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H: Solve the equation, for $p$ prime, $x^{2p}- x^p= [6]$, in $\mathbb {Z}_p$
According to the title, the equation is :
$$x^{2p}- x^p=[6]$$
for $p$ prime, in $\mathbb {Z}_p$.
It's known that $a^p \equiv a \mod p$. So, if the equation was $a^p -a=[6]$ , $a=6$.
I tried to make some manipulations using as tool this theore... |
H: Factorisation of $x^{4}-x^{2}+2x+1$ in $\mathbb Q[x]$?
Hallo can any one tell me what is the idea behind this?
the polynomial $p=x^{4}-x^{2}+2x+1 \in \mathbb Q[x]$ is irreducible, because in in $F_{2}[x]$ it can factored to $(x^2+x+1)^2$ and in $F_{3}[x]$ it can be factored to $(x-1)(x^3+x^2-1)$.
thank you in advan... |
H: What does $X \subsetneqq Y$ mean
What does $X \subsetneqq Y$ mean? I cannot find it on Google, I suppose this means, it is a subset and at the same time it is not the same. I.e. it could be simply written as $\subset$. Pardon my ignorance.
AI: Some people use $\subsetneq$ for strict and $\subset$ for non-strict. Th... |
H: Atiyah and Macdonald Exercise 1.27
Please do not ruin the fun by telling me why $\mu$ is surjective!
I am having trouble understanding the idea of the coordinate functions on the affine algebraic variety $X$. I am trying to understand that $P(X)$ is generated as a $k$-algebra by the coordinate functions. I understa... |
H: How to find what point a wave is reflected off
If a wave is reflected off a surface, the angle of reflection is equal to the angle of incidence. But, how can we use this to find the actual path of the incident and reflected waves if we only know the positions of the wave origin and observer?
It seems clear that th... |
H: Does this normalization of a positive definite matrix alter its positive definiteness?
I have a matrix $A$ that is positive definite. Denoting the elements of $A$ by $a_{ij}$, let $A'$ be a new matrix formed as:
$$A'_{ij} = \frac{a_{ij}}{\sqrt{a_{ii}a_{jj}}}$$
Is $A'$ also positive definite? Note: All diagonal elem... |
H: Difference between internal category and subcategory?
The category SET has an internal category, which is a small category with small objects and small morphisms, and that means that it's a subcollection of the collection of objects in SET.
Is an internal category the same as a subcategory? Are they different only ... |
H: Problem understanding domain of circular relation
I came across an exercise in a book which introduces the circular relation as:
$C$ is a relation from $R \to R$ such that $(x,y) \in C$ means $x^2+y^2 = 1$.
It then says that the domain of $C$ is $R$.
However, as I understand things, the domain can only be the s... |
H: derivative of $f(x)=\frac{x}{x-1}$
I can't figure it out, what is the derivative of $$f(x)=\frac{x}{x-1}$$I have tried it many different ways and I just can't seem to get it figured out.
AI: Of course, you can use the quotient rule, but with this function, we can express it in a manner which makes computing the der... |
H: Stuck on an 'advanced logarithm problem': $2 \log_2 x - \log_2 (x - \tfrac1 2) = \log_3 3$
I'm stuck on solving what my textbook calls an "advanced logarithm problem". Basically, it's a logarithmic equation with logarithms of different bases on either side. My exercise looks like this:
$$2 \log_2 x - \log_2 (x - \t... |
H: Relate to GP 1.3.9 - Differentiating $x_{i_1}, \dots, x_{i_k}$ result span($e_{i_1}, \dots, e_{i_k}$)?
I start to think of this is question when I attempt exercise 1.3.9 on Guillemin and Pollack's Differential Topology
Consider the projection $\varphi: \mathbb{R}^N \rightarrow \mathbb{R}^k$:
$$(x_1, \dots, x_N) \ma... |
H: countability of limit of a set sequence
Let $S_n$ be the set of all binary strings of length $2n$ with equal number of zeros and ones. Is it correct to say $\lim_{n\to\infty} S_n$ is countable? I wanted to use it to solve this problem. My argument is that each of $S_n$s is countable (in fact finite) thus their unio... |
H: Stuck on Elementary Exponentiation
I'm confused. Why is it that for a problem in the form of: $(2^{x+1})(2^{x-1})$ we get $2^{2x}$ instead of $4^{2x}$; Shouldn't we multiply the $2$s by each other..?
Similarly for a problem like: $(2^{x+1})(4^{x-1})$, why do we get $2^{3x-1}$ rather than $8^{2x}$?
