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H: Can my MSE reputation be any positive integer?
As far as I know there are five kinds of vote
$+2$ for an edit
$-2$ for a downvote
$+10$ for an answer
$+15$ for an accepted answer
$+5$ for a question
Suppose that this is true. Can a MSE reputation take any positive integer value? How do you prove it?
AI: Yes. Just... |
H: How to prove this Inverse Property of Group
I am given a Group $G$ with Projection $*:G\times G \Rightarrow G$ and with these properties:
$a*(b*c) = (a*b)*c$
$e*a=a$
$b*a=e$, $b$ is invers Element
$a*b=b*a$
I want to prove $(a*b)^{-1}=b^{-1}*a^{-1}$.
I am stuck not knowing how to interpret $^{-1}$, because it ... |
H: is this operating procedure an Abelian Group?
I have to show if the following procedure gives a (Abelian) Group (G, *).
$G = \{ \textrm{true}, \textrm{false} \}$
$a*b := ( a \leftrightarrow b)$ (which means that $a$ is $\textrm{true}$ if and only if $b$ is $\textrm{true}$)
1.) Closure
For all $a,b \in G$, the re... |
H: what am I doing wrong in the following
Question:
Calculate the equation of tangent at the point $(4,2)$ if $y=\sqrt{X}$
My answer:
$(x_0,y_0) = (4,2)$,
$f'(X) = y^{1/2} y'= 1/2X^{-1/2}$
$f'(4) = (\frac{1}{2})(4)^{-1/2} = 1$
$y-y_0=f'(X)(x-X_0)$
$y-2=1 (X-4)$
$y= x-2$
Correct answer:
$y=\frac{1}{4}x +1$
what am I do... |
H: Calculation of integers $b,c,d,e,f,g$ such that $\frac{5}{7} = \frac{b}{2!}+\frac{c}{3!}+\frac{d}{4!}+\frac{e}{5!}+\frac{f}{6!}+\frac{g}{7!}$
There are unique integers $b,c,d,e,f,g$ such that $\displaystyle \frac{5}{7} = \frac{b}{2!}+\frac{c}{3!}+\frac{d}{4!}+\frac{e}{5!}+\frac{f}{6!}+\frac{g}{7!}$
Where $0\leq b,c... |
H: Find equation of the circular cross section of a unit sphere
I have a unit sphere in Cartesian coordinates:
$x^2 + y^2 + z^2 = 1$
or in spherical coordinates:
$x = \rho \sin(\phi) \cos(\theta)\\
y = \rho \sin(\phi) \sin(\theta)\\
z = \rho \cos(\phi)$
I select a point $P$ on the surface of the sphere, in the coordin... |
H: Choosing Dense sets
More than one option can be correct.
Which of the following sets are dense in $R^2$ with respect to the usual topology.
(a) $\{(x,y)\in R^2:x\in N\}$
(b) $\{(x,y)\in R^2:x+y$ is a rational number$\}$
(c) $\{(x,y)\in R^2:x^2+y^2=5\}$
(d) $\{(x,y)\in R^2:xy\ne0\}$
Natural numbers are not limit p... |
H: Prove the following language is not regular
The set of strings of 0's and 1's, beginning with a 1, such that when interpreted as an integer, that integer is prime.
I'm assuming the best way to move forward is to use the pumping lemma. I'm having difficulty developing a contradiction in this case because typically ... |
H: Total number of divisors is a prime
Which numbers have prime number of divisors?
For example, $16$ has $1$, $2$, $4$, $8$, $16$, a total of $5$ divisors, $5$ being a prime.
I found that primes and the power of primes such that $p^{q-1}$, where $p$ and $q$ are prime numbers, all have prime number of divisors. Is thi... |
H: Linearization by freezing the coefficients of the main part of the PDE
Let $\Omega\subset C^0$ a bounded domian in $\mathbb{R}^2$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a non negative classical solution of
$$
(1+x^2)u_{xx}-2xu_{xy}+(1+u)u_{yy}-(1+u^2)u_x+(1+u_x)u_y-u=1\text{ in }\Omega,\\u(x,y)=\... |
H: How can a single integral equal a triple integral?
Here is part of a discussion about the gravitational potential of a sphere:
Let $dx$ $dy$ $dz$ represent an infinitesimal volume containing matter of density $\rho$ and mass $dm$. Then the potential at distance $R$ from the element will be
$$ dV = -\frac{G\rho}{R}... |
H: The cheapest offcut carpet to buy for two rooms word problem
Correct answer: C
This might come of as a slightly dumby question, but I've been googling for a while and have gone nowhere.
