text stringlengths 83 79.5k |
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H: If $M$ is finitely generated then $M/N$ is finitely generated.
Let $N$ be a submodule of $R$-module $M$. Prove that if $M$ is finitely generated then $M/N$ is finitely generated.
Help me some hints.
AI: Suppose $M$ is generated by $n$ elements, say $\{a_i\}_{i=1}^n$.
Claim : $M/N = (a_i + N \ |\ i \in [1,n])$.
$(\... |
H: Ring made from a Bijection of Positive Integers and Rationals to Reals (or Complex Numbers)
I've been working on this problem in my head and I'm not quite how to phrase this in a way that makes sense since I haven't been doing pure math in some time so I hope this makes sense.
What I'm looking for is something like... |
H: Prove that if $R$ is a principal ideal ring and $S$ is a multiplicatively closed subset of $R$ then $S^{-1}R$ is also a principal ideal ring.
Prove that if $R$ is a principal ideal ring and $S$ is a multiplicatively closed
subset of $R$ then $S^{-1}R$ is also a principal ideal ring.
Thanks for any insight.
AI: ... |
H: Vectors Dependent or not?
Am I doin it right?
$3-2x + x^2,6-4x+2x^2 $ in $P2$ Check if they are dependent or not?
I am taking their determinant as written
$| 3-2x\space\space\space\space\space\space\space\space\space 6-4x |$
$|x^2\space\space\space\space\space\space\space\space\space\space\space\space\space\space... |
H: linearly independence of functions over $\mathbb{R}$
I have been asked to prove that $\{\tan(ax)|a\in \mathbb{R}^+\}$ is linearly independent. I was wondering if there is a generic method/idea for proving linear independence of functions over $\mathbb{R}$.
AI: Hint: Since $\tan \in C^{\infty}(\mathbf{R})$. Pick any... |
H: Why complex conjugate roots of polynomials with real coefficients have the same multiplicity?
Let
$$f(x)=(x-z_1)^{k_1}(x-\overline{z_1})^{l_1} \cdot ... \cdot (x-z_n)^{k_n} (x-\overline{z_n})^{l_n}g(x),$$
where $z_i$ are different complex numbers with nonzero imaginaris parts, $k_i, l_i $ are natural numbers, $g... |
H: Simplify $1 - \cos x + \sin x - \tan x$.
I somehow can't find a way to do this, the expression already seems simplified to me. I've tried factoring, I got $$(\cos x - \sin x)(1 - \cos x) \over \cos x$$ but obviously that's not simplified and I can't see how I could continue.
AI: You're not far off. If you distribut... |
H: Prove the following $(A \cap B') \cup (A\cap C) = A \cap( B' \cup C)$
I need help to prove:
$(A \cap B') \cup (A\cap C) = A \cap( B' \cup C)$
Thanks
AI: Double inclusion. For example:
$$x\in (A\cap B')\cup (A\cap C)\implies (x\in A\;\wedge\;x\notin B)\;\vee\;(x\in A\;\wedge\;x\in C)\implies$$
$$x\in A\;\wedge (x... |
H: Is the following convergence of the series of functions uniform?
For the series f(x) = x + [sum from j=1 to j=infinity, x((1-x)^j)], I already proved that this series converges pointwise for x on [0, 1]: f(x) converges pointwise to 1 if x is on (0, 1]; f(x) converges pointwise to 0 if x=0. Note that I found that fo... |
H: Brownian Motion conditional distribution
Let $\{X(u),u\geq0\}$ be a standard Brownian motion. What is the conditional distribution of $X(t)$ given $\{X(t_{1}),\dots,X(t_{n})\}$, where $0<t_{1}<\cdots<t_{n}<t_{n+1}=t$?
--So far, I have derived the joint pdf of $X(t_{n+1})$ and $X(t_{1}),\dots,X(t_{n})$ using the fac... |
H: How do I find this $\begin{vmatrix} A&B\\ B&A \end{vmatrix}$?
Let $A, B$ be matrices such that $A=(a_{ij})_{n\times n}, B=(b_{ij})_{n\times n}$, $$b_{ii}=2a_{ii},$$ and
$$b_{ij}=\begin{cases}
a_{ij}&i>j\\
-a_{ij}&i<j
\end{cases}$$ for $(i,j=1, 2, 3, \cdots , n)$.
Find $$\begin{vmatrix}
A&B\\
B&A
\end{vmatrix}.$$
My... |
H: Why is the total number of binary relations $2^{n^2}$
Why is the formula for finding the total number of binary relations in a set A equal to $2^{A^2}$ ?
