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H: Circle Chord Sequence
This is my first post, so be nice!
When I was in my first Geometry class in high school, I asked the teacher the following:
Given a circle of radius 2a, find the length of the chord running parallel to the diameter of the circle such that the semicircle cut by the chord is divided into two reg... |
H: Compatibility of pointwise and distributional convergence
Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let $u_k,\, u$ and $v$ be elements of $L^1_{\mathrm{loc}}(\Omega)$. Assume that
\begin{equation}
\begin{array}{ccc}
u_k(x)\to u(x)\quad x\text{-a.e.} &\text{and}& u_k\to v\quad \text{in }\mathscr{D}'.\end{... |
H: Hyperbolic Geometry - reference request
I need some information about Hyperbolic Geometry. For example, Spherical Geometry is a subsection of Hyperbolic Geometry or no?
Can you suggest to me a book or some other reference to help me better understand these notions?
Thanks a lot!
AI: You might want to start with t... |
H: Horizontal bar notation for isomorphisms or bijections
I have seen in many books, particularly on category theory, the use of an horizontal bar to indicate some sort of equivalence, but I have not seen a proper definition in any context.
For example:
$X \to Y^T$
_________
$T \times X \to Y$
(Excuse me for the bad f... |
H: Grothendieck group of a symmetric monoidal category is a lambda ring?
I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a $\lambda$-ring. Now, I have simply seen this (latter)... |
H: show union of two intervals is not connected
Let $X$ be a topological space. $X$ is connected if $X\neq U\cup V$ with open sets $U,V$ and $U\cap V=\emptyset$.
If you consider $A:=(0,1]\cup(2,3)\subset\mathbb R$, $A$ is not connected.
But how can you prove it? Clearly I have to find those open sets like above but ho... |
H: Very symmetric convex polytope
Let $C_n$ be the convex polytope in ${\mathbb R}^n$ defined by the inequalities
(in $n$ variables $x_1,x_2, \ldots ,x_n$) :
$$
x_i \geq 0, x_i+x_j \leq 1
$$
(for any indices $i<j$).
Denote by $E_n$ the set of extremal points of $C_n$. We have a natural action
of the symmetric group $... |
H: Projections onto closed and convex sets
I have to prove that if $A$ is convex and closed set, then $z=P_A(x)$ for all $z\in A$ if and only if $\langle x-z, z-y\rangle \geq 0$ for all $y\in A$
I have following proof which is not much complicated, but I don't understand few things.
If $g(\theta)=||x-((1-\theta)z+... |
H: Vector analysis. Del and dot products
I am trying to prove that
$$\nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A})$$
I've gotten as far as $\nabla(\mathbf{A} \... |
H: Connectedness of the Given Set
How will I find out that $A=\{(x,y) \in\Bbb C^2:x^2+y^2=1\}$ is connected or not in $\Bbb C^2$?
Thanks for any help.
AI: Yes, we have the usual rational parametrization of the conic $C=\{-z_0^2+z_1^2+z_2^2=0\}\subset\mathbb CP^2$:
$$f\colon \mathbb CP^1\to\mathbb CP^2, \quad f([t_0,t_... |
H: Barbalat's lemma for Stability Analysis
Good day,
We have:
Lyapunov-Like Lemma: If a scaler function V(t, x) satisfies the
following conditions:
$V(t,x)$ is lower bounded
$\dot{V}(t,x)$ is negative semi-definite
$\dot{V}(t,x)$ is uniformly continuous in time
then $\dot{V}(t,x) \to 0$ as $t \to \infty $.
Now if w... |
H: Principal ideals and UFD's
Problems 1-6 form a project designed to prove that if R is a UFD and every nonzero prime ideal of R is maixmal, then R is a PID.
Let I be an ideal of R, since {0} is principal, we can assume that $I \not= \{0\}$. Since R is a UFD, every nonzero element of I can be written as $up_1...p_t$ ... |
H: How does standard random variable have variance of 1?
Let X be a discrete random variable and define $Z = \cfrac{X - \mu_x}{\sigma_x}=\cfrac{1}{\sigma_x} \cdot X - \cfrac{\mu_x}{\sigma_x}$ which is a linear transformation of $X$.
How do you get a variance of 1 assuming this? I tried working it out but couldn't g... |
H: Maximal subrings of $\mathbb{Q}$
Consider the sets $$\mathbb{Z}_{(p)}= \left\{ \frac{a}{b} \in \mathbb{Q}\mathbin{\Large\mid} b \notin (p) \right\} $$ Are these all the maximal subrings of the rationals?
