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H: Continuous function from Compact Metric Space to Metric Space is Uniformly Continuous
In Rudin 4.10 he wants us to show that a continuous function from a compact metric space to a metric space is uniformly continuous by deriving that if $f$ is not uniformly continuous then there is an $\epsilon >0$ such that there ... |
H: Integral $\int_0^{\infty} \log(x) e^{-x^2} \mathrm{d}x = -\frac{1}{4}\sqrt{\pi} (\gamma + \log(4)).$
While trying to compute the expected value $E[\log(X)]$ for a normally distributed variable $X$ I found the following integral
$$\int_{0}^{\infty}\log\left(x\right) {\rm e}^{-x^{2}}\,{\rm d}x
=-\,{1 \over 4}\,\,\sqr... |
H: How prove this $\alpha+\beta+\gamma=n\pi$
let $\theta\in R$,and $\alpha\neq\beta\neq\gamma$
and such
$$\dfrac{\cos{(\alpha+\theta)}}{\sin^3{\alpha}}=\dfrac{\cos{(\beta+\theta)}}{\sin^3{\beta}}=\dfrac{\cos{(\gamma+\theta)}}{\sin^3{\gamma}}$$
prove
$$\alpha+\beta+\gamma=n\pi$$
My try: let
$$\dfrac{\cos{(\alpha+\thet... |
H: How to integrate $\frac{4x+4}{x^4+x^3+2x^2}$?
Please could anyone help me to integrate
$\quad\displaystyle{4x + 4 \over x^4 + x^3 + 2x^2}.\quad$
I know how to use partial fraction and I did this:
$$
x^{4} + x^{3} + 2x^{2} = x^{2}\left(x^{2} + x + 2\right)
$$
And then ?.$\quad$ Thanks all.
AI: We have:
$$ \int{\fra... |
H: Integral equations that can be solved elementary
Solve the following integral equations:
$$
\int_0^xu(y)\, dy=\frac{1}{3}xu(x) \tag 1 \label 1
$$
and
$$
\int_0^xe^{-x}u(y)\, dy=e^{-x}+x-1. \tag 2 \label 2
$$
Concerning $\eqref 1$, I read that it can be solved by differentiation. Differentiation on both side... |
H: Definition for the action of a category on a set.
I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one.
If one has a functor $\rho: C \to Set$, one takes the set S as the disjoint union of the $\rho(c)$ for all objects $c$ of $C$, and t... |
H: Finding value of equation without solving for a quadratic equation
How do I go about solving this problem:
If $α$ and $β$ are the roots of $x^2+2x-3=0$, without solving the equation, find the values of $α^6 +β^6$.
In my thoughts: I commenced by expanding $(α +β)^6$, such that:
$$(α +β)^6 =α^6+6α^5β+15α^4β^2+20α... |
H: Quotient space is connected...
Let $X$ be a topological space and $\sim$ be an equivalence relation defined on it. Let $Y$ be the space $X/{\sim}$ and $p :X \rightarrow Y $ be the quotient map, and give $Y$ the quotient topology. If $Y$ is connected, must $X$ be connected as well?
AI: Not necessarily. Let $X$ be ... |
H: Convergence of $\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$
I have to show that the following series convergences:
$$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$
I have tried the following:
The alternating series test cannot be applied, since $\frac{2+(-1)^n}{n+1}$ is not monotonically decreasing.
I tried ... |
H: Where am I wrong in finding area of this triangle?
I was self-reading Mathematics for Economists by Simon and Blume. On page 815, Section 29.4, he has discussed "Norms on Function Space". And here I am stuck:
Let $$f_n =
\begin{cases}
2n^2-2n^3x, & \text{$0\leq x\leq\frac1n$,} \\
0, & \text{$\frac1n\leq x\leq1.$}... |
H: Show that $x|y \Rightarrow x \leq y$
I have to show that $x|y \Rightarrow x \leq y$ where $x,y \in \mathbb{N} \land x,y \neq 0$
Can someone give me a start hint how I can show this? I guess I can proof by induction. Not quite sure where to start
$x|y \Leftrightarrow xn =y$
AI: $y-x=xn-x=x\left(n-1\right)\geq0$
(Her... |
H: Fixed points of self-conformal mappping
Given a conformal self map f (analytic function from unit disc to itslef that is one to one and onto), such that it is not identity.
I need to show that either f has two fixed point on the boundary or one fixed point inside that unit disc.
Thank you beforehand for your help!