And for $(3^{2x}/... |
H: Finding the interval for increase of the function $y =x^2e^{-x}$
Problem :
Find the interval in which the function $y =x^2e^{-x}$ is increasing .
My approach :
We can take first derivative to the find the increase or decrease of function ie.
$y'=2xe^{-x}-x^2e^{-x}$
finding the critical point by putting y'=0
$ ... |
H: $d$ is a metric on $X$ if $d(a,b) = 0 ⇔ a = b$ and $d(a, b) ≤ d(z, a) + d(z, b)$
The following is a question of Metric Spaces by O'Searcoid (pg 19)
Suppose $X$ is a set and $d:X×X→\mathbb{R}$. Show that $d$ is a metric on $X$ if, and only if, for all $a,b,z ∈ X$, the two conditions $d(a,b) = 0 ⇔ a = b$ and $d(a, b... |
H: Will it be a Cauchy sequence?
Let $<x_n>$ be a sequence satisfying
$|x_{n+1}-x_n|\le \frac{1}{n^2}$
Will it be a Cauchy sequence?
AI: We want to show that given any $\epsilon \gt 0$, there is an $N$ such that if $N\le m\lt n$ then $|a_n-a_m|\lt \epsilon$. Note that
$$a_n-a_m=(a_{m+1}-a_m)+(a_{m+2}-a_{m+1})+\c... |
H: Filling a conical tank
I have been working on this problem for about 2 hours and I can't seem to get it, here is exactly what the question reads.
"Water is poured into the top of a conical tank at the constant rate of 1 cubic inch per second and flows out of an opening at the3 bottom at a rate of .5 cubic inches pe... |
H: Need help with a simple math proof
Imagine an infinitely long sequence of squares where one of these squares contains a frog, and another square contains a fly.
For simplicity, let's number all of the (infinitely many) squares by assigning each an integer. We'll say that the frog starts in position 0, and will assi... |
H: The difference with sup and without sup
The difference with sup and without sup，how to judge and choose the use
For example, here is rudin's "Root" Test:
Given $\sum a_n$, put $\color{Green}{\{\alpha =\lim_{n\to \infty}\sup \sqrt[n]{\left|a_n\right|}\}}$
Here is mathworld "Root" Test
I also see SupremumLimit
So, ... |
H: Is $\mathbb{N}$ infinite?
Intuitively the answer is yes. According to the definition, a set $A$ is infinite if there is no bijection between $A$ and some natural number.
Now, I don't know the problem of my reasoning. Accordingly the funcion $f:\mathbb{N}\longrightarrow 0$ is a bijection (Because $f\subseteq \mathb... |
H: Expectation with a "regular" function
I hope this is not a silly question. I know that the expectation of a constant is just a constant (i.e. $E[c]=c$ for $c\in \mathbb{R}$), and that for a function $g$ of a random variable X, $E[g(X)]=\int_{-\infty}^{\infty}g(x)~f(x)~dx$ (in the continuous case).
But, what if yo... |
H: Statements on Cardinality of sets
Which of the statements regarding cardinality of sets are always correct
Let $X$ be an infinite set then
(1)$|\{F|F \subseteq X \;\text {and} \;X \; \text {is finite}\}| > $|X|$
(2) $A \in P(X) $ and $ X\setminus A $ is infinite $\rightarrow |A|<|X|$
(3) $A \subseteq X $ and $|A|<... |
H: Scaling of eigenvalues
Suppose $A_N$ is a positive definite matrix of size $N$ with eigenvalues $\Lambda=\{\lambda_1,\ldots,\lambda_N\}$. Let $D = \text{diag}\{d_1,\ldots,d_N\},\ d_i>0$ be a diagonal matrix. Can the eigenvalues of $A'_N=DAD$ be written in terms of $\Lambda$ and $d_i$?
AI: (Unfortunately) No, the ei... |
H: the continuity of the primitive
Good day!
Need to prove the existence of $t\in [0,1]$ such that
$$\int\limits_{0}^{t} f(x)\,dx = \frac{1}{2} \int\limits_{0}^{1} f(x)\,dx,$$
where $f$ is integrable.
My solution:
$$F(t)=\int\limits_{0}^{t} f(x)\,dx-\frac{1}{2}\left(\int\limits_{0}^{t} f(x)\,dx +\int\limits_{t}^{1... |
H: Iverson bracket help
How can I show that if $G$ is a group and $x,y,m$ are in $G$ then,
$$[xy=m]=\sum_{r\in G}[x=r][y=r^{-1}m]$$
Where $[P]$ is the Iverson bracket.
