text stringlengths 83 79.5k |
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H: Finding a PDF given a (strictly) right continuous CDF.
I have the CDF:
$$
F(x)=
\begin{cases}
0 & \text{if } x < 1 \\
\frac{x^2-2x+2}{2} & \text{if } 1 \le x < 2 \\
1 & \text{if } x \ge 2
\end{cases}
$$
I want to find the PDF and I noticed that $F$ is not continuous (at $x=1$), but it is right continuous. Therefor... |
H: Is the Arens-Fort Space Compact?
thanks in advance. My question is this:
Is the Arens-Fort space $X \; = \; (\mathbb{N} \times \mathbb{N}) \cup \lbrace \omega \rbrace$ compact?
What I have so long is this:
Since we know that if a space is compact then it is locally compact, we know that $(0,0)$ is not locally compa... |
H: Find domain of $\log_4(\log_5(\log_3(18x-x^2-77)))$
Problem:$\log_4(\log_5(\log_3(18x-x^2-77)))$
Solution:
$\log_3(18x-x^2-77)$ is defined for $(18x-x^2-77) \ge 3$
$(x^2-18x+77) \le -3$
$(x^2-18x+80) < 0$ {As it can't be 0}
$(x-8)(x-10)<0$
$8<x<10$
So domain is $(8,10)$
Am I doing right ??
AI: HINT:
For for real ... |
H: Examples of how to calculate $e^A$
I'm trying to learn the process to discover $e^A$
For example, if $A$ is diagonalizable is easy:
$$A =\begin{pmatrix}
5 & -6 \\
3 & -4 \\
\end{pmatrix}$$
Then we have the canonical form $$J_A =\begin{pmatrix}
2 & 0 \\
0 & -1 \\
\end{... |
H: Linear transformation?Rotation question in linear algebra
Given the vector $x=[1, 3, -7]^T$ in the basis $[1, 0, 0]^T$, $[0, 1, 0]^T$, and $[0, 0, 1]^T$ (Cartesian coordinates) perform the following operations:
Rotate 45 degrees about the x-axis
then rotate 30 degrees about the y-axis
then rotate -10 degrees about ... |
H: Convergence of increasing measurable functions in measure?
Let ${f_{n}}$ a increasing sequence of measurable functions such that $f_{n} \rightarrow f$ in measure.
Show that $f_{n}\uparrow f$ almost everywhere
My attempt
The sequence ${f_{n}}$ converges to f in measure if for any $\epsilon >0$ there exists $N\in \m... |
H: Distance between convex set and non-convex set?
So in http://en.m.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma there is some talk about distance between a mintowksi sum and a convex set. But I couldn't get how distance is being defined. Can anyone help here?
AI: In general, the distance between two sets $A$ and... |
H: Problem Solving using Algebra
If Peter is $7$ years older than Sharon and John is twice as old as Peter, work out how old Peter is if the average of their ages is $19$.
Thanks! :)
AI: HINT:
If the age of Peter is $x$ years,
the age of Sharon will be $x-7$ years and that of John will be $2x$ years
So, $$\frac{x-7+x... |
H: Differentiation term by term of Taylor series
Suppose I have A Taylor Series of a function around $z_{0}$ in the complex plane which convergence in a ball of radius $r>0$. Can I differentiate term by term the Taylor series and get the derivative of f?
If so, can you please proof it? else, give an example why is it ... |
H: Limit on a spiral
I was thinking about limits of functions along various spirals and this one stumped me a bit. The limit that needs to be found is ultimately:
$$\lim_{\varphi\to\infty} \coth\varphi\csc\varphi$$
Here is the sprial that it comes from:
The equation for this is $(x,y)=(\tanh \varphi \sin \varphi, \t... |
H: Example of a function that's uniformly continuous on a closed interval but not on an open one
I'm looking for an example of a function that's uniformly continuous on a closed interval [a,b] but isn't on an open one (a,b).
Can such a function exist? If so, can you help me find an example?
AI: I don't think such a fu... |
H: Could we define multiplication of “complex numbers” in this way?
If we define multiplication of complex numbers as follows:
$$z_1 \cdot z_2=(x_1x_2+y_1y_2, x_1y_2+x_2y_1)$$ then it can be shown that it induces a group structure $(G, \cdot)$, because it has inverse elements:
$$z^{-1}=\left(\frac{x}{x^2-y^2},\frac{y}... |
H: Möbius transformation question
Möbius transformation copies the annulus $\{z:r<|z|<1\}$ to the domain between $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$
Please help me to find what is $r$.