... |
H: True/False test: $P(x)=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}.$ Then, $\lim_{n\to\infty}\frac{e^x}{P(x)}=1$
True/False test: $P(x)=1+x+\dfrac{x^2}{2!}+...+\dfrac{x^n}{n!}$ where $n$ is a large positive integer. Then, $\displaystyle\lim_{n\to\infty}\dfrac{e^x}{P(x)}=1$
The paper says it's false but I can see since $P... |
H: Show that diagonal of a square is not a face
Let C be a convex set in $R^n$. We say that F is a face of C if the following condition holds: if $x_1, x_2 \in C$ and $(1-\lambda)x_1+\lambda x_2 \in F$ for some $0\lt \lambda \lt 1$ then $x_1, x_2 \in F$
Based on this definition, how can we show that a diagonal of a sq... |
H: Why isn't $\log(-1)$ defined?
Why isn't $\log(-1)$ defined?
It can be defined as being equal to $i\pi$. Why don't we define the $\log$ function over Complex Numbers as well?
AI: Logarithm is defined for complex numbers too. http://en.wikipedia.org/wiki/Complex_logarithm
It is multivalued function. One of values of ... |
H: The milk sharing problem
I found a book with math quizzes. It was my father's when he was young. I encountered a problem with the following quiz. I solved it, but I wonder, is there a faster way to do it? If so, how can I compute the time (polynomial time) that is needed to solve it? Can we build an algorithm?
The... |
H: $x^3+y^4=7$ has no integer solutions
I am trying to prove that $x^3+y^4=7$ has no integer solutions, but i have no idea how to start, please helps. I have tried to consider mod 7 to restrict the number of possible $x^3$ because $x^3 \equiv -1,0,1 \pmod{7}$, but it is not working.
AI: Consider the equation modulo $1... |
H: Spherical Harmonics expansion of a Dirac Delta at the North Pole
I think all the coefficients for the spherical harmonic expansion of a delta function at the north pole should be a constant (presumably 1), but I'm having difficulties calculating them. Could someone kindly take some time showing me how they are calc... |
H: Line and a triangle never intersect at exactly 3 points - proof verification
Let´s suppose that there is a line l on which lie exactly 3 points of some triangle. Lets assume that none of the points are vertices of the triangle. Then, since no line intersects a side of a triangle at more than 1 (and less than infini... |
H: Show that every polytope is bounded
The definition of polytope is the convex hull of a finite set. Thus:
$$
\parallel\sum_j\lambda _j x_j\parallel\le\sum_j\lambda_j\parallel x_j\parallel\le\sum_j\lambda_j\max_j \parallel x_j\parallel=M\sum_j\lambda_j=M
$$
where $M=\max_j \parallel x_j\parallel$
And how can I conclu... |
H: Drawing a graph of potential energy as a function of displacement $x$.
The water density is changing linearly with the displacement x>0, i.e. $\rho = \rho_0 + kx$, where $\rho$ > 0 and k>0. Also, assume that $\rho_0V < m $.
I know that potential energy, $V = - \int F \ dx $.
I have that my Force, $F = mg - \rho Vg$... |
H: Does one need a countable transitive model of ZFC to force the falsity of CH and "There exists a nonconstructible real" ?"?
In an answer and comment to
Does forcing need a countable transitive model
it was suggested that given a forcing argument using a c.t.m., one could always translate the same argument into a... |
H: show that if $X\ge 0$ , $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
if $X$ is a random variable and also let $X\ge 0$ , I want to show $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
AI: Show $X \le \sum_{n=0}^\infty I_{X > n}$. Note the right hand side is nothing more than $\text{ceil}(X)$, where "ceil" or "ceiling" means round up... |
H: Construction of module on an Abelian group.
Let $M$ be an Abelian group and let $\operatorname{End} M$ be the set of all endomorphims on $M$. Then $(\operatorname{End} M,+,\circ)$ forms a unitary ring. In this setting, if $R$ is a unitary ring and $\mu:R\to \operatorname{End} M$ is a ring homomorphism such that $\m... |
H: Predicate Logic: Difference between 'who' and 'if' in symbolization
Consider the two sentences:
(1) "chessplayers are rich if they are professional"
(2) "chessplayers who are professional are rich"
and the key:
UD: Living things
Cx: x is a chessplayer
Px: x is professional
Rx: x is rich
Now when I write (1)... |
H: Review of solution: Prove $\liminf({a_n}) \ge \liminf({b_n})$
${a_n} \ge {b_n}\forall n \in $
Prove: $\liminf({a_n}) \ge \liminf({b_n})$
I proved it by contradiction. Let's assume $\liminf({a_n}) < \liminf({b_n})$.
