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H: Is there a relationship between $\ln(x\pm y)$ and $\ln(x)\pm\ln(y)$?
I am dealing with points on a 2d space $(x, y)$ where $x$ and $y$ are always positive integers.
In an algorithm, I have pre-computed $\log_2(x)$ and $\log_2(y)$ for given points of interest.
I now need to compute $\log_2(x+y)$ and $\log_2(x-y)$ bu... |
H: Showing that a linear map $\mathbb{R}[x]_{\leq 2} \rightarrow \mathbb{R}^3$ is an isomorphism.
Let $t_1,t_2,t_3$ be distinct real numbers and consider the linear map
$$T : \mathbb{R}[x]_{\leq 2} \rightarrow \mathbb{R}^3, \quad \quad p(x) \mapsto \begin{pmatrix}p(t_1)\\p(t_2)\\p(t_3)\end{pmatrix}$$
I want to show th... |
H: Probability without replacement question
I have a bag full of 10 marbles: 2 black, 1 blue, 1 yellow, 3 green, 1 brown, and 2 purple. I draw 5 marbles one at a time without replacement. What is the probability of a black marble to be in the five drawn?
AI: Think of the marbles as having, in addition to colour, an ID... |
H: Estimates of Gaussian Logarithms
I've been implementing logarithmic number system and I came across these functions called Gaussian logarithms:
$f(x) = \log(1 + e^x)$.
$g(x) = \log(e^x - 1)$ for $x > 0$.
$h(x) = \log(1 - e^x)$ for $x < 0$.
Since I am lazy, I assume $\log$ and $\exp$ aren't expensive to compute, b... |
H: Maximal ideals in $R[x]$
I want to prove the following result:
Let $R$ be a ring and $M$ a maximal ideal in $R$. If $P$ is a prime ideal in $R[x]$ that (strictly) contains $M[x]$, then $P$ is a maximal ideal in $R[x]$.
I have an idea how to prove that, but I'm not quite sure if the argument is totally valid; ma... |
H: Is the set of irrationals separable as a subspace of the real line?
I am trying to find an example of a separable Hausdorff space which has a non-separable subspace. This led me to ask the question in the title: is the set of irrationals, regarded as a subspace of the real line, separable or non-separable?
A spac... |
H: Block inverse of symmetric matrices
Let us assume we have a symmetric $n \times n$ matrix $A$. We know the inverse of $A$.
Let us say that we now add one column and one row to $A$, in a way that the resulting matrix ($B$) is an $(n+1) \times (n+1)$ matrix that is still symmetric.
For instance,
$A = \begin{pmatrix}... |
H: Limit of a complex number over its conjugate, as z approach the infinity.
this question must be pretty easy but i´m just taking my firs course in complex variables, i need to find the limit of Z over it conjugate as z reach the infinity. I need to know how to do it in a rigorous way and informal way, so i need your... |
H: On modules over polynomial rings
Let $\mathbb{A}$ be a polynomial ring in $n$ variables over an algebraically closed field $\mathbb F$. Given a maximal ideal $\mathfrak{m}$ of $\mathbb A$, consider the quotient $\mathbb{A}/\mathfrak{m}$ as a left $\mathbb{A}$-module ($\mathbb{A}/\mathfrak{m}$ is one-dimensional ove... |
H: General questions about equivalence classes and partitions
1) If a set is partitioned into non-overlapping, non-empty subsets, then those subsets are equivalence classes. If each element in a set is unique, how can a set be partitioned into subsets with equivalent elements?
2) What is the criteria for something to ... |
H: Can anyone explain the intuitive meaning of 'integrating on both sides of the equation' when solving differential equations?
For solving differential equations, especially the ones of the form
$$g(x)dx = h(y)dy$$
we solve the equation by integrating on both sides to reveal the solution.
Understanding this for diffe... |
H: A non-square matrix with orthonormal columns
I know these 2 statements to be true:
1) An $n$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^T$.
2) An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I$.
But can (2) be generalised to become "An $m$ x $n$ matrix U has orthonormal columns iff. $U^TU=I=UU^... |
H: Random walk on finite graph
I know that the stationary distribution of a random walk on the graph is given by,
(degree of the node)/($2\times$ total number of links in graph). My question is, how do we get this solution?