AI: Hint: The inverse of this transform is also Möbius and might for simplicity map $1\mapsto 1$, $-1\mapsto-1$, $\frac12\mapsto r$,... |
H: Probability of the last remaining item after removing items
Question:
Mr. J has 3 shirts and he wears them each day with the following probabilities: Green (1/3), White(1/6) and Red(1/2).
Once he decided to clean out his closet, so each evening he independently decides with a prob. 1/5 to throw the shirt he was wea... |
H: Differential Equation of 2nd degree with non constant coefficients
How to solve this equation :
$x''+tx'+x=0$
I tried variable change but no results. Is there a concrete way that works for every equation of this kind. If you can show me step by step that would be great.
Thank you for help !
AI: Note $tx'+x=(tx)'$ ... |
H: $F[x]/\langle p(x)\rangle$ is a field $\iff F[x]/\langle p(x)\rangle$ is an integral domain
$$\color{red}{Is~my~interpretation~correct?}$$
Let $F$ be a field. I know that $p(x)\in F[x]$ is irreducible $\iff \langle p(x)\rangle$ is maximal i.e.
$F[x]/\langle p(x)\rangle$ is a field $\iff p(x)$ is irreducible in $F... |
H: integration with substitution - why is this so?
I have this problem:
$$\int_0^2 \mathrm{(x-1-e^{-\frac{1}{2}x})}\,\mathrm{d}x$$
what I tried:
$t=-\dfrac{1}{2}x \Rightarrow \dfrac{dt}{dx} = \dfrac{1}{2} \Rightarrow dx = \dfrac{dt}{2}$
$$\int_0^2 \mathrm{(x-1)}\,\mathrm{d}x -\int_0^2 \mathrm{(e^{-\frac{1}{2}x})}\,\... |
H: Verifying spectral norm
I was wondering how one could verify the relation that $||A||_2 = \sqrt{\rho(A^HA)}$ for matrices. I mean I have seen this so often, but never found a proof of it. Is there a smart way to do this quickly? Because only referring to the most basic properties of matrices seem to be not a good i... |
H: Proving that $Hom_G (V,W)$ is 1-dimensional when $V,W$ are irreducible
Question: Let $G$ be a group. For any two representations $V,V'$ of $G$ over $\mathbb C$, let $Hom_G (V,V')$ denote the space of all linear maps $h: V\rightarrow V'$ such that $h\rho'_g = \rho_g h\forall g\in G$. I want to prove that if $V$ and ... |
H: manifold diffeomorphic (?) to SO(3)
Consider the set of all pairs $(\boldsymbol{n},\boldsymbol{v})$ of vectors in $\mathbb{R}^3$ such that $\boldsymbol{n}$ is a vector on the unit sphere centered at the origin and $\boldsymbol{v}$ is a unit vector tangent to the sphere at the point $\boldsymbol{n}.$
i. Introduce a ... |
H: Show that a rational number has no good rational approximations
This is homework question.
The teacher proved that if $a$ is irrational, there are infinitely many rational numbers $\frac{x}{y}$ such that $|\frac{x}{y} - a|<\frac{1}{y^2}$.
What we need to prove is:
Let $a$ be a rational number. Show that $a$ h... |
H: How to prove the Archimedian property using the completeness axiom?
The problem follows
"Using the Completeness Axiom for R, prove the Archimedian property of the real numbers
: for any x in R, there is an integer n>0 such that n>x "
I tried to prove it in a reductio ad absurdumd. But I can't....
(I tried in this w... |
H: Algorithm to find the number of numbers which are both perfect square as well as perfect cube
I was teaching indices chapter to my brother when I got this idea to find the number of numbers which are perfect squares as well as perfect cubes. I was wondering whether there is an algorithm to find these numbers betwee... |
H: Relationship between adjoing matrix and inverse function
I am struggling with the following excercise:
Let A be a matrix, then we have for every subspace $U$ that:
$A^*(U ^\perp)=(A^{-1}(U))^\perp$
I do not even know where to start to solve this excercise. Does anybody have a hint for me?
AI: Hints:
We have
$$w\in... |
H: Relationship between an inhomogeneous Poisson process and Markov chain
What type of Markov process relates to an inhomogeneous Poisson process?
A homogeneous Poisson process-- one where the rate, $\lambda$, is constant-- is a pure birth continuous time Markov chain (with a constant birth rate). That is, if $q_{ij}... |
H: Why is the additive identity of a ring always a multiplicative absorbing element?
In problems concerned with finding the units in a ring, my textbook seems to always ignore the additive identity as a possibility. In combination with learning the definition of a field (a ring in which every nonzero element is a unit... |
H: How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?