$a := \liminf({a_n})$
$b := \liminf({b_n})$
So, by the definition of partial limit:
$\eqalign{
& \f... |
H: How prove can't have$ |x|<|y-z|, |y|<|z-x|, |z|<|x-y|$
Question:
let $x,y,z\in R$,show that follow Can't be set up at the same time.
$$\begin{cases}
|x|<|y-z|\\
|y|<|z-x|\\
|z|<|x-y|
\end{cases}$$
My try: if this all is set up,then we have
$$x^2<(y-z)^2,y^2<(z-x)^2,z^2<(x-y)^2$$
so
$$x^2+y^2+z^2<2(x^2+y^2+z^2)-2(x... |
H: Negation of Bayes' theorem.
This is self-learning.
This is very hard to find, there are examples with numbers but none with ven diagrams. This is not homework, I'm studying Markov Chains and have little confidence with conditional probability.
We all know:
$$\mathbb{P}[A|B]=\frac{\mathbb{P}[A\ \text{and}\ B]}{\math... |
H: Convergence of sequence b
Let $b_n$ be a sequence such that $b_{n+1}=\frac{b_n^2+1}{b_n} \ , \ b_1>0$
Is this sequence converging? explain.
I managed to find that the series is monotically incresing, but couldn't show it is not bounded, thus not converging.
Thanks!
AI: In fact the sequence does not have a limit. S... |
H: Drawing a bufircation diagram
$\dot x=x(\mu+x-2)(\mu+2x-x^2)$
The first thing I did was to check the fixed points in the $(\mu,x)$-plane:
$x=0$
$x=2-\mu$ (saddle node at 0 when $\mu$)
$x_{1,2}=1+-\sqrt{\mu+1}$ (no fixed point for $\mu<1$)
Did I specified the type for the bufircation points correctly?
How does the b... |
H: A short question about an e identity
Why is this $\{(1+\frac1{a_n})^{a_n}=e\}$ true when: $a_n \to -\infty$
$a_n$ is a sequence.
Thanks.
AI: Hint: Put $b_n=-a_n,$ and note then that $$\left(1+\frac1{a_n}\right)^{a_n}=\left(1+\frac{-1}{b_n}\right)^{-b_n}=\left(\left(1+\frac{-1}{b_n}\right)^{b_n}\right)^{-1}.$$ What ... |
H: Factorizing $(x-1)(x-3)(x-5)(x-7)-64$
We need to factorize:
$$(x-1)(x-3)(x-5)(x-7)-64$$
We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$.
So this means that if the first part of the polynomial is a square,we can rewrite the whole polynomial as ... |
H: Symmetry property for Laguerre polynomials
The classic type of orthogonal polynomials $H_n(x)$, $P_n(x)$, $T_n(x)$ and $U_n(x)$ have symmetry property.
Does the Laguerre polynomials $L_n(x)$ also has symmetry property?
$L_n(-x)=L_n(x)$ and $L_n(-x)=-L_n(x)$
Pls help me. Thanks
AI: From glancing at the first few, th... |
H: Integration with substitution
I want to integrate (2x+1)/((x^2 - 6x + 14)^3)
I'm guessing you use substitution but im unsure what to substitute, is it best to make u = x^2 or X^2-6x or even x^2 - 6x + 14
I find that it makes turning the top of the fraction into terms of u very difficult
AI: $$\begin{align} \int \fr... |
H: ${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$
I need to prove that $${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$$
I started by writing out all the terms using the formula ${n!\over{k!(n-k)!}}$ but I can't make the two sides equal.
Thanks for any help.
AI: $$ {2n \choose ... |
H: Usage of mixed and scalar product vectors
I'm writing a project and I am stuck on the last part which I have to explain the usage of mixed and scalar product of vectors, where are they useful, what could be done with them in the future and so on. Where can I find information about this?
AI: You can do a lot things ... |
H: integration for x>5
I want to integrate (x^2-25)^0.5 / x
What method is best to use for this?
Also if it doesn't state values for the integral, just that x>5, is there something in particular that needs to be done to accommodate this? thanks
AI: Using Trigonometric substitution, $$x=5\sec\theta$$
$$\int\frac{\... |
H: Unusual Differential Equation
I have the differential equation
$$y''y+n(y')^2=0\tag 1$$
I haven't found any references to equations of this type by searching. Does anyone have a suggestion as to how to proceed?
