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H: What are rational integer coefficients?
I have a question about the following excerpt from Atiyah-Macdonald (page 30):
“A ring $A$ is said to be finitely generated if it is finitely generated as a $\mathbb Z$-algebra. This means that there exist finitely many elements $x_1,\dotsc,x_n$ in $A$ such that every eleme... |
H: Darboux's theorem
In Dunham's "Calculus Gallery", it introduces Darboux's theorem:
If $f$ is differentiable on $[a,b]$, and if r is any number for which $f'(a)<r<f'(b)$ then there exists a c in (a,b) such that $f'(r)=c$
Proof: [summarized] Introduce $g(x)=f(x)-rx$. There is a point $c$ in $[a,b]$ where g takes a m... |
H: Evaluating $\int_{0}^{\infty}\frac{\arctan \sin^2x}{x}dx$
Seems to be a hard nut:
$$I=\int_{0}^{\infty}\frac{\arctan \sin^2x}{x}dx$$ Any hint?
AI: It is not hard to show that
\begin{equation}\int_{0}^{\infty} \frac{\sin^2 x}{x} \; dx = \infty, \quad \cdots \quad (1)\end{equation}
and it is also easy to show that
... |
H: Infinite series: $1/2 + 1/(1\cdot 2 \cdot 3) + 1/(3\cdot 4 \cdot 5) + \ldots$
How do I calculate this: $$\frac{1}{2}+\frac{1}{1\cdot 2\cdot 3}+\frac{1}{3\cdot 4\cdot 5}+\frac{1}{5\cdot 6\cdot 7}+\dots $$
I have not been sucessful to do this.
AI: Hint:
$$ \frac{1}{n(n+1)(n+2)} = \frac{1/2}{n} - \frac{1}{n+1} + \frac... |
H: Minimizing a functional on $L^2$
Let
$$
\mathcal{M} := \left\{f \in L^2([0,\pi]): \int_0^\pi f(x)\cos x dx = \int_0^\pi f(x)\sin x dx = 1\right\}.
$$
Solve this problem:
$$ \tag{P}
\min_{\mathcal M} \int_0^\pi [f(x)]^2dx
$$
Using Cauchy-Schwarz, I get
$$
1 = \langle f(x), \sin{x} \rangle \le \Vert f \Ver... |
H: Proof by Induction $n^2+n$ is even
I'm not entirely sure if I'm going about proving $n^2+n$ is even for all the natural numbers correctly.
$P(n): = n^2+n$
$P(1) = 1^2+1 = 2 = 0$ (mod $2$), true for $P(1)$
Inductive step for $P(n+1)$:
$\begin{align}P(n+1) &=& (n+1)^2+(n+1)\\
&=&n^2+2n+1+n+1\\
&=&n^2+n+2(n+1)\end{... |
H: Set of limit points of continuous functions
Let $x_0$ be an accumulation point of the set $D \subset \mathbb{R}$. We say that $y$ is a limit point of a function $f:D \rightarrow \mathbb{R}$ in $x_0$ iff there exists a sequence $(x_n)$, where $x_n \in D\setminus \{x_0\}$ for $n\in \mathbb{N}$ and $x_n \rightarrow x_... |
H: Is there any graphical explanation of the derivative of $\sin x$?
I'm trying to understand in a practical/graphical view the derivative of $\sin(x)$ (that results in $\cos(x)$).
Is there any animation or illustration explaining that?
AI: MIT OCW's single variable calculus course has a interactive mathlet explainin... |
H: Solving $E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$
$$E=\frac{1}{\sin10^\circ}-\frac{\sqrt3}{\cos10^\circ}$$
I got no idea how to find the solution to this. Can someone put me on the right track?
Thank you very much.
AI: Divide both terms by two and use the fact $\sin(30) = \frac{1}{2}$ and
$\cos(30) = ... |
H: How can I determine which series comparison test to use?
In my textbook, there is a section of questions that's instructions reads "Test for convergence or divergence, using each one of the following tests once," and the test choices it gives me are
nth-Term Test
p-Series Test
Integral Test
Limit Comparison Test
G... |
H: $x$ algebraic over $K$, $v$ a polynomial in $x$ then $v$ algebraic?
