text stringlengths 83 79.5k |
|---|
H: Why in differential geometry tensors are usually defined as multilinear maps?
In multilinear algebra books tensors are usually defined through the universal property. Given a family of $k$ vector spaces $V_1,\dots,V_k$ over the same field $F$ we want to construct a space $S$ and a map $T:V_1\times\cdots\times V_k\t... |
H: Understanding Quillens Theorem A
Let me restate the theorem:
Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor. If $F\downarrow x$ is contractible for every $x\in\operatorname{Ob}(\mathcal{D})$, then $F$ is a homotopy equivalence.
Now let $\mathcal{C}$ be the category with two objects $x$, $y$, and two non-iden... |
H: Law of iterated expectation problem
When $E(u \mid x)=0$, $E(u)=E(E(u \mid x))=0$.
Then why $E(ux)=0$ by law of iterated expectation?
The book says because it is a form of $E(h(x)E(u\mid x)$) but still I can't understand.
AI: You can make the calculations
$$
E(UX) = E( E(UX\mid X))
= E( X E(U\mid X))
=... |
H: Infinite compact subset of $\mathbb{Q}$
Can I find an infinite set in $(\mathbb{Q},\mathcal{T}_e|_\mathbb{Q})$ which is compact?
AI: $$\{0\}\cup\left\{\frac1n:n\in\Bbb Z^+\right\}$$ |
H: Representing complex numbers as matrices, show that $A(z)+A(z')=A(z+z')$
I am doing a task where in which I am representing complex numbers as matrices, so $z=x+iy \in \Bbb C$ is represented by:
$A(z)=\begin{bmatrix}
x & -y \\
y & x
\end{bmatrix}$
Now I have to show for all $z, z' \in\Bbb C$ that $A(z)+A(z')=A(z+... |
H: Homework - countable infinity
I'm trying to solve 2 problems, but I'm having some issues and would appreciate help.
Here are the questions and what I thought could be done:
1) A is the set of all series of numbers, where in an even place there is an even number, and in an odd place there is an odd number (for examp... |
H: How can I do this integral?
How can I compute this integral?
$$\int\frac{x}{\sqrt{1+x^2}-x^2-1}dx$$
I tried to rationalise the denominator and got:
$$\int\frac{x(-\sqrt{1+x^2}-x^2-1)}{x^4+x^2}dx=\int\frac{-\sqrt{1+x^2}-x^2-1}{x^3+x}dx$$
But even still I find separating the integrals does not help me much in determi... |
H: Reference for Fukaya Categories and Homological Mirror Symmetry
What references are there for learning Fukaya categories (specifically, good references for self-study)?
In addition, any references with an eye toward homological mirror symmetry would be greatly appreciated.
AI: I personally recommend Paul Seidel's... |
H: Norm of operator
Let $H := \ell^2(\mathbb N)$ and for $f \in \ell^\infty(\mathbb N)$ definie $T_f:H \to H : g \mapsto f\cdot g$. This is well definied. I want to show that $\| T_f \| = \| f \|_\infty$. It is easy to show that $\|T_f\| \leq \|f\|_\infty$. To show equality I must find a function $g:\mathbb N \to \mat... |
H: How to determine the sum of the series $\,\sum_{n=1}^{\infty}\frac{n+1}{2^n}$
I am stuck on the following problem:
I have to determine the sum of the series $$\sum_{n=1}^{\infty}\frac{n+1}{2^n}$$
My Attempt: $$\sum_{n=0}^{\infty}\frac{n+1}{2^n}=\sum_{n=0}^{\infty}\frac{1}{2^n}+\sum_{n=0}^{\infty}\frac{n}{2^n}=\... |
H: Show that 2 surfaces are tangent in a given point
Show that the surfaces $ \Large\frac{x^2}{a^2} + \Large\frac{y^2}{b^2} = \Large\frac{z^2}{c^2}$ and $ x^2 + y^2+ \left(z - \Large\frac{b^2 + c^2}{c} \right)^2 = \Large\frac{b^2}{c^2} \small(b^2 + c^2)$ are tangent at the point $(0, ±b,c)$
To show that 2 surfaces are... |
H: Where is the error in this integration example?
I've been practicing calculus recently and found these example exercises.
Please look at problem $19$ which says:
$$\text{Integrate }\int\frac{x^3}{(x^2+5)^2} dx.$$
This is the suggested answer which says that the integral is equal to
$$\frac{x^2}{2(x^2+5)}-\frac{\l... |
H: growth and decay finding t formula?
The amount of fish in the main pool increases every month by 5%. Today, we have 6 tonnes of fish in the pool.