It comes about from the following progression:
$$y=(y')^n-\frac 1n\tag 2\\
y'=ny''(y')^{n-1}\implies1=ny... |
H: How find this integral $\int_{0}^{x}\left(\frac{1}{2}-\{t\}\right)dt$
Find the value
$$F(x)=\int_{0}^{x}\left(\dfrac{1}{2}-\{t\}\right)dt$$
where$\{x\}=x-[x]$
my try: since
$$F(x)=\int_{0}^{x}\left(\dfrac{1}{2}-t+[t]\right)dt$$
when
$$k-1\le t<=k,[t]=k-1,k\in Z$$
because $x$ is not integer,and maybe $x\to \i... |
H: Linear Algebra and Set Theory book recommendations.
I would like to studying linear algebra and set theory.
Does anyone have a a good recommendation of books/resources/etc.?
AI: For Linear Algebra, I recommend Linear Algebra Done Right by Sheldon Axler.
$\mathrm{}\\$For Set Theory, I recommend Naive Set Theory by... |
H: We are looking for a function $f(x)$ that satisfies the functional equation of both $f(1)=1$ and $f(x+1)=xf(x)$.
We have the functional equation of both $f(1)=1$ and $f(x+1)=xf(x)$, where $x$ is a real number.
We need to show that this equation has an infinite number of solutions $f(x)$.
I have received a hint that... |
H: Showing that $\lim\limits_{x \rightarrow 0} \frac{1}{x}$ does not exist.
Forgive me if this has been asked before, but I searched and could not find an answer.
I am trying to show that $\lim\limits_{x \rightarrow 0} \frac{1}{x}$ does not exist. If the limit did exist, and was equal to $l$, then for every $\varepsil... |
H: Combinatorics recursion question.
How many binary vectors of length n do not contain a sequence '001' ? Solve by
recursion and explain your solution.
AI: Let $a_n$ and $b_n$ be the number of sequences not containing the sequece $001$ ending in $0$ and $1$, respectively. Given a desired sequence of length $n$, we ca... |
H: integration for |x|<1 for a simple integral
I want to integrate
$$\int_{-1}^{1}\frac{dx}{x^3\sqrt{1-x^2}}.$$
I'm not sure where to begin as I have tried integrating by parts but end up in a continuous circle
AI: Hint: Try the substitution $u=\sqrt{1-x^2}$. |
H: Probability of adjacent pairs on an $N\times N$ board
I have a question about the probability of finding adjacent pairs on a $N\times N$ board.
So there is a $N \times N$ cell surface and $M$ objects are randomly distributed on the surface. Each cell has a maximum of one object on it.
What is the expected value of ... |
H: 9-Rook problem in 3D / weak version of sudoku
So here I have a 9*9 grid, and I have to fill each row and each column with 1~9, but without the constraint about 3*3 boxes as in traditional sudoku. Suppose the grid is blank at first, in how many different ways can I fill the boxes?
Is there a (simple) mathematical w... |
H: proving completness of a metric space
Given $$d(a,b):=\left|\frac{a}{1+|a|}-\frac b{1+|b|}\right|$$
I want to know if $(\mathbb R,d)$ is complete.
My attempt: I think it's complete because given a Cauchy sequence the sequence is bounded. By Bolzano-Weierstrass the sequence has a limit point and and I know a Cauchy ... |
H: Questions regarding probability density functions and distribution functions
So I initially have a pdf defined as:
$f_Y(y)=\left\{ \begin{array}{ll}
cy(1-y^2) & \mbox{$0 \le y \le 1$}\\
0 & \mbox{otherwise}.\end{array} \right.$
$c$ is a constant that I must find. Now have I got it right if I integr... |
H: Probability question..........
Assume the events are independent.
The probabilities that two students will show up for class are 0.5 and 0.7. Find the following probabilities.
(a) At least one shows up for class.
I know this is .85
(b) At least one does not show up for class.
How can I find this?
AI: Take students... |
H: How do I show sets have the same cardinality
How do I show that $\left[a,b \right]$ and $\left(a,b \right)$ have the same cardinality where $a<b$?
AI: Define $\ f:[a,b]\rightarrow\ (a,b)$ with
$f(x)$=$\ \begin{cases}
\frac{(a+b)}{2} &if\ x= a \\
a+\frac{b-a}{n+2} &if\ x=a+\frac{b-a}{n},n\in\mathbb N... |
H: Fourier Series and periodicity
Let $f$ be a $2 \pi$-periodic piecewise continuous function and let \begin{equation} f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx} \right] \tag{*} \end{equation} denote its Fourier series.
