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H: convergence of test functions
Let $\phi\in C_0^\infty(\mathbf R^n)$ such that $0\leq \phi(x)\leq 1$ and $\phi(x)=1$ for $|x|\leq 1$ and $\phi(x)=0$ for $|x|\geq 2$. Defining $\phi_n(x)=\phi(x/n)$. How do we show that $\phi_n\rightarrow \phi$ pointwise as $n\rightarrow\infty$ to conclude that $\phi_nf\rightarrow\phi... |
H: Find an algorithm with O(n) for making Chairlifts on N mountains
Today, I encountered an interesting problem in my textbook. The problem is:
Utopia city has N mountains with height of $$h_{1}, h_{2}, h_{3}, ..., h_{n}$$. We want to make a chairlifts which pass from the top of k consecutive mountain. To do this, we ... |
H: Name of shape with constant distance to a line segment
For a computer graphics problem I have a shape that is defined by a constant distance to a line segment:
I tried to find a name for this shape, but my Google skills have failed me. Does it have a dedicated name?
AI: Maybe it is a ciiiiiiiiircle?
Kidding aside,... |
H: How find this value of $\prod_{1\le i
give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$
where $i^2=-1$
find the vaule
$$\prod_{1\le i<j\le n}(w^i-w^j)^2$$
My try:note
$$w^n=1$$
$$\prod_{1\le i<j\le n}(w^i-w^j)^2=\prod_{1\le i<j\le n}(w^i-w^j)(w^i-w^j)$$
and I kno... |
H: Quadratic twist of an elliptic curve
I found this page:
http://en.wikipedia.org/wiki/Twists_of_curves#Quadratic_twist
which tells me $dy^2=x^3+a_2x^2+a_4x+a_6$ is equivalent to
$y^2=x^3+da_2x^2+d^2a_4x+d^3a_6$. Why is this equivalent (for $d$ as given on that page)?
AI: Put $y = y'/d^2$ and $x=x'/d$. Then
$$d^{-3}y... |
H: Prove $\sqrt{2} + \sqrt{5}$ is irrational
How do you prove that $\sqrt{2} + \sqrt{5}$ is irrational?
I tried to prove it by contradiction and got this equation: $a^2/b^2 = \sqrt{40}$.
AI: Use proof by contradiction. Assume that the sum is rationial, that is $$\sqrt2 +\sqrt5 = {a\over b}$$ where $a$ and $b$ are i... |
H: Hypergeometric die toss
A fair die is tossed until a $2$ is obtained. If $X$ is the number of trials required to obtain the first $2$, what is the smallest value of $x$ for which $P(X \leq x) \geq \frac{1}{2}$?
My thoughts: The general formula for hypergeometric random variables is given by $p(k) = P(X=k) = \frac{C... |
H: Finding inradius given the heights
I'm given the heights of a triangle. Find the inradius. I know that inradius is area/semiperimeter. But then?
AI: Let the area of triangle be $T$, the heights $h_a, h_b, h_c$, and $s$ the semi-perimeter. Then:
$$2T = ah_a = bh_b=ch_c$$
$$a = \frac{2T}{h_a}, b = \frac{2T}{h_b}, c ... |
H: What is $\int \frac1{1+(a\tan x)^2}dx$?
What is $\int \frac1{1+(a\tan x)^2} \mathrm dx$?
This is a difficult integral. If you can, please give a step-by-step solution - I would be delighted.
AI: It's not that hard if you take the suggestion from Clayton:
$u=a \tan{x}$; $x = \arctan{(u/a)}$; $dx=a\, du/(u^2+a^2)$. ... |
H: Double embedding or double restriction
The following generalizes both embedding and restriction for sets $A$ and $B$:
$A \rightleftarrows B = ( A ; B ; \operatorname{id}_{A \cap B})$.
$A \rightleftarrows B$ is considered as a morphism of the category $\mathbf{Rel}$.
It is an embedding $A \hookrightarrow B = ( A ; B... |
H: $A \subset C$ implies $A \cap C \subset B \cap C$
I need help with proving this set of expressions in boolean algebra:
$$
A \subset B \implies A \cap C \subset B \cap C
$$
I tried to solve it like this:
$$
A \subset B \implies A \cap C \subset B \cap C\\
\lnot(A\lor B) \lor (\lnot(A\land B))\lor (B\land C)\\
(\lnot... |
H: What is the moment generating function from a density of a continuous random variable?
Let X be a random variable with probability density function $$f(x)=\begin{cases}xe^{-x} \quad \text{if } x>0\\0 \quad \text{ } Otherwise.\end{cases} $$
Determine the mgf of X whenever it exists.