AI: I you start out in a distribution where the probability of being at a given node is proport... |
H: Trouble with representing power series as polynomials.
I am a math student trying to wrap my head around complex analysis through self-study. I am using Complex Analysis by Serge Lang, but I find myself struggling with some of his power series manipulations and his representations of power series as polynomials. Fo... |
H: A question on calculating probabilities for the random walk
I am currently working on a high school project revolving around the 'Cliff Hanger Problem' taken from ”Fifty Challenging Problems in Probability with Solutions” by Frederick Mosteller.
The problem is 'From where he stands, one step toward the cliff would... |
H: Complex function limit
I asked time ago about the limit of a complex number $z$ over its conjugate, as $z$ goes to infinity.
Now I have a strategy, it is to convert $z$ to trigonometric form, and the limit depends uniquely on the norm of $z$, which can be eliminated from the fraction, so you got a fraction $\cos\th... |
H: How do you prove the trichotomy law for cardinal numbers?
Law of trichotomy is that for any two cardinals $a$ and $b$, exactly one of the
conditions $a<b$, $a=b$, or $a>b$ holds.
AI: First let us prove this only for $\aleph$ numbers.
Suppose that $\aleph_\alpha$ and $\aleph_\beta$ are two cardinals. We take $\omega... |
H: An Interesting Question about Pythagorean Triples
I have recently thought about a interesting question about Pythagorean Triples.
Consider such a right-angled trapezium formed by 3 right-angled triangle. Determine does it exist integral solutions for lengths of sides $AB, BC, CD, DE, EA, AC$ and $CE$.
Here's my ... |
H: Help in understanding integration by changing the variable
I need help better understanding how, and why integration by changing the variable works (I've seen it's related to the derivative of a composite function $f(g(x))$), and generally tips and tricks, an intuitive understanding and exercises.
I sort of underst... |
H: Interpret overall fitness in prediction
I came through this equation and would like to learn more about how to inteprete it.
d is defined as actual value, and $\hat{d}$ is defined as predicted value.
Why does this equation
(a) divides sum of least square error to the sum of $d_{i,j}^2$ ?
(b) takes square root of ... |
H: Switching between $h$ and $-h$ in the definition of derivative of a real function.
Suppose that $f$ is a real function and that $f'(a)$ exists:
$$ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}. $$
By replacing all instances of $h$ with $-h$, I can get an equivalent definition
$$ f'(a) = \lim_{-h \to 0} \frac{f(a-h... |
H: Probability of Bridge hand having at most two suits
I am reviewing MIT 18.440 slides. I am wondering why the question that is worded as How many bridge hands have at most two suits represented? has a given answer of ${4\choose 2}{26 \choose 13} - 8$.
I understand that the ${4\choose 2}$ takes in account choosing th... |
H: Evaluating partial derivatives to apply finite differences
I need to solve the following partial differential equation (Cahn-Hilliard) using finite differences:
$$\frac{\partial c}{\partial t} = \nabla^2h + \cdots$$
where $h = c(1-c)(1-2c)$.
The question I want to ask is, which of
$$\nabla^2 h = \frac{h(x+h) + h(x-... |
H: Given the width and height of an ellipse find n number of points around the ellipse?
I am looking to find $n$ number of points around an ellipse.
They don't necessarily have to be equidistant.
Similar to what this forum is asking:
I found several answers that are similar but I am having a hard time expressing it... |
H: Is there any example of a discontinuous one real variable integral.
I know that an integral from $0$ to $x$ of a function of $x$ is itself a function, so I want to know if is there any condition of it to be discontinuous.
I haven't restrict $f(x)$ to be continuous or not, so I think that one needs to take both opti... |
H: Proving that $2^{2^n} + 5$ is always composite by working modulo $3$
By working modulo 3, prove that $2^{2^n} + 5$ is always composite for every positive integer n.
No need for a formal proof by induction, just the basic idea will be great.
AI: Obviously $2^2 \equiv 1 \pmod 3$.
If you take the above congruence to t... |
H: If $x \geq \min\{y,w\}$ then $x \leq y+w$?
