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H: Are the groups $\Bbb{Z}_8 \times \Bbb{Z}_{10} \times \Bbb{Z}_{24}$ and $\Bbb{Z}_4 \times \Bbb{Z}_{12} \times \Bbb{Z}_{40}$ isomorphic?
This question is taken from "A first course in Abstract Algebra" by Fraleigh 7th edition, section 11 question 18:
Are the groups $\mathbb{Z}_8 \times \mathbb{Z}_{10} \times \mathbb{... |
H: Prove that $(A\cap C)-B=(C-B)\cap A$
$\mathbf{Question:}$ Prove that $(A\cap C)-B=(C-B)\cap A$
$\mathbf{My\ attempt:}$
Looking at LHS, assuming $(A\cap C)-B \neq \emptyset$
Let $x\in (A\cap C)-B$
This implies $x\in A$ and $x\in C$ and $x\notin B$
Looking at RHS, assuming $(C-B)\cap A \neq \emptyset$,
Let $y \in (C-... |
H: Find function given arc length
I'm creating a program that has two points and a cable hanging between them. I feel like modeling the cable using a catenary would be too hard, so I just simplified it to a parabola.
However, I'm still stuck on making sure that the length of the cable is constant. I don't know how to ... |
H: Solving for integral curves of a vector field - how to account for changing charts?
[Ref. 'Core Principles of Special and General Relativity by Luscombe]
Let $\gamma:\mathbb{R}\supset I\to M$ be a curve that we'll parameterize using $t$, i.e. $\gamma(t)\in M$. It's stated that:
If $\gamma(t)$ has coordinates $x^i(... |
H: Show that for any infinite cardinal $\kappa$, we have $\kappa ! = 2^{\kappa}$.
Synopsis
For this exercise, we are asked to show that for any infinite cardinal $\kappa$, we have $\kappa ! = 2^{\kappa}$, where $\kappa! = \text{card}\{f | \text{$f$ is a permutation of $K$}\}$, and a permutation of $K$ is described as ... |
H: Brownian motion mathematical model construction
I quote the construction of a mathematical model of Brownian Motion from Schilling-Partzsch
Consider a one-dimensional setting where a particle performs a random walk (notice that it can move to left or to right with equal probability $\dfrac{1}{2}$). We assume that... |
H: Colimit of a constant functor
Let $\mathcal{C},\mathcal{D}$ be categories and $C\in\mathcal{C}$. Consider the constant functor to $C$, $\Delta_C:\mathcal{D}\rightarrow\mathcal{C}$. In other to show that $(C,(1_{\Delta_C(D)})_{D\in\mathcal{D}})$ is a colimit of $\Delta_C$, is it necessary to assume that $\mathcal{D}... |
H: Show that the sequence is convergent $\frac{(1)(3)(5)\dots(2n-1)}{(2)(4)(6)\dots(2n)}$
$\mathbf{Question:}$ Show that $\frac{(1)(3)(5)\dots(2n-1)}{(2)(4)(6)\dots(2n)}$ is convergent where $n\in \mathbb{N}$
$\mathbf{My\ attempt:}$
Let $a_n = \frac{2n-1}{2n}$ and let $f(n) = a_n$
$$
f(n)=\frac{2n-1}{2n} = 1-\frac{1}{... |
H: Homeomorphism from $\mathbb{N}$ to $T = \{1/n: n \in \mathbb{N}\}$
I'm trying to understand what is probably a basic topological fact and I'm probably overthinking it. I'm consider the map $f: \mathbb{N} \rightarrow T$ where $T = \{1/n: n \in \mathbb{N}\}$ defined by $f(n) = 1/n$, where $\mathbb{N}$ and $T$ have t... |
H: Proving that $\sqrt[n]{a^m}=(\sqrt[n]{a})^m$
I have a doubt on how to solve Exercise B11 of Section 3.4 of Advanced Calculus of Watson Fulks. It only says: prove that $\sqrt[n]{a^m}=(\sqrt[n]{a})^m$. Clearly it's not always true if $a<0$, so I guess I most assume $a\geq0$. And given that the previous sections are a... |
H: Is there more than one definition of homotopic equivalence for two spaces?
Is there more than one definition of homotopic equivalence?
Wolfram.com under the topic Homotopic says the unit circle and a point are homotopic (presumably meaning homotopic equivalent) in the plane. Wolfram also says one must define the “... |
H: Find $\lim_{x \to 1} \cos(\pi \cdot x) \cdot \sqrt{\frac{(x-1)^2}{(x^2-1)}} \cdot \frac{1-x}{x^2+x-1}$ (I need a review of my resolution please :) )
Find the limit:
$$\lim_{x \to 1} \cos(\pi \cdot x) \cdot \sqrt{\frac{(x-1)^2}{(x^2-1)}} \cdot \frac{1-x}{x^2+x-1}$$
This is what I have, i'm not sure about my answer (... |
H: Can these two conjectured relationships between two prime counting functions and the harmonic number function $H(x)$ be proven?