I know that $M(t) = E(e^{tx}) =... |
H: Find all $x,y,z$ satisfying $xy=z-x-y$ and cyclic permutations
Find all ordered pairs $(x,y,z)$ real numbers, which satisfy the following system of equations:
$$xy=z-x-y\\xz=y-x-z\\yz=x-y-z$$
AI: Hint:
$$xy=z-x-y \quad \iff \quad (x+1)(y+1)=z+1.$$
Hence $(x+1)(y+1)=z+1$, $(x+1)(z+1)=y+1$, $(z+1)(y+1)=x+1$. |
H: Show that $\{f_n^2\}\rightarrow f^2$ in measure.
From Royden's Analysis, 4th Edition, Chapter 5, section 2 Problem 7:
Let $E$ have finite measure, $\{f_{n}\}\rightarrow f$ in measure on $E$ and $g$
be a measurable function on $E$ that is finite a.e. on $E$. Prove that $\{f_{n}\cdot g\}\rightarrow f\cdot g$ in... |
H: Using Compact Support to write function as integral
If we have $u \in C_{c}^{1}(\mathbb{R}^{n})$ then why does we require the compact support of $u$ in order to write:
$u(x) = \int_{-\infty}^{x_{i}}u_{x_{i}}(x_{1},…,x_{i-1},y_{i},x_{i+1},..,x_{n})dy_{i}$ ?
Thanks
AI: I think it might be required in order to apply t... |
H: Proof of $(\forall x)(x^2+4x+5 \geqslant 0)$
$(\forall x)(x^2+4x+5\geqslant 0)$ universe is $\Re$
I went about it this way
$x^2+4x \geqslant -5$
$x(x+4) \geqslant -5$
And then I deduce that if $x$ is positive, then $x(x+4)$ is positive, so it's $\geqslant 5$
If $ 0 \geqslant x \geqslant -4$, then $x(x+4)$ is also $... |
H: time speed distance
Two horses start simultaneously towards each other and meet after $3h 20 min$. How much time will it take the slower horse to cover the whole distance if the first arrived at the place of departure of the second $5 hours$ later than the second arrived at the departure of the first.
MY TRY::
Let ... |
H: About definition of "ordered semi-ring"
I need the definition of "ordered semi-ring". Can I use these properties:
$a \preceq b \to a + c \preceq b + c$
$0 \preceq a \wedge 0 ≤ b \to 0 \preceq a \cdot b$ (or: $a \preceq b \wedge 0 \preceq c \to a \cdot c \preceq b \cdot c $)
???
Thanks in advance!
AI: I found thre... |
H: Where's the mistake? Integral of a bounded non-analytic function over a closed curve.
Let $f(z)$ be a bounded non-analytic function defined on a closed curve $\gamma$. What is wrong with the following:
$$\left| \oint_\gamma f(z) dz\right| \le \oint_\gamma \left|f(z) \right| dz \le M \oint_\gamma 1 \cdot dz =0$$
be... |
H: Integral of Infinite Division
In my Calculus II class this question was posed. It is a definite integral, but the function integrated is infinitely dividing. I tried $u$ substitution, but cannot find the correct term to use as my $u$.
$$
\int\limits_0^6 \frac{xdx}{1-\frac{x}{1-\frac{x}{1-\frac{x}{\quad\ddots}}}}
... |
H: Generalizations of pairing function
A pairing function is a one-to-one mapping from $N^2$ into $N$; for example the Cantor pairing function is given by:
$$J(x,y) = \frac{(x+y)(x+y+1)}{2} + 1$$
Another one is: $f(x,y)=2^x(2y+1)$.
They can be easily generalized to encode $n$-tuples (points in a $n$-dimensional space)... |
H: How far does she travel in her Journey?
Ok, I'm doing a Mock End of Unit Test Revision paper and I'm stuck on a few questions.
The first question is:
Susan completes the journey in $2$ stages of her journey. In stage 1 of her journey, she drives at an average speed of $80$km/h and takes $1$ hour and $45$ minutes.
... |
H: Induction with sets: $\forall n \ge 1: \overline{\bigcap_{i=1}^nA_i}=\bigcup_{i=1}^n \overline{A_i}$
I know how to do induction with equations, but for this thing with sets:
$$\forall n \ge 1: \overline{\bigcap\nolimits_{i=1}^nA_i}=\bigcup\nolimits_{i=1}^n \overline{A_i}$$
exactly I don't have an idea. If someone c... |
H: How do I prove inequalities and one-to-one function?
Can anyone please help me with these questions?
1.Given
x + 1 < 0
Prove that:
i) $2x - 1 < 0 $
ii) ${2x-1\over x+1} > 2$
2.For $g(x) = {kx + 8\over 4x - 5}$
i) Find k if gg(x) = x
Is it fine if I just let any value of x for this question?
ii) Find the value of k ... |
H: Real root of a complex equation.