If $x \geq \min\{y,w\}$ then $x \leq y+w$.
This is kind intuitive,very trivial. Or not? It's like triangular inequality, isn't it?
AI: This is wrong. Take $y,w=1$ and $x=3$. |
H: Find all integer solutions to $7595x + 1023y=124$
Find all integer solutions to $7595x + 1023y=124$
Using the Euclidean algorithm I have found the $\gcd(7595,1023)=31$ and found the Bezout identity $31=52\cdot1023-7\cdot7595$ but I'm not really sure how to go about finding all solutions to that equation.
I believe... |
H: An infinite set having "one more element" than another infinite set
A classic example of homeomorphism is between a sphere missing one point and a plane
To see this, place a sphere on the plane so that the sphere is tangent to the plane. Given any point in the plane, construct a line through that point from the "no... |
H: How to simplify the following basic equation
I just wanna know if you can find a way to simplify this following equation :
originalState + ( ( animatedState * ( 100 - ( finishPos - x ) * 100 / (finishPos - startPos) ) / 100 )
Thanks
AI: Being lazy, I'll replace each variable by its first letter, and I get this (wit... |
H: Hahn-Banach theorem
Hope this won't turn out to be a stupid question. If I want to apply the Hahn-Banach lemma to prove the Hahn-Banach theorem, then I want to define a function $p$ that provides an upper bound on my functional defined on a linear subspace $Y \subset X$. Say, I want to extend $f \in Y^\ast$. Then t... |
H: Why do characters on a subgroup extend to the whole group?
As background, I am trying to do exercise 3.10 in Deitmar's "Principles of Harmonic Analysis." I can do most of the problem but I'm stuck on the third part proving surjectivity.
Given a locally compact abelian group $G,$ a closed subgroup $H,$ and a charac... |
H: Help in evaluating $\sum\limits_{k=1}^\infty \frac1{k}\sin (\frac{a}{k})$
I would like to try to evaluate
$$\sum\limits_{k=1}^\infty \frac{\sin (\frac{a}{k})}{k}$$
However, all of my attempts have been fruitless. Even Wolfram Alpha cannot evaluate this sum. Can someone help me evaluate this interesting sum?
AI: T... |
H: Optimization with non-negativity and norm constraint
I am facing the following optimization problem:
$$\min_x w^tx \\
s.t. ||x|| = 1, \forall i: x_i \geq 0
$$
where $w$ and $x$ are real valued vectors. How would I solve this?
My background is not optimization (only some experience with equality constraints via Lagr... |
H: How to find the ratio for a combination of two chemical solutions
Lets say you have a chemical solution $A$ that has a ratio of $4:1$ of water to some chemical substance, and a solution $B$ that has a ratio of $9:1$ of water to some other chemical substance.
If you combine the same amount of each solution, the res... |
H: How would I express this code in math notation?
I'm writing a paper on an application I wrote, and I'm trying to describe mathematically what I did to determine the rotation of an object. What I did is really simple, but my math lags far behind my computer science.
What it does is select a random number between min... |
H: Best Notation for Introducing Finite Numbers of Elements
What is the best way to quantify over a finite number of elements of a set? For instance, suppose that one wished to quantify over $n$ real numbers $a_1,\ldots,a_n$ in order to state a property of their sum. In such a case, I often see authors quantify over t... |
H: Create a new image from an image set's patterns
So, I'm interested in learning how to detect patterns in a set of images and then use those patterns to create a new image of similar style. For example, say there is a group of 20~ish images (hopefully a large enough sample) with some interesting characteristics and ... |
H: Proving that $\mu$ is $\sup S$
Suppose a set $S$ of real numbers is bounded and let $\mu$ be an upper bound for $S$. Show that $\mu$ is the least upper bound of $S$ $\Longleftrightarrow$ for every $\epsilon > 0$ there is an element of $S$ in the interval $[\mu - \epsilon, \mu]$.
My Work
($\Rightarrow$)
If there is ... |
H: Why is $b^x \overset{\mathrm{def}}{=} \sup \left\{ b^t \mid t \in \Bbb Q,\ t \le x \right\}$ for $b > 1$ a sensible definition?