This question assumes the following definitions where $p$ is a prime and $n$ and $k$ are positive integers.
(1) $\quad\pi(x)=\sum\limits_{p\le x} 1\quad\text{(fundamental prime counting f... |
H: Confusion about the underlying function of a Functor.
I was reading about Brower's fixed point theorem and a doubt came to mind about the underlying function of a functor on morphisms. We can think of $\mathbb{S}^1 $ as a subset of $\mathbb{R}^2$, so we get the inclusion map $i : \mathbb{S}^1 \hookrightarrow \mathb... |
H: Why is the expectation of the random variable "W" equal to the limited integral of the survival function of the random variable "Y"?
Ok, so here's the problem. Let the random variable X represent loss in 2005. The density function of X is exponential with mean equal to 1. Let the random variable Y represent loss in... |
H: Any easier way to get the quotient and remainder from a minus number divided by a positive one?
negative number divided by positive number, what would be remainder?
I've read these answers linked above, but I don't feel I'm answered enough.
$$-27 = \underbrace{-6}_q\cdot \underbrace{5}_d + \underbrace{3}_r$$
They s... |
H: One person from a party is selected at random
Here is the question:
John invites 12 friends to a dinner party, half of whom are men.
Exactly one man and one woman are bringing desserts. If one person
from this group is selected at random, what is the probability that it
is a woman, or a man who is not bringing a d... |
H: Prove that a semigroup $(S, +)$ is a group if and only if the only ideal of $S$ is $S$ itself.
Consider a semigroup $(S, +).$ We say that a nonempty subset $I \subseteq S$ is an ideal of $S$ whenever we have that $S + I \subseteq I,$ where $S + I$ is the set consisting of all sums $s + i$ with $s \in S$ and $i \in... |
H: Weakly open sets are unbounded
I was studying the proof that every open set in the weak topology of an infinite dimensional space is unbounded and I came across the following argument. If $(f_1,...,f_n)$ are continuous linear operators defined on an infinite normed space E than the set $\cap_{i=1}^{n}kerf_i$ has a ... |
H: Why does the polynomial splitting implies existence primitive root of unity in $\Bbb{F}_{p^2}$?
This question refers to WimC's answer to this question. Consider the cubic congruence problem:
$$
f(x) := x^3 - x^2 - 2x + 1 \equiv 0 \pmod{p}
$$
We want to know for which $p$ does $f(x)$ splits. The answer to this is $p... |
H: How to find the angle of a non right angled triangle in a cube?
I have to find $\angle MHN$ ($\angle H$ in $\Delta HMN$). It is inside a cube that has side lengths of $12$ cm. $M$ is the midpoint of the diagonal $BD$ and $N$ is the midpoint of edge $GF$. Here's the diagram:
I'm completely lost on how I would find ... |
H: Calculating determinant of $A^n$ given the matrix $A$
Find if $det(A)=det(A^n)$ for $n>1$.
How do I tackle questions like this, in general if the matrix $A$ is provided in the question?
Should I work out with the basic definition of a determinant, which I found very difficult to apply in any question?
I cannot of c... |
H: Number of $3$-digit numbers with strictly increasing digits
A positive integer is called a rising number if its digits form a strictly increasing sequence. For example, 1457 is a rising number, 3438 is not a rising number, and neither is 2334.
(a) How many three digit rising numbers have 3 as their middle digit?
(b... |
H: A jar contains 3 red and 2 white marbles
A jar contains 3 red and 2 white marbles. 2 marbles are picked without
replacement.
(1) The probability of picking two red marbles
(2) The probability of picking exactly one red and one white marble
A. (1) > (2)
B. (1) < (2)
C. (1) = (2)
The question doesn't mention anythi... |
H: about properties of the operator $T_f(g) := f\cdot g$.
let $f \in C([0,1])$ and $T_f: L^2([0,1]) \to L^2([0,1])$ and $T_f(g) := f\cdot g$ prove :
1)$T_f$ is well define , linear and bounded and find $|| T_f ||$ .
2 )if $T_f$ be compact operator then $f=0$
i can prove $T_f$ is linear.
AI: Answer of the second par... |
H: Computing Tor for $R=k[x,y]$
Let $R=k[x,y]$ where $k$ is a field. I want to compute $\operatorname{Tor}_{i}^{R}(R/(y-x), R/(y^2 - x^3))$ for $i \ge 0$.