In the proof of proposition 5.23 Atiyah-Macdonald on page 66 use that if $x$ is algebraic over $K$ and $v = a_n x^n + \dots + a_1 x + a_0$ then $v$ is algebraic over $K$ (where $K$ is the field of fractions of $A$ and $a_i \in A$). I tried to prove... |
H: How to prove the convergence for function of function series
How to prove the convergence for function of function series?
Say, here're two examples
Given $x_1>0, x_{n+1}=\ln(1+x_n)$, Prove $\lim_{n\to\infty}nx_n=2$
Given $0<x_1<1, x_{n+1}=\sin x_{n}$, Prove $\lim_{n\to\infty}\sqrt{n}x_n$ exist, and give this limi... |
H: Bounding Variance of a Convolution
Let $G$ be a group with finite subsets $A,B \subseteq G$. Let $k$ be an integer between $1$ and $|A|$ (or |A|/2 if it helps), and let $C$ be a random subset of $A$ of size $k$, chosen uniformly out of all such sets. We take $\mu_C = \frac{|A|}{k}1_C$ (where $1_X$ is an indicator... |
H: Why do we say the harmonic series is divergent?
If we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we continue to let $n$ increase, we will end up with a number so large in... |
H: Differential equation $d^n/dx^n f(x)=\pm k^2f(x)$
How to solve this differential equation:
$$\frac{d^nf(x)}{dx^n}=\pm k^2f(x)$$
For $n=1,2,3$ and $\forall n\in\mathbb{N}$, and both signs, if this is possible.
I encounter these often in physics, with solutions but no derivations. I would like to know how they are ... |
H: Swatting flies with a sledgehammer
Prompted by a recent exchange with Gerry Myerson, I was wondering if anyone has a favorite example of a relatively simple problem with a rather elementary (though perhaps complicated) answer for which there's another answer that relies on an elegant use of a powerful result that's... |
H: Determine if it is possible to fit 2 circles in a rectangle
I have the following problem:
Given a Rectangle with $L$ length and $W$ width and $2$ circles with $r_1$ and $r_2$ radius, determine if it's possible to fit these two circles inside the rectangle.
I realized that:
If $2r_1 > L$ or $2r_1 > W$ or $2r_2 > ... |
H: Use the Division Algorithm to show the square of any integer is in the form $3k$ or $3k+1$
Use Division Algorithm to show the square of any int is in the form 3k or 3k+1
What confuses me about this is that I think I am able to show that the square of any integer is in the form $X*k$ where $x$ is any integer. For ... |
H: For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2[x]$?
For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2[x]$ ?
E.g. $x+1$, $x^2+x+1$ are irreducibles.
Subcase of this question
Factor by irreducible is field. Does it... |
H: Why do these two methods of calculating the probability of winning a best-of-7 series give the same answer?
I was having a discussion with a friend about the probability, and we came up with very different methods to solve it that lead to the same answer. The problem is pretty simple: you have two teams A and B pla... |
H: The relation between the roots of a polynomial equation and a matrix equation
(I call matrix equation for example this $X^2-X+1=0$. Does it mean so?)
When a polynomial equation is solved for $x$, we get a complex number.
If we solve the "same" equation (same coeficients), such that the unknown is inside a matrix, i... |
H: How to I show that $\lim_{x\to0} \frac{1}{x^2}\left(\frac{\sinh x}{x}-1\right) = \frac{1}{6}$
I can do this limit with a symbolic calculator and get the result.
$$\lim_{x\to0} \left[ \frac{1}{x^2}\left(\frac{\sinh x}{x} - 1\right) \right] = \frac{1}{6}$$
But how would I do it by hand, and show why it is so. I know... |
H: Finding more details about a triangle using the given details.
In the triangle $ABC$ we have
$\tan{\frac{A}{2}}=\frac{1}{3}$
$b+c=3a$
Specify which of the following answers is correct:
$a) m(\angle B)=\frac{\pi}{2}$ or $m(\angle C)=\frac{\pi}{2}$
$b) m(\angle A)=m(\angle B)$
$c) m(\angle A)=\frac{\pi}{2}$
$d) m(\a... |
H: Unbounded element in $R^\infty$
Let $R^\infty$ be the vector space of all sequence $\{a_j\}$ of real numbers. Put
$\|\{a_j\}\|_n:= \sum_{j=0}^n |a_j|$. This collection of semi norms make this as Frechet space.