This problem comes from baby Rudin, Chapter 1, Number 6(c).
For the problem, given the condition $b > 1$, one is asked to show that for rational $r$, $$b^r = \sup B(r),$$
where $B(x) = \le... |
H: Linear Transformation
Let $T:V\rightarrow V$ be a linear transformation and dimension of $V$ be infinite. Then show that either the nullity or the rank of $T$ is infinite.
The problem is that I cannot use rank nullity theorem.
Thanks for any help.
AI: Let $\,\{v_1,...\}\,$ be any (obviously infinite) basis of $\,V\... |
H: Can lines connecting nearest neighbors ever cross?
I don't have any background in graph theory so I am sorry if this is a basic question. I want to know if lines connecting nearest neighbors can cross in a few different cases.
I have convinced myself that if there are no ties for nearest
neighbor it is impossib... |
H: Complex Integration inside a Domain with a singularity
$f(z)= \dfrac{2z+3i}{z^{2}+1/4}$, $C$ is the unit circle centered at zero.
Though the answer is $12\pi$, but my answer is coming to be $4\pi i$
If we decompose it into partial fractions then $f(z)$ reduces to $\dfrac{4}{z-\frac{i}{2}} - \dfrac{2}{z+ \frac{i}{2}... |
H: Use Chebychev's Inequality to find a lower bound.
Let $X$ be the number of heads one would obtain in $140$ flips of a fair coin.
Use Chebychev's Inequality to find a lower bound on the probability $P(60 < X < 80)$.
Okay so Chebychev's Inequality is $P(|X - E(X)| > kσ) \le 1/k^2$ for $ k > 0$, where $σ^2$ is the var... |
H: Intuition behind Snake Lemma
I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even attempt to prove it. I believe I am missing some important intuition that tells me that the lemma “could”... |
H: Fibres in algebraic geometry: multiplicity
Currently I am studying varieties over $\mathbb{C}$ and I know some scheme theory.
My professor mentioned the other day that given a morphism of varieties over an alg. closed field $k$: $f: X \rightarrow Y$, one should count fibres "with multiplicity". I have been trying t... |
H: Parametrization of a conic and rational solutions
How can we parametrize the conic $C$: $x^2+y^2 = 5$, by considering a variable line through $(2,1)$ and hence all rational solutions of $x^2 + y^2 = 5$?
I'm thinking let $x = \sqrt{5}\cos t$, and $y = \sqrt{5} \sin t$, and somehow I need to get the coordinates $(2,1... |
H: Cancellation of addition on convex sets
I recently found a question about a property of the Minkowski sums. However the question was not properly answered (it used a projection argument which might not be true in a general Banach space).
I was wondering whether the following (weaker) statement holds:
Let $X$ be a B... |
H: How long does it take to distribute a file among n computers
So i am at a LAN party where there are $n$ computers of which one $c_0$ computer has a file he wants everybody to have. Let's assume that transferring the file always takes $t$ time between any two computers.
Initially only the one $c_0$ computer has the ... |
H: Notation and naming for two operations with $p$-form valued $n$-forms
While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much.
I'm interested in notion and names of two operatio... |
H: $2$ question on matrix with some condition
we need to tell whether $a,b$ true or false,
I know that there does not exist $n\times n$ $A,B$ such that $(AB-BA)=I$,as if we take trace of both side they are not equal, so $a$ is false?
$b$ I have no idea, could any one give me hint?
AI: (a) Let $C = AB - BA$. Then we h... |
H: Probability distributions - Exam paper question - $\mathrm{Cov}(X,Y)$, PDF
Two people have decided to meet at a certain point in a forest sometime between noon and 2pm.
Their respective independent arrival times are $X$ and $Y$ such that $X \sim \mathrm{Unif}(0,2)$ and $Y \sim \mathrm{Unif}(0,2)$.
Hence the joint ... |
H: Integer coefficient polynomial $p(x)$ has no integer roots of $\,p(0)$ and $p(1)$ are odd [Parity Root Test]
This came up an a training piece for the Putnam Competition and also in Ireland and Rosen.