In our lecture notes, there's the following example problem.
Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 & 0\\0 & -1 & 0 & -1\\1 & 1 & 1 & 2\end{bmatrix}$$
The solutio... |
H: Atiyah-Macdonald, Proposition 2.12, uniqueness of the tensor product.
The following is a result from Atiyah-Macdonald, defining and showing existence and uniqueness of tensor product of modules over a commutative ring.
Proposition 2.12. Let $M, N$ be $A$-modules. There exists a pair $(T,g)$ consisting of an $A$-m... |
H: Symmetric positive definite with respect to an inner product
Let $A$ be a SPD(symmetric positive-definite) real $n\times n$ matrix. let $B=LL^T$ be also SPD. Let $(,)_B$ be an inner product given by $(x,y)_B=x \cdot By=y^T Bx$. Then $(B^{-1}Ax,y)_B=(x,B^{-1}Ay)_B$ for all $x,y$. Show that $B^{-1}A$ is SPD with resp... |
H: $\frac{c}{1+c}\leq\frac{a}{1+a}+\frac{b}{1+b}$ for $0\leq c\leq a+b$
When proving that if $d$ is a metric, then $d'(x,y)=\dfrac{d(x,y)}{1+d(x,y)}$ is also a metric, I have to prove the inequality:
$$\dfrac{c}{1+c}\leq\frac{a}{1+a}+\frac{b}{1+b}$$ for $0\leq c\leq a+b$. This is obvious by expanding, but is there a n... |
H: Can we 'form' infinitely many subspaces out of finite dimensional vector space?
Let $V$ be a vector space over $\mathbb{R}$ of dimension $n$, and let $U$ be a subspace of
dimension $m$, where $m < n$. Show that if $m = n − 1$ then there are only two subspaces of $V$ that contain $U$ (namely $U$ and $V$), whereas if... |
H: Simplify a expression with nested radical signs
Simplify :
$\sqrt{10+6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}}$
I have tried completing square but failed, Can anyone help me please? Thanks.
AI: $$10+6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}=5(2+\sqrt3)+2\sqrt6(2+\sqrt3)=(2+\sqrt3)(5+2\sqrt6)$$
Now, $$5+2\sqrt6=3+2+2\cdot\sqrt2\cdot... |
H: (locally Holder) + (locally Lipschitz) $\Longrightarrow$ Continuity?
Let $f = f(x,y)$ be locally Holder continuous in $x$ and locally Lipschitz continuous in $y$, i.e. given $(z_1,z_2)$ in the domain of definition, there exists a neighborhood $U$ such that for any $(x_1,x_2)$, $(y_1,y_2) \in U$ then $$|f(x) - f(y)|... |
H: Fibonacci sequence: how to prove that $\alpha^n=\alpha\cdot F_n + F_{n-1}$?
Let $F_n$ be the $n$th Fibonacci number. Let $\alpha = \frac{1+\sqrt5}2$ and $\beta =\frac{1-\sqrt5}2$.
How to prove that $\alpha^n=\alpha\cdot F_n + F_{n-1}$?
I'm completely stuck on this question. I've managed to take the equation form of... |
H: Boundedness of $\sum_{m=k}^{\infty} \frac{k}{m^2}$
Is the series $\sum_{m=k}^{\infty} \frac{k}{m^2}$ bounded independently of k?
AI: Write $\sum_{m=k}^{+\infty}\frac 1{m^2}\leqslant \sum_{m=k}^{+\infty}\frac 1{m(m-1)}$, which behave approximatively as $\frac 1k$. |
H: Some matrix calculus (differentiation)
Let $x\in \Bbb R^n$, $f(x)=||Ax-b||_2^2$. I want to show that $grad f(x)=2 A^T (Ax-b)$.
But why is it true? I almost forgot gradient and matrix calculus, though I looked wiki I can't figure it out. My trial:
$$grad f(x)= \frac {\partial f}{\partial x}=2||Ax-b||_2 \frac {\par... |
H: Modular arithmetic: How to solve $3^{n+1} \equiv 1 \pmod{11}$?
Please, I can't solve this equation:
$$3^{n+1}\equiv 1 \pmod{11}$$
for $n \in \mathbb{N}$.
So what should I do please? Thanks.