A set $B$ is bounded if every continuous seminorm is bounded on $B$. That is bounded set will be as $\{... |
H: Compute Hilbert function of a monomial ideal
I'd like to know whether there exist easy methods that compute the Hilbert
function of a graded $k$-algebra, without computer programs. My homework
asks to me to compute the Hilbert function of $R/I$, where $R=k[x_0, \dots,
x_5]$ and $$
I = (x_0 x_3, x_0 x_4, x_0 x_5,... |
H: What kind of book would show where the inspiration for the Laplace transform came from?
I'm trying to find out where to learn about integral transforms and inversions like the Laplace transform and the Bromwich integral.
I'm looking for a book that describes how you can find (derive) that the inverse of the Laplace... |
H: Binomial/Geometric Distribution explanation
I've found the following exercise in my Stats coursework. I only have solutions to it, but no explanation. And I would really like to know how to get to the answer.
An urn holds 5 white and 3 black marbles
If two marbles are drawn at random without replacement and X deno... |
H: Question on Cauchy Criterion of Series
We know that the Cauchy Criterion of a series is as follow (proof is taken as excerpt from an analysis book):
Theorem: A series $\sum_{j=1}^{\infty}a_j$ converges iff for all $\epsilon>0$ there is an $N\in \mathbb{N}$ so that for all $n\ge m \ge N$ we have $|\sum_{j=m}^{n} a_j... |
H: Deriving the formula for the radius of the circle inscribed in an equilateral triangle
I am trying to derive the formula for the radius of the circle inscribed in an equilateral triangle from scratch.
Given
$2*n$ = length of a side
$H$ = the altitude of the triangle = $h + a$
$h$ = the long subdivision (from the c... |
H: Geometric explanation of centroid of triangle
why is the point where the medians of a triangle meet also the center of mass of the triangle.
AI: I think I have it right this time; however the construction is not as simple as I had hoped. The idea is (was) to avoid calculus (explicitly, at least).
The basic idea is ... |
H: equivalence of continuity
Could you give me a hint on this problem?
Show that $f:A\subset\mathbb{R}^n\longrightarrow \mathbb{R}^m$ is continuous if and only if for every subset $B$ of $A$ , $f(A\cap\overline{B}) \subset \overline{f(B)}$.
Currently I know these definitions of continuity:
$(\rm i)$ In terms of pre-... |
H: Books like Grundlagen der Analysis in French
I am looking for some recommendations for a mathematics (text)book written in French. I am hoping to learn to read and write mathematics in French since I expect to take some mathematics courses that will be taught in French next year.
Basically, I would like a book who... |
H: Can someone resolve my confusion about uniqueness of diagonalization?
I am a bit confused about diagonalization. I have $A$ which I know is diagonalizable. I want to find $P$ such that $A = P \Sigma P^{-1}$ where $\Sigma$ is diagonal. Under what circumstances is $P$ unique, if ever? If it is not unique, is it at le... |
H: My book states that $\sum_{n=1}^{\infty}r^{-n} = \frac{1}{r-1}$ for $r > 1$
On page 9 of Edwards' Riemann's Zeta Function, he uses the equality $\sum_{n=1}^{\infty}r^{-n} = \frac{1}{r-1}$ for $r > 1$ to prove an identity connecting the gamma function and the Riemann zeta function. But how can this equality be righ... |
H: What is the order of discontinuity of this function?
Consider the function f(x) such that f(x) = 0 for all rational x and f(x) = 1 for all irrational x. It would seem that the number of 'jumps' up is uncountably infinite and the number of 'jumps' down is countably infinite; or is the other way around? Shouldn't the... |
H: What makes elementary row operations "special"?
This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" (as opposed to "arbitrary")?
It seems that they're always presented ... |
H: For what value of h the set is linearly dependent?
For what value of $h$ set $(\vec v_1 \ \vec v_2 \ \vec v_3)$ is linearly dependent?
$$\vec v_1=\left[ \begin{array}{c} 1 \\ -3 \\ 2 \end{array} \right];\ \vec v_2=\left[ \begin{array}{c} -3 \\ 9 \\ -6 \end{array} \right] ;\ \vec v_3=\left[ \begin{array}{c} 5 \\ ... |
H: Sufficient conditions for a function $g: A \subset \mathbb{R}^n \to \mathbb{R}^n$ to be locally bounded
Let $g: A \subset \mathbb{R}^n \to \mathbb{R}^n$ be an injective continuously differentiable function such that $\forall x \in A, \det g'(x) \neq 0$. Can I say that $g$ is locally bounded? By "$g$ is locally boun... |
H: When antidifferentiating, are we impliclty restricting to an interval?