I was working on a problem from Gamelin; where I was required to find out zeros of $2z^5+6z^1-1$ , in the unit disk (in $\mathbb C$). I applied Rouché's theorem and find out zeros in the unit disk and I got to know that there is only one zero inside it.
Further, I have to show that... |
H: Family with three children. Chances of at least one boy and girl?
Here's my exercise:
A family has three children. What's the probability of event $A \cup C$ where:
$A$- Family has children of both sexes.
$C$- Family has at most one girl.
Well I see two ways to look at this conundrum:
The first one would be to diff... |
H: Divisibility by $9$
Suppose we have a natural number $N$ with decimal representation $A_kA_{k-1}\ldots A_0$. How do I prove that if the $\sum\limits_{i=0}^kA_i$ is divisible by $9$ then $N$ is divisible by $9$ too?
AI: Hint: $\overline{A_kA_{k-1}\ldots A_1A_0} = 10^kA_k + 10^{k-1}A_{k-1} + \ldots + 10A_1 + A_0$.
... |
H: What is the answer to this syllogism? Why is option D incorrect?
Q. a. Some books are not reference books.
b. All books are encyclopedias.
A Some reference books are no encyclopedias
B No reference books are encyclopedias
C All reference books are encyclopedias
D None of the above
AI... |
H: Probability of getting the product divisible by $3$.
What is the probability that the product of two randomly chosen (distinct) numbers between $1$ and $100$ (inclusive) is divisible by $3$?
Now, what I am doing is to find out the number of ways of selecting a multiple of $3$ out of $100$ and then selecting any of ... |
H: How to solve this class of problems?
I was presented with the following problem:
Ricardo, Rogério and Renato are brothers. One of the is a medic, the other one is a teacher and the other one is a musician. It is known that:
Ricardo is a medic or Renato is a medic.
Ricardo is a teacher or Rogério is a musician.
Re... |
H: Rewriting $x^3-3xy^2+2xy+i(-y^3+3x^2y-x^2+y^2 )$ in terms of $z$, with $z=x+yi$
How do I write $f=u+iv$
with:
$u=x^3-3xy^2+2xy$ and
$v=-y^3+3x^2y-x^2+y^2 $
in terms of $z$ with $z=x+yi$?
AI: Have you heard of the binomial theorem? It tells us that (for example) $$(a+b)^2=a^2+2ab+b^2$$ and $$(a+b)^3=a^3+3a^2b+3ab^... |
H: About $S_n=\{(x,y)\mid \lfloor kx\rfloor=\lfloor ky\rfloor,k=1,2,\cdots n;~x,y\in [0,1]\}$
Let $$S_n=\{(x,y)\mid \lfloor kx\rfloor=\lfloor ky\rfloor,k=1,2,\cdots n;~x,y\in [0,1]\}$$
Here are the pictures of $S_1,S_2,\cdots S_{10}$:
We can see that $S_1$ has only one blue region, $S_2$ has $2$ different blue region... |
H: "Moving" a filter from one set to an other set, two equivalent formulations
By filters I mean filters on sets (not on one fixed set but on arbitrary sets).
Informally saying, this question concerns "moving" a filter from one set to an other set.
I define rebase a filter $\mathcal{A}$ (on a set $\mathfrak{A}$) to a ... |
H: Problem applying Cauchy's estimate in proof
Let $f$ is an analytic map from the unit disk to itself.
Taylor series of $f$ centered at 0: $f=\sum_n a_n z^n$ ).
Prove that: $|a_n|\leq 1 \; \forall n$
I've started using Cauchy's estimate:
$|a_n|=\frac{f^{(n)}(0)}{n!}\leq \frac{M_r}{r^n}$ with $M_r=\max\{|f(z)|: |z|=r\... |
H: Prove $X - \bigcup_{C \in \scr{C}}C = \bigcap_{C \in \scr{C}}(X - C).$
Prove $X - \bigcup_{C \in \scr{C}}C = \bigcap_{C \in \scr{C}}(X - C).$
Proof. Assume $a \in X - \bigcup_{C \in \scr{C}}C$.
Then $a \in X$ and $a \not\in \bigcup_{C \in \scr{C}}C$.
Then $a \in X$ and $a \not\in \bigcap_{C \in \scr{C}}C$, since ... |
H: Faithfulness of adjoint representation of Lie algberas
Are there any simple or useful conditions (necessary & sufficient) under which the adjoint representation lie algebra is faithful ? One sufficient condition is semisimplicity, but perhaps this is not necessary.