In how many years will there be 6.6 tonnes of fish in the pool?
So I can solve it like this:
mt = 6.6
m0 = 6
q = 1 + 5/100 = 1.05
t = ?
6.6 = 6 * 1.05 ^ t
6.6 / 6 = 1.1
... |
H: A problem on indefinite integration
$$\int\frac{x^4-2}{x^2\sqrt{x^4+x^2+2}}dx$$
I tried some substitutions, but none succeeded in simplifying the expression. Please help.
AI: Disclaimer I'm cheating, I get the final answer from WA and reverse engineering out the steps. People have any intuition how to get the step... |
H: Orthogonal Complement with Bilinear Map over Different Spaces
Let $\tau:W\times V \rightarrow K $ be a bilinear map where $V$ and $W$ are vector spaces over a field $K$.
Then, let $U$ be a subspace of $V$ and define $S(U) = \{w \in W\ |\ \tau(w,u)=0\ \forall u \in U\}$
My question is this:
Prove that $U \subset S(S... |
H: Can a complex quadratic polynomial have real roots?
Where $a\in \textbf{Z}[i] $ and $a \not\in \textbf{Z}$, suppose for the quadratic formula, $ b^2-4ac = 0 \Rightarrow b^2 = 4 ac \Rightarrow c= \frac{b^2}{4a} $ and $ b=a $, so that $\displaystyle \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}=\frac{-b}{2a}=\frac{-b}{2b}=-\fra... |
H: Determine if a point is contained in the circle in 3d space
I have a problem where I need to determine if a point is contained in the area of a circle in 3d space.
For my circle, I have the radius (R), the position of the center (C) and a normal vector to the circle (N). With that, I can easily draw my circle.
Now,... |
H: Suppose $g(0) > 0$ and $g(1) = 1$. Prove that if $g'(1)>1$, then there exists $t\in (0,1)$ with g(t) = t
Let $g:[0,1]\to [0,1]$ be continuously differentiable (including one-sided derivatives at 0 and 1). Suppose $g(0) > 0$ and $g(1) = 1$. Prove that if $g'(1)>1$, then there exists $t\in (0,1)$ with g(t) = t
Okay, ... |
H: How to solve $7200a+720b+72c=1000x+340+y<10000$?
What is the easiest way to solve $7200a+720b+72c=1000x+340+y<10000$ where all variables are one digit natural numbers?
Trial and error method seems to be tedious.
AI: It’s immediately clear that $a$ must be $1$, which implies that $x$ must be at least $7$.
$7200a+720... |
H: A Question Regarding Diagonalization
For this question I am only considering binary sequences of countably infinite length.
Consider an arbitrary set $S$ of such sequences, $S$ of order type omega, $\omega$. By diagonalization one can construct a binary sequence $s_*$ not contained in $S$. Add $s_*$ to $S$ to fo... |
H: Induction proof with Fibonacci numbers
Prove by induction that for Fibonacci numbers from some index $i > 10$
$1.5^i ≤ f_i ≤ 2^i$
Notice! Because Fibonacci number is a sum of 2 previous Fibonacci numbers, in the induction hypothesis we must assume that the expression holds for k+1 (and in that case also for k) an... |
H: Can we compute $ \mathbf{Pr}[x_{1} < X < x_{2}] $ if we know the cumulative distribution function $ F $?
Assume that we have a cumulative distribution function $ F $. How can we calculate the quantity $ \mathbf{Pr}[x_{1} < X < x_{2}] $?
I know the answer for $ \mathbf{Pr}[x_{1} < X \leq x_{2}] $, but I am not sure ... |
H: Compute $ \lim\limits_{n\to\infty}\sqrt[n]{\log\left|1+\left(\frac{1}{n\cdot\log n}\right)^k\right|}$.
Compute $$ \lim\limits_{n\to\infty}\left(\sqrt[n]{\log\left|1+\left(\dfrac{1}{n\cdot\log\left(n\right)}\right)^k\right|}\right). $$
What I have: $$ \forall\ x\geq 0\ :\ x- \frac{x^2}{2}\leq \log(1+x)\leq x. $$
Ap... |
H: Factoring out an exponential?