AI: First of all, using Fermat's Little Theorem, $3^{10}\equiv1\pmod {11}$ as $(3,11)=1$
Now,as we know if $a^n\equiv1\pmod m... |
H: Prove $n\mid \phi(2^n-1)$
If $2^p-1$ is a prime, (thus $p$ is a prime, too) then $p\mid 2^p-2=\phi(2^p-1).$
But I find $n\mid \phi(2^n-1)$ is always hold, no matter what $n$ is. Such as $4\mid \phi(2^4-1)=8.$
If we denote $a_n=\dfrac{\phi(2^n-1)}{n}$, then $a_n$ is A011260, but how to prove it is always integer?
Th... |
H: Integral equality when $f(x) \ge g(x)$
Given that $f(x) \ge g(x)$ for $x$ in $[0,1]$, I need to find an example of two different functions such that
$$
\int_{0}^1 f(x)\,dx = \int_{0}^1 g(x)\,dx.
$$
Edit:
my answer was to take a function, like f(x) = 1 and g(x) will be the same with discontinuous one point at, say, ... |
H: Ergodic action of a group
What does it mean and how is it defined if the action of a group is meant to be ergodic?
Thank you for your replies!
AI: Here is a context of group action where ergodicity arises naturally. As pointed out by Martin, the definition given in the grey box applies to more general situations wh... |
H: How to integrate a binomial expression without expanding it before?
Let $(3-x^2)^3$ be a binomial expression. What is the integral of such expression?
First I tried integration by substitution, because there is a composition of two functions. But$\displaystyle\frac{d}{dx}(3-x^2)=2x$ and I learned that this method... |
H: What do you use for your basis step when its domain is all integers?
Example: *For all integers $
n
, 4(
n
^2
+
n
+
1)
–
3
n
^2$
is a perfect square what should i use? negative infinity?
I know you can use a direct proof but what if theres an induction question with the same domain?
AI: You need to use induction ... |
H: Find the fixed points
We have $x_{n+1} = ax_n +b$ with $x_0$ given. We have to find the fixed points of this function, and decide for which values of $a$ they are stable.
So I looked it up and found that a fixed point is a point for which $f(x) = x$, so basically intersections of a certain function with the line $y... |
H: Show that the subspace A is the whole Hilbert space H
"Let $A$ be a subset in a Hilbert space $H$, such that $x\in H$ and $x \perp A$ imply $x = 0$.
(1) Show that the closed subspace that is generated by $A$ is $H$.
(2) Let $f(x)$ be a square summable function on $\Bbb R$, such that its Fourier transform is almost ... |
H: Non-convergence of sequence and existence of subsequence with special property
If the sequence $(x_{n})$ doesn't converge to $x_0 \in X$ then there exists subsequence $(x_{n_k})$ so that there is no subsequence of that subsequence that converges to $x_0$.
(or: If the sequence $(x_{n})$ doesn't converge to $x_0 \in ... |
H: $x^4+5y^4=z^2$ doesn't have an integer solution.
I hope to show that $x^4+5y^4=z^2$ doesn't have an integer solution.
You may guess that you can solve it using the infinite descent procedure.
I tried it but I had a trouble in solving it.
What I did :
Observe that we can say $(x,y)$=1.
$$-x^4+z^2=5y^4$$
$$-(x^2-z)(x... |
H: Closed subset of $\;C([0,1])$
$$\text{The set}\; A=\left\{x: \forall t\in[0,1] |x(t)|\leq \frac{t^2}{2}+1\right\}\;\;\text{is closed in}\;\, C\left([0,1]\right).$$
My proof:
Let $\epsilon >0$ and let $(x)_n\subset A$, $x_n\rightarrow x_0$. Then for $n\geq N$
$\sup_{[0,1]}|x_n(t)-x_0(t)|\leq \epsilon$ for some ... |
H: A problem on sequences and series
Let $ (x_n) $ be a sequence of real numbers such that $ \lim x_n =0 $.
Prove that there exists a subsequence $(x_{n_k} )$of $ (x_n) $ such that $ \sum_{k=0}^\infty 2^{k}x_{n_k}$ coverges and $\sum_{k=0}^\infty 2^{k}x_{n_k}$ converges and $\sum_{k=0}^\infty| 2^{k}x_{n_k}|\leq 1$
Did... |
H: Terminological question on "action factors through"
What does it mean that the action of a group on some space factors through the action of another one?
AI: If the other group is a quotient group of the first $G$, it means that the action can be written as composition of the canonical projection (group morphism) f... |
H: Proving a function is continuous on all irrational numbers
Let $\langle r_n\rangle$ be an enumeration of the set $\mathbb Q$ of rational numbers such that $r_n \neq r_m\,$ if $\,n\neq m.$ $$\text{Define}\; f: \mathbb R \to \mathbb R\;\text{by}\;\displaystyle f(x) = \sum_{r_n \leq x} 1/2^n,\;x\in \mathbb R.$$
Pro... |
H: Orthonormalize the set of functions {1,x}
I'm having trouble while doing this exercise, it says:
In the vector space of the continuous functions in [0,1] with the inner product :
$$\langle f,g \rangle = \int_{0}^{1}f(x)g(x)dx$$
a) Orthonormalize the set of functions $\left\{1,x\right\}$
So I assumed $B$ as a... |
H: How can i solve this Cauchy-Euler equation?