What I presumed the question was asking was to add up the two areas of the rooms, and choose the carpet area that is as close to... |
H: How prove this limit $\lim_{n\to\infty}\frac{n^2}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}}=0$
Assume that a positive term series $\displaystyle\sum_{n=1}^{\infty}a_{n}$ converges,
show that
$$\lim_{n\to\infty}\dfrac{n^2}{\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{n}}}=0.$$
My try: since $a... |
H: If a sequnce $(a_n)_n \to L$, $(\sqrt{a_n})_n \to \sqrt L$
How do you prove the following:
If a sequnce $(a_n)_n \to L$, $(\sqrt{a_n})_n \to \sqrt L$
AI: $$|\sqrt{a_n}-\sqrt{L}|= |\frac {(\sqrt{a_n}-\sqrt{L})(\sqrt{a_n}+\sqrt{L})} {\sqrt{a_n}+\sqrt{L}}|=|\frac{a_n-L}{\sqrt{a_n}+\sqrt{L}}|$$
Now, $\sqrt{x} \geq 0... |
H: What condition satisfies the equation.
$f(x) = x^2 - 2x + \sin^2 \alpha$ = 0, if
$-1 \leq x \leq 1$
$\phantom{-}0 \leq x \leq 2$
$\phantom{-}0 \leq x \leq 4$
$-2 \leq x \leq 2$
How do I solve it?
AI: We complete the square, and express the equation as a difference of squares:
$$\begin{align} x^2 - 2x + \sin^2 \al... |
H: How to calculate area of this shape?
I was trying to solve a complicated problem then I came accros to this complicated problem. I believe that there is enough information to calculate the area. Can you help me to find a general formula for the area of this shape, in terms of $x,\alpha,\beta$?
I forgot to write on... |
H: Solving for an $x$ in matrices, with condition $AB=BA$
I'm just starting to learn about matrices, and during one exercise I got a question to which I have no answer; Due to the fact that I haven't learned it yet...
The question is as follows:
Let $A = \left[\begin{matrix}1&x\\2&3\end{matrix}\right]$ and $B = \left... |
H: If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?
If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?
One
Two
Three
Infinitely many
Also, what are these pairs?
AI: Since $a, b$ are zeros, they each satisfy the equation:
$$x=a: \quad... |
H: Showing the expectation of the third moment of a sum = the sum of the expectation of the third moment
Let $X_{1},\cdots,X_{n}$ be independent, each with mean 0, and each with finite third moments. Show that $E\left\{\left( \sum_{i=1}^{n}X_{i}\right)^{3}\right\} = \sum_{i=1}^{n}E\left\{ X_{i}^{3} \right \}$.
Thanks ... |
H: Problem with semisimple ring theorem
Proposition: For a ring $R$ the following statements are equivalent:
(a) $R$ has a simple left generator;
(b) $R$ is simple left artinian;
(c) For some simple $_RT, _RR \cong T^{(n)}$ for some $n$;
(d) $R$ is simple and $_RR$ is semisimple.
My try to understand above proof:
$... |
H: Spectrum of the multiplication operator
Let $B[0,1]$ be the Banach space of bounded complex functions on $[0,1]$ endowed with the supremum norm.
I've have to show that the spectrum of the multiplication operator $T_q: B[0,1] \rightarrow B[0,1]$
$$
(T_q f)(t) : = q(t)f(t), \,\,\, t \in [0,1]
$$
is
$
\sigma(T_q)=\ove... |
H: How do I evaluate left and right limits?
I have this assignment:
$$\lim_{x \to 1} \frac{x^2 - 1}{|1 - x^3|}$$
I do not understand how I should do to separate this into two problems (one for $x \to 1^-$ and one for $x \to 1^+$ and then get rid of the absolute value. How do I do that?
AI: If $x \gt 1$, then $|1-x^3|... |
H: How to integrate this integral,
$\int_{\mathbb{R}^ n} || \mathbf{x} - \mathbf{y} ||^{-k} d\mathbf{x}$? Here $k>0$.
AI: Hint:
$$
\int_{\mathbb R^n} f(\|\mathbf x\|)\,\mathrm d\mathbf x = \omega_n\int_0^\infty f(r)\,r^{n-1}\,\mathrm dr
$$
Where $\omega_n$ is the surface area of the $(n-1)$-dimensional unit sphere. |
H: Help with proof of boundedness of the union of bounded sets
I am trying to prove that the union of a finite number of bounded sets is bounded. A set $A$ is bounded if diameter, $d(A)$ is finite, and $d(A)=\sup_{x,y\in A}\rho(x,y)$ where $\rho(.)$ is metric.
