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H: Maximum Entropy Distribution When Mean and Variance are Not Fixed with Positive Support
I know when the mean and variance of $\ln x$ are both fixed, then the maximum entropy probability distribution is lognormal. When the mean of a random variable is fixed the MEPD is the exponential distribution. My question is, w... |
H: Mathematical news sources.
I'm studying my high school right now but I really like math and it would be great for me if I could find a place where I can find about what is going on in the math world nowadays. About a year ago I was subscribed to the scientific american magazine and one time they put an article in t... |
H: Conversion of $F(x) = {1 \choose 0} (2t^2 - 3t + 1) + {3 \choose 1} (4t - 4t^2) + {1 \choose 2} (2t^2 - t)$
I have a textbook with a calculation step that is pretty unclear to me:
$$F(x) = {1 \choose 0} (2t^2 - 3t + 1) + {3 \choose 1} (4t - 4t^2) + {1 \choose 2} (2t^2 - t)$$
$$= {-8 \choose 0} t^2 + {8 \choose ... |
H: Proving that the graph of a function is closed
Let $F$ be a metric spaces and $(x_n)$ a sequence in $F$ without limit points. Define $f:\{1/n:n\in\mathbb N\}\cup \{0\}\to F$ by letting $f(0)=t$ with $t\in F$ and $f(1/n)=x_n$.
How to prove that the graph of $f$ is closed?
AI: If $\{(1/n, x_n)\}$ had limit points, th... |
H: Prove that the null set is a subset of a set $A$.
Prove that $\; \emptyset \subseteq A$.
The statement seems obvious to me, but how do I prove it?
My instructor said to prove that the statement is vacuously true, but I'm not sure what that means.
AI: Recall that to prove any set $X$ is a subset of another set $Y... |
H: Given the solution for a differential equation, find the corresponding differential equation
The solution of a differential equation is:
$$\left \{ \begin{array}{lcl}
x_1(t) & = & e^{-2t}-e^{-5t} \\
x_2(t) & = & e^{-2t}+e^{-3t}+e^{-5t} \\
x_3(t) & = & e^{-3t} + e^{-5t}
\end{array}\right .$$
with initial conditions
... |
H: Set $A$ has the cardinality of $\aleph_0$. Prove some properties of partial order $ \langle \mathcal P (A), \subseteq \rangle$.
I try to solve the following task:
Set $A$ has the cardinality of $\aleph_0$. The truth is that in the partial order
$ \langle \mathcal P (A), \subseteq \rangle$: (answer true or false... |
H: How to evaluate the derivative$\frac{d}{dx}\left(\ln\sqrt{\frac{4+x²}{4-x²}}\right)$?
How can I evaluate this derivative?:
$$\frac{d}{dx}\left(\ln\sqrt{\frac{4+x^2}{4-x^2}}\right)$$
Thank you.
AI: We'll exploit the properties of logarithms, recalling that $$\ln\left(\frac{a}{b}\right)^b = b \ln\left(\frac ab\rig... |
H: Using $\arctan(x)$ for integrating the following function
We want to find $\int \dfrac{x}{x^2 + 16} dx$
My method was as follows:
Rewrite it to: $\dfrac{\frac{1}{16x}}{{(\frac{1}{4x}})^2 +1}$
Take $u = \dfrac{1}{4}x$. We then have $ \int4u^2 \cdot \dfrac{1}{u^2+1} du = \dfrac{4}{3}u^3 \cdot \arctan(u) + c$.
However... |
H: Suppose A is a set of (x, y)', what is the name of the set that consists of all x in A?
Let $A$ be a set of a vector $(\mathbf{x}',\,\mathbf{y}')$. Here $\mathbf{x}'$ and $\mathbf{y}'$ could both be vectors. Is there a particular terminology for the set of all $\mathbf{x}'$ in the set $A$.
AI: Besides Git Gud's ans... |
H: $\sum c_k^2<\infty$ then $A=\{\sum_{k=1}^{\infty} a_ke_k :|a_k|\leq c_k \}$ is compact
Let $\{e_k\}_{k=1}^\infty$ be an orthonormal set in a Hilbert space $H$. If $\{c_k\}_{k=1}^\infty$ is a sequence of positive real numbers such that $\sum c_k^2<\infty$, then the set:
$$A=\left\{\sum_{k=1}^{\infty} a_ke_k :|a_k... |
H: Proof of closed form expression for a sum
For the following sum
$$
\sum_{i=0}^{n-r} \binom{n-r}{i}s^{r+i}(1-s)^{n-r-i}(1-t)^{i}
$$
Wolfram Alpha finds the following closed form
$$
s^{r}(1-st)^{n-r}.