AI: Yes. It is not too difficult to show that every subring of $\bf Q$ is a localization of $\bf Z$ with respect... |
H: How many ways to divide a population of n members into groups of i members.
Let's say I have a population of 180, to be divided in disjoint groups of 6. In how many different ways can I divide this population? A general formula would help! Thanks.
AI: There are $180!$ ways to line the people up, and we can then gro... |
H: Vector Picking on the Unit Sphere
Imagine a vector from the center of a unit sphere to its surface:
Now imagine a second vector generated in indentical fashion. Given the first vector, how can I generate vectors to uniformally distribute the angle between them (θ).
My first thought was to use spherical coordinat... |
H: derivative of fourier transform
Let $f\in C^k$ and $f^{(k)}$ be absolutely integrable. I want to show for the fourier transform:
$$\widehat{f^{(k)}}(z)=(iz)^k\widehat{f}(z)$$
I want to prove it for $k=1$ and did the following:
partial integration: $$\begin{align*}
\int_a^bf'(w)\exp(-iwz)dw&=f(b)\exp(-ibz)-f(a)\ex... |
H: Find $\lim_{x\to-\infty}{x+e^{-x}}$
I have this exercise in my worksheet:
$$\lim_{x\to-\infty}{x+e^{-x}}$$
I am always ending up with $-∞+∞$ or $\frac{∞}{∞}$. It says the answer is $+∞$, but how can I get that?
AI: Negative numbers make me nervous, so let $t=-x$. We want
$$\lim_{t\to \infty} (e^t-t).$$
The answer ... |
H: Finite group acting freely on Haussdorf space- Topology problem
How to prove the following problem:
It is given Hausdorff space $X$ and finite group $G$ (with neutral $e$) that is acting freely on $X$. For $g\in G$, $\overline{g}:X\rightarrow X$ is homeomorphism.
a) Prove that for every $x\in X$ exists an open nei... |
H: In how many ways can you split a string of length n such that every substring has length at least m?
Suppose you have a string of length 7 (abcdefg) and you want to split this string in substrings of length at least 2.
The full enumation of the possibilities is the following:
ab/cd/efg
ab/cde/fg
abc/de/fg
abc/defg
... |
H: CDFs of generalize beta distribution pdf and standard beta pdf.
Let $f(x)$ be the probability density function (pdf) of the standard beta distribution on $(0,1)$. And let $f_d(x)$ be the pdf of the generalized beta distribution on $(0,d)$. I know that,
$$f_d(x) = d \cdot f(\frac{x}{d})$$
The cumulative distribution... |
H: A problem from <> by Roytvarf, Birkhauser
I got a problem, which turned to be from the book "Thinking in Problems How Mathematicians Find Creative Solutions" by Roytvarf, Chapter One, Jacobi Identities and Related Combinatorial Formulas :
The problem is asking to prove that
$$
\sum_{i=1}^{n}
\frac{x_i^m}{\prod_{j\... |
H: Fourier transform of periodic signal
I have a question that is similar to this one but slightly different.
If I have discrete signal $$s(t) = \sum_k n_k \delta(t-kT_0),\quad k=0,1,\dotsc,$$ where $n_k$ are just some scalar numbers. What is the Fourier transform of $s(t)$? I think it should be some kind of a convolu... |
H: Does Simpsons rule still apply when a < 0?
I am currently working on an assignment where I have to find the answer to the following integral using Simpsons rule:$\int x+1$ (MIN = -1 MAX = 3), I choose to have 6 intervals. I now start calculating:
$delta X = \frac{b-a}{n} = \frac{3--1}{6} = \frac{4}{6} = 0.6666666... |
H: Why is this true? $(\exists x)(P(x) \Rightarrow (\forall y) P(y))$
Why is this true?
$\exists x\,\big(P(x) \Rightarrow \forall y\:P(y)\big)$
AI: Since this may be homework, I do not want to provide the full formal proof, but I will share the informal justification. Classical first-order logic typically makes the a... |
H: Factoring Quadratic Trinomials
I'm currently doing some homework, but I'm COMPLETELY stuck on one problem. I need to factor the following trinomial:
$$5x^2+7xy+2y^2$$
How can I solve this problem? I have no idea what to do because of the different variables.
