac{\partial u}{\partial v} d s_x =0 $
THis $ v $ denotes... |
H: Proper Subsets of Real Numbers
Every non-empty proper subset of Real Numbers is either open or closed. true/false
AI: It is false.
For example, $[0,1)$ is not open, since $0 \in [0,1)$ has not a nbhd $U$ such that $0 \in U\subset [0,1)$. Also, $[0,1)$ is not closed, since the point 1 is accumulation of $[0,1)$, h... |
H: A particular proof that I don't follow.
In Rudin Thm.3.20, along the proof of (d), there is an inequality that says for $n > 2k$
$${n\choose {k}}p^k \ge \frac{n^k}{2^kk!}p^k $$
I simply don't get this, especially why $n$ is restricted as such.
From the looks of it we just need to show
$${n P k} \ge \left(\frac{n}{... |
H: Methods for assessing convergence of $\sum\limits_{k=1}^\infty (-1)^k a_k$ when $a_k$ is complicated
Think of the problem of convergence of the series
$$ \sum_{k=1}^\infty (-1)^k a_k$$
Is it possible to consider the convergence of $\sum\limits_{k=1}^\infty (-1)^k b_k$ if $\lim \limits_{k\rightarrow\infty } \dfrac{... |
H: How has this derivation been achieved?
This is a step in a derivation I am to know, but I can't figure out how to achieve it - specifically, how we end up with $1 + e^{-X\theta}$ instead of $1 + e^{X\theta}$ as the denominator in the applicable terms. Working it through I seem to get the latter.
These steps are giv... |
H: Farkas' lemma application
Farkas' lemma: Let $A$ be $m \times n$ matrix and $b \in \mathbb{R}^m$ $m$-dimensional vector.
Then, exactly one of the following holds:
(a) there exists some $x \in \mathbb{R}^n$, $x \geq 0$, such that $Ax = b$
(b) there exists some vector $y \in \mathbb{R}^m$ such that $y^TA \geq 0$ and ... |
H: Jordan form and an invertible $P$ for $A =\left( \begin{smallmatrix} 1&1&1 \\ 0 & 2 & 2 \\ 0 & 0 & 2 \end{smallmatrix}\right)$
$A = \begin{pmatrix} 1&1&1 \\ 0 & 2 & 2 \\ 0 & 0 & 2 \end{pmatrix}$, find the jordan form and the invertible $P$ such that: $A = P J P^{-1}$.
Now I found the characteristic polynomial and... |
H: How would you best describe the rate of growth of the function $f(x) = cxr^x$?
Consider the function $f(x) = cxr^x$, where both $r$ and $c$ are constants and we have cases: (a) $r<1$, (b) $r>1$. Regarding terminology, how would you best describe the asymptotic growth of $f(x)$ in cases (a) and (b)? Though you cou... |
H: Proving a graph is connected iff it has no isolated points
How do I prove that a graph is connected iff it has no isolated points?
I can do the first half; if the graph is connected, any pair of vertices have a walk between them. Suppose there is an isolated point. Since an isolated point has no edges, it is imposs... |
H: For what values of $a$ $\int^2_0 \min(x,a)dx=1$
I want to check for what values of $a$ the integral of this function is equal to $1$
$$\int^2_0 \min(x,a)dx=1$$
What I did is to check 2 cases and I am not sure is enough :
case 1:
$$a<x \rightarrow \int^2_0 a = ax|^{2}_{0}=2a=1 \rightarrow a=\frac{1}{2}$$
now I dont... |
H: What are two continuous maps from $S^1$ to $S^1$ which are not homotopic?
This is an exam question I encountered while studying for my exam for our topology course:
Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as well.)
The only continuous maps from $S^1$ to $S... |
H: Limit of $\lim\limits_{x\to 0}\left(\frac{1+x\sin(2x)-\cos(x)}{\sin^2(x)}\right)$
I want to evaluate this limit and I faced with one issue.
for this post I set $L`$ as L'Hôpital's rule
$$\lim\limits_{x\to 0}\left(\frac{1+x\sin(2x)-\cos(x)}{\sin^2(x)}\right)$$
Solution One:
$$\lim\limits_{x\to 0}\left(\frac{1+x\sin(... |
H: How to prove that this particular function is continuous?
This was an exam question from the topology course of a previous year.
Let $(X,d)$ be a compact metric space and $f: X \to X$ be a continuous map. How that the map $g: X \to \mathbb{R} , x \mapsto d(x,f(x))$ is continuous.