Set $g(x)=f(x+\pi)$ for all $x \in \mathbb{R}$ and let \begin{e... |
H: Finding eigenvalues of an unknown matrix subtracted by the identity matrix
The question:
If the eigenvalues of $A$ are 0, 1, and 3, find the eigenvalues of $A-I$. Explain how you obtained them.
My intuition is telling me that I just subtract one from each of the eigenvalues since they are related to the diagonal,... |
H: How to prove Linear Independence
How to prove the set $S=\{x,|x|\}$ is linearly independent. Where S is a subset of set of real valued functions on $\mathbb{R}$.
Thank You.
AI: Two non-zero vectors in a vector space are linearly dependent if one is multiple of he other. Do you think that you can find a constant $c$... |
H: Limit of a sum of roots proof
Given the sequence:
$$a_n=\alpha\sqrt{n+a}+\beta\sqrt{n+b}\ with\ \ \alpha,\beta,a,b\in\mathbb{R}\ and\ \alpha,\beta\neq0$$
Prove that $$\lim_{ n\to \infty} a_n = 0\ iff\ \alpha=-\beta$$
I start the proof by supposing that $\alpha\neq\beta\ and\ \alpha\neq-\beta$.Then I have:
$$a_n=\... |
H: First Sum of Zetas
Can I simplify the sum: $\zeta(1) +\zeta(2) +...+\zeta(n)$ .
Whereas $\zeta$ denotes the zeta function.
I want to find the limit involving this sum but need it simplified.
Thanks.
AI: Assuming you meant $\zeta(2) + \zeta(3) + \cdots + \zeta(n)$.
\begin{align*}
\sum_{i=2}^n \zeta(i)
&= \sum_{i=2}^... |
H: Fano geometry - order
Of any three points situated on a line, there is no more than one which lies between the other two.
I suspect this is not true in Fano geometry, but I am not entirely sure (the confustion stems from the fact that I am not sure if I have the right interpretation of this axiom) Does anyone know... |
H: Countability in natural numbers and real numbers.
Let me have the following representation of natural numbers- $n$ will be represented as $\dots 00n$. Now arrange the natural numbers in some sequence $\{x_1,x_2,x_3,\dots\}$.
Say $n=12$. Then index the digits in the following manner- the index of $2$ is $1$, the in... |
H: Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$
Everyone knows about the classic
$$ \sum_{i=1}^{\infty} \dfrac{1}{2^i} = 1 $$
However, is there any way to find
$$ \sum_{i=0}^{\infty} \dfrac{1}{2^{2^i}} = \dfrac12 + \dfrac14 + \dfrac{1}{16} + \dfrac{1}{256} + \cdots $$
AI: This is the Fredholm number as poin... |
H: Isomorphic mapping of $\Bbb C\to \Bbb C$
Prove that the mapping $z\mapsto\bar z$ of $\Bbb C\to \Bbb C$ is an isomorfism of $\Bbb C$ to itself that punctually fixes $\Bbb R$.
That's not so hard to prove, if $\bar z=a-ib=x-iy=\bar w \Rightarrow a=x$ and $-b=-y\Rightarrow b=y\Rightarrow z=w$, so it is inyective. And... |
H: limit of a hyperbolic function
How to evaluate this limit without using hopital rule:
$$\lim_{c\rightarrow + \infty}{\frac{\text{sinh}\sqrt{c}}{2\sqrt{x}}}$$
Here is what I have done so far:
we know that $\text{sinh}(x)= \frac{e^x-e^{-x}}{2}.$
So applying this to the limit we find :
$$L= \lim_{c\rightarrow +\inft... |
H: How many $\alpha \in S_n$ are such that $\alpha^2 = 1$?
This is not for homework, but I am not great at counting arguments and would like some feedback. The question asks
Let $n \in \mathbb{N}$. How many $\alpha \in S_n$ are there such that $\alpha^2 = 1$?
I know that, if $\alpha^2 = 1$, then either $\alpha = 1$... |
H: K-Topology on the real line
The K-topology on the real line by taking as basis all open intervals $(a,b)$ and $(a,b)\setminus B$ where $K=\{1/n:n=1,2,...\}$ and $B\subset K$. According to this topology,the subset $\mathbb{R}\setminus K$ is open in K-topology but not open in usual topology. I wonder that it can be ... |
H: Prove that the series converges to the integral
Prove:
$\int _0^{1}x^{-x}dx$ = $\sum_{n=1}^\infty\frac{1}{n^n} $
I thought of using: $x^{-x}$ = $e^{-x lnx}$ and then using : $e^{-xlnx}$ = $\sum_{n=1}^\infty\frac{(-xlnx)^n}{n!} $ but I'm stuck from here. Help please?
