2
votes
1answer
176 views

Probability with Intersections, Unions, and Complements

I just want to make sure I'm doing this correctly. Here is the problem: Let $A$ and $B$ be sets such that $P(A \cap B)=\frac{1}{4}, P(\tilde{A})=\frac{1}{3},$ and $P(B)=\frac{1}{2}$. What is $P(A ...
2
votes
1answer
54 views

Distributivity of a dot product-like operation

Let $b_1, \ldots, b_n \in \mathbb{N}$. For $x, y \in \mathbb{Z}^n$, define $x \cdot y$ as $\newcommand{\lcm}{\operatorname{lcm}}$ $$x \cdot y = \left(\sum_{i=1}^n (x_i \text{ mod } b_i)(y_i ...
0
votes
1answer
73 views

Constructing border on a garden

So I have this homework question I need help on. There's this rectangular garden that is said dimension ($x \times y$). Someone orders said amount of cement ($c^3$) and wants to make a border of ...
1
vote
1answer
113 views

Shortest radius needed

Ok, so I have this homework problem and I want to know if I did it correctly and if there is an easier way to do so. Question goes: There is a field in the shape of a square with an area of ...
0
votes
1answer
57 views

How to express this exponential equation in terms of $m$?

How can I express the equation $c = 2^{m+1} - 2^m$ in terms of m? t.i.a.
3
votes
3answers
80 views

Prove that $(2-x)^nx^{n-1}$ decreases with $n$ for $0 <x<1$?

How can I show that: $$(2-x)^nx^{n-1}$$ is decreasing with $n$ when $0<x<1$? I think this is generally true, but specifically I am concerned with $n$ as an integer $\geq 2$ and showing that the ...
2
votes
4answers
139 views

Distributivity mod an integer

Let $a,b,c,m \in \mathbb{Z}$, is it always the case that $$a((b+c) \text{ mod } q) \text{ mod } q = (ab \text{ mod } q + ac \text{ mod } q) \text{ mod } q$$
1
vote
0answers
28 views

MLE estimation of parameters, converting normalized observations to integers and back

I am fitting a model's parameters to grouped data by maximizing the likelihood equation: $L(\theta)=N!\prod_{i=1}^{G}\frac{p_i(\theta)^{n_i}}{n_i!}$ $\theta$ is the vector of parameters. $n_i$ is ...
0
votes
4answers
90 views

Simple proof (hopefully) when $0 <x<1$

How can one prove that $x(2-x)^2 \leq 1$ when $0<x<1$ ? Alternatively, of course, this is $4x-4x^2+x^3 -1 \leq 0$ but I don't know where to go from there. Is it enough to show the inequality ...
1
vote
1answer
140 views

Number of ways to arrange $3$ rocks in $6$ boxes [duplicate]

Possible Duplicate: Unique ways to keep N balls into K Boxes? This question may be sound stupid, but we really cant figure it out. We have 3 rocks and 6 boxes. All the rocks have to be in ...
0
votes
2answers
2k views

Prove that every element of a finite group has an order

I was reading Nielsen and Chuang's "Quantum Computation and Quantum Information" and in the appendices was a group theory refresher. In there, I found this question: Exercise A2.1 Prove that for ...
5
votes
2answers
141 views

History of the vocabulary for group extensions

In regular everyday English if you say something like "A was extended by B to get C", to me it means that A was in existence, B was added onto it, and now there is a larger object C. For example, ...
5
votes
3answers
400 views

Powers as a complete residue system modulo $p$?

Question 1. With $0 < a < p$, $p$ prime and $\gcd(a,p-1)=1$, is it true that $0, 1, 2^a, ...,(p-1)^a$ is a complete residue system modulo $p$? If not, will a similar statement hold? Question ...
7
votes
1answer
243 views

Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

It seems that the $n$th cumulant of the uniform distribution on the interval $[-1,0]$ is $B_n/n$, where $B_n$ is the $n$th Bernoulli number. And also $-\zeta(1-n) = B_n/n$, where $\zeta$ is Riemann's ...
1
vote
1answer
646 views

Software for testing relational algebra

Does anyone know of any software to let you test relational algebra queries? By this I don't mean a database such as MySQL, something where the query can be input in some for of mathematical notation ...
7
votes
2answers
249 views

Summing the prime power counting function up to equal some value $n$

I want to find $c_k$ for $n = 1 + c_1 \Pi(n) + c_2 \Pi(\frac{n}{2})+ c_3 \Pi(\frac{n}{3})+ c_4 \Pi(\frac{n}{4})+ c_5 \Pi(\frac{n}{5})+...$, assuming there are such coefficients, where $\Pi(n) = ...
4
votes
3answers
505 views