I was asked this question by a student I am tutoring and I was left a little puzzled because his textbook only defines antiderivatives on intervals (which leads me to believe its author would answer the question in the title in the affirmative).... |
H: Solving the differential equation $\frac{dy}{dx} =\sqrt{7x^3}$
$$\frac{dy}{dx} =\sqrt{7x^3}$$
I need to use substitution on the $7x^3$ but I'm a little stuck. My car broke down on the way to class and I missed the lesson!
I get this far..
$$u =7x^3$$
$$du = 21x^2 dx$$
$$dx = \frac{du}{21x^2}$$
$$dy = \sqrt{u} \fra... |
H: Calculating a number when its remainder is given
I am having difficulty solving the following problem:
Marge has n candies , where n is an integer between 20 and 50.If marge divides the candies equally among 5 children she will have 2 candies remaining . If she divided the candies among 6 children she will have ... |
H: on sequences of Lebesgue measurable subsets of a compact set
In preparation for the real analysis qualifying exam at my grad school, I've been working through the recommended textbook Modern Real Analysis, by Zimmer (it's free online here).
For the last few days, I've been trying to figure out problem 4.27 (on p12... |
H: Why is associativity required for groups?
Why is associativity required for groups?
I'm doing a linear algebra paper and we're focusing on groups at the moment, specifically proving whether something is or is not a group. There are four axioms:
The set is closed under the operation.
The operation is associative.
T... |
H: Closed set in a Hausdorff topological space
Possible Duplicate:
$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed
I'm trying to prove: If $X$ is a Hausdorff topological space and $\Delta \subset X\times X$ such that $\Delta=\{(x,y): x=y\}$, prove that $\Delta$ is closed.
I can not use sequenc... |
H: How to put this problem into equation?
I start with a value A. I decrease it by M each month m. Every year (ie, m mod 12 =0), I calculate the average of what A has been throughout the year, multiply that by t, and add it to A. This goes on until A reaches 0 (t and M are set in such a way that A decreases every year... |
H: Evaluating $\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$
How do I integrate this expression:
$$\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$$.I got this in a book.I do not know how to evaluate integrals of this type.
AI: One uses trigonometric substitution: $t = \tan\left(\frac{x}{2}\right)$. Then
$$
... |
H: Groups such that every finitely generated subgroup is isomorphic to the integers
What are examples of groups such that every finitely generated subgroup is isomorphic to $\mathbb{Z}$?
AI: Suppose any nontrivial finitely generated subgroup of $G$ is isomorphic to $\Bbb{Z}$, and that $G$ itself is nontrivial. Choose... |
H: Why $g(x^{p})=(g(x))^{p}$ in the reduction mod $p$?
In one of the proof in the book "Abstract Algebra'' by Dummit and
Foote (Theorem 41, pg. 554) we have a monic polynomial $g(x)\in\mathbb{Z}[x]$,
and the book claims that $g(x^{p})=(g(x))^{p}\mod p$
Can someone please explain why this is true ? I know that $\foral... |
H: Integrating definite integrals in terms of area
Lately, I've been trying to come up with tricks to solve integrals quickly.
So let's say I have $$\int_{0}^{2\pi} \cos^2 \theta d\theta$$
Now if I were to look at this integral in polar coordiantes, I get
$$\frac{1}{2}\int_{0}^{2\pi} \cos^2 \theta d\theta$$
The integr... |
H: Reference for upper and lower bounds on $e^x$
I'm looking for a reference for deriving the following commonly used upper and lower bounds for $e^x$:
$$1 - x \le e^{-x}$$
and, assuming $x \le 1/2$,
$$1 - x \ge e^{-2x}.
$$
AI: First Inequality: Let $f(x)=e^{-x}-(1-x)$. Then $f'(x)=-e^{-x}+1$. This is $0$ at $x=0$, ... |
H: Proving that $\Phi_{n}$ is irreducible (a problem with the proof)
I am trying to follow the proof in the book Abstract Algebra by Dummit and Foote (Theorem 41, pg. 554) that $\Phi_n$ is an irreducible monic polynomial in $\mathbb{Z}[x]$ of degree $\varphi(n)$.