My problem is this given Cauchy-Euler equation:
$$x^{3}y^{\prime \prime \prime} +xy^{\prime}-y=0$$
My approach: this is an differential equation, so i was looking for a solution with the method of undetermined coefficients. but honestly, i failed.
AI: Besides to another a... |
H: How to find the area of a shape whose sides are made up of line or arc of circle?
The arc and lines form the sides of the shape.
The sides touch each other at end point in such a way that each end point can touch only one shape
and the shape is closed
eg:- line--arc--line--line--line--arc--arc
AI: Hint:
Make them a... |
H: Definition of a monoid: clarification needed
I'm only in high school, so excuse my lack of familiarity with most of these terms!
A monoid is defined as "an algebraic structure with a single associative binary operation and identity element."
A binary operation, to my understanding, is something like addition, subt... |
H: Curiosity about the area between the arclength and the connected line
For example, $f(x)=x^2$
I'm curious if it's possible to find the area, for example, between the function $f(x)$ and the connected line between $f(1)$ and $f(2)$?
AI: Call the line connecting $f(1)$ and $f(2)$ $g(x)$:
$$g(x)-f(1)=\frac{f(2)-f(1)}{... |
H: Infinite average queueing delay for M/M/1 queues
According to queueing theory, the average queueing delay for an M/M/1 queue can be calculated as $\frac{1}{\mu-\lambda}$, where $\mu$ is the average service rate and $\lambda$ is the average arrival rate.
Is there an intuitive explanation for what happens at full uti... |
H: Recursively Defined Functions
I am taking a summer class in discrete math and have done very well up till now. I am nervous because I have reviewed the lecture slides and practice problems but I still don't really understand what to do. The problem example I was given is:
Suppose $f$ is defined by:
\begin{align}
f(... |
H: Simple random sampling
In probability, why when you do random sampling without replacement:
$$\operatorname{Cov}(X_i,X_j)=-\frac{\sigma^2}{N-1}.$$
$N$ is the total population, $\sigma$ is the variance of population.
AI: Because the random variables $X_i$ are identically distributed, the random variables $(X_i,X_j)$... |
H: Does p(a,b,c)=p(a,b)p(a,c) hold when b and c are independent?
I am reading through a thesis, it states that given $b$ and $c$ being independent:
\begin{equation}
p(a\mid b,c) := p(a\mid b) p(a\mid c)
\end{equation}
This would imply just using the definition of a conditional probability:
\begin{equation}
\frac{p(a,b... |
H: Do proper dense subgroups of the real numbers have uncountable index
Just what it says on the tin. Let $G$ be a dense subgroup of $\mathbb{R}$; assume that $G \neq \mathbb{R}$. I know that the index of $G$ in $\mathbb{R}$ has to be infinite (since any subgroup of $\mathbb{C}$ of finite index in $\mathbb{C}$ has t... |
H: Relations between $\|x+a\|$ and $\|x-a\|$ in a normed linear space.
1) Can it happen that $\|x+a\|=\|x-a\|=\|x\|+\|a\|$ when $a\ne0$?
2) How large can $\min(\|x+a\|,\|x-a\|)/\|x\|$ be when $\|x\|\ge \|a\|$?
(For a inner-product space, the answers are no and $\sqrt{2}$ I think.)
AI: 1) This is trivially true if $x=0... |
H: Identity with an alternating binomial sum: $\sum\limits_{i=1}^n(-1)^i{n-i \choose n-k} {k \choose i} = {n-k\choose k}$
I'm learning for the test and:
Prove identity:
$$\sum_{i=1}^n(-1)^i{n-i \choose n-k} {k \choose i} = {n-k\choose k}$$
for all $n,k\in \mathbb{N}$.
This problem is just awful. I was trying to ... |
H: Is This Referring to the Existence of an Antiderivative?
My text gave the following statement for the FTOC:
Let $f$ be an integrable function on $[a, b]$, $F$ be the antiderivative of $f$.
Then $$ \int_a^b f \, du = F(b) - F(a)$$
Why does the book need to specify that $F$ is an antiderivative of $f$?