The question posed was basically:
Let $p(x)$ be a polynomial with integer coefficients satisfying that $p(0)$ and $p... |
H: Is there a relationship between products and integrals? Is there a way to convert a product into an integral?
I know that the Euler-Maclaurin formula establishes a relationship between sums and integrals, but is there some sort of formula that establishes a relationship between products and integrals? I don't mean ... |
H: Convergent Subsequence in $\mathbb R$
Let $\langle a_n \rangle = \dfrac{(-1)^n}{1+n}$ be a sequence in $\mathbb R$.
Considering the limit point(s) of this sequence and the subsequences that converge to this point, I have two subsequences: $$ a_{2k} = \frac{1}{1+2k} \to 0 \text{ and } a_{2k+1} = \frac{-1}{1+2k+1}... |
H: Counterexample Check for Sum of Limit Points of Subsequences
Let $c$ be a limit point of a sequence of real numbers $\langle a_n \rangle$ and $d$ a limit point of $\langle b_n \rangle$. Is $c+d$ necessarily a limit point of $\langle a_n + b_n \rangle$?
My Question:
When considering this question, do I have to sum ... |
H: $F$ closed and connected then...
Let $F\subset X$ be a closed and connected subset of the metric space $(X,d)$ then for every pair of points $a, b\in F$ and each $\epsilon>0$ there are points $a=z_0,z_1,\ldots z_n=b$ in $F$ such that $d(z_{k-1}-z_k)<\epsilon$ for $1\leq k\leq n$.
Is the hypothesis that $F$ is close... |
H: Question about transitivity convergence
Let us define $E^*=\{f:E\rightarrow \mathbb{R} \mid f $ is linear and continuous$\}$, where $E$ is separable space equipped with a norm. Let $A = \{x_1,x_2,\ldots\}$ be a countable dense subset of $E$. Suppose that $(f_n)_{n\in\mathbb{N}}$ is a sequence of functions from $E^*... |
H: How to construct a $2\times 2$ real matrix $A$ not equal to Identity such that $A^3=I$?
How to construct a $2\times 2$ real matrix $A$ not equal to Identity such
that $A^3=I$?
There is a correspondence between the ring of complex numbers and the
ring of $2\times2$ matrices (0 matrix
is included!) i.e.,$$a+ib\left... |
H: Classify all alphabets into homeomorphism classes $\{ M N B H\}$,
classify all alphabets into homeomorphism classes $\{ M N B H\}$,
what does it mean by homeomorphism classes?
They're looking upon letters as drawings of topological spaces, perhaps!?
connectedness distinguishes $B$ from other $3$ as if we remove mid... |
H: Primes Not Dividing $\binom{2n}{n}$
Let $n \geq 3$, show ${2n \choose n}$ is not divisible by $p$ for all primes $\frac{2n}{3} <p\leq n$
Note: This fact along with other facts about ${2n \choose n}$ are used in a proof of Bertrand's postulate.
AI: I think that the idea is rather simple (not verifying Brian's hint..... |
H: Inverse of symmetric matrix $M = A A^\top$
I have a matrix, generated by the product of a non-square matrix with its own transpose:
$$M = A A^\top.$$
I need the inverse of $M$, assuming $\det(M) \neq 0$.
Given the nature of the matrix $M$, are there any specialised computational methods to find its inverse, priorit... |
H: Matrix-product integrals
Suppose you multiply the values of $f$ between $a$ and $b$ at intervals of $\Delta x = (b-a)/n$ and then raise the product to the power $\Delta x$, and take the limit as $n\to\infty$. What you get would bear the same relation to products that integrals bear to sums, and there's a trivial r... |
H: $L^p$ Norm continuity boundedness
I'm studying for a Qualifying exam and can't figure out this problem. I see that the limit must be $f(x)$ and can get the boundedness but had trouble with continuity. Any suggestions?
Let $f\in L^{\infty}(\mathbb{R}^{d})$ and let $\phi:\mathbb{R}^{d} \times (0,\infty)\rightarrow ... |
H: Absolute Continuity, Lipschitz
I'm studying for a qualifying exam and can't seem to solve this problem. Any suggestions would be appreciated!