AI: Fix $x,y,m\in G$. The lefthand side, $[xy=m]$, is $1$ if $xy=m$ and $0$ otherwise. Suppose first that $xy=m$, so that the lefthand side is $1$.
No... |
H: When does $\|x+y\|=\|x\|+\|y\|?$
Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$
Let $\|x+y\|=\|x\|+\|y\|$
Squaring both sides, $\langle x+y,x+y\rangle=\langle x,x\rangle+\langle y,y\rangle+2\|x\|.\|y\|\\\ge\langle x,x\rangle+\lang... |
H: how to compare $\sin(19^{2013}) $ and $\cos(19^{2013})$
how to compare $ \sin(19^{2013})$ and $\cos (19^{2013})$ or even find their value range with normal calculator?
I can take $2\pi k= 19^{2013} \to \ln(k)= 2013 \ln(19)- \ln(2 \pi)=5925.32 \to k= 2.089 \times 10^{5925}$, but it useless.(I can get final answer ... |
H: $n-1$ linearly dependent functions among $f_1',f_2',\ldots,f_n'$
This is a problem from IMC training camp last year.
Given differentiable functions $f_i:\mathbb{R}\rightarrow\mathbb{R}, i=1,2,\ldots,n$ such that $\{f_1,f_2,\ldots,f_n\}$ is linearly independent. Show that there are $n-1$ linearly independent functio... |
H: How to break permutation group in normal subgroup and the quotient group?
could you give steps to show this decomposition
or show in maple code
http://en.wikipedia.org/wiki/Simple_group
AI: Let $G$ be a finite group. The set of all non-identity normal subgroups of $G$ is a non-empty finite set (it's finite because ... |
H: Does a property of Xor like this exist?
Is there a property on Xor that says basically $a = b \oplus (a \oplus b)$? I was thinking associative but I don't think that's correct.
AI: That’s actually a consequence of four properties of $\oplus$: it’s associative, it’s commutative, $x\oplus x=0$ for all $x$, and $0$ i... |
H: Polynomials and Trig
Question:
The equation $x^{2}-x+1=0$ has roots $\alpha$ and $\beta$.
Show that $\alpha ^{n}+\beta ^{n}=2\cos\frac{n\pi }{3}$ for $n=1, 2, 3...$
Attempt:
$x^{2}=x-1 \Rightarrow x^{n}=x^{n-1}-x^{n-2}$ for $n=3, 4, 5...$
$\therefore \alpha^{n}=\alpha^{n-1}-\alpha^{n-2}$
$\therefore \alpha ^{n}+\be... |
H: Over-constrained general solution to wave equation
d'Alembert's formula states that the general solution to the one-dimensional wave equation is $$ u(x,t) = f(x+ct) + g(x-ct).$$ for any well-behaved functions $f$ and $g$. This is a well-known and popular result. The 1D wave equation can be accompanied by two init... |
H: proving continuity
For each n belongs to natural numbers let,
$$f_n(x) =
\begin{cases}
0, & x \in \mathbb{Q} \cap[-1,-1/n] \cup [1/n,1] \\
1, & x \in \mathbb{Q}^c\cap[-1,-1/n] \cup [1/n,1] \\
0, & x \in (-1/n,1/n) \\
\end{cases}
$$
Prove that for each $n \in \mathbb{N}$ $f_n$ is continuous at 0. Find $\lim\li... |
H: A canonical homomorphism of sheaves of modules
Let $\mathcal F$ be a sheaf of $\mathcal O_X$ modules on a scheme $X$. Fix an affine open subset $U$. If $M$ is a module over the coordinate ring of $U$, we let $\tilde{M}$ denote the associated sheaf of modules.
Why do we have a morphism of sheaves
$$\mathcal F (U)^\t... |
H: Is this true? $f(g(x))=g(f(x))\iff f^{-1}(g^{-1}(x))=g^{-1}(f^{-1}(x))$.
Is this true?
Given $f,g\colon\mathbb R\to\mathbb R$. $f(g(x))=g(f(x))\iff f^{-1}(g^{-1}(x))=g^{-1}(f^{-1}(x))$.
I met this problem when dealing with a coding method, but I'm really not familiar with functions. Please help.
Thank you.
AI: Ot... |
H: a fundamental clarification about predicate expression (formula)
I have few foundation questions to be clear about expression involving predicates.
$\forall n\in \Bbb N.p(n) \tag {1.2}$
Here the symbol $\forall$ is read “for all.” The symbol $\Bbb N$ stands for the set of nonnegative
integers, namely, $0, 1, 2, 3,... |
H: If $G$ is a graph with no independent set of size $4$, prove that it is not $4$ colorable
Given $G$ a graph with $n \geq {7 \choose 3} = 35$ vertices, and with no independent sets of size $4$, prove it is not $4$ colorable. (independent set of vertices is a set of vertices that has no edges between any two of the ... |
H: Is Binet's formula for the Fibonacci numbers exact?