Is it suggest... |
H: Geometry: Measurements of right triangle inscribed in a circle
So, I've got a triangle, ABC, inscribed in a circle--Thale's theorem states that it is therefore a right triangle. It is also given that $\overline{BA}$ is the diameter of the circle, and hence angle ACB is the right angle. That's all well and good.
My ... |
H: Series $\{a_nb_n\}$ is not absolutely convergent
Suppose that the sequence $\{a_n\}$ is unbounded. Prove that for some absolutely convergent series $\{b_n\}$, the series $\{a_nb_n\}$ is not absolutely convergent.
Absolute convergence means we must choose $\{b_n\}$ such that series $|b_n|$ converges, but series $|... |
H: Would this be an effective way to study and comprehend text's?
This is probably a grey area question but I am going to test the waters anyway.
What I am thinking of doing would be to basically record myself doing examples from textbooks and making lessons for myself which I can then view later and possibly post the... |
H: $3^{3n+1} < 2^{5n+6} $ for all non-negative integers $n$. Is my induction solution correct?
Show using mathematical induction that $3^{3n+1} < 2^{5n+6} $ for all non-negative integers $n$. I'm not sure whether what I did at the last is valid?
Basis step:
for all non-negative integers
$$P(n) = 3^{3n+1} < 2^{5n+6} ... |
H: Default metrics for $c_0$ and $l^{\infty}$
In my book there is a question like
Let $\{a^{(k)}\}$ be a convergent sequence of points in $l^1$. Prove that $\{a^{(k)}\}$ converges in $l^{\infty}$.
Now I don't see it mentioned anywhere what metric I should use for $l^{\infty}$. The book only mentions the metric $$d(\... |
H: Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series
Probably a simple question, but I wonder about the following:
To prove that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$, I use :
$$\exp(z_1+z_2) = \sum_{n=0}^{\infty}\sum_{k=0}^n\frac{1}{k!(n-k)!}z_1^kz_2^{n-k} $$
by using the binomial expansion.
Now the ... |
H: Why does symmetry allow the assumption that $D(a,c) \ge D(b,c)$?
In Kaplansky, Set Theory and Metric Spaces (pg. 69) there is the following theorem:
Theorem: For any points $a,b,c$ in a metric space, we have:
$$ |D(a,c) -D(b,c)| \le D(a,b)$$
The following proof is provided:
Proof: Because of symmetry in [the above]... |
H: Linear recurence relation
We have the linear recurrence relation $$x_{n+1} = \dfrac{3}{2}x_n - 20$$ with $n = 0,1,2...$ and $a,b$ being constants. Does this equation have a fixed point? Does the equation have a period 2 (a period 2 solution $x_0,x_1$ is a solution where you get from $x_0$ to $x_1$ after 1 iteration... |
H: How many of these four digit numbers are odd/even?
For the following question:
How many four-digit numbers can you form with the digits $1,2,3,4,5,6$ and $7$ if no digit is repeated?
So, I did $P(7,4) = 840$ which is correct but then the question asks, how many of those numbers are odd and how many of them are e... |
H: Given a specefic set $ A$ we need to find $A^\perp$
Suppose we have a set of functions which are an element of $L^2[0,1]$ where if we let f(x) be the function equal to 0 from $0<x<1/2$. If this set A is a subset of the Hilbert space $L^2[0,1]$ then we need to find $A^\perp$.
What my attempt was that we know all fun... |
H: $G$ be a non-nilpotent group and every $2$-maximal subgroup Per with all $3$-maximal subgroup
Let $G$ be a non-nilpotent group. If $|G|=p^{\alpha}q^{\beta}r^{\gamma}$ where $p$,$q$,$r$ are primes (two of them maybe are same) such that $\alpha + \beta +\gamma \leq 3$ then every $2$-maximal subgroup of $G$ permuts wi... |
H: Limit of $\left(2-a^\frac{1}{x}\right)^x$
How do I prove the following limit?
$$\lim_{x \to \infty}\left(2-a^\frac{1}{x}\right)^x = \frac{1}{a}$$
AI: $$\lim_{x \to \infty} \left(2-a^\frac{1}{x}\right)^x
= \exp \log \lim_{x \to \infty} \left(2-a^\frac{1}{x}\right)^x
= \exp \lim_{x \to \infty} \log \left(2-a^\frac{1... |
H: Under which circumstances are there fixed points?
Consider the following equation: $$x_{n+1} = ax_n + b$$
Under which circumstances is there a fixed point solution? Under which circumstances is there a period 2 solution?