I have the following expression
$$\frac{2^{k+1}(k+1)!}{(k+1)^{k+1}}\cdot\frac{k^k}{2^k k!}$$
I get
$$\frac{2(k+1)(k^k)}{(k+1)^{k+1}}$$
But how do I factor out the ${(k+1)}^{k+1}$
AI: It might help if you notice that $(k+1)^{k+1}=(k+1)^k(k+1)$. |
H: Modular arithmetic and one-to-one functions
Let $S = \{0, 1, 2, 3, · · · , 99\}$ . For each of the following functions $f : S \rightarrow S$ , determine
whether it is one-to-one and onto, by computing its values for all $k ∈ S$:
Function 1: $$f(k) = (131k+27)\pmod{100}$$
Well at this point, I've computed the follow... |
H: Demonstration that $\forall\; n>3,\;\;n^2
How do I prove by mathematical induction that$$\forall\; n>3,\;\;n^2<n!$$
I tried, $n=4$ then $4^2<4!$ what is true, because $16<24$.$$$$Hypotesis: $n^2<n!$ $$$$Thesis: $(n+1)^2<(n+1)!$$$$$Show: $$(n+1)^2=n^2+2n+1<n!+2n+1$$ and????
AI: $$(n+1)!=(n+1)\cdot n!>(n+1)\cdot n^... |
H: Historical reason to define a vector dot product the way it is
The dot product of two vectors is defined this way:
$$\begin{pmatrix} a_1\\ a_2 \\ \end{pmatrix}\cdot \begin{pmatrix} b_1\\ b_2\\ \end{pmatrix} = a_1\cdot b_1 + a_2\cdot b_2$$
I know it works exactly like it should, in the Work formula, in physics, ... |
H: Is there a rational number describing the ratio of a volume, as a string, to a surface area?
If you were to take an arbitrary 3-dimensional shape with finite surface area, then look at the volume of that shape, turn the volume into a long cylindrical string bunched up ideally inside the shape with the space between... |
H: The integral of a function that is 0 a.e is 0
I am working on this problem:
Let $f = 0$ for all $x \in [a,b] $ except for $x$ in a set of Lebesgue measure zero. Then $\int_a^b f \,dx = 0$ if the integral exists.
Here are my ideas: Split $f$ into its positive part and negative part so that $f = f^+ + f^- = f^+ - (-f... |
H: Differential equation higher order
Can somebody help me with working out $y'''-4y'=t+\cos t+2e^{-2t}$. I want to solve $y'''-4y'=t$, $y'''-4y'=\cos t$ and $y'''-4y'=2e^{-2t}$ apart. The homogeneous equation I already solved: $y(t)=c_1+c_2e^2t+c_3e^{-2t}$. For the particular solutions I learnt a method called judici... |
H: Show that ${\bf x} \cdot A^t {\bf y} = {\bf y} \cdot A{\bf x}$
Let $A \in \mathcal M_n (R)$ and ${\bf x}, {\bf y} \in R^n$.
How can I show that:
$${\bf x} \cdot A^t {\bf y} = {\bf y} \cdot A{\bf x} \, ?$$
Thanks for any help.
AI: Hint: note that for real vectors $u,v:u\cdot v= u^T v$
Second hint: we can write
$$
x ... |
H: Solving the Expected Value
I have a word problem,
A game is played where a fair coin is tossed until the first tail occurs. The probability tosses will x tosses will be needed is $ f(x) = 0.5^x;x=1,2,3,4,5$. You win $2^x$ dollars if x tosses are needed for x=1,2,3,4,5 but lose $256 if x > 5. Determine the expected... |
H: weak convergence in $L^2$ / $C$ ==> pointwise convergence
I have a sequence of function $B_n \in C([0,1],R)$ and a $B \in C([0,1],R)$, such that:
$\int_{[0,1]} B_n(x) f(x) dx \rightarrow \int_{[0,1]} B (x) f(x) dx$ for all $f \in C([0,1],R)$ which are bounded.
Does this imply point wise convergence of $B_n(x) \righ... |
H: $\sum\frac{k^{k/2}}{k!}$ converge or diverge?
Does the following series converge or diverge?
$\sum\frac{k^{k/2}}{k!}$
I did the ratio test $\frac{a_{n+1}}{a_n}$
I did
$\frac{(k+1)^{(k+1)/2}}{(k+1)!}$ $\frac{k!}{k^{k/2}}$
Then I simplified
$\frac{(k+1)(k+1)^{k/2}}{(k+1)(k^{k/2})}$
I then got
$k\rightarrow\infty$
$(... |
H: Find permutation index of multiple lists where corresponding list indices match
I have several date time values:
Mon 17h10
Tue 20h30
Wed 21h45
that maps to the following lists [Mon Tue Wed] [17 20 21] [10 30 45], a list for days, hours and minutes respectively.