I know several definitions of con... |
H: Normalize a negative range
How can I convert a range of -12 to 12 as a value between 0 and 1 ?
I'm guessing this is a relatively simple problem, I just can't figure it out.
float value=[valuestring floatValue];
float absval =fabsf(value);
if (value<0) {
//0-0.5
... |
H: Prove path P4 and the cycle C5 are self-complementary
I can "show" that the two graphs are in fact self-complementary by making a drawing.
How do I "prove" this?
How can I rigorously put in words that the complement of P4 is P4 itself?
In other words, how is an isomorphism of a graph proven?
Is it possible to do th... |
H: How to show that $\int_{a}^{b}\left \lfloor x \right \rfloor dx+ \int_{a}^{b}\left \lfloor -x \right \rfloor dx=a-b$?
How to show that
$$
\int_{a}^{b}\left \lfloor x \right \rfloor dx+ \int_{a}^{b}\left \lfloor -x \right \rfloor dx=a-b
$$
Where $\left \lfloor x \right \rfloor$ means greatest integer $\leqslant x$.
... |
H: Find the unknown $n$
If $A$ and $B$ are invertible matrices (with 2013 rows and columns) such that
$A^9$ = $1$ and $ABA^{-1}$ = $B^2$, then prove that there exists a natural number
$n$ such that $B^n = 1$. Find the smallest such $n$.
AI: We have $AB = B^2A$, from which it immediately follows that $AB^n = B^{2n}A$. ... |
H: If $\int^{1}_{0} \frac{\tan^{-1}x}{x} dx = k \int^{\pi/2}_{0} \frac {x}{\sin x} dx$, find $k$.
Problem: If $$\int^{1}_{0} \frac{\tan^{-1} x}{x} dx = k \int^{\pi/2}_{0} \frac{x}{\sin x} dx,$$ find $k$.
Solution: If we put $x=\tan t$ in $$\int^{1}_{0} \frac{\tan^{-1}x}{x} dx$$ then integral becomes $$\int^{\pi/4}_{0... |
H: Find the largest number having this property.
The $13$-digit number $1200549600848$ has the property that for any $1 \le n \le 13$, the number formed by the first $n$ digits of $1200549600848$ is divisible by $n$ (e.g. 1|2, 2|12, 3|120, 4|1200, 5|12005, ..., 13|1200549600848 using divisor notation).
Question 1: Fin... |
H: Write the 2nd degree equation which have the following roots
$y_1$=${(x_1+x_2\varepsilon+x_3\varepsilon^2)}^3$
$y_2$=${(x_1+x_2\varepsilon^2+x_3\varepsilon)}^3$
where $x_1,x_2,x_3$ roots for the $x^3+ax^2+bx+c=0$ and $\varepsilon$ = $\frac{-1}{2}$+$i$$\frac{\sqrt{3}}{2}$. ${\varepsilon}^3=1$
AI: Hint: The coefficie... |
H: Is the number of quadratic nonresidues modulo $p^2$, greater than the number of quadratic residues modulo $p^2$?
Let $p$ be a prime. The number of quadratic nonresidues modulo $p^2$, is greater than the number of quadratic residues modulo $p^2$.
Is that statement true or false? Why?
Thank you.
AI: Let's count the n... |
H: A question arising from some misunderstanding involving the Laplace Transform of the Heaviside function
I am trying to compute the Laplace Transform of the Heaviside Step-function.
I define the Heaviside step-function with the half-maximum convention:
$H(t-t_0) = 0$ for $t < t_0$ ; $H(t-t_0) = 1/2$ for $t=t_0$ ; $... |
H: What is cofinality of $(\omega^\omega,\le)$?
Let us consider the set $\omega^\omega$ of all maps $\omega\to\omega$ with the pointwise ordering. By cofinality of $(\omega^\omega,\le)$ I mean the smallest cardinality of the subfamily $\mathcal B$ such that for each $f\in\omega^\omega$ there is a $g\in\mathcal B$ such... |
H: Problem related to GCD
I was solving a question on GCD. The question was calculate to the value of $$\gcd(n,m)$$
where $$n = a+b$$$$m = (a+b)^2 - 2^k(ab)$$
$$\gcd(a,b)=1$$
Till now I have solved that when $n$ is odd, the $\gcd(n,m)=1$.
So I would like to get a hint or direction to proceed for the case when $n$ i... |
H: Multiplying two summations together exactly.