So for the first question I just rewrote it to $x=ax+b$ and I got $ x = \dfrac{b}{1-a}$. So my ... |
H: Derivative using the chain rule
Differentiate $g(x) = (1-x)\left[\cos\left({\pi\over2}x\right){\pi\over2}\right]$
So (...)
$$g'(x)=-\left[\sin\left({\pi\over2}x\right){\pi\over2}\right]+\left[\cos\left({\pi\over2}x\right){\pi\over2}\right](1-x)$$
This is well done?
AI: This question needs both the chain rule and th... |
H: How can evaluate $\lim_{x\to0}\frac{\sin(3x^2)}{\tan(x)\sin(x)}$
I know this:
$$\lim_{x\to0}\frac{\sin(3x^2)}{\tan(x)\sin(x)}$$
But I have no idea how make a result different of: $$\lim_{x\to0}\frac{3(x)}{\tan(x)}$$
Any suggestions?
AI: $$\frac{\sin 3x^2}{\tan x\sin x}=3\cos x\frac x{\sin x}\frac{\sin 3x^2}{3x^2}\f... |
H: Euler-Fermat Theorem
So I am trying to teach myself number theory, and while trying to work on some exercises I got stuck trying to prove that, for all $n \in \mathbb{Z}$,
$$
n^{91} \equiv n^{7} \bmod 91
$$
What I first thought of was applying the theorem directly and then multiply by the number of n's that was ne... |
H: A solvable group with order divisible by exactly two primes contains a normal subgroup of prime index.
$G$ is solvable group then $G$ has a normal subgroup $N$ of $G$ such that $|G: N|$ is a prime.
AI: Hints:
(1) By the correspondence theorem, if $\,H\lhd G\;$ , then any subgroup $\,\overline B\le G/H\;$ is of the... |
H: Confusion in proof that $C(X,Y)$ is separable
From Kechris' Classical Descriptive Set Theory:
(4.19) Theorem If $X$ is compact metrizable and $Y$ is Polish, then $C(X,Y)$ is Polish.
In the proof of separability, we consider
$$
C_{m,n} = \{f \in C(X,Y) : \forall x,y[d_X(x,y) < 1/m \implies d_Y(f(x),f(y))<1/n]\},
$... |
H: How many ways a surface can curve differently in different directions?
How many ways a surface can curve differently in different directions for a n-dimensional embedded submanifolds of $\mathbb{R}^m, m>n$? I think they can curve infinitely many ways but I am not quite certain: don't they have n-dimensional basis? ... |
H: is $[0,1]^\omega$ with product topology a compact subspace of $\mathbb{R}^{\omega}$
Is $[0,1]^\omega$ with product topology a compact subspace of $\mathbb{R}^{\omega}$, where $\mathbb{R}^{\omega}$ denote the space of countably many products of $\mathbb{R}$. Is the subspace locally compact?
AI: By Tychonoff's theore... |
H: Hamilton path and minimum degree
$n$ is a number of vertices in graph $G, n≥4$.
Prove that if $\;\operatorname{min deg}(G) \geq \frac{(n-1)}{2}\,$ then $G$ has a Hamilton path.
AI: Let $G$ be a given graph with $n$ vertices such that $\min \deg G \ge \frac{n-1}{2}$. Create a new graph $G'$ by add another vertex $w$... |
H: Probability (usage of recursion)
In an hour, a bacterium dies with probability $p$ or else splits into two. What is the probability that a single bacterium produces a population that will never become extinct?
AI: Let $x$ (for "extinction") be the probability that the colony dies out. This can happen in two ways: ... |
H: $\delta$ Notation in linear algebra
In this equation below, what is $\delta_{l,q}$ denoting? Is $\delta$ a standard notation, or anything to do with all one's or the basis matrix etc?
$$A_{ij}=\delta_{l,q}\left(\sum_{h=1}^n B_{l,h} + B_{l,q}\right)$$
AI: It's the Kronecker delta function. |
H: Show that if $n$ and $k$ are positive integers, then $\lceil \frac{n}{k} \rceil = \lfloor \frac{n - 1}{k} \rfloor + 1$
This is answer in the back of the book but it doesn't make sense to me:
There is some $b$ with $(b-1)k < n \leqslant bk$.
Hence, $(b-1)k \leqslant n-1 < bk$.
Divide by $k$ to obtain $b-1 < \frac{n... |
H: analysis problem proof with derivative
$f:[a,b] \to \mathbb R$ is a continuous function and $0 < a < b$ and $f$ is differentiable in $(a,b)$ and $\dfrac{f(a)}{a} = \dfrac{f(b)}{b}$.