$$
However, I have been unable to derive this myself. Without the binomial coefficient, it would be straightforward t... |
H: If $m^*(E)=\infty$, then $E=\bigcup_{k=1}^{\infty}E_k$, $E_k$ measurable and $m^*(E_k)<+\infty$
Reading Royden's fourth edition of Real Analysis. I'm working with outer measure defined as
$$m^*(E)=\inf\left\{\sum_{n=1}^\infty l(I_n):\,E\subset \bigcup_{n=1}^\infty I_n\right\},$$
where each $I_n$ is a bounded, open ... |
H: Numbers with no finite representation on paper
It occurred to me that there must be a lot of numbers without any form of finite representation on paper. Is there a name for these numbers?
For example...
Integers and rationals have a very simple representation e.g. 3/4
Irrational numbers obviously can also have a fi... |
H: Incompleteness theorem and $\mathbb{L}$.
Let $\alpha > \omega$ and $u = \{\ulcorner \sigma \urcorner : \sigma \in \mathrm{Th}(\mathbb{L}_\alpha, \in)\} \subseteq \omega$, where by $\mathbb{L}_\alpha$ we denote as usual the constructible sets up to level $\alpha$.
My question is "Can we have $u \in \mathbb{L}_\alpha... |
H: A peculiar binomial coefficient identity
While inventing exercises for a discrete math text I'm writing I came up with this
$$
\binom{\binom{n}{2}}{2}=3\binom{n+1}{4}
$$
It's an easy result to prove, but it got me wondering
Is this pure coincidence, or is it a special case of some more general result of which I'm ... |
H: Homomorphism $f:S_{2009} \rightarrow S_{2009}$
Is there a non-trivial homomorphism $f:S_{2009} \rightarrow S_{2009}$ with $(1,2)$ in the kernel?
Are there more than $2009$ different homomorphisms $f:S_{2009} \rightarrow S_{2009}$ with $(1,2,3)$ in the kernel?
Prime factorization: $2009 = 7^2*41$
Tips and/or answers... |
H: On a proof of dimension of vector spaces
There is a proof in my notes showing that if $U$ is a subspace of a finite-dimensional vector space $V$ then $\dim U \le \dim V$. It proceeds by extending a basis of $U$ to a basis of $V$. Wouldn't it be shorter and neater to say if $\dim U > \dim V$ then there would be at l... |
H: Why does $AX=0$ have only the trivial solution when $A=\left(\int_a^b g_i(x)g_j(x)dx\right)$?
The system is $AX=0$, where
$$A_{m\times m}=\begin{pmatrix}
\int_a^bg_1(x)g_1(x)dx & \cdots & \int_a^bg_1(x)g_m(x)dx \\
\vdots & & \vdots \\
\int_a^bg_m(x)g_1(x)dx & \cdots & \int_a^bg_m(x)g_m(x)dx
\end{pmatrix},$$
$$X=... |
H: Notation to describe the adding of a constant to all terms of a sequence
I've been struggling to get down the proper mathematical notation for sequences. Suppose I have the following sequence:
$$A = (4, 3, 7, 3, 1)$$
How do I describe the addition of a constant to all terms of sequence $A$? For example, if I were t... |
H: Does $a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$ converge?
I want to check if a sequence converges or diverges. The sequence is the following:
$$a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$$
I though of maybe using sandwich theorem, but can I use it, saying that the value will lie betwee... |
H: Help with this definition of $(G:_M I)$
I didn't understand why in this definition $I$ has to be an ideal to make sense.
REMARK
This is from Steps in Commutative Algebra, page 107.
Thanks a lot
AI: The definition doesn't claim that $I$ has to be an ideal, and in fact it doesn't, but if $S$ is any subset of $R$ th... |
H: number theory equation involving GCD
Fix the natural number $b$. How can I solve ?
$$
x+\gcd(x,b) \equiv 0 \mod(b)
$$
Can anyone please give me a reference?
Best
AI: Let $x=cd$ where $d=\gcd(x,b)$, and let $a:=b/d$. Then we have $\gcd(a,c)=1$, and
$$cd+d\equiv 0 \pmod{ad}$$
so that $c+1\equiv 0\pmod a$.