AI: Hint: To evaluate the integral, use the chang... |
H: Geometric Interpretation of members of $\mathrm{O}(2)\setminus\mathrm{SO}(2)$
I recently came across a question which asked to prove the defining properties of the orthogonal matrices (members of $\mathrm{O}(2)$), then to subsequently determine that they can be written in the form:
$$\mathbf{R}(\varphi)=\begin{pmat... |
H: Converting Sum of Trigonometric Functions into Product
I know that $\sin x-\sin y=2\sin(\dfrac{x}{2}-\dfrac{y}{2})\cos(\dfrac{x}{2}+\dfrac{y}{2}).$
I would like to know how to get this formula.
AI: $$
\sin(a+b) = \sin(a) \cos(b) + \sin(b) \cos(a)
$$
and
$$
\sin(a-b) = \sin(a) \cos(b) - \sin(b) \cos(a)
$$
Apply thes... |
H: What is meant when $f:[a,b] \to \mathbb R$ is said to be differentiable?
Sometimes I see an exercise like this:
Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property.
What is usually meant by that? Should $f'$ be defined on $[a,b]$ with one-sided deriv... |
H: Question regarding $A = B^{-1}DB$ and determinants
Consider $A = B^{-1}DB$, where $A$ is a normal matrix represented by unitary matrices $B, B^{-1}$ and the diagonal matrix $D$.
Although $B^{-1}B = BB^{-1} = I_B$ why doesn't $B^{-1}DB$ give you $D$? What special algebraic properties are revealed about two matrices ... |
H: Does every real vectorspace have a symetric positive definite bilinear form?
Does every real vectorspace $V$ (possibly not finite dimensional) have a symetric positive definite bilinear form?
That is a map $s:V \times V \rightarrow \mathbb{R}$ such that:
$$\forall v, w \in V:s(v,w)=s(w,v)$$
$$\forall u,v,w \in V\sp... |
H: Integrate the following integral using partial fractions
I want to integrate
$$\frac{1}{(1-u^2)^2}. $$
I have used the difference of two squares to get
$$\frac{1}{2(1-u^2)} + \frac{1}{2(1+u^2)} $$
and then integrated it to get
$$\frac{1}{2ln|1-u^2|} + \frac{1}{2ln|1+u^2|}.$$
Just wondering if this is correct? tha... |
H: Solution to ODE using power series
The question asks to find relation of the coefficients of the series solution around x=0 for the equation $y'''+x^2y'+xy=0$
Therefore trying:
$$y=\sum_{m=0}^\infty y_mx^m$$
$$\therefore y'=\sum_{m=1}^\infty my_mx^{m-1}$$
$$\therefore y''=\sum_{m=2}^\infty m(m-1)y_mx^{m-2}$$
$$\the... |
H: $X$ has a Gamma distribution with parameters $\lambda$ and $\alpha$. Find $E(X^r)$
$X$ has a Gamma distribution with parameters $\lambda$ and $\alpha$. I must find $E(X^r)$ and $r$ is a positive integer.
How can I do this?
I am guessing I have to use the Gamma function but I don't know how to do this?
AI: You leave... |
H: Which of these compact sets are possible such that they have the following measures?
I am supposed to construct compact sets $K \subset \mathbb{R}$ (if possible) that have the following properties:
lambda is the Lebesgue-measure:
$ \lambda (K^0) = \lambda (K)$. This is easy, just take $K=[-1,1]$
$ \lambda (K^0) < ... |
H: p-adic numbers and group characters
The wiki article on p-adic numbers has this wonderfully charming and pretty graphic:
This is supposed to represent "the 3-adic integers, with selected corresponding characters on their Pontryagin dual group."
And that's the only explanation provided. I've looked, but have been u... |
H: In a normed space, the sum of a Closed Operator and a Bounded Operator is a Closed Operator.
The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following lemma
Let $T:\mathcal{D}(T)\to Y$ be a bounded linear operator with domain $\mathcal{D}(T)\subset X$, where $X$ and $Y$ are no... |
H: Understanding induction proof with inequalities
I'm having a hard time proving inequalities with induction proofs. Is there a pattern involved in proving inequalities when it comes to induction? For example:
Prove ( for any integer $n>4$ ):
$$2^n > n^2 \\ $$
Well, the skipping ahead to the iduction portion, here's... |
H: Prove that the sequence of L-Lipschitz functions converge
$f_n(x): [a,b] \to \mathbb R$ are a sequence of functions that all are $L$-Lipschitz:
means -
$|f_n(x)-f_n(y)| \le L|x-y|$ , ($L$ is for all the functions)
and assume $f_n \to f$ in a pointwise convergence.