My idea is to use induction. So, let $n=2$ and $A_1$ & $A_... |
H: How prove this inequality $\sum_{i=1}^{n}\frac{\sqrt{1}+\sqrt{2}+\cdots+\sqrt{i}}{i^2}\le\sqrt{2n-1}$
show that
$$\sum_{i=1}^{n}\dfrac{\sqrt{1}+\sqrt{2}+\cdots+\sqrt{i}}{i^2}\le\sqrt{2n-1}$$
My try:
$$x\in(n-1,n)\Longrightarrow \sqrt{x}>\sqrt{n-1}$$
so
$$\int_{n-1}^{n}\sqrt{x}dx>\sqrt{n-1}$$
so
$$\dfrac{2}{3}n^{\fr... |
H: Pythagoras numbers and fermats last theorem
I am reading "What Is Mathematics? An Elementary Approach to Ideas and Methods"
And I am stuck here, I don't get it. I have posted a screen shot underlining what my doubt is..
I dont get it when the author says while the pythagoras theorem is : $a^2 + b^2 = c^2$
and then ... |
H: How do I prove that $f(x) = x^2 : (0, 1/2) \to (0, 1/2)$ is not a contraction mapping?
How do I prove that $f(x) = x^2 : (0; 1/2) \to (0; 1/2)$ is not contraction mapping?
I'd like to prove this in $\mathbb{R}$ with the Euclidean metric.
AI: If it was a contraction map, then there would exist some $\lambda<1$ such ... |
H: Could anybody check this integral?
in a lengthy calculation by hand I got that
$$ \frac{2}{\pi \sigma_k} \int_{-\infty}^{\infty} \frac{sin^2(\frac{\sigma_k}{2}(v_gt-x))}{(v_gt-x)^2} dx =1$$
Now I was wondering whether there is anybody who could check this ( with a CAS or by hand )? If you decide to do the last opti... |
H: show $\Bbb Z_6 \cong \Bbb Z_3 \times \Bbb Z_2 $.
I am trying to determine if $\Bbb Z_6 \cong \Bbb Z_3 \times \Bbb Z_2 $. I noticed that $\Bbb Z_6$ has a generator $1$ and $\Bbb Z_3 \times \Bbb Z_2$ has generator $(1,1)$. Now I set up the bijection $f :\Bbb Z_6 \to \Bbb Z_3 \times \Bbb Z_2 $ where
\begin{eqnarra... |
H: Is holomorphic functions on (0, 1) (vanishing at endpoint) dense in $C_0((0, 1))$?
Here is my argument, please let me know if it works or not.
By Stone-Weierstrass Theorem (Complex Version), functions in $C_0((0, 1))$ can be uniformly approximated by polynomials in z and $\bar{z}$ which vanishes at 0 and 1. But on ... |
H: Largest Difference of Two Numbers in an Array
Given an integer sequence $s_1,s_2,...,s_n$, I want to find $\max(s_j-s_i)$, where $j>i$.
I wrote the following algorithm (s is a $1$-based array with n elements):
max = 0
for i from 1 to n - 1
for j from i + 1 to n
if s[j] - s[i] > max
max = s[j... |
H: Interpreting results concerning the global sections ring being finitely generated
Let $A$ be a ring and $X$ be an $A$-scheme.
(Hartshorne exercise II.2.17) Suppose there exist $f_0,\dots,f_n\in \mathcal{O}_X(X)$ such that
a) $X_{f_i}:=\{x\in X: f_i(x)\not=0\}\subset X$ is an affine open for all $i$,
b) $(f_0,\dots... |
H: Conjecture on integer solutions to the equation $ (ab + 1) \mid (a^{2}+b^{2})$
Inspired by the egregious problem in IMO 1988, I simulated the integer solutions to the equation
$$ (ab + 1) \mid (a^{2} + b^{2}) \tag{*}$$
for $1 \leq a, b \leq 3000$ and conjectured that every solution arises as an adjacent pair of num... |
H: Finding a formula for a repeating sequence of 1's and -1's
Is there a simple formula for the sequence $(a_n)$ given by $(1,1,-1,1,1,-1,1,1,-1,\cdots)$
(with the repeating pattern 1,1,-1), starting with $n=1$?
AI: $${ \left( -1 \right) }^{ (n+1)\mod 3 }$$ where $n$ starts from $1$.
Or
$${ \left( -1 \right) }^{ ... |
H: Evaluate a Variable Defined in Terms of its Function
I have a variable x which is defined as follows:
x = 150 / (7 + f(x)) where f(x) = 0.005 * x if x > 200, or 100 otherwise.