My attempt is:
Since $\cdots \rightarrow 0 \rightarrow R \xrightarrow{\times(y-x)} R \rightarrow R/(y-x) \rightarrow 0$ is a projective resolution, $\operatorname{T... |
H: about finite rank operator
let $(X,\|.\|)$ be banach space and $T\colon (X,w)\to (X,\|.\|)$ is linear continuous operator . $(X,w)$ is a banach space with its weak topology. then $dim (rang (T)) < \infty$
AI: Hint: There exists a finite set $x_i^{*}, 1\leq i \leq N$ such that $x_i^{*} (x)=0, 1\leq i \leq N$ impli... |
H: how can I write this sum in sigma notation?
I find it difficult to write this in sigma notation. I tried but couldn't figure out.
$$ \frac{1}{n} \sqrt{1-\left(\frac{0}{n}\right)^2} + \frac{1}{n} \sqrt{1-\left(\frac{1}{n}\right)^2} + \dots + \frac{1}{n} \sqrt{1-\left(\frac{n-1}{n}\right)^2} $$
AI: It is more or less... |
H: Counterexample to Inverse Operator Theorem
Let $Y=(\mathbb{R}^{\mathbb{N}},\|\cdot\|_1)$ where $\|x\|_1=\sum_{n=1}^\infty|x_n|$, and $X=(\mathbb{R}^{\mathbb{N}},\|\cdot\|_X)$ where $\|x\|_X=\|x\|_1+\sup_{n\in\mathbb{N}}n|x_n|$.
Then $\|\cdot\|_X$ is a norm since $\|ax\|_X=|a|\|x\|_X$ for a scalar $a$, $\|x\|_X=0$ i... |
H: $L^{\infty}(\mathbb{R}^{N})$ and smallness of function
Let $(f_{n})_{n\in\mathbb{N}}\subset C(\mathbb{R}^{N})$ be a sequence of real valued function such that $\|f_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\to\infty$ as $n\to\infty$.
Then, I know that there exists $\delta_{1}>0$ and $x_{1}\in\mathbb{R}^{N}$ so that for $n\... |
H: Product of Moore Penrose Inverse and Matrix
Say, we had m x n matrix B. B+ is the Moore Penrose Inverse (pseudoinverse) of the matrix. Would B+B (product of pseudoinverse of B and B) be a projection matrix? How would we prove this?
AI: By the defining properties of Moore-Penrose pseudoinverse, $B^+B$ is Hermitian a... |
H: Inverse map from $\mathbb{Z} \to \mathbb{Z}$
Let $f: \mathbb{Z} \to \mathbb{Z}, \; z \mapsto 3z$ be a map on the integers. I am trying to find a left inverse function, $g: \mathbb{Z} \to \mathbb{Z}$ such that $g \circ f = \text{id}$.
My attempt didn't quite seem to work:
\begin{align*}
(g \circ f)(x) & = g(f(x) = g... |
H: Prove that there exists a language L ⊆ {0}^∗ that is not Turing decidable.
I was revising for my final and got this question. I don't see there exists such a language. Is it really existing?
AI: If $N$ is a subset of $\mathbb{N}$ whose membership problem is not Turing decidable, I think that $\{x \in \{0\}^\ast \mi... |
H: Show that $\lim\limits_{h \rightarrow 0}\frac{f(\xi+h)-f(\xi-h)}{2h}$ exists and is $f'(\xi)$
Let $f:I \longrightarrow \mathbb{R}$ be differentiable in a inner point $\xi \in I$.
Show that $$\lim\limits_{h \rightarrow 0}\frac{f(\xi+h)-f(\xi-h)}{2h}$$
exists and is $f'(\xi)$.
Also give an example which shows, that t... |
H: Maximum volume of cylinder obtained by rotating a rectangle
Question: A cylinder is obtained by revolving a rectangle about the $x$-axis, the base of the rectangle lying on the $x$- axis and the entire rectangle lying in the region between the curve: $y=\dfrac{x}{x^2+1}$ and the $x$-axis. Find the maximum possible... |
H: Question in Proof of theorem 11.22 Apostol mathematical analysis
While self studying mathematical analysis from Tom apostol I have 2 question I proof of above mentioned theorem.
Question 1: How does author deduces $g_{x} $ is measurable on $\mathbb{R}$ ?
Question 2 : why $g_{x} $ belongs to $L^{2}( \mathbb{R}) ... |
H: Is 0 a real number?
I am just curious if 0 is a real number. The definition of a real number is all rational and irrational numbers. And the def definition for rational number is that "$\mathbb{Q}={a\div b|a,b\in\mathbb{Z}}$".
But in done say that $0$ is a whole number and some day that it is not.
In some websites ... |
H: Finding The Number Of The Solutions For $\cos x = \frac{x^2}{100}$
Hello everyone how can I find the number of the solution to the equation:
$\cos x = \frac{x^2}{100}$ in real numbers?