I have looked on the web (including here) and find the answer is because the relation can be in or out. This is the exact thing my professor told me but did not e... |
H: Is $([0, \sqrt 2] \cap \mathbb Q) \subset \mathbb Q$ closed, bounded, compact?
As far as I can tell it is bounded, as it's within $[0, \sqrt 2]$, and is closed as there cannot be an open neighbourhood about 0, and as it's closed and bounded it is therefore compact. However I'm not sure if closed and bounded imply c... |
H: How to show field extension equality
I've seen similar field extension questions on SE, but nothing with a third root, and I'm having trouble adapting any of those solutions to this problem.
So I'm trying to prove that $\mathbb{Q}(\sqrt{2}+{5}^{1/3})=\mathbb{Q}(\sqrt{2},{5}^{1/3})$.
Now, $\mathbb{Q}(\sqrt{2},{5}^{... |
H: Radius of convergence of a Taylor series.
I am looking for the shortest possible way to find out the radius of convergence of the Taylor series expansion about $x = a \in \mathbb{R}$ of the function
$$f(x) = \frac{1}{1 + x^2}$$
Taylor series expansion of the function $f(x)$ about $a$ will be $f(x) = \sum_{n = 0}^{... |
H: Does the following question regarding functions and relations make sense?
I have a function $f: P(\mathbb{N} \times \mathbb{N}) \to P(\mathbb{N} \times \mathbb{N})$ defined: $f(R)=R \cup R^{-1}$
The question I'm struggling with is "Find $f(P(\mathbb{N} \times \mathbb{N}))$". My main problem is that I can't make sen... |
H: Matrix of orthogonal projection
It was required to find the orthogonal projection of the vector $u =(0,1,0,2)$ onto
$W=${$(x,y,z,t) \in \mathbb{R}^4 : x+y-t = 0$} and the matrix of the projection.
First, I've found a basis for $W$ and, use Gram-Schimidt process, an orthonormal basis. With it, I could do the first ... |
H: Equation for this graph
I am trying to find an equation for the graph that crosses the Y axis at (0,100), X axis at (100, 0), is a curve with adjustable degree of "bending" and has an axis of symmetry y(x) = x.
Here are few examples of such graphs:
I've tried quadratic, cubic, quartic equations but I can't adj... |
H: Rationalizing a denominator.
The question instructs to rationalize the denominator in the following fraction:
My solution is as follows:
The book's solution is
which is exactly the numerator in my solution.
Can someone confirm my solution or point what what I'm doing wrong?
Thank you.
AI: Going from the first t... |
H: Isometry of a metric space with proper subset
In Irving Kaplansky's "Set Theory and Metric Spaces", exercise 17 on page 71 asks for an example of a metric space which is isometric to a proper subset of itself. Any infinite discrete space and any $\ell^p$ space are such spaces, for different reasons. I want some kin... |
H: Integral involving logarithm and cosine
For $a,b>0$ I would like to compute the integral $$I=\int_0^{2\pi} -\log{\sqrt{a^2+b^2-2ab\cos{t}}}~dt.$$
Numerical computations suggest that $$I=\min\{-\log{a},-\log{b}\}.$$
How can I prove this? I tried to find the antiderivative using Mathematica, but the result looks awfu... |
H: Real Analysis: Use Intermediate value Thm to prove the functions
Let $f \colon [0,2] \to \Bbb R$ be a continuous function such that $f(0) = f(2)$. Use the intermediate value theorem to prove that there exist numbers $x, y \in [0, 2]$ such that $f (x) = f (y)$ and $|x − y| = 1$.
Hint: Introduce the auxiliary functi... |
H: Does closed imply bounded?
Definitions:1. A set $S$ in $\mathbb{R}^m$ is bounded if there exists a number $B$ such that $\mathbf{||x||}\leq B$ for all $\mathbf{x}\in S$, that is , if $S$ is contained in some ball in $\mathbb{R}^m$.2. A set in $\mathbb{R}^m$ is closed if, whenever $\{\mathbf{x}_n\}_{n=1}^{\infty}$ ... |
H: Why $R=\{(a,b)\mid a=b \mbox{ or } a=-b\}$ is not anti symmetric?
Why $R=\{(a,b)\mid a=b \mbox{ or } a=-b\}$ is not anti symmetric?
I read because this is symmetric so it is not anti symmetric, but $R=\{(a,b) \mid a=b \}$ is both symmetric and anti symmetric.