This is actually a simplified version of a real world problem.
How do I evaluate x?
AI: You have 2 cases. If $x \leq 200$ then $f(x) = 100$ ... |
H: On deductively closed theories
Definition A theory $T$ is a set of sentences. A structure $\mathcal{A}$ is a model of $T$ if $\mathcal{A}\vDash T$.
To better understand the situation, let us recall the classical Galois connection machinery used in these situations. Fix a language $\mathcal{L}$; with a set of formul... |
H: Show that there are infinitely many bijections $f:\mathbb{Z_+} \rightarrow \mathbb{Z_+}$
Show that there are infinitely many bijections $f:\mathbb{Z_+} \rightarrow \mathbb{Z_+}$
Attempt: I dont really know where to start here. A little help is appreciated.
AI: There are lots and lots of ways to do this.
We know ... |
H: Functions between Sets of different Cardinality
Let $\alpha : A \to B$ be a function between finite sets. Show that if $|A| > |B|$, then $α$ cannot be injective, and if $|A| < |B|$, then $α$ cannot be surjective .
AI: Suppose we have $\alpha$ injective with $|A|>|B|$. Then, $\forall b \in B$, $\alpha^{-1}(b)$ must ... |
H: Showing that $\vec{a}\times\vec{c}=\vec{b}\times\vec{c} \implies \vec{c}\cdot\vec{a}-\vec{c}\cdot\vec{b}=\pm\|\vec{c}\|\cdot\|\vec{a}-\vec{b}\|$
For vectors $\vec{a},\vec{b},\vec{c}\in\mathbb{R}^{3}$, how do I show that:
If $\vec{a}\times\vec{c}=\vec{b}\times\vec{c} \implies \vec{c}\cdot\vec{a}-\vec{c}\cdot\vec{b}... |
H: $Y^\varnothing$ has one element; if $X \ne \varnothing$, then $\varnothing^X = \varnothing$
From Section 8 of Halmos' Naive Set Theory.
(i) Show that $Y^\emptyset$ has exactly one element, namely $\emptyset$.
(ii) Show that if $X\neq\emptyset$, then $\emptyset^X=\emptyset$.
I'm not completely sure I understand ... |
H: How can rate of change be with respect to time if you're not differentiating with respect to time?
A high school Calculus textbook asks: Determine the instantaneous rate of change in the surface area of a spherical balloon (as it is inflated) at the point in time when the radius reaches 10 cm.
My solution: The func... |
H: Pushforward commutes with external tensor product?
Let $f:X\rightarrow X'$ and $g:Y\rightarrow Y'$ be morphisms of varieties. Let $\mathcal F$ be a coherent sheaf on $X$ and $\mathcal G$ be a coherent sheaf on $Y$.
Is it true that
$$(f\times g)_* (\mathcal F \boxtimes \mathcal G)=
(f_* \mathcal F) \boxtimes (g_* ... |
H: Integral from zero to infinity of $\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda$
I know that the value of the integral is as follows
$$\int_0^{\infty}\frac{(1-e^{-\lambda z})}{\lambda^{a+1}} d \lambda =z^a \frac{\Gamma(1-a)}{a}$$
However, how exactly the integral is calculated? How one proves t... |
H: Translate a regular grammar to a regular expression
I want to translate the following grammar into a regular expression:
Set of variables
V := {S,T}
Set of terminals
Σ := {a,b}
Set of relations
S → ""
S → aS
S → bT
T → aT
T → bS
Start variable
S
For the regular expression I can only use the following operations:... |
H: Inner Product Space (Continuously Differentiable Functions)
Let $V=C^{2}[-\pi,\pi]$ be the space of real valued twice continuously differentiable functions defined on the interval $[-\pi, \pi]$. Set $$\langle f,g \rangle=f(-\pi)g(-\pi)+\int_{-\pi}^{\pi}f''(x)g''(x)dx$$
Is this an inner product on $V$?
AI: Hint: the... |
H: Show that if $x>0$, then $\ln(x)\geq 1-\frac{1}{x} $
Show that if $x>0$, then
$$ \ln(x)\geq 1-\dfrac{1}{x}. $$
I tried a few things but so far nothing has worked, I could use a hint.
AI: You can use mean value theorem for $\ln$.
$x = 1$, then equality holds.