AI: The adjoint representation of a Lie algebra ha... |
H: Does a discrete set of points in $\mathbb{R}^{n}$ define a locally finite collection of hyperplanes?
Let $v_{1},v_{2},...$ be a discrete set of non-zero vectors in $\mathbb{R}^{n}$. By discrete, I mean that any $v_{i}$ is surrounded by an $\epsilon$-ball not containing any other point $v_{j}$. Equivalently, any com... |
H: If $A$ is a countable subset of $\omega_1$, then there is an $ \alpha < \omega_1$ such that $A \subseteq \alpha$
Prove : if $A$ is a countable subset of $\omega_1$. Then there exists $ \alpha < \omega_1$ with $ A \subseteq \alpha$.
I don't really know where to start, can anyone give a tip first?
AI: HINT: Recall t... |
H: Determination of a uniformly continuous, not locally Hölder-continuous function over an open set
Given an open interval $I=(a,b)$, I would like to exhibit a uniformly continuous function over $I$ that is not locally Hölder-continuous with exponent $\alpha$ for any $\alpha\in(0,1)$.
It is quite easy to exhibit an UC... |
H: proof by contradiction example
Any ideas on how I can use proof by contradiction to show that at least 3 of any 25 days chosen must fall in the same month of the year?
I don't even understand the question.
AI: Pick any $25$ different dates in $2013$ (or any other year), like $13$ May, $27$ June, etc. The claim is t... |
H: Counterexample to in $\mathcal{Mod}_A$ colimits of filtered index categories are exact
This is not a true general fact for any abelian category, as Vakil points out in 1.6.12. He gives the following counterexample, which puzzled for it is in the category of abelian groups, and every abelian group is a module over $... |
H: Proof of $|x^{\alpha} - y^{\alpha}| \le \alpha^{\alpha} |x-y|$ for $\alpha \ge 1, x,y\in [0,1]$
I want to prove
$$
|x^{\alpha} - y^{\alpha}| \le \alpha^{\alpha} |x-y|
$$
for $\alpha \ge 1$ and $x,y \in [0,1]$. For $\alpha \in \mathbb N$ I already got the proof by using the formulae
$$
(x^n - y^n) = (x-y)(x^{n-1} ... |
H: $f: \omega_1 \to \omega_1$. $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $.
Prove: if $f: \omega_1 \to \omega_1$ is an increasing, continuous, unbounded, function, then $\forall$ $\alpha$ $\exists$ $\beta > \alpha $ with $f(\beta)=\beta $.
Can anyone give me a tip?
AI: The use of quantifiers... |
H: Inductive definition with choice for sequence
In topology there is a very common way to define a sequence. This usually go something like:
"Define $\{z_{n}\}$ to be a sequence such that $z_{0}$ is <blah blah blah>, and $z_{n}$ is such that $R(z_{0},z_{1},\ldots,z_{n})$ is true. The sequence is well-defined since th... |
H: Inserting values left to right in a binary search tree
What does it mean to build a binary search tree by inserting values from left to right starting from an empty tree? The "left to right" part confuses me..I know how to build one by normally inserting values from the top..but what does this mean?
Could anyone pl... |
H: Rooted Trees & Induction
So I am a little stumbled upon this question:
A full binary tree is a rooted tree where each leaf is at the same distance from the root and each internal node has exactly two children. Inductively, a full binary tree of depth 0 is the one node tree N, and a full binary tree of depth d+1 is ... |
H: I don't understand how sets can be closed, yet disjoint?
What are some closed, disjoint subsets $A, B$ in $R^2$ where $inf\{d(A, B) = 0 \forall a \in A \forall b \in B\}$?
AI: For example, $A=\{(x,y)\mid xy=1,x>0\}$ and $B=\{(x,y)\mid xy=-1,x<0\}$ |
H: Finding if $\sum\frac{1}{2+3^{-k}}$ divergent or convergent
How would one find if the following series is divergent or convergent.
$$\sum\frac{1}{2+3^{-k}}$$
I did the following
$$\sum\frac{1}{2+3^{-k}}<\sum\frac{1}{3^{-k}}$$
But I am not sure what test I should use the only ones I know are limit comparison and bas... |
H: Finding if $\sum_{}^{}\frac{\ln k}{k}$ converges or diverge
How can I find whether the serie
$$\sum_{}^{}\frac{\ln k}{k}$$
converge or diverge using the basic comparison test or limit test.
The part that is confusing me is how to deal with $\ln$.
AI: You know that the harmonic series $\sum_{k = 1}^\infty \dfrac 1k$... |
H: How to construct co-equalizers in $\mathbf{Top}$?
How to construct co-equalizers in the category $\mathbf{Top}$?