It means, t... |
H: Linear independence of functions: $x_1(t) = 3$, $x_2(t)=3\sin^2t$, $x_3(t)=4\cos^2t$
I want to determine whether 3 functions are linearly independent:
\begin{align*}
x_1(t) & = 3 \\
x_2(t) & = 3\sin^2(t) \\
x_3(t) & = 4\cos^2(t)
\end{align*}
Definition of Linear Independence: $c_1x_1 + c_2x_2 + c_3x_3 = 0 \implies ... |
H: Interpretation of Linear Algebra and superposition
I have been told and have seen that knowing Linear Algebra is foundational in pursuing advanced mathematics. After dealing with it enough I have gotten "used to it" (I still need a lot more practice!) What I would like to know is the interpretation of Linear Algebr... |
H: Uniform Continuity and Cauchy Sequences
Let $(S,d)$ and $(S^*, d^*)$ be metric spaces. If $f:S \to S^*$ is uniformly continuous and if $(s_n)$ is a Cauchy sequence in $S$, then $(f(s_n))$ is a Cauchy sequence in $S^*$.
$f:S \to S^*$ is uniformly continuous: $(\forall \varepsilon >0)( \exists > 0)(\forall s,t \in S)... |
H: Where does the theory of quadratic forms fail in characteristic 2?
Let $V$ be a finite-dimensional vector space over a field $k$, and $Q$ a nondegenerate quadratic form on $V$. If the characteristic of $k$ is not 2, then we can change coördinates on $V$ so that $Q(\vec{x})=\sum_ia_ix_i^2$ for some $a_i\in k.$ Thus ... |
H: Given a matrix $A$ and what it maps two vectors to, is $0$ an eigenvalue of it?
Studying for my Algebra exam, and this question popped out with no solution in a previous exam:
Given a matrix $A$ such that $A \begin{pmatrix} 0 \\ -1 \\ 0 \end{pmatrix} = \begin{pmatrix} -2 \\ -4 \\ 6 \end{pmatrix},\ A \begin{pmatrix... |
H: definitions of the free abelian group
How one can use the universal mapping property of free abelian groups to prove that $\sum_{b\in S}<b>$ is a free abelian group generated by $S$?
AI: Let $A$ be an arbitrary Abelian group and $f:S\to A$ a function.
Then there is a unique way to extend $f$ to a homomorphism $\bar... |
H: The continuity of measure
Let $m$ be the Lebesgue Measure. If $\{A_k\}_{k=1}^{\infty}$ is an ascending collection of measurable sets, then
$$m\left(\cup_{k=1}^\infty A_k\right)=\lim_{k\to\infty}m(A_k).$$
Can someone share a story as to why this is called one of the "continuity" properties of measure?
AI: My underst... |
H: Prove that $v_1^{T}v_2 = [0] $ (the 1 by 1 zero matrix)
This question is from a Linear Algebra past exam paper that I am reviewing as I study for finals. The full question is:
Suppose that $M$ is a square matrix. Recall that $M^T$ denotes the transpose of $M$, the result of interchanging columns and rows of $M$. R... |
H: Constructing a simple isomorphism from $\mathbb{R^{2n}}$ to $\mathbb{C^n}$
I need to construct an isomorphism from $\mathbb{R^{2n}}$ onto $\mathbb{C^n}$ (both over $\mathbb{R}$). I know couple of things:
This isomorphism will be a (bijective) linear map between two vector spaces that have equal dimensions.
The lin... |
H: Conditional Probability P(A ∩ B)
So I found myself in a infinite loop while trying to do some probability. If A and B are independent, calculating P(A ∩ B) is as simple as P(A)P(B). However, how do I calculate P(A ∩ B) if they are dependent? I know P(A∣B)⋅P(B)=P(A∩B).
example:
Charlie and Doug take turns rolling tw... |
H: How many 3 digit positive integers are divisible by 5?
Q1: How many 3 digit positive integers are divisible by 5?
Since it should be divisble by 5 it must end it 5?so
9 * 10 * 1
Is that right? But shouldn't I check if it ends in 0 or 5?
q2: how many odd non repeating 3 digit positive integers are there? It should ... |
H: Cauchy Schwarz Inequality for numbers
The CS inequality is given by
$$x_1y_1 + x_2y_2 \leq \sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}$$
I read that if $x_1 = cy_1$ and $x_2 = cy_2$, then equality holds.
But I reduced the above to $c \leq \sqrt{c^2} = |c|$. So isn't this only true if $c > 0$?
AI: Assume $x_1,\dots,... |
H: Boundary Value Problem $y^{(4)} =-24$ a horizontal beam in theory of elasticity
The equation is $$y^{(4)} =-24$$
boundary values are $y(0) = y'(0) = y(4) = y'(4) = 0$. I integrated the equation four times and got $$y = -x^4 + ax^3 + bx^2 + cx + d$$ but if I used character equation $r^4 + 24 = 0$ I will get complex ... |
H: Area projection from cube to sphere
I have a regular cube of edge length = 2, and a sphere of radius = 1. Each face of the cube has been divided into N*N equal-area squares. How can I compute the projected area of every square on the sphere, so that the area of all the projections sum up to 4*pi.