Prove that there exists $x \in (a,b)$ so that $xf'(x) = f(x)$.
AI: Hint: Apply the mean value theorem (or Rolle's theorem) to $g(x) = ... |
H: I would like a hint in order to prove that this matrix is positive definite
Let $a_{ij}$ be a real number for all $i,j\in\{1,...,n\}$. Consider the matrix below.
$$B=\begin{bmatrix}
\sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & \sum_{k=1}^na_{1k}a_{nk}\\
\sum_{k=1}^na_{2k}a_{1k} & \sum_{k=1}^n(a_{2k... |
H: Evaluating $\lim_{x\to0}\frac{x+\sin(x)}{x^2-\sin(x)}$
I did the math, and my calculations were:
$$\lim_{x\to0}\frac{x+\sin(x)}{x^2-\sin(x)}= \lim_{x\to0}\frac{x}{x^2-\sin(x)}+\frac{\sin(x)}{x^2-\sin(x)}$$ But I can not get out of it. I would like do it without using derivation or L'Hôpital's rule .
AI: $$\lim_{x\t... |
H: Explain this consequence of continuity
A consequence of continuity is the following fact:
if $f(x)$ is continuous at $x=b$ and $\lim\limits_{x \to a} g(x)=b$, then,
$\lim\limits_{x \to a} f(g(x)) = f(\lim\limits_{x \to a}g(x))$
with this fact we can solve the following:
$\lim\limits_{x \to 0} e^{\sin x}= e^{\lim\... |
H: Does $(\{0,1\},*)$ form a group?
I am reading my first book on abstract algebra. I am not enrolled in a class on the subject.
Given $S = \{0,1\}$. Is $(S,\cdot)$ a group?
$S$ is closed under multiplication. $$0\cdot1=0,\,1\cdot0=0,\,0\cdot0=0,\,1\cdot1=1.$$
$S$ has an identity, $1$, I think.
$$0\cdot1=0,\,1\cdot1=... |
H: How to check if a set of vectors is a basis
OK, I am having a real problem with this and I am desperate.
I have a set of vectors $\{(1,0,-1), (2,5,1), (0,-4,3)\}$.
How do I check is this is a basis for $\mathbb{R}^3?$
My text says a basis $B$ for a vector space $V$ is a linearly independent subset of $V$ that gener... |
H: Why are the integers appearing in lens spaces coprime?
I have a past paper question for a first course in algebraic topology, which asks one to calculate the first three homology and homotopy groups for the space $L_n$, defined as follows:
Let $G=\{z\in\mathbb C|z^n=1\}\cong\mathbb Z_n$ act on $S^3=\{(z_1,z_2)| |z_... |
H: In a sequence of $n$ integers, must there be a contiguous subsequence that sums to a multiple of $n$?
Let $x_1, \ldots, x_n$ be integers. Then are there indices $1\le a\le b\le n$ such that $$\sum_{i=a}^b x_i$$ is a multiple of $n$?
AI: Let $$s_k = \sum_{i=1}^k x_i\pmod n.$$ If $s_k$ is zero for any $k$, we have... |
H: Find out the position of an element in a loop according to the number of elements
First of all,
I am terrible at math, and this small problem is giving me a nice headache. I'm sure most here will see this as an easy solution. I will try to be as clear as possible.
I have X number of items
I have 3 positions for th... |
H: Could someone help me to prove that this symmetric matrix is definite positive?
Let $a_{ij}\in\mathbb{R}$ for all $i,j\in\{1,...,n\}$ and $m\in\mathbb{N}$. Consider the matrix below.
$$B=\begin{bmatrix}
\sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & \sum_{k=1}^na_{1k}a_{mk}\\
\sum_{k=1}^na_{2k}a_{1k}... |
H: Is there a rational number between any two irrationals?
Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such a rational number?
[I posted this only so that the useful answ... |
H: polynomial congruence equations
Is there a general method to solve the following equation:
Finding $f(x)$ to satisfy:
$$\left \{ \begin{matrix} f(x) \equiv r_1(x) \pmod{g_1(x)}\\f(x) \equiv r_2(x) \pmod{g_2(x)} \end{matrix} \right. $$
where $f(x),r_1(x),r_2(x),g_1(x),g_2(x) \in \Bbb{F}[x]$ and $\gcd(g_1,g_2)=1$
tha... |
H: A good Problems and Solutions book accompany Baby Rudin?
I'm reading the book Principles of Mathematical Analysis by Walter Rudin, aka "Baby Rudin". The exercises included are very instructive and helpful, but I'd like to find a book that has more problems (with solutions) that could help me build a better understa... |
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