Proving that $2n^2+3n+1=O(n^2)$

For Big-O notation in mathematics, How does $f(n) = 2n^2 + 3n + 1 = O(n^2)$? Does it require any more information for the proof? Edit: ...
1
vote
1answer
113 views

Showing $\bar{y} \rightarrow \mu$ Have I done enough? Converges in probability

From the title I'm supposed to show $\bar{y} \rightarrow \mu$ (converges in probability) where $$y_t = \mu + u_t$$ $$ u_t = \rho u_{t-2} + \epsilon_t$$$$ E(\epsilon) = 0, E(\epsilon^2) = \sigma^2, ...
1
vote
0answers
170 views

Motivation and checking for points being in “general position”

I wanted to know the motivation and the calculation details of checking that a certain number of points are in "general position". Intuitively I was thinking that a set of points being in general ...
3
votes
1answer
409 views

How can I get a huge Linear Programming Problem? Any public data set?

I'm working on a Parallel Simplex Solver using C and nVidia CUDA for my Bachelor Degree in Computer Science. I've already asked one of my teachers to bring me a super linear problem with thousands ...
2
votes
1answer
109 views

Every abelian $p$-group is an image of some direct sum of cyclic $p$-groups

Every abelian $p$-group is an image of some direct sum of cyclic $p$-groups. This is an exercise on page 113 of Derek J.S. Robinson's A course in the Theory of Groups. I don't know why it is ...
0
votes
1answer
105 views

What is the function for this?

I need a function that does the following: x can not be greater than 1 y can not be greater then 2 As x approaches the value 0, then y approaches, but does not reach the value 2. As y approaches the ...
5
votes
3answers
309 views

Properties of Fermat primes

Fermat primes 17 and 257 appear a lot in the prime composition of numbers of the form $a^{2^n}+1$. For example, $11^8+1$ is divisible by 17 and $11^{32}+1$ is divisible by 257. I have verified the ...
3
votes
2answers
213 views

Why don't the limits of integration matter when differentiating both sides of and equation?

Someone asked this question: and I am very interested in the answer, but don't understand it and don't have enough reputation to comment on it directly. The question asks if you can solve something ...
1
vote
1answer
282 views

Showing a Ring of endomorphisms is isomorphic to a Ring

Im trying to show that $\mathrm{End}(\langle \mathbb{Z},+\rangle)$ is naturally isomorphic to $\langle \mathbb{Z},+,\cdot\rangle$, but I'm not quite sure which ring homomorphism to use. Thank you
8
votes
2answers
474 views

Proving an inequality with Cauchy-Schwarz

In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions (link), they make the following claim (inequality (A.4) p. 58) : Given $x$, $\xi$ $\in \mathbb{R}^n$, $A \in \cal{S}(n)$ ...
2
votes
1answer
2k views

How to integrate a composite function with a variable exponent?

Is there a general formula, or any "tools", for solving $$\int f(x)^n \, dx $$ ? I apologize if this is a basic question... but I am trying to remember my calculus, and even with searching online and ...
3
votes
2answers
395 views

Irreducible and aperiodic Markov chain : $P^t(x,y)>0$

Consider a Markov chain $X$ with transition probability $P$ and finite state space $\Omega$. Which of the following statement is true? If $X$ is irreducible then $\exists t>0 \ni P^t(x,y)>0, ...
1
vote
1answer
221 views

Historical reason for calling $\nabla\cdot F$ divergence?

Consider the continuously differentiable vector field in ${\mathbb R}^3$: $$ F:{\mathbb R}^3\to{\mathbb R}^3,\qquad F(x,y,z)=(U,V,W) $$ where $$ U,V,W:{\mathbb R}^3\to{\mathbb R}. $$ According to ...
1
vote
1answer
234 views

Fixing Hasse principle

As everyone know that Hasse principle (I am referring to Hasse Local-Global Principle) doesn't work for cubics, but today my question is concerned about: Is there any method or any known theorem, ...
1
vote
2answers
113 views

Questions about averaging

i have some trouble with averages. Here are two questions rolled in one: why is : $$\frac{\prod _{n=1}^N \left(1-\text{rnd}_n\right)}{N} \neq \prod _{n=1}^N \frac{1-\text{rnd}_n}{N} \mbox{where ...
5
votes
6answers
951 views

Confused between Nested Quantifiers

I am reading nested quantifiers. I am confused in between two cases, ...
5
votes
1answer
894 views