AI: Given:
$\boxed{5x^2+7xy+2y^2}$
We can use a method ca... |
H: How do I solve the weighted normal equations?
I am trying to solve the normal equations for a 3D LSE of a general quadric:
$$ z = ax^2 + bx + cxy + dy^2 + ey + f$$
Write as a vector equation:
$$ \vec{z}= \bf{X}\vec{\beta}$$
where the 'ith row of X is:
$$ Xi = [x^2, x, xy, y^2, y, 1]$$
and
$$ \vec{\beta} =[a, b, c,... |
H: Central Limit Theorem Problem
Let $U_i, i=1,...,300$ be iid r.v's from the uniform distribution on $[-\frac{1}{2},\frac{1}{2}]$. Calculate, using the CLT, $$P(\sum_{i=1}^{300}U_i \le 3).$$
My solution:
By the CLT, for large enough $n$ the random variable $$Z=\frac{\sum_{i=n}^{300}U_i - n \mu}{\sigma \sqrt{n}}$$
is ... |
H: Prove the converse of the Law of Sines
If $\alpha,\beta,\gamma,a,b,c \in \mathbb{R}^+$, $\alpha+\beta+\gamma=180^\circ$, and
$$
\frac{\sin(\alpha)}{a} = \frac{\sin(\beta)}{b} = \frac{\sin(\gamma)}{c} \qquad \text{ (1)}$$
then there exists a triangle in 2-space with angle-side pairs $(\alpha,a),(\beta,b),(\gamma,c)$... |
H: Finding two eigenvalues which add to $1$
$\textbf{Question}$: For $0<t<\pi$, the matrix
$$
\left( \begin{array}{cc}
\cos t & -\sin t \\
\sin t & \cos t \\
\end{array}
\right)
$$
has distinct complex eigenvalues $\lambda_1$ and $\lambda_2$. For what value of $t$, where $0< t< \pi$, is $\lambda_1+\lambda_2=1$?
... |
H: number of vertices in a self-complemntary graph
Problem: Prove that the number of vertices of a self-complementary graph must be congruent to 0 or 1 modulo 4.
I think my starting point would be that P4 and C5 are self-complementary and proceed by induction by adding P4s.
I established the first step in a previous p... |
H: Exercise over metric spaces
Let $A$ a closed subset of a metric space $E$ and let $x\in E-A$.
¿Is posible get two disjoint open sets U, V such that $A\subseteq U$ and $x\in V$?
If $A$ is a compact set I know if it is possible to demonstrate the exercise, but if $A$ If $ A $ is a compact set I know if it is possibl... |
H: Finding the constant term of a degree $3$ polynomial
Let $p(x)=x^3+ax^2+bx+c$, where $a,b,c$ are real constants. If $p(-3)=p(2)=0$ and $p'(-3)<0$, which of the following is a possible value of $c$?
A) $-27$
B) $-18$
C) $-6$
D) $-3$
E) $-\dfrac{1}{2}$
$\textbf{My attempt at this problem}$:
I drew a rough sk... |
H: How i can determine: $\lim_{x\to1} \frac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$?
This is actually a limit tending to 1, if you can help me see how are the steps to multiply the factors, because it seems that there are many multiplications and this confuses me a lot! $$\lim_{x\to1} \frac{x^{\frac{1}{3}}-1}{x^{\frac... |
H: How to prove that $\Delta f = 0$ for $f(x)=\frac{1}{||x||}$?
I am given a function $f:\mathbb{R}^3\rightarrow\mathbb{R}:f(x)=\frac{1}{||x||}$, and I am not really sure that the norm must be Euclidian (anyway it wasn't mentioned in the task), and I have to prove that $\Delta f = 0$ (Laplace operator: $\Delta f(x_1,.... |
H: Vector Parametrization of intersection of a plane and an elliptical cylinder
Plane: $x+y+z= 1$
Elliptical cylinder: $(y/3)^2 + (z/8)^2 =1$
Find the parametrization in which they intersect.
AI: Hint: A point $(x,y,z) \in \mathbb R^3$ is on the zylinder, iff for one $\theta \in [0,2\pi]$ $y = 3\cos\theta$, $z = 8\si... |
H: proof of a tree with two vertices of degree three
This is a practice question from the text.
The Question : Show that a tree with two vertices of degree $3$ must have at least four vertices of degree $1$. I have the answer to PART A.