Is Binet's formula for the Fibonacci numbers exact?
$F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$
If so, how, given the irrational numbers in it?
Thanks.
AI: As others have noted, the $\sqrt 5$ parts cancel, leaving an integer. We can recover the Fibo... |
H: Constant Growth Rate Problem
Say the population of a city is increasing at a constant rate of 11.5% per year. If the population is currently 2000, estimate how long it will take for the population to reach 3000.
How can this be solved using the formula below. I know how to solve when we input x number of years but ... |
H: Measures on Iwasawa decomposition
In the following I present two results, which look very similar, but require different proofs. I'd like to know why the second result doesn't admit the same proof as the first.
Lang $SL_2$ p39: Let $P$, and $K$ be closed subgroups of $G$ such that $G=PK$. Assume that the map $$(p,... |
H: Prove if there are 4 points in a unit circle then at least two are at distance less than or equal to $\sqrt2$
There is a unit circle and 4 points inside the circle. The problem is to prove at least two are at distance less than or equal to $\sqrt2$
AI: An idea: taking the circle $\,S^1:=\{(x,y)\in\Bbb R^2\;;\;x^2+y... |
H: Graphing Basic Exponential Functions
How can we use the graph of $y=2^x$ to sketch the graph of $y=2^{x-1}$?
AI: Note that for any $x,y$ graph: $y = f(x-1)$ is just the graph $y = f(x)$ shifted over one unit to the right. |
H: $\mathbb{S}^2$ as a fibre bundle
I know, by the Hopf fibration, that $\mathbb{S}^3$ is an $\mathbb{S}^1$-fibre bundle over $\mathbb{S}^2$.
Can $\mathbb{S}^2$ be an $\mathbb{S}^1$-fibre bundle over some manifold $M$?
AI: If $F\to E\to B$ is a fiber bundle, $\chi(E)=\chi(F)\cdot\chi(B)$. But $\chi(S^1)=0$ and $\chi(S... |
H: Find equation of plane formed by a point and line
It is required to find the equation of a plane $Q$ formed by point $B\,(5,2,0)$ and the line (d)
of parametric equation
$$
\begin{align}
x&=-2t+1\\
y&=2t-2 \\
z&=t
\end{align}$$
What is the easiest way to find the equation ? Answer is: $x-y+4z-3 = 0$
AI: I think t... |
H: Does this function belong to $\mathcal{S}( \mathbb{R}) $
Let $\mathcal{S}( \mathbb{R}) $ be the Schwartz space, that is the space of all infinitely differentiable functions $f$ such that $f$ an all its derivatives are rapidly decreasing in the sense that
$$ \sup_{x \in \mathbb{R}} |x|^k | f^{(l)}(x)| < \infty .$$... |
H: Check my solution of: H is compact $\iff$ every cover {${E_\beta}$}$_{\beta \in A}$ of $H$ has a finite subcover.
Question:
Let $H ⊆ \Bbb R^n$
Prove that H is compact $\iff$ every cover $\{{E_\beta}\}_{\beta \in A}$ of $H$, where $E_\beta$ 's are relatively open in $H$, has a finite subcover.
Solution:
I did t... |
H: Construction of field extensions
In our lecture notes of Algebra, we have the following construction:
Let $K$ be a field and $P \in K[X]$ be irreducible and monic. Let $L := K[X]/(P)$ and $a:=X+(P)$. Then, $L=K(a)$ is a field extension of $K$ and $P$ is the minimal polynomial of $a$.
I am slightly confused by the... |
H: if $X\sim U(0,1)$ show that $(b-a)X+a \sim U(a,b)$
I'm using the MGF method, this is what I get:
$$
\begin{align}
Y&=(b-a)X+a\\
M_Y(t)&=E[e^{(b-a)X}e^a] \\
&=E[e^{(b-a)X]}e^a &\text{I think this is my error} \\
&= M_x((b-a)t)e^a\\
&=E[\dfrac{e^{(b-a)t}}{(b-a)t}]e^a
\end{align}
$$
and then the $e^a$ messes it up.