Having all permutations of these lists in order, I ... |
H: Proving functions a formal proof
Let $A$ and $B$ be arbitary sets. Let $S_1$ and $S_2$ be arbitrary subsets of A, and let $T_1$ and $T_2$
be arbitrary subsets of $B$. For each of the following state whether it is True or False. If
True then give a proof. If False then give a counterexample: $f(n) = n^2$
$f(S_1 ∩ ... |
H: Function Proof that deals with Set Theory
How to prove that this does not hold if $H$ is not injective or how to show that this equation is true just for injective functions?
$$H(X\cap Y)=H(X)\cap H(Y)$$
AI: If $H$ is not injective, that means that there are two points $x,y$ such that $H(x) = H(y)$.
Then let $X=\{x... |
H: Simple recurrence relation - 1D
I know this is a very simple recurrence relation, but how would you go on solving it?
$$x(n+1)=\frac{x(n)}{1+x(n)}$$
AI: Hint: Let $y(n)=\frac{1}{x(n)}$. The recurrence for $y(n)$ is very pleasantly simple. |
H: Number of ways of expressing $n$ as a sum of positive integers
a) Let $s_n$ denote the number of ways of expressing $n$ as a sum of positive integers. Thus $s_1=1$, $s_2=2$, and $s_3=4$ (the four ways are $3$, $2+1$, $1+2$, and $1+1+1$). Prove that $s_n=s_{n-1}+s_{n-2}+\cdots+s_1+1$. Hence calculate $s_{10}$. Find... |
H: Polynomial fullfilling certain derivative properties
I have to solve the following task:
For an n $\in \mathbb N$, find a polynomial f(x), s.t. $f^{(k)}(1) = 0$ for $\forall$ k < n and $f^{(n)}(1)=1$.
I have tried out a couple of variations - without success. Is there a systematic way to solve this problem?
Thanks
... |
H: Find the limit: $\lim \limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right)$
Find:
$$\lim\limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right)
$$
Without using L'Hospital or Taylor approximations
Thanks in advance
AI: Note that $$\log(x)=\int_1^x \frac{dt}{t}$$
and that for $1 ... |
H: Prove or disprove a set $F$ is closed.
This is an example in my book that talks about $F$ being precompact;
Let $F$ be the subset of $C([0,1])$ that consists of functions $f$ of the form
$$f(x) = \sum_{n=1}^{\infty}a_n\sin(n\pi x) \hspace{.5cm} \text{ with } \hspace{.5 cm} \sum_{n=1}^{\infty} n|a_n|\leq 1.$$
I kn... |
H: compute the following integral using Cauchy Integral Formula
Prove that $\int_{0}^{\pi}{e^{k\cos t}\cos (k\sin t)}=\pi$. Using Cauchy Integral Formula. But I don't know how. I want to rewrite the integral as a line integral first.
AI: First of all, note that one may exploit symmetry in the integral and rewrite as
$... |
H: Order of $g^i$ in cyclic group of order $n$
I am very new to groups and still have problems with proving some basic facts. I stumbled on such theorem:
Let $g$ be a generator of cyclic group of order $n$. Then $g^i$ has order $\frac{n}{\gcd(n,i)}$
And I'm frustrated with no ideas how to prove it. Clearly $g$ has o... |
H: Equation of a plane containing a point and perpendicular to a line
Find an equation of the plane containing the point $(1, 1, -1)$ and perpendicular to the line through the points $(2, 0, 1)$ and $(-1, 1, 0)$
This is what I have:
I first find the vector between the points: $\vec{n}^{\ } = (2,0,1) - (-1, 1, 0) = (3,... |
H: Evaluating $\lim\limits_{n \to \infty} \sum_{i=1}^{n}\sqrt{\frac{i}{n^{3}}}$ using Riemann Sums and FTC
Factoring out the $\frac {1}{n}$ out of the sigma, we get:
$$\lim_{n \to \infty} \sum_{i=1}^{n}\sqrt{\frac{i}{n}}\cdot\frac{1}{n}$$
which looks awfully similar to
$$\lim_{n \to \infty} \sum_{i=1}^{n}\sqrt{x^{*}_... |
H: Question on Exact sequence
Let $$0 \rightarrow A \rightarrow B \rightarrow C\rightarrow 0$$ be an exact sequence, with $f:A \rightarrow B$, and $g: B \rightarrow C$. Let $Q$, $P$ be two submodules of $B$.
I want to determine whether the following fact is true:
$g(Q)=g(P)$ and $f^{-1}(Q)=f^{-1}(P)$, implies $P=Q$.
I... |
H: ZFC and irrational numbers
I understand how integers and rationals are expressed/derived in ZFC. But what about the irrational numbers? Can they also be expressed?
If not, are there other axiomatic set theories able to express them?