Let $f:[a,b] \rightarrow \mathbb R$ be absolutely continuous. Show, for each $\epsilon>0$, that there is a uniformly Lipschitz function(global) $g:[a,b] \rightarrow \mathb... |
H: Fundamental Theorem of Calculus problem
I don't understand the intuition behind this. Why can we just plug in $x$ for $t$ here and that gives us the result? I thought I was understanding the Fundamental Theorem of Calculus, but I don't see how it applies here. I thought the Theorem mainly stated that the area un... |
H: About the sequence satisfying $a_n=a_{n-1}a_{n+1}-1$
"Consider sequences of positive real numbers of the form x,2000,y,..., in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x does the term 2001 appear somewhere in the sequence?
(A) 1 (B)... |
H: A question on iterated sums
We are to prove that the set
$$\Bigg\{ \sum^m_{i=1}\sum^n_{j=1}|a_ib_j| :m,n\in \mathbb N \Bigg\}$$
is bounded. Also, We are to use this to show that the iterated sum $\sum^\infty_{i=1} \sum^\infty_{j=1}|a_ib_j|$ converges.
I could prove that the set is bounded, however, I can't solve th... |
H: Prove that $ \frac{n}{\phi(n)} = \sum\limits_{d \mid n} \frac{\mu^2(d)}{\phi(d)} $
I am trying to show that:
\begin{equation}
\frac{n}{\phi(n)} = \sum_{d \mid n} \frac{\mu^2(d)}{\phi(d)}
\end{equation}
where $\phi(n)$ is Euler's totient function and $\mu$ is the Möbius function.
The identity clearly holds for $n=1$... |
H: Example of a quasinilpotent operator
Can anybody please give me an example of a quasinilpotent operator $T$, i.e. an operator such that $\sigma(T)=\{0\}$ on $l_2$ such that it has finite dimensional but non-trivial kernel and is not compact?
This is probably easy and well known but I just can't figure it out it a... |
H: Compact operators and uniform convergence
Suppose $T: H \rightarrow H$ is a compact operator, $H$ is a Hilbert space, and let $(A_n)$ be a sequence of bounded linear operators on $H$ converging strongly to $A$. Show that $A_nT$ converges in operator norm to $AT$.
I'm attempting to prove this by contradiction: That ... |
H: Windowed Fourier transform of Gaussian distributed random time series
If I have a discrete time series $x(t_i)$, and each of the $x(t_{i})$ are normally distributed, i.e., come from a Gaussian distribution with mean zero and variance one, would a windowed finite Fourier transform of $x(t_0)$ through $x(t_{N-1})$ al... |
H: Residue Rings of Polynomial Rings over $\mathbb F_p$
Let $\mathbb Z$ be the ring of integers, $p$ a prime and $\mathbb F_p = \mathbb Z/p\mathbb Z$ the field with $p$ elements. Let $x$ be an indeterminate. Set $R_1 = \mathbb F_p[x]/(x^2-2)$, $R_2 = \mathbb F_p[x]/(x^2-3)$. Determine whether the rings $R_1$ and $R_2$... |
H: A question about convolution of two distributions
Generally,when taking convolution of two distributions,at least one of which is supposed to be of compact support.
But when u,$v\in S'(\mathbb{R})$ ( temperate distributions) have suports on the positive half axis,then $u\ast v \in S'(\mathbb{R})$
how to prove this... |
H: Calculus: volumes
My tutor gave me this question and I don't know how to tackle this type of question.
Consider the solid of revolution obtained when the region surrounded by $x=2$, $x=3$, $y=1$ and $y=\frac{x}{x+3}$ is rotated about the line $y=1$. Find an expression for the cross-sectional area of this solid at ... |
H: Convention of digit grouping after decimal point
I read that different cultures have different ways of grouping digits before the decimal point for readability e.g. 1234567890 can be grouped as 1 234 567 890 (English), 12 3456 7890 (Chinese) or 1 23 45 67 890 (Indian), but is there an established convention for gro... |
H: Probability Density Function: Small question
Let $X \sim N(\mu, \sigma^2)$ and $f_{X} {(x)} = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( -\frac{(x-\mu)^2}{2\sigma^2} \right)$, where $-\infty < x < \infty.$
Find the PDF of $X^3$.