I tried to convert it to function $y = \frac{x^2}{100} -\cos x$
and find all the cutting points with the x axis by find all the ext... |
H: $(G,\cdot)$ is abelian $\Longleftrightarrow$ $\forall a,b \in G: (ab)^{-1}=a^{-1}b^{-1}$
Show that for a group $(G,\cdot)$ the following statements are equivalent:
$A:$ $(G,\cdot)$ is abelian
$B:\forall a,b \in G: (ab)^{-1}=a^{-1}b^{-1}$
$A \Longrightarrow B:$
Since $(G,\cdot)$ is a group:
$\forall a,b \in G: (ab)... |
H: What does $\{0,1\}^X$ mean?
I got this notation in a question without prior explanation, I suspect it's something related to the power set but I am not sure.
AI: As Angina said in the comments, $\{0, 1\}^X$ would be the set of all functions $f:X \to \{0, 1\}$. However, as you suspected, it is very closely linked to... |
H: If $TS$ = $I$ then $S$ and $T$ need not be invertible
Let $T$ be a linear operator on a vector space $V$ over a field $F$. Suppose there is a linear operator S on $V(F)$ such that TS = I where I is identity operator on $V(F)$.
Now I am looking for an example where neither of $T$ or $S$ is invertible.
As $TS$ is one... |
H: Perms and Coms - what am I doing wrong??
The question is:
Find the number of ways that 6 different coloured balls can be placed in 3 non-identical urns so that no urn is empty.
Here's my working:
Let $A$ be the event Urn $A$ is empty
$B$: Urn $B$ is empty
$C$: Urn $C$ is empty
$n(A^C \cap B^C \cap C^C)$
\begin{alig... |
H: How to find a second solution for ODE $xy''+3y'+x^3y=0$?
I need some help with this problem. I need to solve the differential equation $$xy''+3y'+x^3y=0$$ using power series. I used Frobenius method to expand about $x=0$ since it is an singular regular point. So I assumed a solution $y(x)=\sum_{j=0}^\infty a_jx^{s+... |
H: Prove that $\mathbb{Q}[x,y]/ \langle x+y \rangle \cong \mathbb{Q}$
My direction: Set mapping $\varphi:\mathbb{Q}[x,y] \to \mathbb{Q}$ that $\varphi$ is surjective homomorphic and $\ker{\varphi}=\langle x+y \rangle$. Since Noether Theorem, I have $\mathbb{Q}[x,y]/\langle x+y \rangle \cong \mathbb{Q}$.
I have try man... |
H: $\int_0^1f(x) dx =0$, $\int_0^1xf(x) dx =0$. How to show that f has at least two zeros?
$f:[0,1]\to \mathbb{R}$ is a countinous function.
$$\int_0^1f(x) dx =0 \qquad \mbox{ and } \qquad \int_0^1xf(x) dx =0. $$
If $f \ge 0$ ($f\le0$) were true then $\int_0^1f(x) dx \ge0$ ($\int_0^1f(x) dx \le0$). This is a contradic... |
H: Simplify $\frac{4\cos ^2\left(2x\right)-4\cos ^2\left(x\right)+3\sin ^2\left(x\right)}{4\sin ^2\left(x\right)-\sin ^2\left(2x\right)}$
Simplify:
$$\frac{4\cos ^2\left(2x\right)-4\cos ^2\left(x\right)+3\sin ^2\left(x\right)}{4\sin ^2\left(x\right)-\sin ^2\left(2x\right)}$$
After the substitution as $\cos(x)=a$ and... |
H: Derivative of the $dh(x)Ax$
Given a function $h:\mathbb{R}^n \rightarrow \mathbb{R}^n $ and a matrix $A$ what it is the derivative of
$dh(x)Ax$ ? That is how calculate $d(dh(x)Ax)$?
AI: Define $\phi(x) := Dh_x(Ax)$ (the subscript is simply to indicate the base point of the derivative). Think of this like a "product... |
H: Is the statement "$(m, n)=d$ if and only if there exist integers $r$ and $s$ such that $r m+s n=d$" problematic?
My textbook Groups, Matrices, and Vector Spaces - A Group Theoretic Approach to Linear Algebra by James B. Carrell said that
$(m, n)=d$ if and only if there exist integers $r$ and $s$ such that $r m+s ... |
H: If $A$ is a deformation retract of $X$, then is $H_n(X,A) \cong H_n(A,A)$?
If $A$ is a deformation retract of $X$, then is $H_n(X,A) \cong H_n(A,A)$? I know this is true for usual homology grouups but does it also hold for relative homology groups?