Consider the integral: $$\int_0^1 \frac{\sin(\pi x)}{1-x} dx$$ I want to do this via power series and obtain an exact solution.
In power series, I have $$\int_0^1 \left( \sum_{n=0}^{\infty} (-1)^n \frac{(\pi x)^{2n+1}}{(2n+1)!} \cdot \sum_{n=0}^{\infty} x^n \right)\,\,... |
H: Cumulative distribution function (of NHPP inter-arrival time) not tending to 1?
According to this website, for a non-homogeneous Poisson process with mean $m(t) = \int^t_0 \lambda(u) \, dt$, the cumulative distribution function (CDF) for the inter-arrival time to the first event is,
$$F_0(t) = 1 - e^{- m(t)}.$$
As ... |
H: Interval of convergence for $\sum_{n=1}^∞({1 \over 1}+{1 \over 2}+\cdots+{1 \over n})x^n$
What is the interval of convergence for $\sum_{n=1}^∞({1 \over 1}+{1 \over 2}+\cdots+{1 \over n})x^n$?
How do I calculate it? Sum of sum seems a bit problematic, and I'm not sure what rules apply for it.
Thanks in advance.
AI... |
H: probability question
a class of 60 students, 15 students failed in exam A, 25 students failed in exam B ,8 students failed in both , what is the probability of a student passing A and failing B ?
when I solve it using Venn diagrams the probability is 17/60
but when I solve it using P(Passing A & failing B)=P(A' int... |
H: Probability of consecutive dice rolls
This is probably quite a simple question but here I go..
Suppose you are going to roll a six-sided (fair) die N times, what is the probability that you will get at least one set of three consecutive numbers in a row?
Hopefully that's clear enough but as an example, in nine roll... |
H: $\mathbb{C}[f(x)]$ is not a maximal subring of $\mathbb{C}[x]$
Prove that $\mathbb{C}[f(x)]$ is not a maximal subring of $\mathbb{C}[x]$ for all $f\in\mathbb C[x] $.
I managed to prove it in a straightforward way by taking $f(x)=a_0 + a_1x+\cdots+ a_nx^n$ and by means of contradiction. But I am looking for a mor... |
H: Wild automorphisms of the complex numbers
I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could explain in the simplest possible way (please) how I could imagine such wil... |
H: Proving an equality on $\arctan x$
To demonstrate the following equality
\begin{equation}
\arctan x=\pi/2- \arctan(1/x)
\end{equation}
I have proceeded in this way. I know that
\begin{equation}
\arctan x= \int \frac{1}{1+x^2} dx + C
\end{equation}
But:
\begin{equation}
\int \frac{1}{1+x^2} dx=\int \frac{1}{x^2(1+1/... |
H: An inequality for symmetric matrices: $ \mbox{trace}(XY)\leq \lambda(X)^T\lambda(Y) $.
Let the vector of eigenvalues of a $n\times n$ matrix $U$
is denoted by
$$
\lambda(U)=\big(\lambda_1(U),\lambda_2(U),\ldots,\lambda_i(U),\ldots\lambda_{n-1}(U),\lambda_n(U)\big)^T.
$$
where the eigenvalues are ordered as $\lambd... |
H: Find the legs of isosceles triangle, given only the base
Is it possible to find the legs of isosceles triangle, given only the base length? I think that the info is insufficient. Am I right?
AI: You are correct that it is impossible. Given only the base length of an isosceles triangle, we cannot determine the leng... |
H: Find the limit : $\lim\limits_{n\rightarrow\infty}\int_{n}^{n+7}\frac{\sin{x}}{x}\,\mathrm dx$
I have this exercise I don't know how to approach :
Find the limit : $$\lim_{n\rightarrow\infty}\int_{n}^{n+7}\frac{\sin x}{x}\,\mathrm dx$$
I can see that with $n\rightarrow\infty$ the area under the graph of this fun... |
H: prove this $\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$
let $f\in C^1[0,2]$,and such $\int_{0}^{2}f(x)dx=0,f(0)=f(2)$,
show that
$$\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$$
I think we must use $Cauchy$ inequality
my idea:I have see this
let $f(x)\in C^1([a,b],R)$,and $f(a)=f(b)=0$,show that：$$\displaystyle\... |
H: Derivatives constants basic
I'm struggling with basic rules for derivatives.