AI... |
H: Isomorphic varieties
I just want to see if my approach for this problem is fine:
Show $W=\mathbb{P}^1 \times \mathbb{P}^1$ is not isomorphic to $W'=\mathbb{P}^2.$
Well $V= \{ [0:1] \} \times \mathbb{P}^1, V' = \{ [1:0] \} \times \mathbb{P}^1$ are closed subvarieties of $W$ each isomorphic to $\mathbb{P}^1$ so ea... |
H: Under what conditions are the variables $a, b$ satisfying $A \cap B = \emptyset$ and $A \cap B = \{0\}$
I am struggling with the following question:
You have the two sets $A={x∈R; |x−a|≤1}$ och $B={x∈R; |x−b|≥2}.$ Give the conditions for when the variables: $a$ and $b$ are satisfying $A \cap B = \emptyset$ and $A \... |
H: where this series converges
Given the series $$\sum_{j=0}^{\infty}\frac{1}{6j^2-5j+1}$$
I am completely stuck and do not understand the answer from my book which is $\pi^2/36-1$. I need explanation and different approach how this result is gained. Thanks
AI: Rewrite your series as :
\begin{align}
\tag{1}\sum_{j=0}... |
H: Locally exact differential in a disk is exact
I'm reading through Ahlfors' Complex Analysis text for self study, and I found difficulty with a proof.
In chapter 4 he defines a locally exact differential as a differential who is exact in some neighborhood of every point of its domain.
In the proof of theorem 16 howe... |
H: How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent?
How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent?
I applied integral test and found the series is divergent. I wonder if there exist easier solutio... |
H: Holomorphic function with bounded real part
Suppose that $f(z)$ is holomorphic over $|z| \leq R$, for some positive $R$, and that $f(0)=0$. Further, suppose that $Re(f(z)) \leq C$ for all $|z| \leq R$. How do we show that $|f(z)| \leq \dfrac{2Cr}{R-r}$ for all $|z| \leq r$, for any $0 < r < R$? Any help or hint wou... |
H: Why is the well ordering principle counter-intuitive?
I read here that while 'The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians'. I don't understand why this is so. According to Wikipedia, 'the well-ordering principl... |
H: Determine the lengths of the sides of a right triangle
The positive real numbers $a,b,c$ are such that $a^2+b^2=c^2$, $c=b^2/a$ and
$b-a=1$. Determine $a,b,c.$
AI: From
$a^2+b^2=c^2$ , $c=b^2/a$ and $b-a=1$,
$a^2+b^2 = b^4/a^2$,
so
$a^4+a^2b^2 = b^4$.
We could substitute
$b = a+1$
(if we do,
I get
$a^4-2a^3-5a^2-... |
H: Shannon's entropy in a set of probabilities
Let $P = p_1, \ldots, p_N$ be a set of probabilities (i.e., $0 \leq p_i \leq 1$).
I can compute the Shannon's entropy as follows:
$$
H(P) = -\sum_{i=1}^N p_i \log_2 p_i
$$
Now, suppose I perform the following operations:
I select some $p_i \in P$, and create the set $P_{... |
H: When solving an ODE using power series method, Why do we need to expand the solution around the singular point?
When solving a differential equation using series expansion method, if it has the following form :
$$y''+\frac{p(x)}{x}y'+\frac{q(x)}{x^2}y=0$$
; where $p$ and $q$ are analytic at $x_0$; if we want to fin... |
H: Inequality involving definite integral
Just wondering, what may be the best way to show that $$\int_0^1 xf(x)dx \leq \frac{1}{2}\int_0^1 f(x)dx,$$ provided that $f(x) \geq 0$ over the interval $[0,1]$ and that $f(x)$ is monotonically decreasing?
Thanks!
AI: $f$ and $x\mapsto x$ are of opposing monoticity, thus by C... |
H: Directional derivative, gradient and a differential function
Let there be some function $f$, some point $(x_0,y_0)$ and some vector $u$.
Is $D_{u}f(x_0,y_0)=∇f(x_0,y_0)⋅u$ always correct? Even if the the function is not differentiable at the point? Or in more general, I didn't quite understand when that statement ... |
H: How find this value $\frac{a^2+b^2-c^2}{2ab}+\frac{a^2+c^2-b^2}{2ac}+\frac{b^2+c^2-a^2}{2bc}$
let $a,b,c$ such that
$$\left(\dfrac{a^2+b^2-c^2}{2ab}\right)^2+\left(\dfrac{b^2+c^2-a^2}{2bc}\right)^2+\left(\dfrac{a^2+c^2-b^2}{2ac}\right)^2=3,$$
find the value
$$\dfrac{a^2+b^2-c^2}{2ab}+\dfrac{a^2+c^2-b^2}{2ac}+\dfra... |
H: How can I calculate the expected change of my function?