On first sight, the proof that works for usual homology seems to ge... |
H: Derivation of density of $X^2$ based on distribution of $X$
I have the problem following :
Let's say we have random variable with density function $f_x=\begin{cases} \frac{x+1}{2} &for \; x \in[-1,1] \newline 0 & for \; x \in(-\infty,1)\cup(1,+\infty)
\end{cases}$
And we want to derive density function of $X^2$
M... |
H: Point spectrum of a particular operator
Let $X=C[0,1]$ and we define
$$T\colon X\to X\quad\text{as}\quad(Tf)(t)=g(t)f(t)\quad\text{for all}\quad t\in [0,1]$$ with $g\in C[0,1]$ fixed. I have proved that $\sigma(T)=g([0,1])$.
Question 1. If I prove that $\lVert T \rVert =\lVert g \rVert_{\infty}$ I can coclude tha... |
H: How to integrate $\int_{-\infty}^{\infty} xe^{-2\lambda |x|} dx$?
I need to integrate
$$\int_{-\infty}^{\infty} xe^{-2\lambda |x|} dx$$
We are given that $\lambda$ is positive and real
This is my attempt
$$\int_{-\infty}^{0} x e^{2\lambda x} dx+ \int_{0}^{\infty} xe^{-2\lambda x} dx$$
Using u-sub $u = {2\lamb... |
H: Deformation retracts induces deformation retract on quotient
I'm reading the following proof in Hatcher:
How does a deformation retract of $V$ onto $A$ give a deformation retract of $V/A$ onto $A/A$?
So, let $r: V \to A$ be a retract and $1_V \sim r$. How do I get a retract $V/A \to A/A$? Is it simply mapping all ... |
H: Bivariate normal distribution and law of $Z$
Let $(X,Y)$ a bivariate normal variable $f(x,y)=\frac{1}{2\pi \sqrt{1-\rho^2}}e^{\frac{1}{2(1-\rho^2)}(x^2-2\rho xy+y^2)}$. Let $Z=\frac{Y-\rho X}{\sqrt{1-\rho^2}}$.
Find the law of $Z$.
Say if $X$ and $Z$ are independent or not.
Find $\mathbb{P}(X>0,Y>0)$.
Already... |
H: Every locally integrable function defines a Radon measure
I have just started with Measure Theory and I have read several times that
Every locally integrable function defines a Radon measure.
I understand this statement in the sense that if we have $f\in L^1_{loc}(\mathbb{R}^N)$ (respecto to Lebesgue measure) , t... |
H: Prove $\sup \{x^2|x \in S \}$ exists, and equal to $s^2$, given $S \subset \Bbb{R}, S \neq \emptyset$, with sup $s$, inf $t$ and $s \geq -t$
Question
Suppose $S$ is a non-empty set of real numbers, with supremum $s$ and infimum $t$, and also that $s \geq -t$.
a) Show that $-s \leq x \leq s, \forall x \in S.$
b) Sho... |
H: Tautological 1-form on the cotangent bundle is intrinsic using transformation properties
I'm following this lecture in symplectic geometry and I'm trying to show the result stated at 31 minutes that the canonical 1-form on the cotangent bundle $M = T^*X$ is well defined regardless of which coordinates we choose, th... |
H: Power Series ODE Question - Final Step
I just started learning how to use the power series method to solve ordinary differential equations and this is one of the first questions we were asked. I've managed to get it right up to the last step (assuming I've made no silly errors which I don't think I have as my final... |
H: Conjecture If f is surjective then there exists x $\in$ (a, b) such that $|f'(x)| = 1$
Conjecture
Let $f$ be a continuous function from [a, b] to [a, b], and is differentiable
on (a, b).
If f is surjective then there exists x $\in$ (a, b) such that $|f'(x)| = 1$
Any counter example for this conjecture ?
**Addition ... |
H: Finding The Complex Roots Of $4z^5 + \overline z^3= 0$?
Hello everyone how can I find all the complex roots of:
$4z^5 + \overline z^3= 0$?
I tried to mark $a+bi = z , a-bi = \overline z$ and
$4(a+bi)^5 +(a-bi)^3 = 0$
But I don't know how to continue.
AI: Use the exponential form: if $z=r\mathrm e^{i\theta}\enspace ... |
H: Is there a function which satifies this condition?
Is there a function $f:\Bbb R\to\Bbb R$ which satisfies
$$\prod_{n=0}^{\infty} (f(x)^2-(2n+1)^2)=0$$
for all $x\in\Bbb R$? In other words, for all $x$, $f(x)$ is an odd integer.
AI: $$ y = 1+2\lfloor x\rfloor. \tag{1}$$ |
H: Notation for the union between the finite input alphabet and the empty string. What is the standard notation?
In the book “Introduction to the Theory of Computation”, the author writes this.
For any alphabet $\Sigma$ we write $\Sigma_{\varepsilon}$ to be $\Sigma \cup \{\varepsilon\}$.
In my formal lecture notes a... |
H: Continuous spectrum of operator.