What I understand is that if it is not irreducible, tha... |
H: Help with set notation?
I want to describe the set of all words in the following format: a0w1
where a represents EITHER 0 or 1, and w represents {0,1}*
So 00011 is valid as is 1010011, etc. etc.
I'm really new to set notation, so I'm not sure what I can do.
Is L = {a,0,w,1 | a = 0 or 1, w $\in$ {0,1}*} valid for w... |
H: Group of groups
The product $\times$ of two groups is associative and commutative and there's a neutral element $\{1\}$. Let's say I create "virtual groups" which are inverses with respect to $\times$ (like getting $\mathbb{Z}$ from $\mathbb{N}$). Then I have a group $G$ whose elements are all groups.
This isn't ... |
H: Is there a monotonic function discontinuous over some dense set?
Can we construct a monotonic function $f : \mathbb{R} \to \mathbb{R}$ such that there is a dense set in some interval $(a,b)$ for which $f$ is discontinuous at all points in the dense set? What about a strictly monotonic function?
My intuition tell... |
H: When is this quotient by an action on the product of a variety with itself non-singular
Let $X$ be a smooth projective geometrically connected variety over a field $k$.
Let the cyclic group $G=\{e,a\}$ with two elements act on $X \times X$ via $a\cdot (x_1,x_2) = (x_2,x_1)$.
When is the quotient $X\times X/ G$ non... |
H: A question about a weak form of Hilbert's Nullstellensatz
Corollary 5.24 on page 67 in Atiyah-Macdonald reads as follows:
Let $k$ be a field and $B$ a finitely generated $k$-algebra. If $B$ is a field then it is a finite algebraic extension of $k$.
We know a field extension $E$ over $F$ is algebraic if it's finite,... |
H: Can two collections of different size have same A.M., G.M. and H.M?
After following
Can two sets have same AM, GM, HM? and the sublime answer of Micah, I am tempted ask the solution of the same question when size of these two sets are not same.
AI: Sure, why not? The general statement that my answer was a special... |
H: physics related question
i am trying to calculate simple problem from physic,but i am getting somehow wrong answer.problem is that
what is a mass of bag which is hold by child with mass $50$KG,if there is force of heaviness on both which is equal $600$N
so in shortly,we know that on child and bag,there wo... |
H: Proof of error propagation formula?
In my course we have stated and used the error propagation formula:
$$|y-y_0|\approx|f^\prime(x)|\cdot|x-x_0|$$
But it was presented with no proof and I wonder if you can help me understand the formula holds?
AI: Possibly the best way to understand it is via the mean value theore... |
H: How many such squares can be formed?
$S$ is a set of all points $(a, b)$ such that $0 ≤ a$, $b ≤ k$. How many squares are there such that all the $4$ vertices are from set $S$?
For diagonal squares, a square must contain odd points on its side.
so that we can join the mid points of each side.
suppose we take $5$ po... |
H: Special dot-product
I have been wondering if the following dot product definition for the $n$-coordinate vectors $a$ and $b$ has a name:
$$<a\backslash b> = \sum_{i=1}^{n} a_i*b_{n-i+1},$$
rather than the classical dot product:
$$<a\backslash b> = \sum_{i=1}^n a_i*b_{i}.$$
Did you already seen it use somewhere?
Tha... |
H: Is there an unique "minimal enclosing group" for any two groups?
I'm not sure I'm using the correct terms, therefore let me define what I mean:
Given a set of groups $G_i$, $i\in I$, I call an enclosing group of those groups any group $G$ so that for all $i\in I$ there exists a subgroup $H_i$ of $G$ which is isomor... |
H: Using the substitution $p=x+y$, find the general solution of $dy/dx=(3x+3y+4)/(x+y+1)$.