As for Dedekind cuts, from my understanding (maybe wrong), any irrational in questi... |
H: Proving reflexivity and transitivity
I want to show that if $R$ is reflexive and transitive then $R^{-1}$ is also.
Transitivity:
$$(a,b)\in R^{-1} \wedge (b,c)\in R^{-1} \Rightarrow (b,a)\in R \wedge (c,b)\in R \Rightarrow
(c,a)\in R \Rightarrow (a,c) \in R^{-1}$$
Reflexivity:
$$(a,a)\in R^{-1} \Rightarrow (a,a) \... |
H: What shape is traced out by this animation?
Found this animation circulating online, and was wondering what shape the rod's end traces out. It seems to be an ellipse, but can that be proved somehow?
$\quad\quad\quad\quad\quad\quad\quad\quad\ $
AI: Call the two pivots moving on a cross $a$ and $b$. $a$ moves on $\{(... |
H: What does this notation mean? $f(x|\theta)=\frac{3x^2}{\theta^3}I_{(0,\theta)(x)}$
Particularly I want to know what the meaning of $I_{(0,\theta)(x)}$ is here.
AI: It's called the indicator function. It means:
$$
I_{(0, \theta)}(x)=\begin{cases}
1, \quad \text{if } 0<x<\theta\\
0, \quad \text{otherwise}
\end{cases}... |
H: Calculating $\int_{- \infty}^{\infty} \frac{\sin x \,dx}{x+i} $
I'm having trouble calculating the integral
$$\int_{- \infty}^\infty \frac{\sin x}{x+i}\,dx $$
using residue calculus. I've previously encountered expressions of the form
$$\int_{- \infty}^\infty f(x) \sin x \,dx $$
where you would consider $f(z)e^{iz... |
H: Proving that the Intersection of any Collection of Compact Sets is Compact
I was trying to do this problem this way:
Let $\mathcal{B}=\{B_i\}$ be a collection of compact sets. By Heine-Borel, each of the $B_i$'s are closed and bounded. We already know that the intersection of a collection of closed sets is once aga... |
H: Combinatorics homework
Here's a question from my homework. First 2 questions I solved (but would appreciate any input you can give on my solution) and the last question I'm just completely stumped. It's quite complicated.
On the shelf there are 5 math books, 3 science fiction books and 2 thrillers (all of the book... |
H: Sum of self power
Is there a formula to calculate the sum of a number to the power of this same number, like:
$$1^1 + 2^2 + 3^3 + 4^4 + 5^5 + ... + n^n$$?
or
$$x^x + (x+1)^{(x+1)} + (x+2)^{(x+2)} + ... + (x+n)^{(x+n)}$$
AI: No. There isn't. But we do know that it is of the order $n^n$, and that all other terms, sa... |
H: How to prove if n>2 is a prime number, then n is odd?
I understand why this is true, but I have no idea on how I'd go about proving it by writing a detailed structured proof.
Since a prime number is a integer, I started off like this:
Assume n in Z
Assume n > 2 and is a prime
Then...?
I also know that ... |
H: How would I solve this?
I don't understand this question. Or more precisely how they derived the answer for the examples given. Can someone explain? Thanks.
E.g. All integers can be represented using the base B =-10 using the digits 0, 1, 2...9 and without using a negative sign in front of the number.
For example, ... |
H: Equivalent definitions of atom in a Boolean Algebra
I want to show that the following conditions are equivalent for a nonzero element $a$ in a Boolean algebra $\mathcal{B}$:
1) for all $x\in\mathcal{B},a\leq x$ or $a\leq x'$
2) for all $x,y\in\mathcal{B},a\leq x\sqcup y\Rightarrow a\leq x$ or $a\leq y$
3) $a$ is mi... |
H: I reach a half-dead end when trying to find the minimum possible value of an equation
I have the equation $x^2 + 4xy + 5y^2 - 4x - 6y +7$ and I'm supposed to transform it to look like this:
$[x + 2(y - 1)]^2 + (y + 1)^2 + 2$
First I transformed it into:
$x^2 + 4x(y - 1) + 5y^2 - 6y + 7$
and then completed the squa... |
H: How to identify coefficients with the binomialcoefficient
i tried to identify the coefficients $\gamma n \nu \mu $ in
$$(a+b+c)^n = \sum_{\nu=0}^{n} \sum_{\mu = \nu}^{n} \gamma n \nu \mu ~a^\nu b^{\mu- \nu}c^{n- \mu}.$$
I used the Binomial Theorem, but didn't succeed, can you help me?
Binomial Theorem:
$$(a+b)^n = ... |
H: Integration: product containing square root.
How can I integrate the following expression?