I came to the foll... |
H: What do I do in the case of this double series?
I am student who learns from examples, and I have yet to see what happens when two sums such as this one,
$$\displaystyle\sum\limits_{a=1}^2 \displaystyle\sum\limits_{b=a+1}^2 4\left[\left(\frac{1}{b-a}\right)^8 - \left(\frac{1}{b-a}\right)^4\right]$$
be solved. I am ... |
H: About Baire spaces
I'm having difficult to solve this:
Determine whether or not $\mathbb{R}_l$ is a Baire space.
I tried to aplly the following lemma: "X is a Baire space iff given any countable collection $\mathbb{U}_n$ of open sets in X, each of which is dense in X, their intersection $\bigcap{U}_n$ is also dense... |
H: A Joint Density Problem Involving Change of Variable
A point $(X,Y)$ is picked at random uniformly in the unit circle.
Find the joint density of $R$ and $X$, where $R^2 = X^2 + Y^2$.
So the question is asking for $f_{R, X} (r, x) = \mathbb P(R=r, X=x)$.
Now, if I integrate$f_{R, X} (r, x)$ over all values of $r$... |
H: Solving this system
I have been using substitution as the method to solve this. It doesn't work and I also do some trial-and-error but it takes a lot of time before I found the solution. Now, Is there a better way of solving this system of equation? Thank You.
$$\begin{cases} y =2000( 1 + 0.2x) \\ z= 2000(1.2)^x \... |
H: SOA Exam P Question: $P$ is a random point on the Cartesian Coordinate Plane. Find the variance of the area of a circle formed by $P$.
Caution: This problem was "passed down" to me and I think the wording was altered or lost along the way. I will post the problem as I have it and then make suggestions on what I th... |
H: Check solutions of vector Differential Equations
I have solved the vector ODE: $x\prime = \begin{pmatrix}1& 1 \\ -1 &1 \end{pmatrix}x$
I found an eigenvalue $\lambda=1+i$ and deduced the corresponding eigenvector:
\begin{align}
(A-\lambda I)x =& 0 \\
\begin{pmatrix}1-1-i & 1 \\-1& 1-1-i \end{pmatrix}x =& 0 \\
\begi... |
H: Why should $|2^\mathbb{N}|>|\mathbb{N}^2|$ be true?
I've been thinking a bit about infinite things lately, and this question I had wondered about came back to me.
One of the classic expository demonstrations of Cantor's work is the two equally surprising facts that there are as many rationals as natural numbers, bu... |
H: (1)Determination of local and global maximum/minimum points
Be $I=[-2,2]\subset \mathbb{R}$ and a function $f:I \Leftarrow \mathbb{R} $ (continuous and differentiable on $I$)
$$ f(x) =
\begin{cases}
xe^{x-1} & \text{if $x \leq 0$}\\
xe^{1-x} & \text{if $x>0$ }
\end{cases}$$
Determine the local & global ... |
H: $f''$ bounded implies $f'$ is bounded
Suppose that $f''$ exists on $[0,1]$ and that $f(0)=0=f(1)$. Suppose also that $|f''(x)|\le K$ for $x\in(0,1)$. Prove that $|f'(1/2)|\le K/4$ and that $|f'(x)|\le K/2$ for $x\in [0,1]$.
The mean-value theorem might help, but I can't see how.
AI: Suppose $f'\left(\dfrac12\righ... |
H: If $\mathcal A$ is diagonalizable, determine $\mathcal S$ and $\Lambda$ such that $\mathcal A=\mathcal S\Lambda\mathcal S^{-1}$
$$\mathcal A=\pmatrix{-5.25&-4.125&8.25\\-15&-0.5&15\\-4.75&-0.375&7.75}$$
If $\mathcal A$ is diagonalizable, determine $\mathcal S$ and $\Lambda$ such that $\mathcal A=\mathcal S\L... |
H: Simple algebraic logic question
$2x^2 + 3y^2=0$. This is possible only when both the value of $x$ and $y$ are zero. But the thing is I fail to understand the significance of this equation. Why would such an equation exist or why we create them?
AI: As you note, the only solution to $\;2x^2 + 3y^2 = 0$, is indeed $(... |
H: The minimum possible value of $\,\,|w|^2+|w-3|^2+|w-6i|^2$
I am stuck on the following problem :
What is the minimum possible value of $\,\,|w|^2+|w-3|^2+|w-6i|^2\,\,,w \in \Bbb C,i=\sqrt{-1}\,\,$?