Do you literally just sub $x^3$ into the function given?
AI: Let $Y=X^3$. Find first an ex... |
H: Do these two constructions lead to the same set?
Be $A=\bigcup_{n=0}^\infty A_n$ where $A_0=\emptyset$, $A_{n+1}=P(A_n)$.
Be $B=\bigcup_{n=0}^\infty B_n$ where $B_0=\{\emptyset\}$, $B_{n+1}=\{P(X):X\in B_n\}\cup\{X\setminus Y:X,Y\in B_n\}$.
Question: Is $A=B$?
Note: Here $P(X)$ denotes the power set of $X$.
AI: Cle... |
H: (Continued:) finiteness of étale morphisms
I was writing a question, it became too long, and i decided to split it into two parts. I hope posting two questions at the same time is not a problem.
First question: Checking flat- and smoothness: enough to check on closed points?
Now let $f: X \rightarrow Y$ be a morphi... |
H: Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $
Let $f$ be a nondecreasing, integrable, function defined on $[0, 1]$. Show that $$\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x \,f(x)^2\,dx $$
AI: Fact: Let $a$ and $b$ denote two nondecreasing functions on $[0,1]$, then $\int\limits_0^1ab\g... |
H: Inverse of symmetric matrix M = A*At
Possible Duplicate:
Inverse of symmetric matrix $M = A A^\top$
I have a matrix, generated by the product of a non-square matrix with its own transpose:
M = A * A^T
I need the inverse of M, assuming det(M) != 0.
Given the nature of the matrix M, are there any specialised comp... |
H: Checking flat- and smoothness: enough to check on closed points?
I am currently studying varieties over $\mathbb{C}$, i know some scheme theory.
Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition only at the closed points of $Y$? If yes, could you ... |
H: Why does this text insist on changing the variable name here?
In What is mathematics? by Courant, Robbins, and Stewart, "5. An important inequality", the authors change $n$ in this example:
$$(1+p)^n\geq1+np$$
to $r$ in this example:
$$(1+p)^r\geq1+rp$$
In other examples given by the book's author, he also switches... |
H: Intuition on proof of Cauchy Schwarz inequality
To prove Cauchy Schwarz inequality for two vectors $x$ and $y$ we take the inner product of $w$ and $w$ where $w=y-kx$ where $k=\frac{(x,y)}{|x|^2}$ ($(x,y)$ is the inner product of $x$ and $y$) and use the fact that $(w,w) \ge0$ . I want to know the intuition behind ... |
H: Normal subgroup of a group of automorphisms
let $A$ be a non empty set with a given structure, and let $G$ be the group of automorphisms of $A$, that is, the set of bijective maps from $A$ to itself that preserve its structure, with composition as group law.
Is it true that for any $a\in A$, the subgroup $G(a)$ of ... |
H: $f(n)=7^{6n} - 6^{6n}$, where $n$ is a positive integer.
$f(n)=7^{6n} - 6^{6n}$, where $n$ is a positive integer. find the divisors of $f(n)$ for odd and even values of $n$. Is there a general solution for the divisors.
$$f(1)=7^6-6^6=(7^3)^{2}-(6^3)^{2}$$
$$f(1)=(7-6)(7^2+(7)(6)+6^2)(7^3+6^3)$$
$$f(1)=(1)(127)(7^3... |
H: $(x-1)^{p-1} = 1+x+\dots+x^{p−1}$ mod p?
It is well known that in characteristic p we have the "freshman dream" $x^p -1 = (x - 1)^p$. Some experimentation seems to suggest that the heuristic computation $(x-1)(x-1)^{p-1} = (x-1)^p = x^p - 1 = (x-1)(1 + x + \dots + x^{p-1}) \implies (x-1)^{p-1} = 1 + x + \dots + x^... |
H: Totally ordered set with greater cardinality than the continuum
Does there exist a totally ordered set $S$ with cardinality greater than that of the real numbers? Sequences are continuous functions with domain $\mathbb{N}$ and paths are continuous functions with domain $\mathbb{R}$; both of them are very important ... |
H: Proving that a sequence such that $|a_{n+1} - a_n| \le 2^{-n}$ is Cauchy
Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy.