I have tried using u-substitution, but I am having problems with integrating the entire expression. So far, I have the following:
Any help on this is highly appreciated!
AI: You have it. $u=1+e^{-kt}$ and $du=-ke^{-kt}dt$... |
H: Prove that if the set B is in the finite union of the sets Ai, if p is limitpoint of B then p is limitpoint for at least one Ai.
I am not that familiar with writing proofs and would like some feedback checking if the steps in my proof are valid.
Problem statement,
Let $B_n = \bigcup _{i=1} ^n A_i, \quad Ai \in X $ ... |
H: Proving the limit of a function of a sequence is equal to the function of the limit of that sequence
Suppose $f$ is a continuous function at $x = c$ in $[a,b]$. Prove that for any sequence ${x_n}$ in $[a,b]$ converging to $c$, the sequence $\{f(x_n)\}$ converges to $f(c)$. That is, $$ \lim_{n\to\infty}f(x_n)= f\l... |
H: What are some good introductory books on complex analysis?
I am looking for self study books or general interest (above the layman level) books on complex analysis.
AI: A few of my favourites:
Stewart and Tall Complex Analysis - does not demand massive pre-requisites.
Needham Visual Complex Analysis - fantastic fo... |
H: A continuous, bijective, function such that two different metrics have the same open sets.
Let $ f: \Re \rightarrow \Re$ be a continuous and strictly increasing function such that $f(0)=0$. Suppose $d_1$ and $d_2$ are two different metrics on the nonempty set $X$ and that $d_1(x,y)=f(d_2(x,y))$. Show that $(X,d_1)$... |
H: How would I prove $|x + y| \le |x| + |y|$?
How would I write a detailed structured proof for:
for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$
I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ and $y<0$, $x<0$ and $y \ge0$, and $x,y \ge 0$. But I'm not sure how I'd go about ... |
H: Homework basic abstract algebra
My question is as follows:
$R$ is a ring such that for all $x \in R, x^2=x$
$p$ is a prime ideal of $R$.
Show that $R/p$ (R modulu p) has exactly 2 elements.
What I did:
$x^2=x$
$x^2-x=0$
$x(x-1)=0$
So the equivalence class of $x(x-1)$ is equal to the equivalence class of $0$ in $R/p... |
H: Quotient of maximal ideals by its power giving simple modules
So I understand that $R/m$ where $m$ is a maximal ideal would give simple module. My question is, would $m/m^2$ also give a simple module?
My progress thus far: My first approach was to realize that inside $R$, and only prime ideal containing $m^2$ is $m... |
H: Check if a system admits solutions of period 2
I have the following problem.
Let $r \geq 0$ be a parameter in the discrete time system $x(k + 1) = r − rx(k)$.
Verify whether there exist $r \geq 0$ such that this system admits solutions of period 2.
I do not quiet understand how I should solve this problem. I alre... |
H: If $f(a) = g(a)$ and $f'(x) < g'(x)$ for all $x \in (a,b)$, then $f(b) < g(b)$
Assume that $f$ and $g$ are continuous on $[a, b]$ and differentiable on $(a, b)$.
Prove that if $f(a) = g(a)$ and $f'(x) < g'(x)$ for all $x \in (a,b)$, then $f(b) < g(b)$.
I understand that if $f$ and $g$ start at the same point, and... |
H: Meeting point for 5 people with least distance travelled (interview question)
I had an interview today and I'm completely stumped on what they asked me. Essentially: if you are given 5 people on a 2D grid, and you need to meet at a point with the least amount of distance travelled, how would you calculate it?
What ... |
H: Probabilities for $1$-in-$n$ events over $n$ trials
I know there are lots of related questions on here, but I can't seem to find what I'm looking for.
Given some event with, say, a $1$ in $1{,}000{,}000$ probability (e.g., $7$ being chosen randomly as a number between $1$ and $1{,}000{,}000$), I'd like to get a rou... |
H: Solving $4^{667} ≡ x \pmod{13}$ without Eulers totient theorem or CRT
Does anyone know any efficient ways to solve this without Euler's Totient Theorem or Chinese remainder theorem?
AI: As $5*13=65$ we have
$$4^3 \equiv -1 \pmod{13}$$
Even without this observation, you can calculate $4,4^2, 4^3, ... \pmod{13}$ and... |
H: Is there an easy formula for this sequence?
It is the sequence which represents the maximum number of cycles in an undirected graph with n nodes, n>=3. These graphs have all nodes connected to every other node.
How would I count the number of cycles in such a complete graph.