The options are $\,\,15,45,20,30.$
I have no idea how to tackle it effectively. Some detailed explanation will be of... |
H: Is "have the same cardinality" a equivalence relation?
A relation is a subset of the Cartesian product of two sets, if "have the same cardinality", denoted as $R$, is a relation, then there must exist set $A, B$, such that $R \subset A \times B$. What are $A, B$ then? They cannot be "set of all sets", because there... |
H: $f(x)=x$ if $x$ irrational and $f(x)=p\sin\frac1q$ if $x$ rational
Define the real-valued function $f$ on $\mathbb{R}$ by setting $f(x)=x$ if $x$ is irrational, and $f(x)=p\sin\frac1q$ if $x=\frac{p}q$ is written in lowest terms. At what points is $f$ continuous?
I'm pretty sure $f$ is continuous at no point. For... |
H: If the product of two integers is odd, then both are odd
Prove $ \forall x, y \in \mathbb{Z}$ if $x\times y$ is odd then $x$ and $y$ are odd.
Is this valid proof?
Proof by contrapositive. The contrapositive of the implication is: $ \forall x, y \in \mathbb{Z} $if $x$ and $y$ are not both odd then $x \times y$ is... |
H: Show that $7 \mid( 1^{47} +2^{47}+3^{47}+4^{47}+5^{47}+6^{47})$
I am solving this one using the fermat's little theorem but I got stuck up with some manipulations and there is no way I could tell that the residue of the sum of each term is still divisible by $7$. what could be a better approach or am I on the right... |
H: Are these combinations and permutations correct?
I wanted to know if I did these 3 questions correctly:
There are 100 distinct people in line. How many arrangements are there?
Ans: Combination - 100!
There are 30 distinct objects. How many (unordered) selections of 6 objects can be obtained?
Ans: Combination - (30... |
H: Logical Implications Quiz Help
These are fill in the blank questions. I scored an 18 out of 25, but I don't see where I am making mistakes. Please help me understand these conditional logic statements!
The answer options are:
IMPLIES (-->)
IS IMPLIED BY (<--)
IFF (<-->, both "IMPLIES" and "IS IMPLIED BY")
NONE O... |
H: The order of a group with two generators
Let $G$ be an abelian group generated by $x$ and $y$ such that the order of $x$ is $16$, the order of $y$ is $24$ and $x^2=y^3$. What is the order of $G$?
The elements of $G$ are of the form $x^ny^m$ with $n=1,3,5,7,\cdots,15$ and $y=1,2,3,4,\cdots,23$. But I don't know ho... |
H: Is it possible for a function(f) to be $O(f)$ but not $o(f)$?
Is it possible for a function(f) to be $O(f)$ but not $o(f)$? or $o(f)$ but not $O(f)$?
I guess it might be possible for a function that is not monotonically increasing.
Is there an example of this case?
Added: Is it correct if I say subtracting $\thet... |
H: Can one show me how to plot this graph by hand (composition).
Can one show me how to plot this graph by hand (composition).
$$\frac{1}{1-x^2}$$
Plot[1/(1 - x^2), {x, -5, 5}]
Output result see picture in the end
Fine, thanks, I got it by plot 1/(1 + x) and 1/(1 - x)
Plot[{1/(1 - x), 1/(1 + x), 1/(1 - x) 1/(1 + x... |
H: Need helping solving this reflexive, symmetric, and transitive closure question
I'm working on a sample exam with no solutions and I've googled examples of similar problems but can't understand them either. Can someone show me a clear cut way to solve this problem:
Let R be a relation on { a,b,c,d,e } defined by R... |
H: How to show that $\log_{10} n$ is not a rational number if $n$ is any integer not a power of $10.$
How to show that $\log_{10} n$ is not a rational number if $n$ is any integer not a power of $10.$
If not, let $\log_{10}n=\dfrac{p}{q}$ for some $p,q(\ne0)\in\mathbb Z$ where $(p,q)=1.$
Then $q\log_{10}n=p\implies... |
H: Piecewise linear function close to continuous function
A continuous function $\phi$ on $[a,b]$ is called piecewise linear provided there is a partition $a=x_0<x_1<\ldots<x_n=b$ of $[a,b]$ for which $\phi$ is linear on each interval $[x_i,x_{i+1}]$. Let $f$ be a continuous function on $[a,b]$ and $\epsilon$ a posit... |
H: How do I understand the meaning of the phrase "up to~" in mathematics?
I am reading a book that explains elementary number theory: Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks and Erica Flapan.