Well, do co-equalizers in $\mathbf{Top}$ exist at all?
AI: If $f,g:X\to Y$ are two parallel morphisms, then the coequalizer $c:Y\to Z$ of $f,g$ will be the quotient map to $Z=Y/\sim$ where "$\sim$" is th... |
H: Give the transformation matrix that send $\vec i$ on $ [1,1/2,0]$
I'm asked to give the linear transformation that sends $\vec i$ on $[1,1/2,0]$ , $\vec j$ on $[-1/3,1,0]$ , $\vec k$ on $[1/4,1/4,1]$.
Is it a translation where I need to use the actual $i,j,k$ vectors or is it just changing the $i,j,k$ vectors to th... |
H: sum over primes less than 'x
is there a function $ f(x) $ so
$$ \sum_{p\le x}f(p)=S(x)$$
where $ S(x)=g(f(x), \pi(x) $
this means that the sum S(x) depends on the function $ f(x)$ but also on the prime number counting function
the only case is $ f(x)=0$ but can be another alternatives ?
for example, the integral ... |
H: For sets $A,B$, $A \cap (B \setminus A) \subseteq \varnothing$
I need help with this proof:
Let $A$ and $B$ be sets. Prove that $A\cap (B\setminus A)\subseteq \varnothing$.
My problem: Since $B\setminus A$ = $B\cap A^{c}$, we can say that if $x \in B$ then $x\in B\cap A^{c}$and $x \in A^{c}$ as well. So $x\notin ... |
H: Using Gauss's lemma to show if p(a) = p(b) = p(c) = p(d) = 5, there is no integer k with p(k) = 8
I'm attempting to prove this:
Let p(x) be a monic polynomial with integer coefficients. Suppose that there are distinct integers a, b, c, d with p(a) = p(b) = p(c) = p(d) = 5. There is no integer k with p(k) = 8.
I h... |
H: Simple tax puzzle
I recently saw some post on facebook whining about taxes. Simplifying it (and changing numbers, facts, etc.), this was saying:
For each dollar an employer wants to pay you:
20% go in taxes that your employer pays
out of the remaining 80%, you pay 40% of various taxes (from income taxes, to sales ... |
H: Volume of $n$-ball in terms of $n-2$ ball
Let $V_n(R)$ be the volume of the ball with radius $R$ in $\mathbb{R}^n$. This page says
$$V_n(R)=\int_0^{2\pi}\int_0^RV_{n-2}(\sqrt{R^2-r^2})r\,dr\,d\theta$$
I don't really understand the explanation given in there. Could someone explain it in the case $n=3$ (so $n-2=1$... |
H: Strong induction definition clarification: is the hypothesis that it is true for all $k$ with $0 \le k < n$ or for all $k$ with $0 \le k \le n$?
I have a general question about strong induction:
Assuming that the base case is 0, if I let my inductive hypothesis be that for all 0 <= k < n some statement is true, and... |
H: Why is the gradient of F a constant multiple of a parallel vector?
Problem 14:
Solution:
I don't understand why the gradient of F is a constant multiple of a parallel vector. Why is the equation in the black box true? I understand why it would be true if the gradient of F is a stretched version of (27, 8, 1) but ... |
H: $\limsup_{n\rightarrow\infty} a_n^{1/\log n}<1/e$ and $a_n>0$ then $\sum a_n$ converges
If $\limsup_{n\rightarrow\infty} a_n^{1/\log n}<1/e$ and $a_n>0$ then $\sum_{n=1}^{\infty} a_n$ converges.
$$0<a_n < e^{-\log n}=\frac{1}{n}$$
Also, $\exp\{\frac{1}{\log n}\log a_n \} \le \exp(-1)$ so $\displaystyle\frac{\log a_... |
H: solutions of a linear equation system
The following matrix is given over $\mathbb R$
A=$\begin{pmatrix} 1 & -1 & -1 \\ 1 & 0 & a \\ 1 & a & 0 \end{pmatrix}$
The linear equation System $Ax=\begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}$ has exactly one solution for all $a$ except $a=-2$ which has no solution. Is it right... |
H: Find a L-sentence which is true in a structure $M$ iff the universe $A$ of $M$ consists of exactly two elements
Find a L-sentence which is true in a structure $M$ iff the universe $A$ of $M$ consists of exactly two elements, where the language L consist a unary function $S$ and $2$-ary predicate $<$.
This is an exe... |
H: which one? permutation or combination?
Let say we have a bookshelf that can fit 6 books, we want 4 computer science books and 2 physics books but computer books should be together and physics books also should be together, we have 8 computer and 6 physics book in total in how many ways we can do that?