AI: It’s not antisymmetric because $\langle 1,-1\rangle... |
H: How to compress data dimension
I have n points on the plane $(x_1, y_1) \ldots (x_n, y_n)$ - it's the points of one stroke. I want to reduce number of points without significant information loss.
I read some information about PCA, but I don't sure if it's what I need.
Can anyone recommend methods for doing wh... |
H: Limit of $n/\ln(n)$ without L'Hôpital's rule
I am trying to calculate the following limit without L'Hôpital's rule:
$$\lim_{n \to \infty} \dfrac{n}{\ln(n)}$$
I tried every trick I know but nothing works. You don't have to prove it by definition.
AI: Every time you make $n$ twice and big, you increase $\ln n$ by l... |
H: Why does $\operatorname{Spec}(\prod_1^\infty \Bbb F_2)$ have connected components that are not open?
To be honest, I don't even know how to describe all prime ideals in $\prod_1^\infty \Bbb F_2$. I know we get one for each $n \in \Bbb N$ corresponding to the set of elements that are zero in the $n$-th coordinate, a... |
H: Find number of triangles.
This is an extremely basic question posed to my brother in fifth standard and i was only able to find $20$ triangles in this figure however the answer according to the answer key was 24.
$Side -1: 12 $
$Side -2: 6 $
$Side -3: 2 $
Is there any way to conclusively prove that there are... |
H: Prove or refute contingent: If A implies B is contingent, then B is too
The question is:
If $A, A \to B$ are contingent, then so is $B$
$A, A \to B$ (implies) is a contingent, but how exactly to show «so is $B$»?
If I'm using a truth table, how should I show that $B$ is also contingent?
AI: This is not necessaril... |
H: If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen.
If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen.
This is exercise 3.6P of Vakil. I can see that a union of connected components is closed. This is so because connected com... |
H: How to find dim(S∩T)
I have S and T in the subspace of $R^4$ and defined by
$S={[s, t, 0, 0]|s, t ∈ R}$
$T={[0, s, t, 0]|s, t ∈ R}$
Then I am asked to find $dim(S)$, $dim(T)$, and $dim(S∩T)$. From what I have gathered, the dimensions of the first two are 4 each, but I'm lost on how to find the dimension of an inte... |
H: Solving for upper limit of integration, given a lower limit and integrand
Consider $f$ a real integrable function, usually we want to evaluate the integral $\int_a^bf(x)dx$ for some $a<b$ given. Now, suppose we know $a$ but we don't know $b$, further, we know the value of this integral, let $\int_a^bf(x)dx=\lambda$... |
H: $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$
Let $A$ be a finite abelian group of order $n$ and let $p^k$ be the largest power of the prime $p$ dividing $n$. Then $\Bbb{Z}/p^k \Bbb{Z} \otimes_{\Bbb{Z}} A $ is isomorphic to the Sylow $p$-subgroup of $A$.
Hints on proving ... |
H: Is binary-relation $\left\{\left(a,b\right)\mid a,b\in\mathbb{N}\wedge a,b \text{ are even numbers}\right\}$ reflexive?
I'm a novice in set theory and I'm not clear about reflexive relation.
My question is the title.
Is binary-relation $R:=\left\{\left(a,b\right)\mid a,b\in\mathbb{N}\wedge a,b \text{ are even numbe... |
H: Why is this set a Borel set on $R^2$?
Consider a measurable space $([0,1]\times [0,1], \mathcal{B}([0,1]) \times \mathcal{B}([0,1]))$, and a subset $A:=\{(x,y):x=y\}$ (the diagonal). According to the text book, $A \in \mathcal{B}([0,1]) \times \mathcal{B}([0,1])$. Can any body tell me the reason and in general, how... |
H: How do i find the lapalace transorm of this intergral using the convolution theorem?
$$\int_0^{t} e^{-x}\cos x \, dx$$
In the book, the $x$ is written as the greek letter "tau". Anyway, I'm confused about how to deal with this problem because the $f(t)$ is clearly $\cos t$, but $g(t)$ is not clear to me.
Please he... |
H: a question on : why sin(x) = sin(180-x)
I was trying to follow a proof from another question in this forum:
Why is $\sin(x) = \sin(180^{\circ}-x)$
I was following the geometric proof given by forum poster : egreg.
I could follow his proof for the cosine.
But I think the proof for the sine contains an error.