Part B) Show that the result of Part (a) is the best possible i.e. a tree with t... |
H: Finding the root of a degree $5$ polynomial
$\textbf{Question}$: which of the following $\textbf{cannot}$ be a root of a polynomial in $x$ of the form $9x^5+ax^3+b$, where $a$ and $b$ are integers?
A) $-9$
B) $-5$
C) $\dfrac{1}{4}$
D) $\dfrac{1}{3}$
E) $9$
I thought about this question for a bit now and can anyo... |
H: Celsius to Fahrenheit back and forth conversion with rounding.
Recently I've encountered some problem with conversion Celsius and Fahrenheit scales.
Let's assume that I have value of 44 degrees in Fahrenheit scale, I convert this to the Celsius which gives me 6,6667 degrees in Celsius. I'm rounding it to the 7 deg... |
H: Decreasing sequence in a normed space
Consider by $p\geq 1$ the set $l^p=\{(x_n):x_n\in\mathbb{R},\,\,\sum |x_n|^p<\infty\}$. If defined by $x\in l^1$ $$||x||_p=\left(\sum_{n=1}^{\infty} |x_n|^p\right)^{1/p}$$ How to prove that the sequence $(||x||_p)_{p\geq 1}$ is decreasing?
AI: Suppose $x\in \ell^p$ with $\|x\|... |
H: Smooth function which is not continuous
I have seen it mentioned that in certain infinite dimensional topological vector spaces it is possible to have a smooth curve which is not continuous, but I've never seen an explicit example. Can anybody point me towards a reference for this?
AI: The mentioner might have been... |
H: Finding all possibilities as the fourth vertex of a parallogram
Consider the points $A=(-1,2), B=(6,4)$, and $C=(1,-20)$ in the plane.
For how many different points $D$ in the plane are $A,B,C,D$ the vertices of a parallelogram?
A) none
B) one
C) two
D) three
E) four
$\textbf{My attempt}$: One parallelogram can be... |
H: Open neighborhoods of a $G_\delta$ set
This may have a simple answer, but I couldn't find it so far either in textbooks or in math.stackexchange. Let $X$ be a metric space, and $$A=\bigcap^\infty_{n=1}A_n$$ a $G_\delta$ subset of $X$, where $A_n\subset X$ is open for each $n\in\mathbb{N}$. We assume for simplicity ... |
H: The probability of getting more heads than tails in a coin toss
A fair coin is to be tossed $8$ times. What is the probability that more of the tosses will result in heads than will result in tails?
$\textbf{Guess:}$ I'm guessing that by symmetry, we can write down the probability $x$ of getting exactly $4$ heads... |
H: simplifying $\min(\max(A,B),C) $
In a larger problem, I have to make use of the following
$$\min(\max(A,\ B),\ C)$$
Please how do I simplify?
AI: If you wanna use the notation in stochastic caculus I saw sometimes, $max(A,B) = B + (A-B)_+$, then $$min(B+(A-B)_+, C) = C - (C - B - (A-B)_+)_+$$ |
H: Determine if vectors are linearly independent
Determine if the following set of vectors is linearly independent:
$$\left[\begin{array}{r}2\\2\\0\end{array}\right],\left[\begin{array}{r}1\\-1\\1\end{array}\right],\left[\begin{array}{r}4\\2\\-2\end{array}\right]$$
I've done the following system of equations, and I th... |
H: Why is the line between two points called the line of the "secant"?
The definition of the slope of the line of the secant is:
slope = $\frac{y2-y1}{x2-x1}$
The definition of the slope of the tangent line is:
$\lim_{h->0}\frac{f(x+h)-f(x)}{h}$
I understand why they call it the tangent line since the angle to the x a... |
H: Every bounded non countable subset of $\mathbb{R}$ has a two-sided accumulation point.
Inspired on proving that every compact set of the Sorgenfrey line is countable.
Trying to prove any of these in $\mathbb{R}$: 1) Every bounded non countable set has a both-sided accumulation point. That is, I supposed a stronger ... |
H: The number of continuous real-valued functions on $[-1,1]$ such that $(f(x))^2=x^2$ for all $x$ in the domain
I've been thinking about the following problem:
How many continuous real-valued functions $f$ are there with domain $[-1,1]$ such that $(f(x))^2=x^2$ for each $x$ in $[-1,1]$?