I believe tha... |
H: Elementary question regarding binomial coefficient
I would like to show that $ {n \choose j}$ is a multiple of $n$ if and only if $n$ and $j$ are coprime. Thanks for the help!
AI: You do get one implication: suppose $j$ and $n$ are coprime. Then: $$\binom{n}{j} = \frac{n!}{j!(n-j)!} = \frac{n (n-1) \ldots (n-j+1)}{... |
H: Prove a function is continuous
Discuss the continuity of the function $f:\mathbb{R} \to \mathbb{R}$
if
$f(x) = 0$ if $x$ is not rational
$f(x) = \frac1q$ if $x = \frac{p}{q}$ where $p,q$ are integers with no
common divisors other than $\pm 1$.
I know that the function is not continuous so I must show that f... |
H: Inequality with condition similar to Schwarz lemma
Suppose $f(z)$ is holomorphic and $|f(z)|\leq 1$ for $|z|\leq 1$. Show that $$\frac{|f'(z)|}{1-|f(z)|^2}\leq \frac{1}{1-|z|^2}.$$
If I also have the condition $f(0)=0$, I would be able to use the Schwarz lemma to conclude that $|f(z)|\leq|z|$ and $|f'(0)|\leq 1$.... |
H: Tiny question about the proof of $L^p \cap L^r \subset L^q$
I'm reading about $L^p$ spaces as a sort of self-studying thing. And usually when I read proofs, I'll try to fill in the steps that the author skips myself, but I'm having trouble with this one:
The proof is for: if $1 \leq p < q < r \leq \infty$, then $L^... |
H: Defining matrices in MATLAB
I'm completely new to MATLAB, and I can't figure out how to do the following: I have
A=[-1,0.4,0.8;1,0,0;0,1,0];
b=[0;0.3;6];
What I want to do is define, for $N$ fixed but arbitrary, a $3\times N$ matrix C by having the $i$th column of C be $A^{(N-i)}b$. Without doing the calculations ... |
H: Contingency table on Bayers Rule
The question is this:
A medical test for the disease StatsLove will correctly state that a diseased student suffers from StatsLove 95% of the time. The medical test correctly states that a non-diseased student does not suffer from StatsLove 99% of the time.
From prior terms it is k... |
H: Number of roots of $x^n+ax+b$
Let $P(x) = x^n + ax+b$ with $n\geq 2$ and $a,b\in\mathbb{R}$. Then one of these is true:
the number of distinct real roots of $P$ can be any number between $0$ and $n$.
the number of distinct real roots of $P$ is less than or equal to $3$.
the number of distinct real roots of $P$ is... |
H: Question about a proof that The Cantor set is uncountable.
I am reading a proof from a paper I found online, and it goes like this: We want to show that there exists a surjection $f$ from the cantor set $\mathfrak{C}$ to the interval $[0,1]$ then we can show that $|\mathfrak{C}|\geqslant [0,1]$, but since $\mathfr... |
H: Meaning and Intuition behind product $\sigma$ algebras , Folland Definition
I am currently studying Real Analysis using Folland's book by the same name, but I have troubles in understanding the product $\sigma$-algebras definition which is as follows:
Let $\{ X_{\alpha} \}_{\alpha \in A}$ be an indexed collection o... |
H: $f^{\prime}(x)\rightarrow 0$ as $x\rightarrow\infty$. $g(x)=f(x+1)-f(x)$, then $g(x)\rightarrow 0$ as $x\rightarrow\infty$
If $f$ is a defined and $\forall x>0, f^{\prime}(x)\rightarrow 0$ as $x\rightarrow\infty$. $g(x)=f(x+1)-f(x)$, then $g(x)\rightarrow 0$ as $x\rightarrow\infty$.
Note that
$$g(x)=\frac{f(x+1)-f... |
H: If $P$ is a prime ideal in a commutative ring $R$, is the ideal $P\times P$ a prime ideal in $R \times R$
If $P$ is a prime ideal in a commutative ring $R$, is the ideal $P\times P$ a prime ideal in $R \times R$
AI: As Matemáticos Chibchas points out, it will not be a prime ideal (assuming $R$ has a multiplicative ... |
H: How to prove $\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1$
Prove the following equation.
\begin{eqnarray}
\\\cos ^6x+3\cos ^2x\space \sin ^2x+\sin ^6x=1\\
\end{eqnarray}
I can't prove it by many methods I use.
Please give me some hints.
Thank you for your attention
AI: It seems one sign in your equality is wron... |
H: Idea of how to approach Exponential Functions Questions Please
Can someone help me on how to approach this question?