By now I know that $f$ is $L$-Lipschitz as well.
... |
H: Show that the matrix is invertible
let $A \in M_n(F)$ be a n by n matrix with values from an unknown field $F$.
$P_A(t)$ is the characteristic polynomial of $A$, and $g(t) \in F[t]$ a polynomial of an unknown degree.
assume that $gcd(P_A(t),g(t))=1$
Show that $g(A)$ is invertible.
I don't understand why this is tr... |
H: Is the empty set a relation?
Is the empty set is a relation?
In Enderton's book A Mathematical Introduction to Logic, a relation is defined as a set of ordered pairs. If the empty set is a relation, why is that? In the text, there is an example of a function $\varnothing \to A$. This function is of course is the em... |
H: Limit $\mathop {\lim }\limits_{n \to \infty } n({1 \over {{{(n + 1)}^2}}} + {1 \over {{{(n + 2)}^2}}} + \cdots{1 \over {{{(2n)}^2}}})$
Without using integrals, how to find this limit:
$$\mathop {\lim }\limits_{n \to \infty } {a_n} = n\cdot\left({1 \over {{{(n + 1)}^2}}} + {1 \over {{{(n + 2)}^2}}} + \cdots{1 \over ... |
H: examples of toy non-Euclidean geometries
Today I was working on a problem in euclidean geometry, and I found it immensely useful to compare with Fano geometry, for contrast. Are there any other toy geometries like Fano geometry? I think it could be useful to have a repertoire of those.
AI: Felix Klein considered no... |
H: Given $\frac{y+a}{x+a}=b$, is there a solution for $\frac{x}{y}$?
I have an expression of the form:
$$\frac{y+a}{x+a}=b$$
I know there is an infinity of solutions for x and y, but what I'm looking for is a solution for x/y, unique or otherwise.
AI: We have $$ \dfrac {y+a}{x + a} = b \implies y + a = b \cdot (x+a) =... |
H: Find a graph with an adjacency matrix consisting $0$
(okay, I'm not learning math in english, so please don't be harsh with me for not using the correct terminology here, but I hope you can understand my problem. also feel free to correct me)
Find a connected graph for every $n\geq4$ , for which is true, that his a... |
H: Evaluate $\lim\limits_{x \to a} \frac{x^m-a^m} {x-a}$
How to evaluate this:
$$\lim\limits_{x \to a} \frac{x^m-a^m} {x-a} ; m\in \mathbb{N}$$
if i take:
$ m = 1 $
$$\lim\limits_{x \to a} \frac{x^1-a^1} {x-a} =1 $$ Is this correct?
but how evaluate limit where $ m = 2 $
$$\lim\limits_{x \to a} \frac{x^2-a^2} {x... |
H: Properties of Order of a Group
Having real trouble getting started on this question, even though it doesn't seem hard:
Let $g,h \in G$ where $G$ is an Abelian group.
Then assume that $ord(g), ord(h)$ are finite with $hcf(ord(g),ord(h)) = 1$.
Prove $ord(g+h) = ord(g)*ord(h)$
I have tried showing that $ord(g+h)$ is t... |
H: Proving a triangle is isoceles given two points on the sides
Here is the problem
A $\triangle NML$ is given. On the sides $NM$ and $ML$ respectively points $A$ and $B$ are chose so that $ {NA \over AM} = {MB\over BL} = 2 $ and $ \angle NLM = 2\angle MBA$. Prove that $\triangle NML$ is isosceles.
I'm pretty much stu... |
H: Existence of a basic sequence with basis constant $1$ in a Banach space
I am interested in a problem motivated by the following theorem. A proof (and the relevant definitions) can be found in many textbooks about Banach space theory, see for example Corollary 1.5.3 in Topics in Banach Space Theory by Albiac and Kal... |
H: Proof by induction that if $a_0 = 0, a_1 = 1, a_n = \frac{a_{n-1} + a_{n-2}}{2}$ then $a_n = \frac23 \left( 1 + \frac{(-1)^{n+1}}{2^n} \right)$
let $a_0 = 0, a_1=1,a_n = \frac{a_{n-1} + a_{n-2}}{2} $
prove with induction that $a_n = \frac23 \left(1 + \frac{(-1)^{n+1}}{2^n} \right) $
i assumed $a_k$ was equal to ... |
H: Graph theory & Feynman integrals
I am attending a course in Graph Theory and I am interested learning something about applications of this subject to Physics, especially I would like to learn something about Feynman integrals. Could you suggest me some books related to this topic? A friend suggest me to look at "Gr... |
H: Find $P(T>t)$ and thus find the cdf for $T$ - exponential distribution
I have a word problem question where a "man" is waiting for two buses, bus A or bus B.