Using the substitution $p=x+y$, find the general solution of $$\frac{dy}{dx}=(3x+3y+4)/(x+y+1)$$
Here are my steps:
Since $p=x+y$, $$\frac{3x+3y+4}{x+y+1}=\frac{3p+4}{p+1}=\frac{1}{p+1}+3$$
Therefore, integrate both sides
$$y=\... |
H: A question about a proof of a weak form of Hilbert's Nullstellensatz
I'm trying to prove the following (corollary 5.24 page 67 in Atiyah-Macdonald):
Let $k$ be a field and let $B$ be a field that is a finitely generated $k$-algebra, i.e. there is a ring homomorphism $f: k \to B$ and $B = k[b_1, \dots , b_n]$ for $b... |
H: homeomorphism of a subset of $GL_3(\mathbb{R})$ with $GL_2(\mathbb{R})$ and connectedness
Suppose I denote $G_3$ be the set of all $3\times 3$ matrices with positive determinant, and consider the map $\pi:G_3\rightarrow \mathbb{R}^3\setminus\{0\}$ define by $\pi(g)=ge_1$ where $e_1=(1,0,0)$ then i want to know abou... |
H: Integral-Summation inequality.
The following question was in an entrance exam:
Show that, if $n\gt0$, then: $$\int_{{\rm e}^{1/n}}^{\infty}{\frac{\ln{x}}{x^{n+1}}\:dx}=\frac{2}{n^2{\rm e}}$$ You are allowed to assume $\lim_{x\to\infty}{\frac{\ln{x}}{x}}=0$. Hence explain why, if $1\lt a\lt b$, then: $$\int_{b}^{\i... |
H: Are there an infinite set of sets that only have one element in common with each other?
In a card game called Dobble, there are 55 cards, each containing 8 symbols. For each group of two cards, there is only one symbol in common. (The goal of the game being to spot it faster than the other players, which is not the... |
H: Transition Kernel of a Markov Chain
Supposing $X_t$ is a Markov Process, can the transition kernel be defined by
$$K_t(x,A):= P(X_{t+1} \in A | X_t = x)?$$ Assume that $X_t : \Omega \to \mathbb{R}^n$.
The issue is that under the normal definition of conditional probability, r.h.s is defined as $$P(X_{t+1} \in A | X... |
H: Numerical Methods for Linear Matrix Equation
How can I solve (numerically) the linear equation $AB=0$. where $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$?
How much is the computational cost?
AI: You may treat $B$ one column at a time:
\begin{equation}
B = \Bigg[b_1\;\;b_2\;\;\ldots\;\;b_m\Bigg]
\e... |
H: Closure of the set of all polynomial with variable $x\in [0,1]$
Let ${P}$ denote the set of all polynomial with variable $x\in [0,1]$, I need to know what is the closure of ${P}$ in $C[0,1]$?
Well, Stone-Weierstrass theorem says: If $f\in C[0,1]$ then there exists a sequence of polynomials $p_n(x)$ which converges ... |
H: What is the Least Common Multiple of $(a-b)$ and $(b-a)$?
What is the Least Common Multiple of $(a-b)$ and $(b-a)$?
The question is simple. What's the answer? I'm an Engineering student, but looks like I forgot my basics.
AI: $|a-b|$ is the LCM of $(a-b), (b-a).$ Recall LCM is the smallest positive integer that i... |
H: Sequence of continuous functions with bounded derivative
Let $f_n$ be a sequence of continuous functions on $[0,1]$, and continuously differentiable on $(0,1)$. Assume $|f_n|\le 1$ and $f_n'\le 1$ $\forall x\in [0,1]$ and $n$. Then
$f_n$ is a convergent sequence in $C[0,1]$
$f_n$ has a convergent subsequence in $C... |
H: An inequality of red and black balls
We have a box containing red and black balls. If we draw two at random the probability of getting both of them red is $1/2$. Which basically means:
\begin{equation}
\frac{r}{r+b} \cdot \frac{r-1}{r+b-1} = \frac{1}{2}
\end{equation}
Then we have for a positive number of red and b... |
H: Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$
Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show,
$$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$
when $X$ has : a) a discrete distribution, b) a continuous distribution.
I assumed that f... |
H: Equation involving an integral depending on two parameters
I have some difficulty to find possible solutions of the following equation:
$$\int_0^\tau dx \frac{1}{x^\alpha+1}=\beta$$
where $\tau \gt 0,$ $\alpha\in \mathbb N$ ($\alpha=1,2,3,\dots$) and $\beta$ a given real valued constant.
