2
votes
1answer
220 views

Dense subset of the Cantor set

Prove that the set of endpoints of removed intervals in the Cantor middle thirds set is a dense subset of the Cantor set. Attempt at proof: Since each subinterval is of length $(1/3)^n$, any ...
1
vote
2answers
195 views

Modular Arithmetic on Circle

On the unit circle, what does the set $\left \{ n \bmod{2\pi}:n \in \mathbb{N}\right \}$ represent? What is the subsequentieal limits of $\left \{ \sin(n) \right \}_{n\in \mathbb{N}}$? I am probably ...
0
votes
0answers
38 views

characteristics proyections of a PDE

Let $u(x,y)$ by the integral surface of the equation: $a(x,y)u_x+b(x,y)u_y+u=0$ Where $a,b$ are positives differential function in the hole plane. Let $D=\{(x,y)||x|<1 ,|y|<1\}$ How do I ...
0
votes
0answers
131 views

Maximum number of truths in an optimized truth table.

I have a math-related question: I have a set of predicates that need to be evaluated. These predicates can have two kinds of operators; AND/OR. When such an expression is constructed my code builds ...
0
votes
1answer
89 views

Solutions of Diophantine equations ${x^y} = {y^x}$ [duplicate]

Possible Duplicate: $x^y = y^x$ for integers $x$ and $y$ How to prove that $(2,4)$ and $(4,2)$ are the only solutions of Diophantine equations ${x^y} = {y^x}$ for $x \ne y$?
0
votes
0answers
91 views

$x^n-x$ and some irreducible factors properties over $K[x]$

Let $K$ be a field , $a\in K$ , let $d$ be the greatest common divisor, of all the irreducible factors of $x^n-a$ in $K[x]$. $i)$ Prove that $ d|n$ , and there exist $b\in K$ , such that $a^d =b^n$. ...
3
votes
1answer
35 views

Chracterizing quartic polynomials F such that $F, F',F''$ have only real rational roots.

When designing friendly problems for a calculus class one comes up with such a question. (The cubic case is relatively easy.) Of course one can generalize: Characterize degree $n$ polynomials such ...
1
vote
1answer
57 views

Stability in PDE with the $L^2$ norm

$u_t +au_x = f(x,t)$ for $ 0<x<R$ and $t>0$ $u(0,t)=0$ for $t>0$ $u(x,0)=0$ for $0<x<R$ I have manage to get: $\int_0^R \! u^2(x,t) \, \mathrm{d} x \leq e^t \int_0^t \int_0^R \! ...
0
votes
1answer
36 views

In these 2 graphs, is x=2 an asymptote?

Do these 2 graphs have asymptotes at x=2? http://puu.sh/1aihI In the second graph, IS THERE an asymptote? Thanks!
1
vote
2answers
152 views

Proving algebraic numbers are countable? Simply stated…

Let $n$ a positive number, and let $A_n$ be the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree $n$. Using the fact that every polynomial has a finite ...
0
votes
2answers
45 views

Theorem: For any sequence , $b_n >0$ then $b_n \to 0$ iff limit of $\frac{1}{b_n} = \infty$ as $n \to \infty$

Theorem: For any sequence , $b_n >0$ then $b_n \to 0$ iff limit of $\frac{1}{b_n} = \infty$ as $n \to \infty$. I need help with this problem proof! Thank you very much!
2
votes
1answer
110 views

How to define subtraction in lattice and how to define complementation of an element in a lattice?

The story is that in a lattice, there are some elements incomparable, and there exits some elements have overlapping elements. for example, suppose the partial order is defined on inclusion, then ...
0
votes
2answers
75 views

Sets problem with upper bounds

I have come across this problem in my discrete mathematics class and I have no clue how to go about it since I haven't dealt with upper bounds before in sets. If anyone could help me out, I'd greatly ...
2
votes
1answer
74 views

Prove the tail end theorem: For any positive integer $M$, if $b_n \to b$, then $b_{n+M} \to b$

Prove: The tail end theorem: For any positive integer $M$, if $b_n \to b$, then $b_{n+M} \to b$. I do not know how to get started :(
2
votes
1answer
258 views

Convergence of an infinite product

Does this product converge? $$\prod_{n=1}^{\infty}\frac{1}{n^{2}+1}$$ any hint?
0
votes
1answer
282 views