AI: The model is, to put it nicely, implausible. It assumes that the seeds have been hidden in sets of $100$, perhaps by a mathematically minded squirrel. It also assumes that searching for Type 1 a... |
H: Convergence of a sequence whose even and odd subsequences converge
Suppose $\{a_n\}$ is a sequence such that the subsequences $\{a_{2n−1}\}$ and $\{a_{2n}\}$ converge to the same limit, say $a$. Show that $\{a_n\}$ also converges to $a$.
AI: Hint: Take any $\epsilon>0.$ Since $a_{2n-1}\to a,$ then there is an $N_1$... |
H: Solution of $ 1+ 5 * 2^m =n^2$ Equation and some other question from this equation
I want to find all integer Solutions of $ 1+ 5 * 2^m =n^2$ Equation . From this eqn I have to answer the following answer .
I have to find an expression for $ n^2-1 $
Are $ (n+1) $ and $ (n-1) $ both even or both odd or is one even... |
H: Show that if x divides a power of 2, then x is a power of 2
I'm trying to prove that if $x$ divides $2^a$ for some integer $a \geq 0$, then $x = 2^b$, where $a \geq b$. In other words, if $x$ divides a power of 2, then $x$ is a power of 2. This makes sense, since the all the factors of a power of 2 are also powers ... |
H: How to solve $e^{ax}+e^{bx}+e^{cx}+d=0$
How to solve an equation like $e^{ax}+e^{bx}+e^{cx}+d=0$ (i.e. to write $x=...$) where $a,b,c,d$ are fixed non-zero real numbers.
I have tried assuming that $x=ln(y)$ for $y>0$ but it goes nowhere.
$$e^{ln(y)a}+e^{ln(y)b}+e^{ln(y)c}+d=y^a+y^b+y^c+d=0$$
AI: $\newcommand{\+}{^{... |
H: Possibility of making diagonal elements of a square matrix 1,if matrix has only 0 or 1
Let $M$ be an $n \times n$ matrix with each entry equal to either $0$ or $1$. Let $m_{i,j}$ denote the entry in row $i$ and column $j$. A diagonal entry is one of the form $m_{i,i}$ for some $i$. Swapping rows $i$ and $j$ of the ... |
H: $S$ is a closed subset of $E'$, then $f^{-1}(S)$ is a closed subset of $E$
Let $E,E'$ be metric spaces, $f: E\to E'$ a continuous function. Show that if $S$ is a closed subset of $E'$, then $f^{-1}(S)$ is a closed subset of $E$.
I need to show that $C(f^{-1}(S))\subset E$ is open. Since $S$ is a closed subset of ... |
H: $f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.
$f$ defined on $[1,\infty )$ is uniformly continuous. Then $\exists M>0$ s.t. $\frac{|f(x)|}{x}\le M$ for $x\ge 1$.
I know f uniformly continuous $\implies \forall\varepsilon > 0s.t.\forall x,y\in [1,... |
H: Calculating sample size
A city Humane Society wishes to determine the life expectancy of adopted dogs from their shelter.
How many of the dog owners do they need to contact in order to be 90% confident of being within 0.29 of the true mean life expectancy? A preliminary survey indicates that the ages are approximat... |
H: Solve $\lfloor{x}\rfloor+\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$
Solve for $x$
$$\lfloor{x}\rfloor+\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$$
I tried to put $x$=$I$+$f$ where $I$ is integer part and $f$ is fractional part but that didn't wor... |
H: Convergence of exponential matrix sum
Let $A$ be an $n\times n$ matrix. Consider the infinite sum $$B=\sum_{k=1}^\infty\frac{A^kt^k}{k!}$$ Each term $\dfrac{A^kt^k}{k!}$ is an $n\times n$ matrix. Does the sum $B$ always converge? (i.e. does the sum for each of the $n^2$ entries always converge?)
AI: This is just $\... |
H: Polynomial convergence to zero
Let $f_k $ be a series of n degree polynomials, that converges to $0$ uniformly in $[-M, M] $ for every $M$.
Say $f_k = a_{(0,k)} + a_{(1,k)}x +... + a_{(n, k)}x^n$
Prove that for every i, $a_{(i, k)} $ converges to 0.
It seems very very obvious to me, but I can't prove it formally.... |
H: Conditional Probability - Two methods
There are two methods, A and B, to finish a work.
Method A succeeds with probability $1/3$, but if it fails one tries method B with probability $3/4$ or method A again with probability $1/4$.
Method B succeeds with probability $1/4$, but if it fails one tries method A with pr... |
H: Prove that $(A,*)$ is a monoid
$F$ is a field.
$A$ is defined as the group of all the functions from $\mathbb{N}_0=\{0,1,2,3,...\}$ to $F$.