Egreg s... |
H: If $L\cdot\{\epsilon,0\}$ regular language, is $L$ regular?
I've encountered a question during my studies:
If $L\cdot\{\epsilon,0\}$ regular language, is $L$ regular?
I thought to disprove it by using $A\subseteq 2\mathbb{N}, L=\{w\in\{0\}^*:|w|\notin A\}$ but I need to prove that L is irregular and using the pumpi... |
H: Application of the Dominated Convergence Theorem
$$lim_{n \to \infty} \int_0^1(1 - e^{\frac{-x^2}{n}})x^{-1/2}dx$$
I want to use the Dominated Convergence Theorem to solve this.
Let $f_n = (1 - e^{\frac{-x^2}{n}})x^{-1/2}$.
Step 1: Determining convergence of $f_n$
Fix $x$ to be some constant number.
Now, bringing t... |
H: Find the vector so that it is parallel
Find a vector with length 7 and is parallel to the line $y=\frac{12}{13}x-1$
Is there some more advanced mathematics behind this than elementary maths or?
Could use a nudge.
AI: The line $y=12/13 x$ is parallel to the line $y=12/13x -1$. Consider a vector in it, for example $(... |
H: Arbitrary meaning two different things in two different settings
The arbitrary in induction:
Induction step: We assume that P(k) is true and then we need to show that P(k+1) is true as well.
k is arbitrary and means "any one"
Consider this example of arbitrary used in the rule of Universal Instantiation
Rule:
(FOR ... |
H: Why $\sum_{k=1}^{\infty}\frac{\ln(k)}{k}$ diverges?
So my solution manual says this diverges since it is a p-series and $p=1 \le 1$
I understand why that p-series converges, but I don't understand conceptually how they got there using the comparison test (as the instructions advise).
Again, here is my series:
$$\su... |
H: Radical extension over $\mathbb{Q}$
Let $K=\mathbb{Q}(\sqrt[n]{a})$, where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d$. Prove that $E=\mathbb{Q}(\sqrt[d]{a})$. Hint: Consider $N_{K/E}(\sqrt[n]{a})\in E$.
Attempt:
I proved that the norm is in $E$ ... |
H: Undetermined coefficients guessing particular solution's shape
I try to find general solution for:
$$y'' - y = 8 t e^t$$
I find comp. as $(D-1)(D+1)=0$ and general solution of it as $y_g= C_1 e^t + C_2 e^-t$
at this point.
I know particular solution shape is:
$$y= A t e^t+ B t^2 e^t$$
But what is the way to find i... |
H: Finding the Maximum Likelihood Estimation (Self learning)
I'm trying to learn the subject of Maximum Likelihood Estimation by myself.
I'm facing one of the first questions in the textbook which is:
`The waiting time (in minutes) on a queue to the dentist is the random variable $X$ with the following pdf:
$$f(x) = \... |
H: A self-convolution formula that counts bracket expressions
Problem: Consider an alphabet of size $m+2$, consisting of the two bracket symbols $\ [ \ ] \ $ plus $m$ non-bracket symbols ($m \ge 0$). Define $f_m(n)$ to be the number of length-$n$ strings on this alphabet, in which brackets may occur only in the usual ... |
H: Why is this binary-relation symmetric?
From the example of binary-symmetric-relation demonstrated in Wikipedia, how can they say the relation "$x$ and $y$ are odd numbers" is symmetric without stating any set of $x$, $y$? If such set is $\mathbb{N}$, then the relation is not symmetric because relation set does not ... |
H: Finding the velocity vector
Am I finding the equation of the slope of the tangent line at c(t)?
$\frac{dy/dt}{dx/dt}$ = $\frac{2t}{3t^2-8}$
AI: The position of the particle at any particular time will be given by the vector
r(t) = (x(t),y(t))
You want to differentiate r(t), i.e,
v(t) = dr(t)/dt = (dx(t)/dt,dy(t)... |
H: Prove that the following sequence has a finite partial limit
Let $ a_n $ be a sequence.
Prove that if the sequence $|a_n| $ does not converge to $\infty$, $a_n$ has a finite partial limit.
AI: As $|a_n|$ doesn't converge there must be a subsequence $b_i$ such that $b_i \in [-K,K]$ for some $K \in \mathbb{R}$ (If s... |
H: Prove that [0,1] is equivalent to (0,1) and give an explicit description of a 1-1 function from [0,1] onto (0,1)
The problem is stated as follows:
Show that there is a one-to-one correspondence between the points of the closed interval $[0,1]$ and the points of the open interval $(0,1)$. Give an explicit descript... |
H: Quotient groups and dihedral groups
I came across a question asking for me to find all the possible quotient groups for the dihedral group $D_6$. How must I go about this?