Let $X=C[0,1]$ and we define
$$T\colon X\to X\quad\text{as}\quad(Tf)(t)=g(t)f(t)\quad\text{for all}\quad t\in [0,1]$$ with $g\in C[0,1]$ fixed.
I must find the continuous spectrum.
Now, if $\lambda\in g([0,1])$, then $\lambda=g(t_0)$ for some $t_0\in [0,1]$. As a consequence we hav... |
H: Calculating $\lim _{n\to \infty }\left(\frac{1\cdot n + 2\cdot(n-1) + 3\cdot (n-2)+ ... +1\cdot n}{n^2}\right)$?
Hello everyone how can I calculate the limit of:
$\lim _{n\to \infty }\left(\frac{1\cdot n + 2\cdot(n-1) + 3\cdot (n-2)+ ... +1\cdot n}{n^2}\right)$?
My direction was to convert it to something looks lik... |
H: If a finite group $G$ of order $n$ has at most one subgroup of each order $d\mid n$, then $G$ is cyclic
I'm reading the proof of a theorem in Fundamentals of Group Theory An Advanced Approach by Steven Roman.
(Characterization by subgroups) If a finite group $G$ of order $n$ has the property that it has at most on... |
H: An Interesting Question I Posed to Myself About $\pi$ as an Average.
Prove or disprove:
There is a sequence $x$ with each $x_i\in\{1,2,3,4\}$ so that $\pi$ can be written as the average $$\pi = \lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{x_i}{n}$$
I am sure that this question would be trivial using advanced number... |
H: Find number of triangles formed by lines( given:angle along x-axis)
i came across this problem in a competitive coding class :
A number of lines (extending infinity) in both directions are drawn on a plane. the lines are specified by the angle (positive or negative) made with the x axis(in degrees,constrained to -8... |
H: Show that for $n>3$, there is always a $2$-regular graph on $n$ vertices. For what values of $n>4$ will there be a 3-regular graph on n vertices?
Show that for $n>3$, there is always a $2$-regular graph on $n$ vertices. For what values of $n>4$ will there be a 3-regular graph on n vertices?
I think this question is... |
H: Convergence. Cauchy and uniform
I know that if function is uniformly convergent ($ |f_n(x)-f(x)|<\epsilon. \forall n > N(\epsilon)$), it is Cauchy convergent ($ |f_n(x)-f_m(x)|<\epsilon. \forall n,m > N(\epsilon)$)
So my question is: if sequence is Cauchy convergent does this imply uniform convergence? I think the... |
H: Calculate the volume of the region using triple integration.
A region bounded by the planes $x=0 , y=0 ,z=0 ,x+y=4, x=z-y-1$
I want to calculate the volume of the region using triple integration.
so what bounds should i use for the triple integration.
AI: You can just start by any one dimension although some are ea... |
H: Solve the equation: $\left|3^x - x\right|\left|3^x + x - 4\right| = 49$
I want to solve the equation in $\mathbb{R}$:
$$ \left|3^x - x\right|\left|3^x + x - 4\right| = 49 $$
My attempt:
The above equation is the same as:
$$ \left(3^x - x\right)\left(3^x + x - 4\right) = \pm 49 $$
Case 1: $\left(3^x - x\right)\left(... |
H: Infinite Prisoners dilemma
Please help me understand the idea of solving this problem.
There are infinitely repeated game $G( \infty, \sigma)$.
$$\begin{array}{|c|c|c|}
\hline
&c&d \\
\hline
c&(0,0)&(7,-3) \\
\hline
n&(-3,7)&(4,4) \\
\hline
\end{array}$$
Strategies with punishments in t... |
H: Prove $\lim\limits_{n\to\infty}n[\int_a^b f(x)dx-\sum_{k=1}^n f(a+k\frac{b-a}{n})\frac{b-a}{n}]=-\frac{1}{2}(b-a)\int_a^b f'(x)dx$
Assume $f(x)$ is a continuous function on $[a,b]$, and its first-order derivative $f'(x)$ is also continuous on $[a,b]$, prove the identity:
$$\lim\limits_{n\to\infty}n\left[\int_a^b f(... |
H: Module vs. absolute value
Can I use a term "module" as an alternative for a term "absolute value"? For example, could this phrase be used:
"We need to raise the module of the amplitude to the second degree"
for this expression:
$p = |a|^2$
($a$ in this expression can be a real or complex number)
AI: The correct... |
H: For $f:N\rightarrow N$, where $f(x)=x-(-1)^x$, then prove that $f$ is one-one and onto
Taking the first derivative
$$f’(x)=-x(-1)^{x-1}$$
Depending on the value of $x$, the slope of the graph changes from positive to negative. Thus it cannot be one-one
However, if we simplify the original function
For $x$ is even
$... |
H: $\det(I+A)=1+\operatorname{Tr}(A)$ if $\operatorname{rank}(A)=1$
Let $A$ be a complex matrix of rank $1$. Show that $$\det (I+A) = 1 + \operatorname{Tr}(A)$$ where $\det(X)$ denotes the determinant of $X$ and $\operatorname{Tr}(X)$ denotes the trace of $X$.