Direction Field for System of Equations

I have a question that asks to draw the direction field for the set of linear systems. $$\begin{align} \frac{d\,x}{dt} &= -x + y + 1 \\ \frac{d\,y}{dt}&=x+y+3\end{align}$$ My attempt: What ...
1
vote
1answer
146 views

minimal polynomial of $x$ over $ K\left(\frac{p(x)}{q(x)}\right) \subset K(x) $

Let $K$ be a field , let's consider the field of rationals functions over x , $k(x)$. Let $t\in k(x)$ be the rational function $\frac{p(x)}{q(x)}$ , where $P,Q$ have no common factors. I have to prove ...
0
votes
2answers
3k views

Calculating missing data points from standard deviation and mean

I have to figure out what to missing data points are from a set of 10 scores. The mean of the 10 observed scores is 20.0 and the standard deviation is 6.0. The observed scores are listed below, with ...
0
votes
2answers
57 views

Inverse of a Particular Matrix

Let $a>0$. Let $A$ be the $n\times n$ matrix with $a+1$ on the diagonal and $a$ in all other entries. How can one compute $A^{-1}$ as a function of $n$?
1
vote
2answers
2k views

Limit Supremum and Infimum. Struggling the concept

I am struggling to understand what is the limit supremum/infimum. I've been told that it is not the same thing as "the limit of a supremum of a set" (which makes sense since the supremum/infimum is ...
1
vote
1answer
106 views

Bounded sequences and lim inf

Let $a_n$ and $b_n$ be bounded sequences. Prove that lim inf $a_n$ + lim inf $b_n \leq$ lim inf$(a_n + b_n)$ I have no idea where to begin.
1
vote
1answer
47 views

Calculus Limit: mathematical syntax for multiple variables

Is this the correct way of writing it? $$\lim_{\substack{t \to 0^+ \\ \text{and } x \to 2^+}} \frac{x}{t}$$ Thanks!
0
votes
2answers
311 views

Is the set of infinite ordered tuples of distinct natural numbers countable?

One such tuple is [2, 45, 342 ... onwards to infinity]. Another such tuple is [0, 1, 2, 3 ... onwards to infinity]. I think this set is countable because we can lay all of these sets out one below ...
2
votes
1answer
544 views

Integral involving Matrix Exponential to solve LTI system equation

I am given that for $A$ that is $n \times n$ matrix of full rank, $$\int_{0}^{t}e^{A\sigma}d\sigma = (e^{At}-I)A^{-1}$$ Then I am using this to solve LTI system $$\dot{x}=Ax+Bu$$ Here, $x(0) = ...
2
votes
2answers
325 views

How to show that this logical argument is valid?

I am asked to show the following argument is valid: I know you need to use the rules of inference like modus ponens/converse fallacy but I'm confused because it doesn't look like any of the forms ...
2
votes
2answers
2k views

Epsilon Delta Limit Proofs at and going to infinity.

So I understand the concept of epsilon delta limit proofs with linear functions, easy enough, and I am still shaky about doing it with non linear but I am slowly understanding that. I don't quite ...
0
votes
1answer
184 views

Diagonalising a matrix

I've got a matrix $A = \begin{bmatrix}1&-1\\ 2&-1\end{bmatrix}$ and wish to diagonalise it. I find the eigenvalues as below. $$\det(A - xI) = 0 = \det\begin{bmatrix}1-x&-1\\ ...
1
vote
2answers
91 views

finding rational roots of polynomials

Could someone please explain me how to apply the Descartes's Criterion? For example , how do I find the rational roots of $ x^3 -x +1$. I've been looking at some examples, but I get confused.
1
vote
1answer
430 views

How many degrees of freedom?

I have another question on my stats homework, relating to degrees of freedom. However, I have NO idea how this question even relates to degrees of freedom! I understand degrees of freedom on their ...
2
votes
1answer
102 views

Difference equation with trigonometric term

Trying to find the general solution to this homogeneous difference equation: $$y_k - 2\cos\theta y_{k-1} + y_{k-2} = 0.$$ The characteristic equation is $$\lambda^2 - 2\cos\theta \lambda + 1 = 0.$$ ...
3
votes
0answers
72 views

Calculate sums of logs in precision

I am encountering a situation where I cannot calculate exact sum of a seris of logorithms in calculating entropy. Suppose we have a series of numbers $p_i$ and we want to calculate $\sum_ilog(p_i)$, ...
1
vote
1answer
113 views

Characterization of primitive roots modulo a prime

Let $p$ be a prime. Given a divisor $d$ of $p-1$, there always exist numbers with multiplicative order $d$ modulo $p$, and there are $\phi(d)$ such numbers. If for example $a$ has order $d$, then ...
2
votes
2answers
223 views

Exponential growth function

Under ideal conditions a certain bacteria population is know to double every three hours. Suppose that there are initially 100 bacteria. How would you go about formulating a function for this?
0
votes
2answers
163 views

How do I account for opposite answers when calculating city block distances?