AI: I assume that your definition of $D_6$ is
$$D_6=\{1,r,r^2,s,sr,sr^2\}$$
where $r^3=s^2=rsrs=1$.
A quotient set $D_6/N$ is a group if and onl... |
H: Adjunctions in Category Theory
My question is how can we see that if $F$ and $F'$ are both left adjoints of $G$, there is natural isomorphism between $F$ and $F'$? So how can we show that adjoints are unique upto isomorphism?
AI: If $F \vdash G$, $F' \vdash G$, then we have units/counits $\epsilon:I \rightarrow FG,... |
H: Sum of two subspaces
Let $$U_1 := \{(\gamma_1, \gamma_2, \gamma_3) \in \Re^3 : \gamma_1 + \gamma_2 + \gamma_3 = 0\}$$ $$ U_2 := \{ (\lambda, \lambda, \lambda) : \lambda \in \Re \}$$
be subspaces. Show that $U_1 + U_2 = \Re^3$.
The sum of the two subspaces $U_1 + U_2$ is by definition $\{u_1 + u_2 : u_1\in U_1, u_2 ... |
H: What does the author (Ahlfors) mean here (modular $\lambda$ function)
In Ahlfors' complex analysis text, page 281 it says
By reflection the region $\Omega'$ that is symmetric to $\Omega$ with respect to the imaginary axis is mapped onto the lower half plane, and thus both regions together correspond to the whole p... |
H: Is half of the area of a Fibonacci triangle a congruent number?
A Fibonacci triangle is a triangle with integer area and sides whose lengths are Fibonacci numbers. An example of a Fibonacci triangle is the triangle whose sides have lengths (5,5,8). Is half of the area of a Fibonacci triangle a congruent number ? (A... |
H: A question on combinatorics
Bob and Laura have bought an apartment, and are going to carpet the floor in the kitchen.
The kitchen has size $2 \times n$, where $n$ is a positive integer. In how many ways can
Bob and Laura complete it using carpet pieces of two types: of size $1 \times 2$ and of size $2 \times 2$? So... |
H: Asymptotic behavior of a solution to the quadratic equation
I have a quadratic equation of real $x$,
$$ x^2 - 4(1+2y)x + 8(y+1) = 0 $$
for $ x>0, y>0$ and the solution is
$$ x(y) = 4y + 2 - \sqrt{4(1+2y)^2 - 8(y+1)} $$
$$ = 4y + 2 - 2\sqrt{4y^2 + 2y -1} $$
I found the solution approaches to 1 for large $y$, if I p... |
H: How to show the space is totally bounded?
the space the Real line with bounded metric (i.e. d/(1+d), d: euclidean)
we know that totally boundedness means that there exists a finite epsilon-net.
we first approached to question by directly try to find finitely many points s.t. their metric balls cover the space. then... |
H: Recovering ring information from localizations
I'm curious if such a statement is true:
Let $R$ be a commutative ring with unity. Then if $x \in R$ and $x = 0$ in $R_p$ for all primes $p$, where $R_p$ is the localization of $R$ at prime $p$, then $x = 0$ in $R$.
AI: Yes this is true. You can even restrict to maxi... |
H: Prove $x^n-1$ is divisible by $x-1$ by induction
Prove that for all natural number $x$ and $n$, $x^n - 1$ is divisible by $x-1$.
So here's my thoughts:
it is true for $n=1$, then I want to prove that it is also true for $n-1$
then I use long division, I get:
$x^n -1 = x(x^{n-1} -1 ) + (x-1)$
so the left side is div... |
H: Help me evaluate this integral
$\displaystyle \int \frac{\ln x}{x^2} \mathrm dx$
I just can't seem to figure this one out. I tried integrating by parts but I'm stuck.
AI: Integration by parts is not u-substitution. You split the integrand into two parts, $u$ and $dv$. Choose $u$ so that it will simplify when you ta... |
H: Clarification on Rules Differentiation and First Principles Derivatives
My grade 11 class has just started differential calculus, the one area seemingly glazed over in our book. We have covered some simple rules of differentiation, like
f(x) = n x^(n-1), and have applied these with subsequent rules like the consta... |
H: What is the easiest way to find the inverse Laplace of F(s)?
$$
F(s)= \frac{1}{(s-1)^2(1-1/s^2)}
$$
Do I have to multiply by $s^2/s^2$ and then use partial fractions or is there a way to use the convolution theorem?
