1
vote
0answers
16 views

conditional probability, logical product

I was working my way through Kruschke's textbook and got to Chapter 9 and the result on factoring out conditional probabilities for hierarchical models, seemed similar to something in Feller Vol1 ...
1
vote
0answers
27 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
0
votes
0answers
10 views

Example of a non-regular curve which has the same geometric image as a regular curve parametrized by arclength

Give an explicit example of: $(a)$ a regular curve parametrized by arclength; $(b)$ a non-regular curve which has the same geometric image as the previous one. Could someone please help me with ...
2
votes
3answers
30 views

Differential equations (second edition) - William E. Boyce & Richard C. Diprima (#$31$, page $142$)

In many physical problems, non-homogeneous term may differ from one time interval to another. By example, determine the solution $y = \phi(t)$ at $$y'' + y = \begin{cases} t ...
2
votes
1answer
28 views

How to determine if this musical exercise is valid: will the pattern complete?

I'm hoping that math has an answer to a question arising out of a musical exercise. In music terms, the exercise is: Choose two arpeggios (sets of notes) of equal (or roughly equal) span (number ...
0
votes
0answers
13 views

Number of ways, modulo a prime $p$, to write $n$ as a sum $n = x_1^k + x_2^k + \cdots + x_s^k$

Removing the restriction on $p$, this is known as Waring's problem. The circle method has been used successfully to tackle this. Using analysis, nice estimates can be given. I wonder what analytic ...
1
vote
1answer
11 views

Independence of two non-negative integer valued random variables

Let $X,Y$ be two non-negative integer valued random variables defined on a probability space $(\Omega,\cal F, \Bbb P)$. The question is, If $\Bbb P\{X=i,Y=j\}=\Bbb P\{X=i\}P\{Y=j\}$ for every ...
0
votes
0answers
20 views

Showing set is undecidable with Turing Machines

I'm given the set $T = \{\langle M, w\rangle : M $ is a Turing Machine that accepts $w$ reversed whenever it accepts $w \}$ and I want to show it's undecidable but recognizable. (I'm using the bracket ...
0
votes
0answers
19 views

Is this how to do matrix representation?

Say, $$f: \mathbb{Q}[t]_{4} \to \mathbb{Q}[t]_{4}$$ $f(q)=3q'''+2q''$ And we have the base $B=\{1,t,t^2,t^3,t^4\}$ and we wanted to find $[f]_{B}^{B}$ Then is this what would we do; $$f(1)=0$$ ...
0
votes
0answers
16 views

Travelling Salesperson MTZ

I have been solving a $10$ city travelling salesperson problem. Having solved the assignment based relaxation problem, I have $5$ subtours: $1 \rightarrow 10 \rightarrow 1$ $2 \rightarrow 8 ...
0
votes
0answers
8 views

Proving linear operators are Markov Generators

I am trying to do the following question from Liggett. Let $A$ be a linear operator define on the space of continuous functions $C(E)$ for a compact set $E$. Define $A$ by $A=T-I$ where $T$ is a ...
1
vote
0answers
26 views

Finding convolution of two functions?

1. Continuous Functions $x_1(t)$ and $x_2(t)$ definitions' link How to evaluate $(x_1∗x_2)(t)$ at $t = −T, 0, +T$ in terms of $T$ 2. Discrete Functions $x_1[n]$ and $x_2[n]$ definitions' link ...
0
votes
0answers
8 views

Square wave in the limit of infinite frequency

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity? Does it depend on how the function is written down (e.g. defined as ...
1
vote
1answer
18 views

Finding MLE of uniform distribution with actaul example values

I'm watching this video and going to part I am stuck at here https://youtu.be/XaAtkCzdjLE?t=6m2s Following the example in the video, I assume that $\theta$ will be between $14$ and $501$. Now I ...
1
vote
1answer
34 views

Formulate quadratic equation

Here is an equation $$r^3-5r^2+8r-4=0$$ Is there a way I can formulate a quadratic equation from this? Sorry if the question seem dumb, I am stuck and I can't figure a way out.
2
votes
0answers
8 views

Can I still uses the method of logarithmic differentiation to simplify complicated functions if the the range of that function includes 0?

The method of log differentiation refers to taking the natural log of both sides of an equation to simply an complication functions evolving lots of multiplication and divisions and exponents. But ...
0
votes
1answer
19 views

Point to Plane Distance Questions

I'm reading from Marsden Vector Calculus 6th Edition and this picture is from page 43. I'm having difficulty understanding how they get to $$ \text{Distance} =|\vec v \cdot \vec n|$$ The way I ...
0
votes
0answers
9 views

Probability of maximum degree in On random graphs I by Erdos

In The Maximum Degree of a Random Graph by RIORDAN et al., the authors commented that the study of the distribution of the maximum degree $d^{max}(G)$ of a random graph $G$ was started by Erdos and ...
2
votes
1answer
38 views

Why can entire function be written as exponential, and why is it bounded in this way?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose now that $f(x) \in \sigma(x)$ for every $x \in A$ where $\sigma(x)$ denotes the spectrum of $x$. Now, let $x\in A$ and ...
2
votes
1answer
25 views

Hyperbolic area

Define Hyperbolic area of a subset $E$ of the unit disk $D$ to be $\displaystyle 4\int \int_E \frac{dx dy}{(1-|z|^2)^2}$. Show that the hyperbolic area is invariant under conformal self maps of ...
1
vote
2answers
18 views

The covariance between $X$ and $Y$.