$0 < x < 1$:
$$\ln x \ge \frac{x-1}{x} \\
\frac{\ln x... |
H: What is the relationship between saying "a Taylor series converges for all $x$" and "a Taylor series converges to a function, f(x)"
Given the following Taylor series:
$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$
We know that:
It converges for all of $x$
It converges to the function $\cos ... |
H: Show that F(t) is an immersion
I've got here an exercise that says: "Show that the map $F:\mathbb{R}\rightarrow \mathbb{R^2}$ defined by $F(t)=(\cos t, \sin t)$ is an immersion".
$F$ is an immersion if $dF_x:T_x\mathbb{R}\rightarrow T_{F(x)}\mathbb{R^2}$ is injective. Now $dF_x$ is $(-\sin x, \cos x)^t$ (am I wrong... |
H: What is zero choose one?
Algebraically it comes out to be undefined- but if I have zero elements, and I'm asked to pull elements from it, this should just be zero, right?
AI: Combinatorically speaking, you are absolutely right. There is no way to choose $1$ element from an empty set, so $\binom01=0$. It's a good id... |
H: Homogeneous Equations and Such
"Consider the linear system $\begin{bmatrix} 1 & -2 & 3\\2 & 1 & 4\\1 & -7 & 5\end{bmatrix} * \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0\end{bmatrix}$ and find a general solution of the homogeneous system."
So a homogeneous system is just a system of ... |
H: Let R be a relation on set A. Prove that $R^2 \subseteq R <=>$ R is transitive $<=> R^i \subseteq R ,\forall i \geq 1$
this is my first question here. I'm still relatively new to more advanced
mathematics and don't have much experience with proofs yet. I'm self-studying
at the moment and therefore have no one to ch... |
H: When does "positive expected value" imply "positive conditional expectation with positive probability"?
Let $(\Omega,\mathcal G,\mathbb P)$ be a probability space, $\mathcal F\subseteq\mathcal G$ a sub-$\sigma$-algebra and $X:\Omega\rightarrow\mathbb R$ $\mathcal G$-measurable map. Assume that $\mathbb E_\mathbb P[... |
H: Without using any Sylow theorem, if every element is a $p$-element then $G$ is a $p$-group
How can we prove the following theorem without using any Sylow theorem?
Let $p$ be a prime. In a finite group $G$, if every element is a $p$-element then $G$ is a $p$-group.
Or is it possible to generalize the following theor... |
H: Equation of a line through a point on a plane
Find the line that passes through the point $(2, 5, 3)$ and is perpendicular to the plane $2x - 3y + 4z + 7 = 0$
My only real problem with this is how to shift the line
My first step is to find the norm of the plane which is $\vec{n}^{\ } = (2,-3,4)$
Is the answer as si... |
H: Is $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ a normal extension?
I want to reason as to whether $K=\mathbb{Q}(\sqrt{2}, \sqrt{3})$ is a normal extension over $\mathbb{Q}$. There are competing definitions and so my question is re-phrased as:
Is it true that for every irreducible polynomial $f \in \mathbb{Q}[X]$ which has (at... |
H: Why is it important to define that a logarithm and exponential function is one-to-one?
I'm currently studying the properties of logarithm in an open source pre-calculus textbook that can be found here (Page 438). Before the text goes on to the Algebraic properties of exponential and logarithmic functions it defines... |
H: Expected value proof : E(cx)=?
Let $E(x)$ be the expected value of random variable $x$. $c$ is a constant, then what will be the $E(cx)$ in simplest form and why?
AI: For the discrete case:
$$
\mathbf{E} \varphi(X)=\sum_{k=0}^{\infty}c k p_k= c \sum_{k=0}^{\infty}k p_k = c \mathbf{E}X
$$
that's if I understand your... |
H: trigonometric equation opening
Solve:
$$ \sin x + \sin 3x + \sin 5x = 0 . $$
Attempt at a solution: applying formulas for summation of sine we get after a series of operations: $ \sin x(8 \cos x \cos 2x \cos x + 1) = 0$ equaling sine to $0$ we get one solution $180k$. comparing the other factor we eventually g... |
H: Uniform convergence
How can I prove the sequence of functions $x(1-x), x^2(1-x), ...$ converges uniformly on $[0,1]$?
I know from my earlier question here that I can prove it by taking the derivative. But how can I prove it without taking the derivative since we still haven't defined "derivative" yet in my class? I... |
H: Showing that if $x_k \rightarrow x \implies f(x_k) \rightarrow f(x)$, then $f^{-1}(C)$ is closed for any closed set
As part of proofs on continuity, I should show that
(i) $\forall x \in \mathbb{R}^n,$ if $x_k \rightarrow x \implies f(x_k) \rightarrow f(x)$
implies
(ii) $f^{-1}(C)$ is closed for any closed set $C \... |
H: In this diagram, why does the limit exist on this interval?