Also is there was a formula to this sequ... |
H: Proof: $2^{n-1}(a^n+b^n)>(a+b)^n$
If $n \in \mathbb{N}$ with $n \geq 2$ and $a,b \in \mathbb{R}$ with $a+b >0$ and $a \neq b$, then $$2^{n-1}(a^n+b^n)>(a+b)^n.$$
I tried to do it with induction. The induction basis was no problem but I got stuck in the induction step: $n \to n+1$
$2^n(a^{n+1}+b^{n+1})>(a+b)^{n+1} $... |
H: How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?
So I'm tutoring at the library and an elementary or pre K student shows me a sheet with one problem on it:
Put 9 pigs into 4 pens so that there are an odd number of pigs in each pen.
I tried to solve it and failed! Does anybody know ... |
H: Calculating probabilities of events of different time periods
The average probability of an event occurring is 3 times in a year. What is the probability of:
1) an event occurring in any specific month; and
2) 10 events occurring in any specific month?
AI: We use a Poisson model: the number $X$ of events per year ... |
H: equality of two objects depending on conditions.
I want to state that two objects $t_i, t_j$ are equal if some conditions hold. Can this be done by writing: $t_i = t_j \rightarrow$ some conditions?
AI: An example of such a condition is the axiom of extensionality found in most set theories. It states that if $x$ an... |
H: How to compute the fundamental group of a necklace of $\mathbb{S}^1$' s?
I was trying to compute $\pi_1 (X)$ where $X =$ "necklace of $n$ $\mathbb{S}^1$'s". At first, I tried using Van Kampen theorem however I could not find open sets $U$ and $V$ such that $U \cap V$ is path connected. I tried using the covering sp... |
H: Why is the sequence $ a_n = \left(1+\frac{1}{n}\right)^n $ Cauchy?
I was looking at the post:
Cauchy Sequence that Does Not Converge
And the top answer was this sequence:
$ a_n = \left(1+\frac{1}{n}\right)^n$. I understand that this sequence converges to $e$, which is not a rational number, and that $a_n$ is a seq... |
H: Group theory: Let $H=\{0,\pm 3, \pm 6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$.
I am having trouble understanding the following homework question,
Let $H=\{0,\pm 3, \pm6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$.
I know the answer is $H$, $1+H$, and $2+H$ but I am having difficu... |
H: help with nand circuit
I tried to make a circuit from this expression but it's not working right.
Here's the expression and circuit:
Expression
$$\overline{\overline{P_1.S_1}.\overline{P_2.\left(\overline{\overline{P_1}.\overline{\overline{S_1}.\overline{S_2}}}\right)}}$$
AI: Everything looks right to me, perhaps t... |
H: Expected value of a number of events
If I have a set of random events say
${\{A_{1}, A_{2}, ... A_{n}\}}$ and a number ${N=\sum_{1}^{n} I(A_{i})}$ where ${I(A_{i})}$ is the indicator function (basically ${N}$ is the number of events that happen after a random experiment).
Can anybody please tell me how can I find t... |
H: Sum of absolute values and the absolute value of the sum of these values?
I'm working on a proof and I need some help with this:
I determined that for some situations ($x$ or $y$ are negative but not both):
$|x| + |y| > x + y$
How can I conclude using that statement the fact that:
$|x| + |y| > |x + y|$
AI: You can ... |
H: Find integer solutions to linear congruence
Find all integer solutions of $2n \equiv 12 \bmod 19$
So I have re-arranged to: $2x-19y=12$ and by the extended Euclidean Algorithm, I get $$x=1 \ $$
$$y=-9$$
However, this is how far I was able to get to and not sure what follow past this point? What exactly are we looki... |
H: What are some physical, geometric, or otherwise useful interpretations of divergent series?
I don't understand what ideas such as Abel, Cesàro summation or other types of sum 'regularization' help us describe. What is the practical application to discussing the 'sum' of sequences that are not convergent in the usua... |
H: What is the relationship between the spectrum of a matrix and its image under a polynomial function?
Clearly if $\lambda$ is an eigenvalue of $A$ then $p(\lambda)$ is an eigenvalue of $p(A)$ where $p$ is a polynomial. And there are cases where $A$ may have eigenvalues other than these.
Is there a general rule for ... |
H: How to solve $(2x+2, 6x) = x+1$
I'm looking at an old discrete mathematics test preparing for my test tomorrow and one question says solve for $x$ and gives the above. I'm thinking that this is the $\gcd(2x+2, 6x)$ but have never seen this type of question before.