Any hint, please. I do not get how to combine the ideas ... |
H: T $\models \varphi$ iff $T \cup \{\lnot \varphi\}$ inconsistent, proof verification
I would like to show that:
T $\models \varphi$ iff $T \cup \{\lnot \varphi\}$ inconsistent
My attempt:
$(\Rightarrow)$ Suppose that $T \models \varphi$. Then $\varphi$ is true in every model of $T$. Since $\varphi$ and $\lnot \varph... |
H: Inequality conservation about convergence in law
I am trying to prove a claim about probability, and it is concluded if I can prove the following statement:
Suppose $X_n \rightarrow X$ in law, and $\exists\, M>0$ s.t. $\forall\, n$, $E[|X_n|] \leq M$. Then $$E[|X|] \leq M$$
Is this statement true? And how can I p... |
H: Proof of Euler's Theorem using Lagrange
Theorem : If $a,n \in \mathbb{N}$, $\gcd(a,n) = 1$ then $a^{\phi (n)} \equiv 1 \pmod n$
I am going through the proof that uses Lagrange's theorem
In the proof, we use the fact that if $G$ (s.t. $o(G) < \infty$) is a group and $a \in G$, then $a^{o(G)} = e$. The proof of this ... |
H: Function representing a Taylor Series
Find a function represented by the Taylor series $\sum_{k=0}^{\inf}\left(-1\right)^{k}\left(\frac{3^{2k+1}}{\left(2k+1\right)!}\right)x^{2k}$.
After taking a bunch of derivatives, I figured out $f^n(x)=\frac{3^{2n+1}}{2n+1}$, which further simplified to $\frac{3(3x)^2k}{(2k+1... |
H: (Maple) Linear Combination of Matrices
Problem: [This problem is intended to be done with Maple] Suppose that:
$$\textbf{u}_1=\begin{pmatrix}
356\\ -185\\ -580\\ -918\\ 147\\ 468\\ 504\\ 594
\end{pmatrix},
\textbf{u}_2=\begin{pmatrix}
573\\ 230\\ -950\\ -877\\ 69\\ 677\\ 323\\ 486
\end{pmatrix},
\textbf{u}_3=\begin... |
H: Upper bound for the sum of the nth powers of a digit
I am doing Project Euler # $30$. I currently want to find all the numbers that can be written as the sum of fifth powers of their digits. I solved this using a generic upper bound of a million. However, this isn't very precise and so I want to find out an upper b... |
H: Cayley-Hamilton-Theorem - Possible characteristic polynomial
Let $A: \mathbb{R}^3 \to \mathbb{R}^3$ s.t.
$A^3-2A^2+A= 0$
The Cayley-Hamilton-Thm. states that if I put $A$ into its characteristic polynomial it'll equal $0$.
But am I allowed to conclude from the given equation $A^3-2A^2+A= 0$ that $\lambda^3-2 \lambd... |
H: Difference between $S^{1}$ and $\mathbb{S}^{1}$
Whats the difference between the two? Or are they both the unit circle? My lecture notes for a topology course use $S^{1}$ for the "circle" in one example and $\mathbb{S}^{1}$ in the next example for the "boundary circle". Is it just a notational inconsistency?
EDIT: ... |
H: Inclusion-Exclusion Permutations
Taken from finals on discrete mathematics;
How many permutations of the set $\{a,b,c,d,e,f,g,h,i,j\}$ are there such that:
a) Each of the patterns ab, de, gh and ij appears
b) None of the patterns ab, de, gh and ij appears
c) At least one of the patterns ab, de and gh appears.
d) Ex... |
H: Derivative of the complex norm as commonly used in physics
On the one hand, I read that the derivative of the complex conjugate $C[z]=\overline{z}$ is not differentiable anywhere (for instance see here). (see 1, below)
On the other hand, I see in physics taking the derivative of a complex scalar field to obtain the... |
H: Question on a problem regarding improper integral
I'm studying improper integrals with Paul's Online Notes as a reference. Sorry if I'm quoting it here, but the website has the following problem:
Determine if the following integral is convergent or divergent. If it is convergent find its value.
$$\int_{-2}^{3} \fr... |
H: Prime numbers equation
Could you please help me for this proof :
Prove that there is no triplet of integers (x, y, z) prime to each other such that: $$x²+y² = 3z²$$
I tried to make a proof by contradiction...