I thought that there are tw... |
H: Must $X$ be lindelöf if it has countable network?
A network is like a base, except that its members need not be open sets.
A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exists an $M \in \mathcal N$ such that $x\in ... |
H: What is the importance of examples in the study of group theory?
When I study topics in group theory (I am currently following Dummit and Foote) I don't care about examples so much. I read them, try to understand the applications of the theorems and corollaries on the examples. Most of the examples are about $D_{2n... |
H: Probability of being matched against a pair of people
$\textbf{Question:}$ Suppose you are playing a game in which two teams of five people, call them Team A and Team B, compete. Each of the ten people is randomly assigned a unique role (no two people share the same role) from a pool of 100 different roles, call th... |
H: Weakly inaccessible cardinals and Discovering Modern Set Theory
So I've been trying to teach myself some set theory and I've come across some exercises in Just and Weese's Discovering Modern Set Theory. To whit:
Pg. 180
Definition 20: A cardinal $\kappa$ is called weakly inaccessible if $\kappa$ is an uncountable r... |
H: Counting the number of elements in the set $\{ x^{13n}:n \mbox{ is a positive integer}\}$ under certain conditions
A cyclic group of order $15$ has an element $x$ such that the set $\{x^3,x^5,x^9 \}$ has exactly two elements. How many elements are in the set $\{ x^{13n}:n \mbox{ is a positive integer}\}$?
I feel li... |
H: determine whether graph is planar
This is not a HW question just a practice exercise in the text.
The question is to determine whether its planar or not. I dont think its Planar and I cant find a subgraph that is homeomorphic to $k_{3,3}$ or $K_5$. I have a feeling I might be wrong and if I am right i.e the graph... |
H: probability of who will be selected
"In an office there are 3 secretaries, 4 accountants, and 2 receptionists. If a committee of 3 is to be formed, find the probability that one of each will be selected?
Attempted Solution:
First attempt: (3/9)(4/9)(2/9) = 8/243
Second attempt: (3/9)(4/8)(2/7) = 1/21
Don't if eithe... |
H: Mutually Exclusive Events (or not)
Suppose that P(A) = 0.42, P(B) = 0.38 and P(A U B) = 0.70. Are A and B mutually exclusive? Explain your answer.
Now from what I gather, mutually exclusive events are those that are not dependent upon one another, correct? If that's the case then they are not mutually exclusive sin... |
H: Content of a polynomial
Define the content of a polynomial (over an arbitrary commutative ring $A$) to be the ideal generated by its coefficients, denoted $c(f)$. I want to show that
$$c(fg) = c(f)c(g).$$
(I was told this is true.)
What I was able to show was that $c(fg) \subseteq c(f)c(g)$ (this is obvious), ... |
H: Volume of a Parallelpiped, Homework Check
I've not sure if I am doing these types of questions properly, so I figured I would ask here. Find the volume of the parallelpiped with sides u, v, and w:
$u = 3, 1, 2$
$v = 4, 5, 1$
$w = 1, 2, 4$
(v x w) =
$\begin{align}
\begin{bmatrix}
4 & 5 &1\\
1 &2 & 4
\end{bmatrix}
\... |
H: Solving a Linear System
I'm reading through my textbook, and it gives the linear system:
$a+2b-c+d=0$
$2a+3b+c+d=0$
$3a-b+2c+d=0$
It doesn't explain how this is solved, they just provide the answer which they come up with:
$a = -\frac{9}{16}t$
$b = -\frac{1}{16}t$
$c = \frac{5}{16}t$
$d = t$
I don't understand how ... |
H: How to derive the expected value of $X^\alpha\log X$
Let $X$ follow a Weibull distribution, with density $$f(x)=\frac{\alpha}{\theta}x^{\alpha-1}e^{-\frac{x^{\alpha}}{\theta}}\quad x>0 .$$
How can I find the following expectation?
$$E[X^{\alpha}\log X]$$
The answer given in the paper by Debasis Kundu "Estimation o... |
H: How Would You Interpret This Question?
A man has 5 coins, two of which are double-headed,
one is double-tailed, and two are normal.
He picks a coin at random and tosses it.
What's the probability that the lower face is a tail?
P(H) = P(H|2H)P(2H) + P(H|2T)P(2T) + P(H|N)P(N) = $\dfrac {3}{5}$
He sees that the c... |
H: Need help proving the determinant of a particular sum of matrices.