The authors say,
"We express this idea in the statement o... |
H: linear algebra (matrix representation of a linear mapping) problem
Find the matrix representation $A$ of a linear mapping $T$ : $\mathbb R^2 \to \mathbb R^2$ that rotates points $\pi$ radians clockwise, then reflects points through the line $x_2 = - x_1$. Determine the range of $T$. Determine if $(1,1)$ is in th... |
H: Trigonometry: Law of Cosines
How to solve using rule of cosines? I can solve using law of sines but trying to check using rule of cosines is tripping me up, can anyone help clear things up?
AI: By writing down the law of cosines for this situation, we have
$$ 5^2=a^2+4^2-2\cdot a\cdot 4\cdot \cos 65^\circ$$
i.e.
$$... |
H: If $f$ is twice differentiable in $(-1,1)$ and $f(0) = f'(0) =0$, does $\sum\limits_{n=1}^\infty f(1/n) $ converge?
In calculus, I'm trying to understand if the following forces the convergence of the sum. I think it does, but I have no clue as how to prove it.
Similar to subject line, here is the question:
If $f$... |
H: Numerical Analysis and Big O
How can I show that $e^x -1$ is not $O(x^2)$ as $x\to0$
I'm not sure where to start. We can use Taylor's Theorem with remainder:
\begin{equation}
e^x = \sum\limits_{k=0}^n\dfrac{x^n}{n!} +\dfrac{f^{n+1}(\xi)}{(n+1)!}x^{n+1}
\end{equation}
Where $\infty < x < \infty$ and $\xi$ is betwee... |
H: Linear algebra - Coordinate Systems
I'm preparing for an upcoming Linear Algebra exam, and I have come across a question that goes as follows: Let U = {(s, s − t, 2s + 3t)}, where s and t are any real numbers. Find the coordinates of x = (3, 4, 3) relative to the basis B if x is in U . Sketch the set U in the xyz-c... |
H: Factor $x^4 - 11x^2y^2 + y^4$
This is an exercise from Schaum's Outline of Precalculus. It doesn't give a worked solution, just the answer.
The question is:
Factor $x^4 - 11x^2y^2 + y^4$
The answer is:
$(x^2 - 3xy -y^2)(x^2 + 3xy - y^2)$
My question is:
How did the textbook get this?
I tried the following methods ... |
H: If $|f(z)|\le1$ if $|z|\le 1$ then $|a_k|\le1$
Given the polynomial $f(z)=a_nz^n+a_{n-1}z^{n-1}+ \dots+a_0$ is bounded by $1$ on a unit disc, which means $|f(z)|\le1$ if $|z|\le 1$. Prove that $|a_k|\le1$ for all $k$.
I haven't found any idea for this problem.
AI: Hint: Notice that $a_k= \frac{1}{k!}f^{(k)}(0)$ an... |
H: How can an oblique asymptote be $y = x$ , as $x\to \infty$?
In my Calculus book, an oblique asymptote defined as:
Oblique Asymptote:
the function $y = f(x)$ has an oblique asymptote $y = mx + n$, if:
$$\lim_{x\to \infty} {f(x) \over x} = m$$ where $m$ is a finite number.
$$\lim_{x\to \infty} [{f(x) - mx}] = n$$
... |
H: Calculating 7^7^7^7^7^7^7 mod 100
What is
$$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$
I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.
AI: $7^4 = 2401 \equiv 1 \pmod{100}$, so you only need to calculate $7^{7^{7^{7^{7^7}}}} \pmod{4}$. We know that $7 ... |
H: Is my understanding of limit points wrong?
From Munkres p. 163:
"The notion of compactness is not nearly so natural as that of connectedness. From the beginnings of topology, it was clear that the closed interval $[a, b]$ of the real line had a certain property that was crucial for proving such theorems as the maxi... |
H: Prove that $(a-b) \mid (a^n-b^n)$
I'm trying to prove by induction that for all $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}$, that $(a-b) \mid (a^n-b^n)$. The base case was trivial, so I started by assuming that $(a-b) \mid (a^n-b^n)$. But I found that:
\begin{align*}
(a-b)(a^{n-1}+a^{n-2}b + a^{n-3}b^2+...+b^{n-1})... |
H: Air Strike Game
This is an Air Strike Game with the solution, I have added some questions regarding the solution and I would appreciate if someone could answer them.
Army $A$ has a single plane with which it can strike one of three possible targets. Army $B$ has one anti-aircraft gun that can be assigned to one of ... |
H: A question regarding propositional logic
Good day, I'm currently studying for an exam and need to learn about propositional logic. Well, since I'm not good at English I'll just write what I've done so far:
$(A \land (B \rightarrow \neg A)) \rightarrow \neg B$
I started like this:
$(A \land (\neg B \lor \neg A)) \ri... |
H: How to show that the index $[G: H]$ is invertible?