Is it possible to find valu... |
H: Iterating the transform $(a,b)\mapsto(a+b+\sqrt{a^2+b^2} ,a+b-\sqrt{a^2+b^2})$
Assume that $a_0=-2$, $b_0=1$, and that, for every $n\ge0$,
$$a_{n+1}=a_n+b_n+\sqrt{a^2_n+b^2_n}
\qquad
b_{n+1}=a_n+b_n-\sqrt{a^2_n+b^2_n}$$
How to find $a_{2012}$?
AI: Here is a 7-steps plan:
Stop asking questions with no indication wh... |
H: Subgroups between $S_n$ and $S_{n+1}$
Lets look at $S_n$ as subgroup of $S_{n+1}$.
How many subgroups $H$, $S_{n} \subseteq H \subseteq S_{n+1}$ there are ?
AI: None.
Let $S_n<H<S_{n+1}$ and suppose that $H$ contains a cycle $c$ involving $n+1$, say
$$
c=(a,\cdots,b,n+1).
$$
Then by composing to the left with a su... |
H: All valuations equal one : unit?
Let $F$ be a global field with integers $o$, and let $x \in F$. Does $|x|_v =1$ for all non-archimedean valuations of $F$ imply that $x \in o^\times$.
AI: Let $x\cal O$ the (possibly fractional) ideal generated by $x$. If it is non-trivial, i.e. $\cal O\neq x\cal O$, it must be a pr... |
H: How can i compute the probability that a cloud of points was created by a probability distribution?
I have a number of probability distributions that describe a number of points. like this:
Now if i have draw a one point out of each distribution, i get a bunch of points randomly set on the 2-dimensional surface. H... |
H: Short matrix algebra question
$A$ is a square matrix, and there is a matrix $D$ such that $AD=I$.
I need help to prove either (a) or (b) below:
(a) Show that for each $\vec b$, $A\vec x=\vec b$ has a unique solution.
OR
(b) Show that $A$ is invertible.
For (a), $AD\vec b=I\vec b = \vec b$, so obviously the given e... |
H: Proving an elementary integral inequality (in early Dirichlet space material)
I'm reading up on Dirichlet spaces using this document here and, on page two, am stumped by a particular integral inequality. If this is trivial and/or I'm missing something blatant, I apologize.
First we define, for any analytic function... |
H: pseudo numbers and surreal numbers
A surreal number $\{x_L\|x_R\} \in No$ is a number when for all $\xi\in x_L$ and all $\eta \in x_R$ we have $\eta > \xi$. All the things $\{x_L\|x_R\}$ which are not of that form are called "pseudo-numbers" and usually ignored (except in certain game-theoretic applications).
Quest... |
H: Calculating the minimum of $\cos x \sin y$
I am about to start university in October, to study computer science, and have been asked by my university to complete a number of problem sheets. I have become stuck on the following question, and therefore would appreciate any help possible.
The numbers $x$ and $y$ are ... |
H: proving convergence for a sequence defined recursively
The sequence $\left \{ a_{n} \right \}$ is defined by the following recurrence relation:
$$a_{0}=1$$ and $$a_{n}=1+\frac{1}{1+a_{n-1}}$$ for all $n\geq 1$
Part 1)- Prove that $a_{n}\geq 1$ for all $n\geq 0$
Part2)- Prove that the sequence $\left \{ a_{n} \right... |
H: Proof of Hilbert's Nullstellensatz
I'm working through my notes and I'm stuck in the middle of the proof of Hilbert's Nullstellensatz.
(Hilbert's Nullstellensatz) Let $k$ be an algebraically closed field. Let $J$ be an ideal in $k[x_1, \dots , x_n]$. Let $V(J)$ denote the set of $x$ in $k$ such that all $f$ in $J$... |
H: Algebraic Topology pamphlets?
I'm looking to self-learn some Algebraic Topology and have found the books I've looked at so far (ie. Hatcher) to be rather tome-like for my tastes. Does anyone know of a good slim lecture notes style book (or, indeed, an actual set of lecture notes) I can have a look at?
I need to ca... |
H: Why is $B[x]/M$ algebraic over $B/m$?
Let $B$ be a subring of some field $K$, $x$ some element in $K$, $m$ a maximal ideal in $B$ and $m[x]$ the extension of $m$ in $B[x]$ and $M$ a maximal ideal in $B[x]$ such that $m[x] \subset M $ and $M \cap B = m$.
Why is $B[x]/M$ algebraic over $B/m$? Thank you.
AI: The $(B/\... |
H: proving existence of a sequence such that the limit exists?