I'm just learning how to use Mathematica and I was screwing around with it and I noticed that the following expression holds for a bunch of numbers that I threw into it.
I was wondering if someone could help me prove/disprove this?
$det(cJ_n+I)=cn+1... |
H: probability hypergeometric distribution
An HR manager estimates that 35% of married employees in a large office complex have spouses whose employers provide dental insurance and 65% have spouses whose employers provide extended medical and drug insurance. Of those whose spouses have dental insurance, 90% have also ... |
H: Solve the following in non-negative integers: $3^x-y^3=1$.
Solve the following in non-negative integers: $$3^x-y^3=1$$
Of course $(x,y)=(0,0)$ is a trivial solution. After seeing that I proceeded like this:
$$3^x-y^3=1$$$$\implies3^x-1=y^3$$$$\implies2(3^{x-1}+3^{x-2}+ \cdots +3^1+1)=y^3$$$$\therefore2|y$$
So let $... |
H: Using polynomials as recursions
I made this observation in my discrete math course a while back. I explored it further online, so not all the ideas contained are mine alone. I still am confused about some things, though. Consider the equation $x^2-x-1=0$ whose positive root is the golden ratio $\phi$. Rewriting... |
H: Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$
I cannot think of a non-constant smooth function which maps all real numbers into rational numbers.
Can anyone give a simple example ? The simpler, the better !
AI: No. There isn't even a non-constant continuous function $\mathbb R \rig... |
H: Given that $\int_{a}^{b} f(x) dx\le M$ for all $a,b\in\mathbb{R}$
Given that for a continuous $f$, $\int_{a}^{b} f(x) dx\le M$ for all $a,b\in\mathbb{R}$
Then
i) $\int_{0}^{\infty} f(x) dx$ exists if $f\ge 0$
ii) $\int_{0}^{\infty} f$ exists if $f$ is differentiable
iii) $\int_{0}^{\infty} f$ exists if $f$ is diff... |
H: A set is a finite chain if every subset has a top and bottom element
I am presently attempting Exercise 2 in Kaplansky, Set Theory and Metric Spaces
Exercise 2: Let $L$ be a partially ordered set in which every subset has a top and bottom element. Prove that $L$ is a finite chain.
Proof: Denote a subset of $L$ by $... |
H: Where is the fallacy in the argument using Prime Number Theorem
I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem:
Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say $\log(y)\rightarrow\log(x)$ and if I apply antilog I ge... |
H: Probability Theory (Pinochle deck of cards)
-=Attempts added=-
A Pinochle deck is a special deck of cards with 48 cards in total. it consists of two copies of each of the 9, 10, J, Q, K and Ace of all four suits (so there are 2 nine of clubs, 2 nine of diamonds, 2 nine of hearts, two nine of spades, and so on for e... |
H: Lebesgue vs. Riemann integrable function
While trying to learn the difference between Lebesgue and Riemann integrals, I came across the following example:
$$\int_{0}^{1}t^\lambda\,\mathrm dt$$
What I know so far:
only for $\lambda>0$ the integral exists as a Riemann integral,
only for $\lambda>1$ it exists as an... |
H: Evaluating $\lim\limits_{n\to\infty}(a_1^n+\dots+a_k^n)^{1\over n}$ where $a_1 \ge \cdots\ge a_k \ge 0$
Need to find $\lim\limits_{n\to\infty}(a_1^n+\dots+a_k^n)^{1\over n}$ Where $a_1\ge\dots\ge a_k\ge 0$
I thought about Cauchy Theorem on limit $\lim\limits_{n\to\infty}\dfrac{a_1+\dots+a_n}{n}=\lim a_n$ and someth... |
H: Affine transformation, if $L_1, L_2 - $ skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective
Could you tell me how to prove that if $f$ is affine transformation, $L_1, L_2 $ are skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective?
AI: Suppose $L_i = a_i + \mathbb Rv_i$ are th... |
H: Is a continuous map between smoothable manifolds always smoothable?
Let $X$ and $Y$ be topological manifolds and $f:X\to Y$ a continuous map.
Suppose $X$ and $Y$ admit a differentiable structure (at least one).
My question: is it always possible to choose a differentiable structure on $X$ and one on $Y$ in such a w... |
H: Eigenvectors orthogonal to $j$
I'm studying the proof of the following statement:
$Spec(K_n) = (n-1)^1(-1)^{n-1}$
At some point I have:
By the Spectral Theorem, when looking for eigenvectors $v$ we can assume they are orthogonal to $j$.