I am reading the book Elements of representation theory of associative algebras, volume 1. I have a question on page 176, the proof of Corollary 5.2. In order to applied (5.1)(b), we have to show that the index $[G: H]$ is is invertible as an element of $A=K$. In t... |
H: Negative binomial with a rational power?
I didn't quite understand the expansion of, for instance $1 \over (1-x)^\alpha$, for $\alpha \in \mathbb Q$, for instance for $\alpha = {1\over 2}$ using the binomial coefficients. I know that for $\alpha \in \mathbb N$, ${1 \over (1-x)^\alpha}=\sum_{n=0}^∞ {n+\alpha-1 \choo... |
H: How to show that $K[t]/(t^d)$ is indecomposable as a $K[t]$-module and $\operatorname{End}_{K[t]} (K[t]/(t^d)) \cong K[t]/(t^d)$?
How to show that
(1) $K[t]/(t^d)$ is indecomposable as a $K[t]$-module?
(2)$\operatorname{End}_{K[t]} (K[t]/(t^d)) \cong K[t]/(t^d)$?
I think that if (2) is true, then $\operatorname{E... |
H: Find area of the surface obtained by rotating curve around x-axis?
I got a curve $y=a \cdot cosh \frac{x}{a}$ where $|x|\le b$. The task is to find area of the surface obtained by rotating curve around x-axis.
Here is my solution. Unfortunately the result is not identical with the result of the textbook. Would yo... |
H: How can I know the time difference between two cities almost at the same latitude?
Well I know that's the earth rotation speed is:
$v=1669.756481\frac{km}{h}$
I have two cities New York, Madrid almost at the same latitude and the distance between them is:
$d=5774.39$ $km$
I know that's :
$\Delta t=\frac{d}{v}=... |
H: $w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$
I am stuck on the following problem:
Let $w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$.Then the triangle in the complex plane with $w_1,w_2,-1$ as vertices
must be isosceles ,but not necessarily equ... |
H: Cardinality of Hausdorff Space
Here is a theorem which proof I don't understand (taken from R. Engelking, General Topology).
Theorem: For every Hausdorff space $X$ we have $|X| \le \operatorname{exp}\operatorname{exp} d(X)$. ($d(X)$ denotes the cardinality of the smallest dense subset of $X$)
Proof: Let $A$ be a de... |
H: Contour integration: $\int_0^{\infty} x^p /(x^4+1) dx$ where $-1 < p < 3$
I want to calculate $\int_0^{\infty} x^p /(x^4+1) dx$ where $-1 < p < 3$. My first guess is to let $f(z) := \frac {z^p}{z^4+1}$ and integrate this over $\gamma_R$ where
$$
\gamma_r = [-R,R] \cup \{Re^{i\theta} : \theta \in [0,\pi] \}
$$
Th... |
H: An Exercise in Kunen (A Model for Foundation, Pairing,...)
This is exercise I.4.18 in Kunen's Set Theory.
Derive $\forall y (y \notin y)$ from the Axioms of Comprehension and Foundation. Don't use the Pairing or Extensionality Axioms. Then find a 2 element model for Foundation, Extensionality, Pairing, and Union,... |
H: Question on step in the proof of Itō's formula (along the book of Revuz and Yor)
I am working through the proof of Itō's formula contained in the book "Continuous Martingales and Brownian Motion" by Revuz and Yor and am stuck at a point in the proof.
Theorem (Itō's formula). Let $X = (X^1, \ldots, X^d)$ be a conti... |
H: $q$ be a real polynomial of real variable $x$ of the form $q(x)=x^n+a_{n-1}x^{n-1}+....+a_1x-1 .\,\,$
I am stuck on the following problem:
Let $q$ be a real polynomial of real variable $x$ of the form $q(x)=x^n+a_{n-1}x^{n-1}+....+a_1x-1 .\,\,$ Suppose $q$ has no roots in the open unit disc and $q(-1)=0.$
Then... |
H: Limit-preserving point-wise maximum
Let $F$ denote the set of monotonic functions in $N^{\infty} \rightarrow N^{\infty}$ (where $N^{\infty}$ includes 0 and $\infty$) ordered by the point-wise application of $\le$ i.e., for $f,g \in F$
$$
f \le g \iff \forall n \in N^{\infty} :~ f(n) \le g(n)
$$
then I'd like t... |
H: How to understand the limit in the generalized real system?