I am attempting to model the following equation with 3 variables (H, M, and S) and 1 constant (C):
D = .05*H*M + .05*S*(M+C)
or, when factored
D = .05*M (H+S) + .05*C(S)
It'd be a simple matter to calculate the value of D at any point in the formula (just ... |
H: Projection onto plane given by matrix without full rank
This should be a really simple question, but I can't seem remember my linear algebra. Suppose I have a complex $m\times n$ matrix $A$, $m > n$, that may not have full rank (hence $A^{*}A$ may not be invertible). Consider the linear equation system $Ax = y$. Gi... |
H: Supremum over dense subset
I'm interested about the following supremum:
$$\sup_{g\in A}E[-gf]$$
where $A\subset\{g\in L^1:g\ge 0,E[g]=1\}$ and $f\in L^\infty$ is fixed. $E$ denotes the expectation, i.e. it is the integral with respect to a finite measure. If I have a dense subset $B$ of $A$ in the $L^1$ sense. Is t... |
H: What is the definition of the slope of a linear function in the context of economic graphs?
I only ask this because of the fact that economists tend to plot the dependent variable on the horizontal axis and the independent variable on the vertical, which is opposite to the "normal" way of doing things in math. This... |
H: three distinct positive integers $a, b, c$ such that the sum of any two is divisible by the third
I need to determine three distinct positive integers $a, b, c$ such
that the sum of any two is divisible by the third.
I tried like with out loss of generality let $a<b<c$
As, $a\mid (b+c)$ so $b+c=ak_1$ for some $k_1\... |
H: induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$
I encountered the following induction proof on a practice exam for calculus:
$$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$
I have to prove this statement with induction.
Can anyone please help me with this proof?
AI: If $P(n): \sum_{k=1}^nk^2 = \frac{n(n+1)... |
H: Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$
I have to resit a calculus exam and for some reason set proofs were never my best friend...
Anyway, on a practice exam I encountered the following proof:
$$A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$$
When I draw a Venn-diagram it seems quite obvious but I couldn't ma... |
H: Finding parameter of equation
$$x^4 + 1 = kx; k>0$$
The question is at what k the equation has 1 solution.
I understand that it could rephrased as follows: at what k $f(x) = x^4 - kx + 1$ has 1 $x:f(x) = 0$. but I've no idea how to solve quadratic equation.
Also, the problem may be seen as finding point of interse... |
H: Is there an intuitive explanation for the formula for the number of observations in an average given two averages and a marginal observation?
First, apologies for the long-winded title!
I'm helping my 10 year old son with math, and we had a set of problems based on the following scenario: Given an average of a set,... |
H: Relations between radius of convergence
Let $R$ denote the radius of convergence. Then $\sum_{n=0} a_{2n} x^{2n}$ has $R = 2$, and $\sum_{n=0} a_{3n} x^{3n}$ has $R = 3$. How to prove that $\sum_{n=0} a_{n} x^{n}$ has $R \leq 2$?
For simplification suppose that $a_n > 0\quad \forall n$. Then the following could be ... |
H: Is't possible to define such a real inner product?
Consider the $\mathbb R-$linear space $B(X,\mathbb C)$ of all bounded functions from $X\to\mathbb C$ for some $X\ne\emptyset.$ Then $\displaystyle||.||:B(X,\mathbb C)\to\mathbb R:f\mapsto\sup_{x\in X}|f(x)|$ is a norm on $B(X,\mathbb C).$
Is't possible to define a... |
H: Check whether or not the series $\sum_{n=1}^{\infty}\sqrt{4n^2-4n+9}-(2n-1)$ converges
I want to check whether or not the series converges.
the series is:
$$\sum_{n=1}^{\infty}\sqrt{4n^2-4n+9}-(2n-1)$$
The first thing I thought of is to do multiply by his compliment and the result I get is:
$$\sum_{n=1}^{\infty} \f... |
H: If Sam's age is twice the age Kelly was two years ago, Sam's age in four years will be how many times Kelly's age now?
If Sam's age is twice the age Kelly was two years ago, Sam's age in four years will be how many times Kelly's age now?
(A) .5
(B) 1
(C) 1.5
(D) 2
(E) 4
So say at -2 years Sam's age is 12 and Kell... |
H: $(A\cup B)\cap C = A\cup(B\cap C)$ if and only if $A\subset C$
I tried to prove this statement:
$$[(A\cup B)\cap C = A\cup(B\cap C)]\iff A\subset C$$
I did it in the following way, can anyone tell me if it's correct what I've done?