AI: This can be rewritten as (another form is also possible):
$$F(s) = \dfrac{s^2}{(s-1)^3 (s+1)}$$
... |
H: Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$
I am trying to calculate the following limit without Stirling's relation.
\begin{equation}
\lim_{n \to \infty} \dfrac{n!}{n^n}
\end{equation}
I tried every trick I know but nothing works. Thank you very much.
AI: By estimating all the factors in $n!$ except th... |
H: How many transitive relations on a set of $n$ elements?
If a set has $n$ elements, how many transitive relations are there on it?
For example if set $A$ has $2$ elements then how many transitive relations. I know the total number of relations is $16$ but how to find only the transitive relations? Is there a formula... |
H: Show that the set $\{\frac{1}{n^2}: n \in N \} \cup \{0\}$ is closed
I want to show that $Y=\{\frac{1}{n^2}: n \in N \} \cup \{0\}$ is closed.
I think I can do so by arguing that $Y$ contains its upper and lower bounds, $1$ and $0$, and hence is closed. But I think I might need to argue that $Y^c$ is open, which i... |
H: How to prove that $(\sec x-\cos x)(\csc x -\sin x)=\tan x/(1+\tan^2x)$?
I've tried solving from both sides but can't seem to get them equal. Any tips on which side to start on and what to do after?
AI: Outline: $$\sec x - \cos x = \frac 1 {\cos{x}} - \cos x = \frac{1 - \cos^2 x}{\cos x} = \frac {\sin^2 x}{\cos{x}}$... |
H: How to find $f(2013)$ if $f(5)=45$ and $f(m)+f(n)= f(m+n)$ for all $m,n\in\mathbb N$?
$f: \mathbb N\to\mathbb N$, $f(m)+f(n)=f(m+n)$ for all $m,n\in\mathbb N$, and $f(5)=45$. Find $f(2013)$.
I messed up my original posting, its fixed now.
I changed $m+ n$ to $f(m+n)$.
AI: The question now says that $f$ is linear... |
H: Laplace transform of two functions
I know how to do a basic laplace transform, but how does one deal with transforming complex combination of functions?
For example, how would we handle:
$$\mathcal{L}\left( \ \sqrt{\frac{t}{\pi}}cos(5t) \right) = ... $$
From a table of laplace transforms it is known that:
$$\mathca... |
H: Construction of two graphs
I would like to know if it is possible to construct two graphs $G,H$ such that $|G|=|H|, e(G)=e(H)$ (means that the two graphs have the same number of vertices and edges) and $\chi(G)>\chi(H)$ where $\chi()$ represents the chromatic number, such that there are more ways to colour $G$ than... |
H: Why arithmetic series never sums to a fraction
Sometimes with series we find a solution in a form of a fraction which does not a priori obviously take only integer values. On the other hand from the sum it is pretty obvious that the sequence of partial sums can take only integer values. (arithmetic and geometric se... |
H: Evaluate $\tan\left(2\sin^{-1}\frac{\sqrt{5}}{5}\right)$ without using a calculator
Evaluate without using a calculator:
$\displaystyle{\tan\left(2\sin^{-1}\left(\sqrt{5} \over 5\right)\right).}$
So I built my triangle hyp=$5$, adj=$2\sqrt{5}$, opp=$\sqrt{5}$.
$$
\tan\left(2\theta\right) = 2\sin\left(\theta\right)\... |
H: Eigenvector and Its Span
Let $V$ be a vector space over the field $F$ and let $T$ be a linear transformation from $V$ to $V$. Let $v\in V$ such that $v\neq 0$, let $W=span\{v\}$. Prove that if $T(W) \subset W$, then $v$ is an eigenvector for $T$.
So, my idea for this problem was since $v\in V, v=1.v\in span\{v\}=W... |
H: Path counting discrete
I need help understanding a simple concept. Lets say you're given a problem where you start at (0,0) on a 2D grid and want to count the number of paths to (8,4). Would I be following the combinations problem where (m,n) = (m + n) choose m? And would this be the same as (m + n) choose n? To bu... |
H: Find all solutions of the equality $y^2=x^3+23$ for integers $x,y$
Find all solutions of the equality $y^2=x^3+23$ for integers $x,y$
I guess, that x cannot be even. because we can apply mod 4 test
say $x=2k$ then $y^2=8k^3+23\equiv3\mod(4)$
but, this is not possible, since the square of an integer is congruent ... |
H: Removing the square root in $\sum_{n=1}^{\infty}\frac{\sqrt{n+3}}{4n^2+n+4}$
I know I can use the comparison test to examine this series, but say I wanted to take the limit of this series, how do I handle square root in the series?
$$\sum_{n=1}^{\infty}\dfrac{\sqrt{n+3}}{4n^2+n+4}$$
For example, if there wasn't a s... |
H: Is $O(n \log n)$ always smaller than $O (m)$ for $n-1 < m < n^2$?