Let's define a binary operation $*$ on $A$ as follows:
For all $f,g\in A$ the function $f*g\in A$ is defined by: $(n)f*g=\sum_{k=0}^n (k)f(n-k)g$ for every $n\in \mathbb{N}_0$... |
H: Proof of A is orthogonal $\Leftrightarrow \|Ax\|=\|x\|$
I want to proof this property.
A is orthogonal $\Leftrightarrow \|Ax\| = \|x\|$
I tried to elaborate from this, but cannot see how to get any further:
$\|A\| \times \sqrt{<x,x>}$
I really appreciate your answer!!!
AI: You should add to your equivalence... |
H: To derive an inequality of the form $f(r) \leq cr^\alpha$
Let $f$ be a continuous positive function on $[0,1]$ which satisfies
$$ f(s/2) \leq \theta f(s)$$
for all $s\in [0,1]$, where $\frac{1}{2} < \theta < 1$. Can we show that
$$f(s) \leq Cs^\alpha \ \ \forall s\in [0.1]$$
for some positive $C$ and $\alpha\in (... |
H: Find the coefficient of a power of x in the product of polynomials - Link with Combinations?
I came across a new set of problems while studying combinatorics which involves restrictions to several variables and use of multinomial theoram to evaluate the number of possible combinations of the variables subjected to ... |
H: How find this limit $I=\lim_{n\to\infty}(1+\sin{(\sqrt{4n^2+1}\cdot\pi)})^n$
find this limit
$$I=\lim_{n\to\infty}(1+\sin{(\sqrt{4n^2+1}\cdot\pi)})^n$$
My try: let
$$I=e^{\lim_{n\to\infty}n\sin{(\sqrt{4n^2+1}\pi})}$$
AI: $$\sin((\sqrt{4n^2+1})\pi)=\sin(2\pi n(\sqrt{1+\frac{1}{4n^2}}))=\sin(2\pi n+2 \pi n\frac{1}{8... |
H: If $a^3 + b^3 +3ab = 1$, find $a+b$
Given that the real numbers $a,b$ satisfy $a^3 + b^3 +3ab = 1$, find $a+b$.
I tried to factorize it but unable to do it.
AI: There are a continuum of solutions to
$$
a^3+b^3+3ab=1
$$
Suppose that
$$
x=a+b
$$
then
$$
\begin{align}
1
&=a^3+(x-a)^3+3a(x-a)\\
&=x^3-3ax^2+3a^2x+3ax-... |
H: Number of models for some theory
Let $\mathcal L = \{ E(\_,\_) \}$ and $T$ be the $\mathcal L$-theory that says that $E$ is an equivalence relation with an infinite number of infinite classes. (I find this statement not clear, because it does not seem to specify the number of finite classes. In the following I assu... |
H: meaning of math symbol in probability expression?
anyone knows the meaning of the marked symbol in the pic and its Latex syntax?
AI: It is an indicator function that is 1 if its argument is true and 0 if it is false. The latex syntax is $\mathbb{I}$ (\mathbb{I}). |
H: Proving that roots of a quadratic lie between two values
To prove that one of the roots of a quadratic $f(x) = ax^2 + bx + c$ with real coefficients lies between two values $x_1, x_2$ is it enough to prove that:
$$f(x_1) < 0 < f(x_2) $$
Can this be generalized to a polynomial with an arbitrary degree?
AI: Any polyn... |
H: How to find this integral $I=\int_{-\pi}^{+\pi}\frac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx$?
Find the integral
$$I=\int_{-\pi}^{+\pi}\dfrac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx$$
My try:
let
$$I=\int_{-\pi}^{0}\dfrac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx+\int_{0}^{+\pi}\dfrac{x\sin{x}\arctan{e^x}}{1+\cos^2{x}}dx=I_{... |
H: Is $\mbox{lcm}(a,b,c)=\mbox{lcm}(\mbox{lcm}(a,b),c)$?
$\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?
I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number...
AI: Let the highest power of prime $p$ in $a,... |
H: Local sections of Hopf fibration $(S^3,\pi,S^2)$
In the lecture we showed the local triviality of the Hopf fibration $(S^3,\pi,S^2)$ as a principal-$S^1$-bundle by constructing local sections
$$s_1:S^2\setminus\{\infty\}\cong\mathbb{C}\to S^3,\qquad s_1(z):=\frac{1}{\sqrt{1+\lvert z\rvert^2}}\left(z,1\right)$$ and ... |
H: a basic question on continuity
Suppose $f:\Bbb R \to \Bbb R$ which is not injective and continuous. I need to find an open set in the domain which does not get mapped to an open set in the range. how ?
AI: Consider a value $y_0 \in \mathbb{R}$ that gets mapped to by several $x\in \mathbb{R}$. Now one of the followi... |
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