As stated on the title:
How to understand a sequence limit in the generalized real system $[-\infty,\infty]$?
AI: Exactly as in the typical case of $\mathbb{R}$. The only difference is that now, having a limit of $\infty$ or $-\infty$ counts as convergin... |
H: What does $K^{1/p}$ for a field $K$ mean?
In the proof of the finite generation of the invariant ring of a finite group acting on $k[x_1,\dots,x_n]$, at one time there is a symbol I don't understand. The situation is as follows.
$k$ is a field of characteristic $p<\infty$, and $P$ its prime field. Suppose that $k$ ... |
H: How to describe a n-tuple of sequences
When I write computer programs, I often use something called a multidimensional array. I think the concept would be equivalent to an n-tuple of finite sequences.
Suppose I have the following four sequences and 4-tuple called $t$:
$$A = (0)$$
$$B = (3,4,2)$$
$$C = (6,7,2,5)$$
$... |
H: Vector bundle definition
Is the condition
$$ \pi \circ \varphi (x,v) = x $$
in the definition of a vector bundle needed? In Milnor/Stasheff "Characteristic classes" the definition is given without it.
AI: It's not strictly necessary, because it is a consequence of the following
the map $v \mapsto \varphi (x, v)$ i... |
H: Plotting in maple/MATLAB
How do you plot the following parametric equation (equation of an ellipse) in the same graph
$$ x = a\cos{t} $$
$$ y = b\sin{t} $$
with varying value of $a$ and $b$ in either maple or MATLAB ?
Many thanks.
AI: And in MATLAB:
min_a = 1; max_a = 10;
min_b = .5; max_b = 5;
t = linspace(0,2*pi... |
H: Doubling a point on an elliptic curve
I've a programming background and am just about to get into a project where Elliptic Curve Cryptography (ECC) is used. Although our libraries deal with the details I still like to do background reading so started with the ECC chapter of Understanding Cryptography. Everything wa... |
H: Fourier Transform of Derivative
Consider a function $f(t)$ with Fourier Transform $F(s)$. So $$F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \ dt$$
What is the Fourier Transform of $f'(t)$? Call it $G(s)$.So $$G(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f'(t) \ dt$$
Would we consider $\frac{d}{ds} F(s)$ ... |
H: A sufficient and necessary condition of a continuous map
Prove that a map $f:X\to Y$ is a continuous iff $$\text{cl}f^{-1}(A)\subseteq f^{-1}(\text{cl}A）$$ for each $A\subset Y$, where $X$ and $Y$ are both topological spaces and $\text{cl}$ denotes the closure.
In fact,for any $A\subset Y$,we have $A\subset \text{c... |
H: Prove that: the center of any group is characteristic subgroup .
Let $G$ be any group , $Z(G)$ is the center of the group $G$ , prove that :
$\forall \tau \in Aut(G) , \tau [(Z(G)] = Z(G)$
My first trial was to prove that the center of any group is the unique subgroup of its order hence is a characteristic . but i ... |
H: Showing that the mean of translations of a function approaches 0 in $L_p$
Given $p \in (1,\infty)$, $f \in L^p(\Bbb R)$ and $T: \Bbb R \to \Bbb R,x \mapsto x+1$. How do I show that for $n \to \infty$ $$\frac{1}{n}\sum_{k=0}^n f \circ T^k \to 0$$ in $L^p$? I see that for very large $N$ $f$ and $f\circ T^N$ have thei... |
H: Making a function in $W^{1,2}$ continuous
Let $\Omega$ be an open domain in $\mathbb{R}^n$, $u\in W^{1,2}(\Omega)$ and assume that for any $y$ in $\Omega$
$$\lim_{\varrho \to 0} \operatorname{osc}(u,B(y,\varrho)) \rightarrow 0 , \varrho \rightarrow 0$$ where $$\operatorname{osc}(u,B(y,\varrho))= \sup_{x\in B(y,\v... |
H: Limit on the expression containing sides of a triangle
To find the bounds of the expression $\frac{(a+b+c)^2}{ab+bc+ca}$, when a ,b, c are the sides of the triangle.
I could disintegrate the given expression as $$\dfrac{a^2+b^2+c^2}{ab+bc+ca} + 2$$ and in case of equilateral triangle, the limit is 3.
Now how to pr... |
H: if f(x,y) = 1 for all (x,y) where R has a nice shape like a rectangle or triangle what is another geometric interpretation of....?
If $f(x,y)$ is greater than or equal to $0$ on region $R$ in the plane, then $\iint_R f(x,y)dA$ can be interpreted geometrically as the volume of the solid under the surface $z=f(x,y)$ ... |
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