For a stats class, I have a question in which an occupational therapist uses a checklist about meal preparation. The checklist consisted of five statements to which persons responded using the ...
2
votes
2answers
149 views

Equivalence Relation problem?

I need help with this problem: Suppose $\sim$ is a relation on a set $S$ which is both symmetric and transitive. Let $A = \{x∈S\ \vert \text{ for some }y∈S, x\sim y\}$. Prove that $\sim$ is an ...
0
votes
1answer
51 views

Show that $A \cap B$ is in $F$

I just have a quick probability question. Let $A$, $B$ be in $F$. Show that $A \cap B$ is in $F$ using $(A \cap B)^c$. Any ideas on how I can solve this?
2
votes
2answers
182 views

Prove: from the definition of the limit if $a_n \to L$ then $|a_n| \to |L|$

Prove: from the definition of the limit if $a_n \to L$ then $|a_n| \to |L|$ What is the strategy for a proof of this nature? There are similar proof's of the type "given a sequence converges to $L$ ...
3
votes
1answer
265 views

Reflexivity of a Banach space

I've run into a few problems in which reflexivity of a Banach space is given as a hypothesis. These problems are sometimes of the type where the banach space is specific/concrete, and sometimes it is ...
0
votes
1answer
434 views

composition of two measurable function

I've come across this problem in my studies, where the book (Real Analysis (4th Edition) by Royden) gives an example of two measurable functions whose composition is nonmeasurable. The two functions ...
0
votes
1answer
195 views

Null recurrence in Discrete Time Markov Chains

Is it possible to have null recurrent states if the number of states is finite? If so, I would appreciate a small example (a 2x2 or 3x3 transition probability matrix would be nice).
2
votes
2answers
101 views

Expectation and Lebesgue integration question

How can I show: If a random variable $Z$ has finite expectation $E(Z)$ (i.e., $Z$ is Lebesgue integrable), then $nP(|Z|>n) \to 0$ as $n \to \infty$?
3
votes
0answers
236 views

On the bounded derived category of a finite dimensional algebra with finite global dimension

Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) ...
1
vote
1answer
95 views

Trying to solve a related rates problem

I am trying to solve a related rates problem. The problem states: If y = 4x -x^3 and the x-coordinate is increasing at the rate of 1/3 unit/sec. How fast is the slope of the graph changing at the ...
5
votes
1answer
273 views

Find entire functions that satisfy certain conditions

1) Find all entire functions that are uniformly continuous on $\mathbb{C}$. 2) Find all entire functions $f(z)$ such that such that for every integer $n \geq 1$, $$\oint_{\partial\mathbb{D}} ...
-2
votes
2answers
776 views

{Thinking}: Why equivalent percentage increase of A and decrease of B is not the same end result?

original post the examples here are, the most important word -- fundamentally -- the same. example1: the most abstract way to present this example. Why equivalent % increase of A in event1 and % ...
1
vote
2answers
825 views

Unions and intersections of compact subsets

I'd really appreciate some input on these two proofs regarding unions and intersections of compact subsets (under additional necessary conditions). Namely, are my proofs valid? Could they be ...
2
votes
1answer
84 views

Find an ellipse whose length is the same as the outer rim of the monkey saddle

Given the monkey saddle $z=x^3-3xy^2$ over the unit circle $x^2+y^2 \leq 1$, find an ellipse whose length is the same as the length of the outer edge of the monkey saddle. I've already found a ...
0
votes
0answers
335 views

Simplex Tableaux Problem

I have the following LP which I need to solve using the simplex method. I know there are no feasible solutions as there are constricting constraints. How do I use the Tableaux method to show this? ...
1
vote
1answer
128 views

On the eigenvalues of the square of a real matrix $A$

I just read this snippet in a textbook "The eigenvalues of a symmetric real matrix are real (The proof follows by noting that if $A$ is symmetric, the eigenvalues of $A^TA$ are the ...
1
vote
1answer
135 views

Set theory equivalent statement proof

If $A\cap B = B\cap C$, then $(A-B)\cup C = A\cup (C-B)$. Are the two statements at the end equivalent?

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