AI: If $\gcd(2x+2,6x)=x+1$ it means that $\def\div... |
H: Positive and negative integer that is congruent to 0 (mod 5) and incongruent to 0 (mod 6)
I'm kind of confused by this because I thought 0 mod 5 = 0, and 0 mod 6 = 0 as well. So what's an integer that is congruent to one but not the other?
AI: "$m$ is congruent to $n$ modulo $r$", typically written
$$m\equiv n \pmo... |
H: on exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$
if $A$is a Noetherian ring, $M$ a finitely generated module,$I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know $\hat{A}\otimes_{A}M\cong\hat{M}$.
Also on Atiyah&Macdonald, there is a remark on Pag... |
H: Combinations: Poker hands, full houses
Reading through my Probability book brushing up on some stuff:
What is the probability that a poker hand is a full house? (A full house is defined as a hand with three cards of one denomination and two cards of another denomination; ex. three Queens and two 4's.)
The solution ... |
H: Why this is false?!
$\Gamma \models (\alpha \vee \gamma)$ iff $(\Gamma \models \alpha$ or $\Gamma \models \gamma)$
[$\Longrightarrow$]
If $\Gamma \models (\alpha \vee \gamma)$ then ($\Gamma \models \alpha $ or $\Gamma \models \gamma$)
Suppose $\Gamma \models (\alpha \vee \gamma)$
By def. of logic consequence: $\fo... |
H: Help showing that this ideal is principal.
This is not for homework, and I would really like a hint please. The question asks
If $P = \{ 2a + (1 + \sqrt{-5})b : a, b \in \mathbb{Z}[\sqrt{-5}] \}$ is an ideal in $\mathbb{Z}[\sqrt{-5}]$, show that $P^2$ is the principal ideal $(2)$.
I have shown that $P^2 \subsete... |
H: Proof of the limit of a sequence of functions.
The questions is this.
Let $f_n(x)=\frac{1}{n}sin(nx).$ Each $f_n$ is a differentiable function. Show that
(a) $\lim f_n(x)=0,\forall x \in R$
(b) but $\lim f'_n(x)$ need not exist [at $x=\pi$ for instance].
Proof of (a)
Let $\epsilon > 0$ and $N = \frac{1}{\epsilon}.$... |
H: The distance between two disjoint compact subsets $A,B$ of a metric space $X$ is positive
Please tell me whether my argument for the following result is true:
The distance between two disjoint compact subsets $A,B$ of a metric space $X$ is positive:
$d:X\times X\to \mathbb R$ is continuous$\implies d|_{A\times B}$... |
H: How to find a matrix $X$ such that $X+X^2+X^3 = \begin{bmatrix} 1&2005\\ 2006&1 \end{bmatrix}$?
Find a matrix $X \in M_{2}(\mathbb Z)$ such that $$X+X^2+X^3=\begin{bmatrix} 1&2005\\ 2006&1\end{bmatrix}$$
My try:
Let
$$X=\begin{bmatrix}
a&b\\
c&d
\end{bmatrix}$$
where $a,b,c,d\in Z$
then
$$X^2=\begin{bmatrix}
a^... |
H: Find the Maclaurin series of the function $f(x) = 7 x^2 \sin 2 x$
Find the Maclaurin series of the function
$f(x) = 7 x^2 \sin 2 x$
$(f(x) = \sum_{n=0}^{\infty} c_n x^n) $
That is what is given on the question, we have to fill in $5$ blanks $c_3$ to $c_7$
The homework is past due, so I have the answers $(14, 0, -9... |
H: How to make a piecewise function differentiable?
I have the following question:
Suppose $$f(x) = \left\{\begin{array}{cc}x^2 & \text{if }x\leq 2 \\ mx+b& \text{if }x>2\end{array}\right.$$
If $f$ is differentiable everywhere, then what are the values of $m$ and $b$?
How exactly would I be able to get the values to b... |
H: Proof by induction of Sylow's theorem
Prove directly that if $p$ is a prime and $p^{\alpha}\ | \ o(G)$, then $G$ has a subgroup of order $p^{\alpha}.$
How can I prove this, by induction on the order of the group $G,$ without using the existence of a $p$-Sylow subgroup?
AI: CLAIM Let $k\geqslant 0$, $p$ a prime an... |
H: Find a matrix $X$ given $X^4$
Find the matrix $X$ such that
$$X^4=\begin{bmatrix}
3&0&0\\
0&3&1\\
0&0&0
\end{bmatrix}$$
This problem I can't work,and I think let the matrix the eigenvalue is $\lambda$,then
$\lambda^4$ is $$\begin{bmatrix}
3&0&0\\
0&3&1\\
0&0&0
\end{bmatrix}$$
eigenvalue?Thank you for your help.
AI:... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.