Let $T_1(T_2)$ be the time till the next bus A(B) arrives.
$T_1$ and $T_2$ are independent continuous variables. They can be defined as follows:
$T_1$~$Exp(\l... |
H: Math induction sum of even numbers
I need to prove by induction this thing:
$2+4+6+........+2n = n(n+1)$
so, this thing is composed by sum of pair numbers, so its what I do, but I'm stucked.
$2+4+6+\cdots+2n = n(n+1)$
$(2+4+6+\cdots+2n)+(2n+2) = n(n+1) + (2n+2) $
$n(n+1)+(2n+2) = n(n+1)+(2n+2) $
$n^2 + 3n + 2$
$n(n... |
H: Get angle from any point
How can I get an angle of a point positive or negative? When I use tangent the angle only appears in the first two quadrants.
So I need to do some magic and make it so that I can get the angle of any point on a two dimensional plane.
So for example, (-10 ~ x, -10 ~ y) must be 225 degrees, ... |
H: Is it possible to do a Hasse Diagram for divisibility on the following set
{2, 3, 5, 6, 10, 15, 25}
In Hasse Diagrams of divisibility am I allowed to cross edges? If so then I believe I have a solution, if not then here's where my problem lies.
AI: Yes, you can create a Hasse Diagram. Any finite poset can be made... |
H: Product and sum of positive operators is positive
I want to show that for $S,T\in B(H)$ bounded operators on Hilbertspace with $S\geq 0,T\geq 0$ and $ST=TS$, we have $S+T\geq 0$, and $ST\geq 0$.
$T\geq 0$ means $(Tx,x)\geq 0$.
To me it seems that $((S+T)x,x) = (Sx,x)+(Tx,x)\geq 0$.
But for $ST$ i like some help.... |
H: How many four-vertex graphs are there up to isomorphism;
Let us call graphs $G = (V,E)$ and $G' = (V', E')$ fundamentally different if they are not isomorphic. How many fundamentally different graphs are there on four vertices?
This is a question on my homework. I'm thinking that I need to exhaust all the possible ... |
H: Convex Quadrilateral: $ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + \cot D $
Problem
Let $ABCD$ be a convex quadrilateral with no right angles. Show that $$ \dfrac {\tan A + \tan B + \tan C + \tan D}{\tan A \tan B \tan C \tan D} = \cot A + \cot B + \cot C + ... |
H: What, if anything, does it mean to be neither finite nor infinite for real numbers?
I have this book which is said to be notoriously bad by my professor and the graduate T.A's and in a section titled "Series with Nonnegative Terms" the following statement appears:
Every series $\Sigma{a_n}$ with $a_n \ge 0$ has a ... |
H: Modular Arithmetic Homework
Find an integer $ m \ge 2 $ so that the equation $ x^2 \equiv 1 $ in $\mathbb{Z}/ m$ has more than two solutions.
In a previous part I proved that there are two solutions $x=1,-1$ when $m$ is prime. I'm not sure if that is of any relevance here?
I'm not sure how to even start this, I wa... |
H: Do non-square matrices have eigenvalues?
I've looked at this and it doesn't help because I don't know anything about SVD. Can someone dumb it down for me please?
AI: It is not exactly true that non-square matrices can have eigenvalues. Indeed, the definition of an eigenvalue is for square matrices. For non-square... |
H: Proof that a field is an integral domain
Here is my attempt at proving this.
Let $F$ be a field and let $a \in F, \ a \neq 0$.
Then $a$ is a unit and hence $\exists \ b \in F$ such that $ab = 1$
Now let $c \in F, \ c \neq 0$
Let $a \cdot c = 0$
Then $b \cdot a \cdot c = b \cdot 0$
$\implies 1 \cdot c = c = 0$
This... |
H: Find all possible abelian groups of order $120$.
Find all possible abelian groups of order $120$.
If someone could walk me through how to do this, that would be great.
AI: Start by taking $120$ and decomposing it into its prime factorization.
$$120 = 2^3\times 3 \times 5$$
Then apply the Fundamental Theorem of Fin... |
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