So $\dfrac{d}{dx} 2x = 2$
Because you factor out the constant to $2\times \dfrac{d}{dx}x$ and that is $2\times1 = 2$
But $\dfrac{d}{dx}2 = 0$
Again factor out to $2\times \dfrac{d}{dx}1(*)$ and that is $2\times0 = 0$
(*) Is that the right ... |
H: Cauchy inequality $\Rightarrow$ Schwarz's integral inequality. Why can't the limit of the prior be used to deduce the latter?
Given the Cauchy-Schwarz inequality and the Riemann definition for the integral,
$$\sum_{k=1}^{n}a_kb_k\le\sqrt{\sum_{k=1}^{n}a_k^2}\sqrt{\sum_{k=1}^{n}b_k^2}$$
$$\int_a^bf(x)dx=\lim_{n\righ... |
H: Collinear points in $\mathbb{K}^n$, $k_1k_2k_3 = (k_1-1)(k_2-1)(k_3-1)$
Could you tell me how to prove that given three non collinear points in $\mathbb{K}^n$: $A, \ B, \ C$, the following three points:
$A_1 = k_1B + (1-k_1)C, \ \ B_1=k_2C + (1-k_2)A, \ \ C_1=k_3A + (1-k_3)B$
are collinear $\iff \ \ \ k_1k_2k_3= (k... |
H: Space $H^1([0, T], H^{-1}(U))$
My teacher uses the space $H^1([0, T], H^{-1}(U))$, where $H^1 = W^{1,2}$ (Sobolev space) and $H^{-1}(U)$ is the dual space of $H_0^1(U)$.
So we have $u \in H^1([0, T], H^{-1}(U))$ if $u(t)$ is in $H^{-1}(U)$ for all $t \in [0, T]$, $u \in L^2([0, T], H^{-1}(U))$ (which means that $\d... |
H: $m$ balls into $n$ urns
Assume that there are $m$ balls and $n$ urns with $m\gt n$. Each ball is thrown randomly and uniformly into urns. That is, each ball goes into each urn with probability $\dfrac1n$.
What is the probability that there are exactly $r$ urns with at least one ball in it? In other words, what is t... |
H: Verify my attempt to diagonalize matrix
Decide whether matrix $A$ is diagonalizable. If so, find $P$ such that $P^{-1}AP$ is diagonal.
We are given: $A = \begin{bmatrix}1 & 0 & 0 \\ 1 & 2 & 1 \\ 0 & 0 & 1\end{bmatrix}$
We set up and and solve: $|A - \lambda I| = 0$, which yields:
$$\left|\begin{matrix}1-\lambda & 0... |
H: On the definition of cofinite.
I am having some difficulty comprehending the definition of a cofinite set. I am seeking confirmation of whether my understanding is correct and some clarification on the definition.
Wikipedia provides the following definition:
Definition: A cofinite subset of a set $X$ is a subset $A... |
H: Optimization of entropy for fixed distance to uniform
Suppose that I know that a probability distribution with $n$ outcomes is very close to being uniform (that is: $\forall i,p_i=\frac{1}{n}$), and in particular for $n\epsilon\ll 1$ the distribution verifies
$$\sum_{i=1}^n|\frac{1}{n}-p_i|=\epsilon$$
Now consider... |
H: open subset of $G\times G$
If $O$ be an open symmetric subset of topological group $G$ such that $e\in O$, then is $V_O=\{(a,b)\in G\times G: a^{-1}b\in O\}$ open in $G\times G$?
AI: Yes. $f\colon G\times G\to G$, $(x,y)\mapsto x^{-1}y$ is continuous, hence $V_O=f^{-1}(O)$ is open. (We do not need $e\in O$ or symme... |
H: solving for an argument of arctan()
In a course of writing a software to do computer vision I'm trying to "calibrate" the assembly with as little user interaction as possible. As a result I'm getting a few numbers from the point of view of my hardware and then trying to analyze those and figure out the relative spa... |
H: What is the notation for the set of all $m\times n$ matrices?
Given that $\mathbb{R}^n$ is the notation used for $n$-dimensional vectors, is there an accepted equivalent notation for matrices?
AI: If $A$ is an $m\times n$ matrix, then $$A \in \mathbb{R}^{m\times n}$$ |
H: Question involving absolute function
I saw this interesting problem in a math puzzle forum:-
Find all integral values of $t$ such that the equation $|s-1| - 3|s+1| + |s+2| = t $ has no solutions.
How does one approach these kind of problems?
AI: Divide into regions like so:
Case 1: Assume $s\ge 1$
Your equation r... |
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