I am writing an algorithm that needs to finish in $O(m)$.
The problem is for a graph $G( V, E )$, where $m = |E|$ and $n = |V|$.
$m$ can be in the range of $n-1$ to $n^2 - 1$.
If I do some processing that takes $O\left( (n-3) \log(n-3)\right)$ time,... |
H: Kronecker's Theorem - what can be deduced if $f$ is reducible?
I am happy with the statement of Kroncker's Theorem as follows. Let $\:$ $ f \in K[X]$. Then there exists a simple field extension $L = K(\alpha)$ of $K$ with $f(\alpha) = 0$. If $f$ is irreducible over $K$, then the extension $L$ is unique in sense tha... |
H: Prove that free modules are projective
Prove that free modules are projective.
Help me.
AI: Definition: $M$ is projective iff for every surjective morphism $f:A\rightarrow B$, and every morphism $g:M\rightarrow B$ there is morphism $h:M\rightarrow A$ such that $fh=g$.
Now if $M$ is free with base $\{x_i\}$ then w... |
H: Show $\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = \int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$
I read in an article about Laplace's equation that
$$-\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = -\int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$$
Could someone explain to me why this is? I understand that one can mo... |
H: What is this formula called? summation
Does people know what this formula called? I want to google its properties and read about it more - if it has official name
AI: It is called a "weighted average" or "weighted mean". The numbers $\mu_i$ are called the "weights". |
H: equivalence relation-showing that an operation is well-defined
Define $f: \mathbb{Z}_n \to \mathbb{Z}_n$ as $f([a]) = [a^2]$. Show that $f$ is a well-defined function. I am confused as to how I could show this.
AI: You need to show that if $[a]=[b]$ in $\Bbb Z_n$ (i.e., $a\equiv b\pmod n$, i.e. $n\,|\,b-a$), then $... |
H: How do you prove a piece of the Short Five Lemma?
Let $\alpha, \beta, \gamma$ be a homomorphism of short exact sequences, in that order. Then if $\alpha, \gamma$ are injective, then so is $\beta$.
Let the sequeces be:
$$
\begin{matrix}
0 & \to & A & \xrightarrow{\psi} & B & \xrightarrow{\phi} & C & \to & 0 \\
\ & ... |
H: Prove that the set of Real numbers R can be partitioned into a denumerable collection of uncountable sets
Prove that the set of Real numbers R can be partitioned into a denumerable collection of uncountable sets
My thoughts on the problem were to show the break down of the set of real numbers into subsets such as r... |
H: Code book for Hamming Code
For a natural nuber $d$ there are $N=2^d-1$ non-zero vectors in $\mathbb F_2^d$. Now I take these as the columns of a $d\times N$ matrix $S$. The kernel of $S$ is then the code book for a Hamming code $c:\mathbb F_2^{N-d}\rightarrow \mathbb F_2^N$ (Is this a clear result or is there anyth... |
H: Jordan Normal Form
I'm asked to find the Jordan Normal form of $A\in M_5(\mathbb{C}^{5x5})$ with the characteristic polynomial: $p(A)=(\lambda-1)^3(\lambda+1)^2$ and minimum polynomial $m(A)=(\lambda-1)^2(\lambda+1)$
I got so far:
$$m_A(x)=(x-1)^2(x+1)\;\;\;:\;\;\;\;\begin{pmatrix}1&1&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0... |
H: Existence of uncountable set of uncountable disjoint subsets of uncountable set
"Can you to any uncountably infinite set $M$ find an uncountably infinite family $F$ consisting of pairwise disjoint uncountably infinite subsets of $M$?"