AI: RHS is divisible by $3$. If either $x$ or $y$ are $1$ or $-1$ modulo $3$ then $3$ will not divide the L... |
H: Could this integral be estimated with a positive constant?
Let $\Omega$ be an opend bounded subset of $\mathbb{R}^n$ and let $p, q$ be two real numbers such that $p, q\geq 1$. Let $(w_n)_n\subset W_0^{1, p}(\Omega)$ and $(z_n)_n\subset W_0^{1, q}(\Omega)$ such that $\exists w\in W_0^{1, p}(\Omega)$ such that
$$ w_n... |
H: Why is there a hierarchy of interest between associativity and commutativity
In mathematical structures, there are among other things : groups.
Among their particular properties of the group, the groups have the property of associativity.
Within the various groups, there are commutative (abelian) and non commutativ... |
H: The antiderivative of $\sum_{n\gt 0}\frac{x}{n(x+n)}$
I tried to calculate $\int\sum_{n\gt 0}\frac{x}{n(x+n)}\, \mathrm dx$:
$$\begin{align}\int\sum_{n\gt 0}\frac{x}{n(x+n)}&=\sum_{n\gt 0}\frac{1}{n}\int\left(1-\frac{n}{x+n}\right)\, \mathrm dx \\&=\sum_{n\gt 0}\left(\frac{1}{n}\int\mathrm dx -\int \frac{\mathrm dx... |
H: Why $M/\mathfrak{a}M \oplus M/\mathfrak{b}M \simeq M/(\mathfrak{a \cap b})M$?
Let $M$ be an $A$-module and let $\mathfrak{a}$ and $\mathfrak{b}$ be coprime ideals of A.
I must show that $M/ \mathfrak{a}M \oplus M/ \mathfrak{b}M \simeq M/ (\mathfrak{a \cap b})M$.
My attempt is the following:
Let $x \in M/ \mathfrak{... |
H: How to find an integrating factor?
I am trying to understand the integrating factor technique starting with a simple case to see how it develops into more complicated structures.
Suppose I have a differential equation I want to solve of the for $y'(x) = g(x)y(x)$ where $g(x)$ is known and $y(x)$ is not.
Now althoug... |
H: Prove or Disprove : There exists a continuous bijection from $\mathbb{ R}^2$ to $\mathbb{R} $
This question was asked to me by a mathematics undergraduate to me and I was not able to solve it. So, I am asking it here.
Prove or Disprove : There exists a continuous bijection from $\mathbb{ R}^2$ to $\mathbb{R} $ .
... |
H: Area of a rectangle using congruency
I don't understand the lecturer, solving the question, says that the rectangles are congruent each other, so the result can be obtained by proportioning them to each other. However, AFAIK two quadlaterals are congruent to each other such that sides an interior angles of one quad... |
H: What is $\lim\limits_{b\to a}\frac{e^{-\frac{x}{a}}-e^{-\frac{x}{b}}}{a-b}$?
I'm trying to evaluate $$\lim\limits_{b\to a}\frac{e^{-\frac{x}{a}}-e^{-\frac{x}{b}}}{a-b}$$
I know that the limit exists. The limit of the numerator and denominator are both zero when $b\to a$, so I tried to apply L'Hospital's Rule for $\... |
H: Finding the area between ${y=x^2}$ and ${y=2x+8}$
The two equations are ${y=x^2}$ and ${y=2x+8}$
I got the result ${-64}$ multiple times, but the proper answer seems to be different
AI: The answer is supposed to be an area; therefore, it cannot be negative.
The intersection points of those two curves are $(-2,4)$ a... |
H: If for invertible matrices $A$ and $X$, $XAX^{-1}=A^2$ then eigenvalues of $A$ are $n^{th}$ roots of unity.
Question: Let $A$ and $X$ be two complex invertible matrices such that $XAX^{-1}=A^2$. Show that there exists a natural number $n$ such that each eigenvalue of $A$ is an $n^{th}$ root of unity.
I can say from... |
H: How to transform a regular expression into a context free grammar with 2 variables?
I'm tasked with transforming this regular expression $((0+1)(0+1)^*(0+1))^*$ into a context free grammar. As an added constraint I'm must do so with a maximum of 2 variables. This is what I did :
S -> VWV | S | ɛ
W -> 0 | 1 | W | ɛ
... |
H: $\Vert f \Vert^{2}$, where $f: [0,1] \to \mathbb{R}$ continuously differentiable.
Let $f: [0,1] \to \mathbb{R}$ be continuously differentiable with $f(0)=0$. Prove that $$\Vert f \Vert^{2} = \int_{0}^{1} (f'(x))^{2}dx$$
Here $\Vert f \Vert$ is given by $\sup\{|f(t)|: t \in [0,1]\}$.
I see how to prove that $(\int_{... |
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