Does the $\lim_{x \to x_0} f(x)$ exist at every point $x_0$ in $(-1,1)?$
I answered False, but the correct answer is True. Why?
My thoughts:
$f(x)$ is not the same number as $x \rightarrow 1$ from the right and from the left, and as $x \rightarrow -1$, o... |
H: Isomorphism functorially
I was reading the lecture notes of Pierre Schapira
http://www.math.jussieu.fr/~schapira/lectnotes/AlTo.pdf
I am not able to understand one thing. Please help.
In page 75, theorem 4.6.1, the author says that 'quasi-isomorphism from X to $\lambda(X)$ is functorial in C. What does it mean. ... |
H: Necessary and sufficient conditions for a polynomial $f(x) \in \mathbb{Z}[x]$ to be irreducible.
This is not for homework, and I am not totally convinced I understand the question entirely. Also, I'm not allowed to use the fact that $\mathbb{Z}[x]$ is a UFD. The question asks
Show that $f(x) \in \mathbb{Z}[x]$ i... |
H: Proof of the limit of a sequence
The sequence is:
$a_n$ = $\frac 1n$ [$(\frac 1n)^2 + (\frac 2n)^2 + (\frac 3n)^2...(\frac nn)^2$]
The objective is a proof of the limit from 1 to infinity. Just from toying around with a few examples, I speculate the limit is (1/3), but I'm not sure. This is how far I am into the p... |
H: $\sum_{k=0}^{n-1}\sin(k\frac{\pi}{n}+\theta)$
I'm trying to find the closed form of the above formula. This link shows the solution of tan version. Sum of tangent functions where arguments are in specific arithmetic series
Though I'm trying to find the way to apply the argument used in the link to this case, it be... |
H: Prove that class of regular languages is closed on operation
Let's have an operation $$\odot(L)=\{w\in L \; | \; |w|=2k \land k>0\}$$Show that result of this operation will be regular.
PS: It's not homework, it's from last year's exam.
AI: The intersection of two regular languages is regular.
Let $E = \{ w : |w| \... |
H: Product of two random variables
How can one show that the product $X \cdot Y$ of two real-valued random variables $X,Y$ is again a random variable?
We can fix some set generating the Borel sigma algebra on the real line, then take for instance an arbitrary open interval, and consider $(X \cdot Y)^{-1}((a,b))$. We n... |
H: Linear density variation on a 2D plane
Here's my problem:
on a 2-dimensionnal plane, I know the $x$, $y$ coordinates of 3 points $A, B, C$
each point comes with an associated density $T$, that can vary from infinite minus to positive infinite
$T$ values vary in a linear fashion depending on the location of the poi... |
H: Are Base Ten Logarithms Relics?
Just interested in your thoughts regarding the contention that
the pre-eminence of base ten logarithms is a relic from
pre-calculator days.
Firstly I understand that finding the (base-10) logarithm of positive real numbers without a calculator can be reduced to finding the (bas... |
H: On the definition of an exact sequence in an abelian category
I am slightly confused about the notion of exactness in a general abelian category (I want to stay clear of anything related to the Mitchell embedding theorem). Here are two definitions that I have seen:
A sequence $A \xrightarrow{f} B \xrightarrow{g} C$... |
H: What exactly is a stationary point?
I am asked to find the stationary points of the function $y=5+24x-9x^2-2x^3$. When I looked on the wikipedia page for the definition of stationary points, I read that a stationary point is a point where the derivative equals zero.
However, when I looked at the same article in a d... |
H: What is the least element of this set
What is the least element of the set
$$A=\{a\in \mathbb{R} \,\,| \,\,1<a<2\} $$?
Is incorrect to write $$1,000...1 $$?
AI: The open interval $(1,2)$ has no least element. For every element $a \in (1,2)$ there is an element $b \in (1,2)$ with $b< a$; for example, the poi... |
H: Prove that $(|u-s|+|x-y|)^2\leq 2|u-s|^2+2|x-y|^2$.
Prove that $(|u-s|+|x-y|)^2\leq 2|x-y|^2+2|u-s|^2$.
My professor used this inequality for a proof last week. How would one prove this? I thought about using the Cauchy-Swartz inequality.
This is not a homework question. I am interested in the proof just for self-... |
H: If two submodules are isomorphic, so is their quotient… conditions on the ring!
In this question Isomorphic quotients by isomorphic normal subgroups it is shown that if we have isomorphic normal subgroups, their quotients need not be isomorphic.