$\leftarrow$ Assume $A\subset C$, then $\forall x \in A$, $x\in C$
Then, $\forall x ... |
H: If $\sin a+\sin b=2$, then show that $\sin(a+b)=0$
If $\sin a+\sin b=2$, then show that $\sin(a+b)=0$.
I have tried to solve this problem in the following way :
\begin{align}&\sin a + \sin b=2 \\
\Rightarrow &2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)=2\\
\Rightarrow &\sin\left(\frac{a+b}{2}... |
H: Topological Conditions Equivalent to "Very Disconnected"
Definition: Let $(X, \mathcal{T})$ be a topological space, where the set $X$ has more than one element. Suppose that for every pair of distinct elements $a, b \in X$, there exists a separation $(A,B)$ of $X$ such that $a \in A$ and $b \in B$. Then we say $(X,... |
H: Why does this sequence happen like this?
The other day I sent my girlfriend this text
<3
she sent me back
<3<3<3
not to be one upped I responded with
<3<3<3<3<3<3<3
this got very silly very quickly.
After our "<3" battle was over I got to thinking about the pattern we were forming. Since we were doubling the number... |
H: Finding a point on a circle
I have a circle that I am trying to find series of points on. I know the radius and horizontal tangent point at the top of the circle. I need to find a point that lies on the circle's circumference that is $x$ distance below the top point.
AI: The vertical coordinate of that point will... |
H: $A$ is a subset of $B$ if and only if $P(A) \subset P(B)$
I had to prove the following for a trial calculus exam:
$A\subset B$ if and only if $P(A) \subset P(B)$ where $P(A)$ is the set of all subsets of $A$.
Can someone tell me if my approach is correct and please give the correct proof otherwise?
$PROOF$:
$\Big(\... |
H: Filter properties: Downward directed set and finite intersection property
I have a list of questions concerning the properties of filters:
(1) If a finite subset of a poset is downward directed is it necessarily closed under finite intersection? At the very least, if said subset is downward directed, is the interse... |
H: Free $\mathbb{Z}$ module question
Is $M=\left\{\frac{a}{2^n}| a \in \mathbb{Z}, n \in \mathbb{N}\right\}$ a free $\mathbb{Z}$-module? I am struggling with free modules and am not sure how I would check this? I know that the basis cannot be finite from another problem I have worked before. However, I am not sure ... |
H: Input expression with summations in Wolfram Alpha
Wolfram understand this expression , but i need to do the limit when n tends to infinity of that expression .
As you can see in the wolfram web itself, in the last link it fails to understand the query.
How can i do that?
AI: As suggested in the comment, at the ve... |
H: Counting number of elements in the empty set
When is was making some exercises I encountered the following exercise:
$Exercise$:
Let $P(A)$ denote the set of all subsets of an arbitrary set $A$.
List first the elements of $P(\emptyset)$, then the elements of $P(P(\emptyset))$.
Finally, check in two steps whether y... |
H: Direct sum of orthogonal subspaces
I'm working on the following problem set.
Let $\mathcal{H}$ be a Hilbert space and $A$ and $B$ orthogonal subspaces of $\mathcal{H}$.
Prove or disprove:
1) $A \oplus B$ is closed, then $A$ and $B$ are closed.
2) $A$ and $B$ are closed, then $A \oplus B$ is closed.
I could prove 1)... |
H: Convergence of binomial to normal
Problem: Let $X_n \sim \operatorname{Bin}(n,p_n) $ where $p_n \xrightarrow{} 0$ and $np_n \xrightarrow{} \infty$. What I need to show is that
$$\frac{X_n - np_n}{\sqrt{np_n}} \xrightarrow{d} N(0,1) \text{ as } n\xrightarrow{} \infty.$$
My thoughts: My first thought was to set
$$... |
H: The set of "variations" of an unconditionally convergent series is compact
Proposition: If $\sum x_i$ is an unconditionally convergent series in a Banach space $X$, then $S=\{\sum \varepsilon_ix_i:\varepsilon_i=\pm1\}$ is compact.
Proof:
1) $\{-1,1\}^{\mathbb N}$ is compact in the pointwise topology.
2) $f:\{\varep... |
H: Concave function properties
Given a concave function $f(x)$, $\,f(x)$ decreases as $\,x\,$ increases.
That is, $\;f(x_1)\gt f(x_2)\,$ if $\,x_2\gt x_1$
For $\;f(x_1)+f(x_2)\;$ and $\;\large\left(\frac{f(x_1+x_2)}{2}\right)^2,\;$ which one is larger?
For $\;(1-f(x_1))(1-f(x_2))\;$ and $\;1\large-\left(\frac{f(x_... |
H: Can a bipartite summation graph have a unique solution?