Intuitively, I feel like it should be possible for the real numbers at least: you... |
H: proving something is a well-defined function
1)
Define ~ S4 as follows: for f, g element of S4 f~g if and only if f(4) = g(4) this is easily seen to be an equivalence relation on S4 (you don't have to show this) let X = S4/ ~ be the set of all equivalence classes under ~. Define * X as [f] * [g] = [f o g]. Is * a w... |
H: Solving a integro-differential equation
I'm trying to solve the following problem, but my solution doesn't seem to be correct. Could I be rewriting the integral incorrectly?
$$ \frac{dy}{dt} + 25 \int_0^t{y(t-w)e^{-10w}dw} = L[4]; y(0)=0$$
$$ sL[y(t)] - y(0) + 25(L[y(t)]*L[e^{-10t}])=L[4] $$
Let $L[y(t)] = Y(t)$:
$... |
H: Examples of canonical projections that are not epimorphisms and canonical injections that are not
Although in $\mathsf{Set}$, canonical projections from a product are surjective and canonical injections to a coproduct are in fact injections, there seems to be nothing forcing this to be the case elsewhere, and indee... |
H: Solving ODE, reduce to exact differential equation
Solve the equation:
$xdy=(x^5+x^3y^2+y)dx$
My attempt at a solution:
I've tried to check if this was an exact differential equation or if I could reduce it to one, so
$xdy=(x^5+x^3y^2+y)dx \iff xdy-(x^5+x^3y^2+y)dx=0$. If I call $M(x,y)=-(x^5+x^3y^2+y)$ and $N(x,y)... |
H: Direct sum of projective module
Let $\left\{ P_{i}\right\} _{i\in I}$ be a family of $R$-module. I know that if each $P_{i}$ is projective then $\oplus_{i\in I}P_{i}$ is projective. Is the converse true, i.e if $\oplus_{i\in I}P_{i}$ is a projective module, is $P_{i}$ projective module for all $i\in I$?
AI: Yes its... |
H: Primitive Roots Proofs
I am stuck on how to prove these two questions:
(1) Let r be a primitive root of the prime $p$ with $p$ congruent to $1$ modulo $4$. Show that $-r$ is also a primitive root.
(2) Let n be a positive integers possessing a primitive root. Using this primitive root, prove that the product of all ... |
H: Is it possible to flip tails indefinitely?
If someone makes the argument that it is impossible to flip tails $n$ times on a two-sided coin, then we can argue there is a $1$ in $2^n$ chance. There is not a definable point at which it becomes impossible.
Is the following statement true or false?
It is possible to fl... |
H: Double complement law proof
I'm having difficulty proving that $S-(S-A) = A$ iff $A \subseteq S$, where "$-$" is the set difference.
For the "only if" part, I found that I can prove that
$S \subset A \Longrightarrow S-(S-A) \neq A$
by letting $x$ be an element in $A$ which is not in $S$. Then I show that $S-(S-A)$... |
H: Power Set Difference Proof P(A-B)=P(A)-P(B)
I need help proving or disproving this! If anyone could show me the sequence of steps I can use to help me show the left is equal to the right I would be grateful.
$$\mathcal P(A-B)=\mathcal P(A)-\mathcal P(B)$$
AI: HINT: $\varnothing\in\wp(A\setminus B)$, $\varnothing\in... |
H: What is the probability of picking Exactly 1 red marble and than not 1 red marble? without rep.
A urn has 3 red marbles, 2 blue marbles, 1white, 1 black 1 brown.
What is the probability of getting exactly 1 red marble than not 1 red marble?
What is the probability of getting at least 1 red marble?
I could not find... |
H: Separating Partial Differential Eq
I have a PDE:
$$
\frac{\partial^2\phi(r,\theta)}{\partial r^2} + \frac{1}{r}\frac{\partial\phi(r,\theta)}{\partial r} + \frac{1}{r^2}\frac{\partial^2\phi(r,\theta)}{\partial\theta^2} + C^2\phi(r,\theta)=0
$$
I need to separate the PDE (just functions of r,theta) and show the relat... |
H: What are the reasons for using a semi-circle in upper half plane of $\mathbb{C}$ for contour integration?
Why is it that when one in considering contour integration of a real function, such as $$
\int_{-\infty}^{\infty} \frac{dx}{1+x^2}$$ the contour in the complex plane used is the following:
Furthermore, what ar... |
H: Find point between two intervals which equals the average value.
Find the point in the interval [5,9] at which the function $f(x)=17e^{3x}$ equals its average value on that interval.
So I've got my function as follows $f(x)=\frac{1}{4}(\frac{17}{3}(e^{27}-e^{15})$
So I've simplify it to $\frac{17e^{12}}{12}$
Now... |
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