Probability that two random numbers are coprime

This is a really natural question for which I know a stunning solution. So I admit I have a solution, however I would like to see if anybody will come up with something different. The question is ...
0
votes
1answer
292 views

Simplify water salinity equations

I recently became interested in calculating water salinity from conductivity and temperature. I failed to find any theoretical model to how this is done, only empirical models derived from fitting to ...
0
votes
1answer
122 views

Recommendation of Books of Survival Analysis

I'd like a book of survival analysis at undergraduate level, if it's autocontained better, what do you suggest?
31
votes
1answer
920 views

Automorphisms inducing automorphisms of quotient groups

Let $G$ be a group, with $N$ characteristic in $G$. As $N$ is characteristic, every automorphism of $G$ induces an automorphism of $G/N$. Thus, $\operatorname{Aut}(G)\rightarrow ...
2
votes
1answer
256 views

Properties of the greatest common divisor and least common multiple

Let $a$, $b$, $c \in \mathbb{N}$. $[a, b]$ denotes $\mathrm{lcm}(a, b)$ and $(a,b)$ denotes $\gcd(a, b)$ Show that $(a,[b,c]) = [(a,b),(a,c)]$. $[a,(b,c)] = ([a,b],[a,c])$.
5
votes
1answer
133 views

Prove ranks are uniformly distributed

We have n IID random variables $X_1, X_2, \ldots, X_n$. Let $R_i$ be $X_i$'s rank in the set $\{X_1, X_2, \ldots, X_3 \}$ when we order from large to small. How to prove $R_i, \forall i \in \{1, 2, ...
14
votes
6answers
1k views

If multiplication is not repeated addition

What is it? How do you define multiplication? All of us had to memorize the multiplication table in elementary school, but how did they come up with it if so many people claim that it is wrong to ...
1
vote
0answers
94 views

Infinite number of primes in the sequence $1+t^2$? [duplicate]

Possible Duplicate: Primes of the form $n^2+1$ - hard? $1, 2, 5, 10, 17, \ldots$ Are there an infinite number of primes in this sequence $1 + t^2$, $t$ being an integer?
5
votes
2answers
213 views

Evaluate the integral $\int_0^\infty \frac{t}{e^t-1}\mathrm dt$

An integral related to the zeta function at the point $2$ is given by $$\zeta(2) = \int_0^\infty \dfrac{t}{e^t - 1}\mathrm dt$$ How to calculate this integral?
1
vote
1answer
670 views

Proving Cauchy-Schwarz inequality question

This is for a first year calculus course. Everything I can find online about Cauchy-Schwarz inequalities involves real analysis and vectors etc. I've only just begun calculus. $x_1$, $x_2$, $y_1$, ...
1
vote
4answers
647 views

What is the Conjunction Normal Form of a tautology?

I have a tautology and I need to write its CNF(Conjunction Normal Form). Since its a tautology CNF will not have any element. So ...
18
votes
2answers
1k views

Why is lambda calculus named after that specific Greek letter? Why not “rho calculus”, for example?

Where does the choice of the Greek letter $\lambda$ in the name of “lambda calculus” come from? Why isn't it, for example, “rho calculus”?
0
votes
1answer
36 views

Quick check re subspace

Following my previous question (not necessary to refer to), let the number of automorphisms of an $n$-dimensional vector space be $k$. Then am I right in thinking that the number of $m$-dimensional ...
6
votes
2answers
215 views

Integer matrix with particular Jordan's form

For teaching purposes I would like to find integer matrices with a particular Jordan's form. Is there some kind of technique to find nice examples? For example for ...
1
vote
3answers
222 views

How to define $-\infty$?

I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one ...
2
votes
1answer
164 views

Counting automorphisms

How does one count the number of automorphisms of a vector space? If a vector space over $\mathbb F_p$ has $n$ ordered bases how many are there? I think I should be considering the mappings of a set ...
4
votes
4answers
566 views

Difficult Integral: $\int\frac{x^n}{\sqrt{1+x^2}}dx$

How to calculate this difficult integral: $\int\frac{x^2}{\sqrt{1+x^2}}dx$? The answer is $\frac{x}{2}\sqrt{x^2\pm{a^2}}\mp\frac{a^2}{2}\log(x+\sqrt{x^2\pm{a^2}})$. And how about ...
4
votes
1answer
191 views

A Boolean function with total influence 1 must be a dictatorship

Let $f:\{-1,1\}^n\to\{-1,1\}$ be a boolean function. Define the influence of the $i$'th coordinate of $f$ as follows: $$\operatorname{Inf}_i(f)=\Pr_{x}[f(x)\neq f(\hat x_i)]$$ where $x$ is uniformly ...

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