Now, what if we take finite dimensional vector spaces, instead of grou... |
H: Building a function to display of the sum of matrices at different powers in Matlab
I'm trying to write a for loop for the sum A+A^2+A^3+..+A^n.
Here is my code:
function [ x ] = Untitled2( A , n )
for k=1:n,
x =sum(A^k)
end
The problem I'm having with this is this function is listing the matrix A to each power ... |
H: angles of polynomials
Here is an improved question that was asked before.
Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Let our inner product be defined by:
$$
\langle p,q\rangle = p(1)q(1)+p'(1)q'(1)+p''(1)q''(1)+p'''(1)q'''(1).
$$
Find the angle between the p... |
H: Find height of a triangle given length of three sides?
How can I find the height of a triangle given the length of all three sides?
The only solution I could find was to use Heron's formula to find area then $A=\frac{1}{2}bh$ to find height. Is there an easier way to do this?
I'm going to be using this in a piece s... |
H: Solve the equation $\tan \theta = 2\sin \theta$.
Solve the equation $\tan \theta = 2\sin \theta$.
What I did was rewrite it to the form $$\sin \theta = 2 \sin \theta \cos \theta$$ You'll get $$\sin \theta = \sin\ 2 \theta.$$
How am I supposed to solve this when I have $\sin$ on both sides? My main problem with... |
H: If there exist open, disjoint sets $A, B$ in $X, d$, then is $\bar A \cap B = A \cap \bar B = \emptyset$?
Clearly $A, B$ are separated in $X, d$ if $A, B$ are closed and disjoint since $(A \cup A') \cap B = \emptyset$ and vice versa. For disjoint open sets, I cannot come to a conclusion. I tried constructing a set ... |
H: Help with this integral? I can't figure out the substitution.
I've been struggling to figure out this integral.
$$\int \frac {1}{x\sqrt{5-x^2}}$$
I'm almost certain it has something to do with this fact:
$$\int \frac 1{\sqrt{1-x^2}} = \sin^{-1}(x) + C$$
But I can't figure out how to use that to my advantage. No ob... |
H: Closed unit disk homeomorphic to $\mathbb{R}^2$?
I've already shown the existence of a homeomorphism between the open unit disc and $\mathbb{R}^2$ and now I'm trying to work out whether the closed unit disc is homeomorphic to $\mathbb{R}^2$ or not. Clearly the only real difference is the fact it includes the points... |
H: Determine number of function given two sets and properties
Let A={1,2,3,4,5,6,7} and B={v,w,x,y,z}. Determine the number of functions $f:A \rightarrow B$ where (i) f(A)={v, x}; (ii) |f(A)|=2
For (i) the answer key gives 2!S(7, 2) and (ii) $\binom{5}{2}[2!S(7,2)]$
I don't understand where these answers come from. So... |
H: $f=g$ almost everywhere $\Rightarrow |f|=|g|$ almost everywhere?
Suppose $(X, \mathcal{M}, \mu)$ is a measure space. Assume $f: X\to\overline{\mathbb{R}}$ and $g=X\to\overline{\mathbb{R}}$ are measurable maps. Here $\overline{\mathbb{R}}$ denotes the set of extended real numbers. My question is:
If $f=g$ almost ev... |
H: Proving a relation's inverse's properties by knowing the original's.
I'm getting fairly confused with two exercises related to proving a relation's inverse's properties by knowing the original's. I couldn't do either. Any hint is appreciated.
If $R$ is a symmetric relation over $A$ with $A \not = \emptyset$,
pr... |
H: How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$?
How to compute the integral $\int_{-\infty}^\infty e^{-x^2}\,dx$ using polar coordinates?
AI: Hint: Let $I=\int_{-\infty}^\infty e^{-x^2}\,dx.$ Then $$I^2=\left(\int_{-\infty}^\infty e^{-x^2}\,dx\right)\left(\int_{-\infty}^\infty e^{-y^2}\,dy\right)=... |
H: Modular congruence rules
May take exponent of both sides of a modular congruence?
For instance, may I write
$$n^2\equiv-1\mod p \quad \rm as \quad(n^2)^{2k+1}\equiv (-1)^{2k+1}\mod p ?$$
AI: If $p|(a-b)$ does $p|a^k-b^k$? |
H: Proving that a certain set is convex
I'm trying to prove that the set $W = \{x \in \mathbb{R}^2 : 2x_{1}^{2} + 3x_{2}^{2} \leq 4\}$ is convex. I've been trying to do this using the definition of W being convex when for all $x, y \in W $ and every $\lambda \in [0,1] $ we have $\lambda x + (1-\lambda)y \in W $.