My Work
So by the definition of a Cauchy sequence, for all $\varepsilon > 0$ $\exists N$ so that for... |
H: Value of functions when approximating area using Riemann sums
When approximating the area beneath the curve $f(x)=x^2+1$ on the interval $[0,3]$ using a left Riemann sum with 4 rectangles, I calculated that the height of each rectangle, in order from left to right, would be
$f(0)$, $f(\frac{3}{4})$, $f(\frac{3}{2})... |
H: A problem for Matlab solver
I can numercally solve a system of two ODEs for x and p:
dy1 = -y1 + y2
dy2 = -1 + y2
with the following Matlab code:
%main.m
[t y]=rk4sys(20,0.05);
[t' y]
%rk4sys.m
function [t,y] = rk4sys(n,h)
C = [0;-1];
M = [-1,1;0,1];
y(1, :) = [0, 0];
f =@(t,y)(C+M*y')';
t(1) = 0;
for j = 1 : n... |
H: Question about Sobolev embedding theorem
The Sobolev embedding theorem as stated in my notes says that if we have $k > l + d/2$ then we can continuously extend the inclusion $C^\infty(\mathbb T^d) \hookrightarrow C^l(\mathbb T^d)$ to $H^k(\mathbb T^d) \hookrightarrow C^l(\mathbb T^d)$.
We define $H^k$ to be the cl... |
H: (ZF) Equivalent statements to Baire Category Theorem
So far, I have proved following two for a polish space $X$;
1.If $\{F_n\}$ is a family of closed subset of $X$, where $X=\bigcup_{n\in \omega} F_n$, then at least one $F_n$ has a nonempty inteior.
2.If $G_n$ is a dense open subset of $X$, then $\bigcap_{n\in \ome... |
H: What is the number of triangles with integer sides, given the length of the longest side?
Suppose $a,b,c \in\mathbb N$, and the value of $c$ is known and fixed, while $a$ and $b$ are unknown and are both smaller than $c$. What is the total number of unique triangles possible with $a, b$ and $c$ as its sides?
AI: Wh... |
H: What is the outer measure of Vitali set?
Possible Duplicate:
Vitali-type set with given outer measure
Given that the construction of Vitali set is based on the axiom of choice. How can the outer measure of this set be calculated?
AI: On Vital's request:
There isn't just one Vitali set: each choice of representat... |
H: An inequality involving the $\sup$ and $\inf$ of a bounded function and its absolute value
Let $f:[a,b]\to\mathbb{R}$ be a bounded function and set $$M=\sup_{[a,b]}f(x)\,,\; m=\inf_{[a,b]}f(x)\,,\;M^*=\sup_{[a,b]}|f(x)|\,,\;m^*=\inf_{[a,b]}|f(x)|\,.$$ Prove that $M^*-m^*\le M-m$.
First, since $f$ is bounded, all ... |
H: $\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$. Find the positive integer solutions (x,y).
$\frac{1}{x}+\frac{4}{y} = \frac{1}{12}$, where $y$ is and odd integer less than $61$.
Find the positive integer solutions (x,y).
AI: There is a neat trick for solving such equations in ... |
H: Inequality. $3(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)\geq (x+y+z)^2(xy+yz+zx)^2$
Let $x,y,z$ be positive real numbers. We have to prove that:
$$3(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)\geq (x+y+z)^2(xy+yz+zx)^2.$$
Thanks :)
AI: Note that $x^2+xy+y^2=\frac{3}{4}(x+y)^2+\frac{1}{4}(x-y)^2\ge \frac{3}{4}(x+y)^2 $
Repeating... |
H: Trigonometry problem involving oblique triangle
How would I solve the following problem?
A ship sails $15$ miles on a course $S40^\circ10'W$(south 40 degrees 10 minutes west) and then $21$ miles on a course $N28^\circ20'W$(north 28 degrees 20 minutes west). Find the distance and direction of the last position from ... |
H: Understanding algebraic method of successive approximations to solve quadratic equations
I have straight away copied and pasted a worked example from john birds higher engineering mathematics {page 80 problem 4} which reads as follows.
Use an algebraic method of successive
approximations to determine the value
of t... |
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