$j$ being the all-1 vector.
I don't understand which part of the Spectral T... |
H: Limit of double sum: $\lim\limits_{n\to\infty}n^{-2}\sum\limits_{k=1}^n\sum\limits_{m=k+1}^n\left(\frac{n-2k}{n+2k}\right)^2\frac{n-2m}{n+2m}$
Who is so kind to enlighten me about the steps I need to follow?
$$\lim_{n\to\infty}\frac{1}{n^2}\sum_{k=1}^n\sum_{m=k+1}^n\left(\frac{n-2k}{n+2k}\right)^2\frac{n-2m}{n+2m}... |
H: Combinatorics question with infinite sets
I have made a claim that I now am trying to prove. The claim is:
if $f: S \times S \to P(S) $ is an injective function mapping pairs to subsets then there exists a pair $(a,b)$ with the property that $f(a,b) = s$ and $c \in s$ with $c \neq a$ and $c \neq b$.
I'm quite sure ... |
H: Representation of $j$ in an orthonormal basis
I'm studying a proof and I have $j$ (the all-1 vector) represented in the basis of orthonormal vectors $\{v_1, ..., v_n\}$ such that
$$j = \sum_ic_i v_i$$
I don't understand why I then have:
$$j^Tv_i = c_i$$
and
$$\sum_ic_i^2 = n$$
AI: Note that $$j^Tv_i=\left(\sum_i c_... |
H: How can I evaluate $\int_0^\infty \frac{\sin x}{x} \,dx$? [may be duplicated]
How can I evaluate $\displaystyle\int_0^\infty \frac{\sin x}{x} \, dx$? (Let $\displaystyle \frac{\sin0}{0}=1$.)
I proved that this integral exists by Cauchy's sequence.
However I can't evaluate what is the exact value of this integral.
A... |
H: variance inequality
Show that. for any discrete random variable X that takes on values in
the range [0,1]. Var[X] $\le$ 1/4.
I translate it into a inequality like this:
$x_1, x_2, x_3 \cdots ,x_n$ where $0 \le x_i \le 1$, and $p_1, p_2, p_3 \cdots ,p_n$ where $p_1+ p_2+ p_3 \cdots +p_n = 1$, prove that $\sum _1^nx... |
H: Does $X$ have countable network if it has countable extent?
Let $X$ have a $\sigma$-discrete network and have countable extent.
Does $X$ have countable network?
A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exis... |
H: Largest eigenvalue of a graph
I have $\lambda_1$ the largest eigenvalue of a graph, with $x = (x_v)_{v \in V(G)}$ the corresponding eigenvector.
$x_u$ is the entry of $x$ with maximum absolute value.
I don't understand why I then have:
$$\lambda_1 x_u = \sum_{v \in N(u)} x_v$$
$N(u)$ being the neighborhood of $u$.... |
H: Easiest example of rearrangement of infinite leading to different sums
I am reading the section on the rearrangement of infinite series in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press.
As an example, the author shows that
is a rearrangement of the sequence
\begin{align} \f... |
H: Pole set of rational function on $V(WZ-XY)$
Let $V = V(WZ - XY)\subset \mathbb{A}(k)^4$ (k is algebraically closed). This is an irreducible algebraic set so the coordinate ring is an integral domain which allows us to form a field of fractions, $k(V).$ Let $\overline{W}, \overline{X}$ denote the image of $W$ and $X... |
H: Which accompanying text do you suggest on these topics of finite fields?
Please take a look at pages 80-85 (Section 2.6 Finite fields) of this handbook, Handbook of Applied Cryptography.
I am trying to learn the mathematics enumerated in these pages. I do not need the algorithms. I will also skip any proof of the... |
H: Set Unions with repeating elements
the union of {1,2,3,4} and {2,3,4,5} is {1,2,3,4,5}
but the union of {2,2,2,3} and {2,3,3,5} is {2,3,5}
Is there another Union concept which takes account of the number of occurences of repeating elements. i.e.
the SOME_TERM of {2,2,2,3} and {2,3,3,5} is {2,2,2,3,3,5}. basically ... |
H: Choosing disjoint representatives from two sets of squares
There are 3 red axis-aligned interior-disjoint squares.