Suppose we define a summation graph $G$ as follows:
Each vertex $v \in G$ has a unique but unknown value ascribed to it. Each edge $e \in G$ is labelled with the sum of the values of the two vertices it joins.
This construction corresponds to a system of equat... |
H: Check my answer: Prove that every open set in $\Bbb R^n$ is a countable union of open intervals.
I have a question. I have solved this but please can you check my solution? Thank you.
If there are any mistakes or something is missing and so on, please tell me.
This is important to me. Is this proof enough to get ... |
H: Fundamental Theorem of Calculus in Multivariate Case
From the FTC we have, for continuously differentiable $f: \mathbb{R} \to \mathbb{R}$,
$$
f(a) - f(b) = \int_b^a \frac{d}{dx} f(x) dx
$$
I'm trying to write the difference between a vector function in similar terms, that is, given $g : \mathbb{R}^d \to \mathbb{R}... |
H: Differentiability of the supremum norm in $\ell^{\infty}$
Let $\ell^{\infty}=\{x\in \mathbb{R}^{\mathbb{N}}: x\,\, \text{is bounded}\}$ and $E=\{x\in \ell^{\infty}:x_n\rightarrow 0\}$ with the norm $||\cdot||_{\infty}$ and let $f(x)=||x||_{\infty}$. How to prove that:
a) If $x\in E$ then there exists $m\in\mathbb{N... |
H: How to find the limit $\lim\limits_{m\to\infty}\frac{m^{m-2}}{(m-1)^{m-2}}$?
I am trying to evaluate the limit of this:
$$\lim_{m\rightarrow \infty} \frac{m^{m-2}}{(m-1)^{m-2}}$$
That is just basic calculus I think but I forget those methods for finding the limit. I think the limit is just $e$. Anyone could prove t... |
H: Convergence of $\sum_{n=1}^{\infty}(-1)^n*\frac{n^2}{(\sqrt{n^7+n+2})^{1/3}}$
I`m trying to check if this series is convergent and would like to get some advice.
$$\sum_{n=1}^{\infty}(-1)^n*\frac{n^2}{(\sqrt{n^7+n+2})^{1/3}}$$
1) I need to multiply in his compliment?
2) I need to make the comparison test?
Thanks!... |
H: Is an inverse Laplace Transform always solvable?
I just read on Wikipedia that if we got a certain Laplace Transform
$$\mathcal{L}\{f(t)\}= \frac{A}{s-\alpha_1} + \frac{B}{s-\alpha_2} + ... $$
can be solved like this:
$$f(t)= A e^{\alpha_1 t}+ Be^{\alpha_2 t}+ ...$$
My question now is: Given that we can always use... |
H: How to know the flattening factor for a ellipse?
I want to know how can I get the flattening factor for a ellipse by knowing its semi-major and semi-minor axes ?
Actually I tried this formula:
$f=\left(\frac{a}{b}-1\right)$
While $f$ is the flattening factor, $a is the semi\,major\,axes$, $b is the semi\,minor\,a... |
H: "universe" in set theory and category theory
In all applications of the theory of sets, all sets under investigation take place in the context of the universal set $U$. What exactly is the purpose of the universal set in set theory? $U$ is infinite, why do we need to mention this infinite context in regards to set ... |
H: For polynomial $f$, does $f$(rational) = rational$^2$ always imply that $f(x) = g(x)^2$?
If $f(x)$ is a polynomial with rational coefficients such that for every rational number $r$, $f(r)$ is the square of a rational number, can we conclude that $f(x) = g(x)^2$ for some other polynomial $g(x)$ with rational coeff... |
H: Find $m$ so that $2^x+m^x-4^x-5^x\ge0$
Let:
$$f:\mathbb{R}\to \mathbb{R}, \ \ f(x)= 2^x +m^x -4^x -5^x \text{ with } \ m>0$$
Find $m$ so that $f(x)\ge0$ for all $x$ in $\mathbb{R}$.
I tried proving that its ascending for $x>0$ and descending for $x<0$ but it didn't work
AI: You want the minimum of $f(x)$ to be at ... |
H: Proving if $|a|_p=1$ then $a$ is invertible in $\mathbb{Z}_p$
I decided to take a look $p$-adic integers. I am trying to show that
$$a \in \mathbb{Z}_p \text{ is invertible if and only if } |a|_p=1$$
where
$$|x|_p= \left\{
\begin{array}{ll}
p^{-n} & \mbox{if } x\neq0 \\
0 & \mbox{if } x=0
\end{array}
\ri... |
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