What ... |
H: Proving that $\mathbb{A}=\{\alpha \in \mathbb{C}: \alpha \text{ is algebraic over } \mathbb{Q} \}$ is not a finite extension
It is true that all finite field extensions are algebraic. It is not true however that all algebraic extensions are finite.
In lectures we were given the example of the field extension
$$\ma... |
H: What is the name for this operator, and how can it be applied to multiple variables within the same equation?
My question is in two parts; the first is, what is the $|$ operator called? Here's an example of it in use:
$$(x + 5)|_{x=3} = 8$$
My second question is, how do I use this operator for more than one variabl... |
H: What other definite integrals can be computed in a manner similar to $\int_{-\infty}^\infty e^{-x^2}dx$?
The technique for computing $\int_{-\infty}^\infty e^{-x^2} dx=\sqrt{\pi}$ by computing the integral squared using polar coordinates is well known. Are there any other integrals that can be computed in a similar... |
H: Why is $\mathbb R^n$ under the Zariski topology not a topological group?
Reasons that $(\mathbb R^n, +, \mathcal Z)$ is not a topological group: Given any two distinct points $\vec{p},\vec q \in \mathbb R ^n$ let $P$ be the unique hyperplane through $\vec p$ which is perpendicular to the vector $\vec p- \vec q$. $P... |
H: Find the series expansion of 2 multiplied functions
The first three terms in the series expansion of $(1+x)^m$ are $1 + mx + \dfrac{m(m-1)x^2}{2}$.
Find the first 3 terms in the series expansion of $(1+x)^{m+1}(1-2x)^m$.
I don't really know how to do this, because I have always done these binomial expansion using... |
H: How many telephone numbers have no $0$ in the prefix (first three numbers)
I got that the total number of telephone numbers: $10^{10}$, but should I do the number of one $0$ in prefix, number of two $0$ in prefix and number of three $0$ in prefix, and subtract them. But I don't know how to count the number of $0$ i... |
H: Integration by Substitution problem
I was given an integration problems sheet...with answers too but how a certain answer is to be obtained is obviuosly not stated.
Using integration by substitution integrate the following:
$$ \int \dfrac{5x +3}{\sqrt{3-x^2}} \, dx$$
And the answer at the back is:
$$ -5 \sqrt{3-x^... |
H: Non-empty interior
I would like to verify a certain condition for a theorem. It requires that $\{(\theta,\rho\theta):\rho>0,\theta>0\}$ has a non-empty interior. I have trouble visualizing such sets can you guys help me see if this has a non-empty interior (and if it does, why?)
AI: The set is the first quadrant wi... |
H: An integrable Functions is almost everywhere finite
An integrable Functions is almost everywhere finite
Attempt: Let $X = \{ x : f(x) \, \text{ is infinite}\} $. We must show $m(X) = 0$. Suppose $m(X) > 0 $. then on $X$, we have
$$ \int\limits_X f \, dm > \infty$$
which implies $f$ cannot be integrable: contradict... |
H: Check to see if my isomorphism is correct
Is multiplication modulo $10$ isomorphic to addition modulo $4$?
$U(10) = \{1,3,7,9\}$,
the identity is $1$,
it is a cyclic group of order $4$, with generator $3$.
$\Bbb Z_4 = \{0,1,2,3\}$,
the identity is $0$,
it is a cyclic group of order $4$
\begin{gather*}
1 \mapsto ... |
H: Why isn't $2\log(-1)$ real?
In high school we learn that a $a\log[(x)] = \log (x^a)$
From this I would assume $2\log(-1) = \log [(-1)^2]$
However, the first is not real and the second is, according to my calculator and textbook. Why is this?
AI: The formula $a\log x=\log x^a$ requires that $\log x$ exists. Here you... |
H: Use implicit function theorem to show $O(n)$ is a manifold
In class today our teacher mentioned that one can use the implicit function theorem to show that $O(n) \subseteq \mathbb{R}^{n^2}$ is a submanifold...that is, map $A \mapsto A^* A$, and set it equal to the identity matrix. There should be $n(n+1)/2$ equatio... |
H: Logical representation of a prime number
Is it correct to represent a prime number like this?
$$\exists k \in \mathbb N,\, \exists n\in \mathbb N\, \Big((n\mid k) \land (n=k \lor n=1)\Big)$$
AI: What you have claimed is that there are two natural numbers, $n$ and $k$, such that $n$ divides $k$ and $n=k$ or $n=1$. ... |
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