There are 3 blue axis-aligned interior-disjoint squares.
Is it always possible to find a pair of 1 red square and 1 blue square, such that they are interior-disjoint?
I tried many combinations, and it ... |
H: Proof of the 2 pointer method for finding a linked list loop
The linked list with a loop problem is classical - "how do you detect that a linked list has a loop" ? The "creative" solution to this is to use 2 pointers, one moving at a speed of 1 and the second one at the speed of 2. If the two pointers meet then the... |
H: A question of the property of positive definite matrix
Let $A$ be a positive definite $n$ by $n$ matrix.
Let $c_1$ be the smallest eigenvalue of $A$ and $c_2$ be the largest eigenvalue of $A$.
Then how can I show that
$$ | \langle z, Aw \rangle |^2 \le \langle z, Az \rangle \langle w, Aw \rangle$$
and
$$ c_1 |z|^2... |
H: Eigenvalues of power of matrices
How come if $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$?
And is its multiplicity necessarily the same?
AI: Let $v$ be an eigenvector corresponding to $\lambda$. Then, by induction
$$ A^k v = A^{k-1}(Av) = \lambda A^{k-1}v = \lambda^k v $$
hence $A^... |
H: Understanding a limit in standard Borel probability space
There is an exercise in lectures:
Let $(X, \Sigma, \mu)$ be a standard Borel probability space and $(B_n)_{n \in \mathbb{N}} \subset \Sigma$.
Show that
\begin{align*}
μ(\cup_{n \in \mathbb{N}} B_n) = \lim_{N \to \infty} μ(\cup_{n
\leqsl... |
H: Determinant of Matrix of Matrices
My question concerns a situation where you are looking for a determinant of a matrix which is in itself composed of other matrices (in my example, all the inner matrices are square and of equal dimensions).
Say we have matrix $A_{cl}$:
$$
A_{cl}=
\left[\begin{matrix}
0 & I\\
-kL... |
H: Showing symmetry involving a matrix and its transposed matrix
I'd appreciate if someone could find a better title for this question, for I'm short of ideas right now.
Given a matrix $A \in R^{n,n}$, show that
$$
\frac{1}{2}(A + A^t)
$$
is symmetric.
I see that it's symmetric and it seems obvious, but I don't really... |
H: Prove that det(BA) = 0 under some circumstances
How to prove that:
$ det(BA) = 0 $ Assuming:
$ m < n, A \in M_{mxn}, B\in M_{nxm}$?
AI: rank$(BA)\le $min(rank(A),rank (B))$= m<n$
Now as $AB$ is a $n\times n$ matrix and as its rank is less than $n$ its rows(columns) are not independent vectors implying $det(BA)=0$
... |
H: given point (2,6) and a line passes through point (3,0)
The question is: does the distance between the point $(2,6)$ to the that line could be $5$?
is there a solution to the problem without computing?
i would glad to know.
thanks.
AI: since the line $y=0$ has distance 6 from the point $(2,6)$ and there is a line w... |
H: Probability to select all 3 male mouses from 10 selected at random
In a cage there are 100 mouses from which 3 are male.
Compute the probability of selecting all 3 males from a group of 10 mouses selected at random.
I have this intuition:
$$
P(male)=0.03
$$
and number of all possibilities of selecting all 3 males m... |
H: Boolean simplification, 5 variables
I'm currently learning for my maths exam, and in the part about boolean algebra I came across an exercise that I can't seem to solve. I probably only need the first few steps to get started.
$$ (xyz + uv)(x+\overline{y}+\overline{z}+uv) $$
Usually, if I get into trouble, I can fa... |
H: Trace of a diagonalized matrix
Why do I have: $Tr(SDS^{-1})=Tr(D)$?
AI: Note that for any matrices $A$ and $B$ we have $\def\tr{\mathop{\rm Tr}}\tr(AB) = \tr(BA)$. To see this, one can argue as follows:
\begin{align*}
\tr(AB) &= \sum_i (AB)_{ii}\\
&= \sum_i \sum_j A_{ij}B_{ji}\\
&=\sum_j \sum_... |
H: What is a "maximal" object?
The idea of a "maximal" graph was introduced in a proof for Ore's Condition.
I didn't quite get the idea, and I would like more detailed explanations.
The theorem and proof are as follows.
Suppose G is a graph with v vertices ($v \ge 3$), and for every pair of non adjacent vertices $x$ a... |
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