Can anyone prove the existence of a sequence $(n_{k})_{k\in \mathbb{N}}$ of distinct positive integers such that the limit: $\lim_{k\rightarrow \infty }\sin(n_{k})$ exists in $\mathbb{R}$
I can definitely construct a sequence $(n_{k})_{k\in \mathbb{N}}$ su... |
H: Which of the following are compact sets?
Which of the following are compact sets?
$\{\operatorname{trace}(A): A \text{ is real orthogonal}\}$
$\{A\in M_n(\mathbb{R}):\text{ eigenvalues $|\lambda|\le 2$}\}$
Well, orthogonal matrices are compact, but the trace of them may be any $x\in\mathbb{R}$, so I guess 1 is n... |
H: Under which conditions does $a^n \equiv 1\mod(b) \Rightarrow\ a^{n^m} \equiv 1\mod(b) $? Can you prove it?
Under which conditions does $a^n \equiv 1\mod(b) \Rightarrow\ a^{n^m} \equiv 1\mod(b) $?
What about viceversa?
What is the strongest result(s) that can be proved regarding this kind of thing?
I'm kind of ge... |
H: Prove that a monotonically increasing continuous function is invertible.
Let $f:[a,b]\rightarrow[f(a),f(b)]$ be strictly increasing continuous function (i.e $x>y \implies f(x)>f(y)$). Prove that f is invertible.
Proving that the function is one-to-one was simple enough. I need some guidance on proving it's onto. I'... |
H: Units in number fields with complex embeddings
Assume that we have an algebraic number field with integers $o$, and with a complex embedding $\iota$.
What can be said about the image $\iota( o^\times)$ under $\iota$?
Is it discrete? Is infinite?
AI: As has been noted in comments, the important result here is Diric... |
H: Is R with $j_d$ topology totally disconnected?
Let consider the topological space $R_j=(\mathbb{R},j_r)$ where $j_r$ is generated by right side open intervals, i.e $[a,b)$ for $a,b \in \mathbb{R}$; note that this topology includes the euclidean topology.
$R_j$ is not a connected space, because given $x \in \mathbb{... |
H: Is the space of continuous function with given order of decrease closed?
For example, denote $O^1$ the space of continuous function with property $\lim_{x\to\infty}{|x|f(x)}=0$ or $f(x)=o\left(|x|^{-1}\right)$ as $x\to\infty$. It's obviously an intermediate vector space between $C_c$ and $C_0$. Is it closed?
Assume... |
H: Extension of a continuous function in $\mathbb{R}^2$
Well, for (a) I have no idea how to extend, I feel that there will be a continuous extension. For example I can define $f=g$ when $f$ takes values from upper boundary of the disk and upper half plane and same way for lower boundary and lower half plane.
For (b) ... |
H: which of the following are homeomorphic?
well, I have forgotten how to identify ellipse, hyperbola,circle straightline from the general equation of conic, so is there any other way to identify these homeomorphic or not?
a) B is an ellipse, b) B is an hyperbola, c) B is an complement of a closed ellipse.
please hel... |
H: What are the differences between class, set, family, and collection?
In school, I have always seen sets. I was watching a video the other day about functors, and they started talking about a set being a collection, but not vice-versa. I also heard people talking about classes. What is their relation? Some backgroun... |
H: a question on normal subgroup of $GL_n(\mathbb{C})$ and $GL_n(\mathbb{R})$
I am really sorry that I am not able to solve this one, thank you for your help.
AI: $a$ and $c$ are true by abstract group theory.
The first one shoudn't be too hard. Just write it out.
The third one basically says that the path component ... |
H: Linear dependence of a set for what h?
I asked the same question yesterday, but this one is a bit different in terms of computations. It is from my exam I took an hour ago.
For what $h$ the columns of this matrix are linearly dependent? $$\begin{bmatrix} 1 & -3 & 4 \\ -4 & 7 & h\\ 2 & -6 & 8 \end{bmatrix}$$
Attem... |
H: Prove that $X´$ is a closed set
Let $X\subset \mathbb{R}$.I have to prove,for all $X$ that $X´$,i.e, the set of accumulation points, is a closed set.
Well,I know , by definition, that every accumulation point is a point of closure.
How a set is said to be closed if $X = \overline X$ ,and how the accumulations point... |
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