Suppose that $X$ and $Y$ are both continuous random variables that have a joint probability density that is uniform over the rectangle given by the four $(x,y)$ coordinates $(0,0)$ , $(2.46,0)$ , ...
0
votes
1answer
20 views

What is this graphed function with asymptotes at π/2 and -π/2?

I've come across this function with asymptotes at π/2 and -π/2, which crosses the axis at y=1. It doesn't seem to be polynomial or exponential—can anyone figure out what it is? The asymptotes and ...
1
vote
0answers
29 views

What is my special quadratic?

Start with $f(x)=x^2+bx+c$. Then, attempt to solve for $x$ in $f(x)=x$. It is easily found that $x=\frac{1-b\pm\sqrt{(b-1)^2-4c}}{2}$. Then, start again with $f(x)=x$ and apply the function $f$ to ...
0
votes
1answer
42 views

When is the order important in Combinatorics?

In a shop five different type of chocolates are sold. How many different ways 6 chocolate bars can be chosen in such a way that at least 3 chocolate bars must be of type one and at most one of type ...
0
votes
1answer
20 views

Under what conditions can we move the limit symbol through the logarithm symbol?

I was reading the derivation of the derivation of a log function. And saw this: $$\frac{d}{dx}[\log_b x]= \frac{1}{x}\lim_{v \to 0} [\log_b(1+v)^\frac{1}{v}]$$ Then, the limit notation gets moved ...
0
votes
1answer
19 views

does $f(n) = O(g(n))$ implies $(f(n))^{log(n)} = O((g(n))^{log(n)}) ?$

if $f(n)$ and $ g(n)$ are monotonically increasing, and $f(n) = O(g(n))$. Does it imply that $(f(n))^{log(n)} = O((g(n))^{log(n)}) ?$ Well I had a go at it saying I need to show that ...
0
votes
3answers
38 views

Prove that $\frac{a^2}{b}+\frac{c^2}{d}\ge\frac{(a+c)^2}{b+d}$

I'm looking for hints, not for a complete solution: prove that for $a,b,c,d\in\mathbb{R}_{+}$ the following inequality holds: $\frac{a^2}{b}+\frac{c^2}{d}\ge\frac{(a+c)^2}{b+d}$
0
votes
1answer
27 views

E Lebesgue Measurable implies E^2 Lebesgue Measurable?

Suppose $E \subset \mathbb{R}$ is Lebesgue measurable. Define $$ E^2 = \{x^2 : x \in E\}. $$ Is $E^2$ Lebesgue measurable as well? I believe the answer is yes, but I am struggling to prove it. I ...
0
votes
0answers
15 views

Give an example of limits that misbehave under conjugation of function

My quest: Find real valued functions $f(x)$ and $g(x)$ such that $f \rightarrow b$ as $x\rightarrow a$ and $g\rightarrow c$ as $x\rightarrow b$ but $g(f(x)) \nrightarrow c$ as $x\rightarrow a$ I ...
3
votes
0answers
16 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
1
vote
0answers
32 views

The closure of the closed ball is a closed [on hold]

In how many ways you can show that $\overline{\overline{B}(x,r)}=\overline{B}(x,r)$ where $\overline{B}(x,r)=\lbrace y \in \mathbb{R}^n : d_e(x,y) \leq r \rbrace$, and $d_e$ is the euclidean metric ? ...
0
votes
1answer
32 views

Natural deduction proof / Formal proof : Complicated conclusion with no premise

Find a formal proof for the following: $\vdash [(\neg p \land r)\rightarrow (q \lor s )]\longrightarrow[(r\rightarrow p)\lor(\neg s \rightarrow q)]$ As you can see. No premise to use. We have to use ...
0
votes
0answers
35 views

Prove that these two fields are isomorphic.

I want to prove that the field $\bar{K}[V]/M_p \sim \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and: $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V$$ and $I_V$ is the ideal attached to ...
0
votes
1answer
17 views

Find $\lim_a \frac{Tr^n \left( ({\bf x}-a\cdot{\bf y}) \cdot ({\bf x}-a \cdot{\bf y})^T \right)-Tr^n \left( {\bf x} \cdot {\bf x}^T \right)}{a}$

Let ${\bf x}, {\bf y} \in \mathbb{R}^{ m \times 1}$ and $a \in \mathbb{R}$. How to find the following limit \begin{align} \lim_{a \to 0 } \frac{Tr^n \left( ({\bf x}-a\cdot{\bf y}) \cdot ({\bf x}-a ...
1
vote
0answers
14 views

On an inequality involving primorial numbers.

Let $N_k$ denote the $k-th$ primorial number. That is, the product of the first $k$ primes and $\phi(n)$ be the Euler totient function. How can one show that there exists a constant $\theta>1$ ...
1
vote
1answer
19 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
0
votes
1answer
17 views

How to find isoclines of the following system?

If I had a system of equations such as: $$\dot{x}=(1+2x+y)x$$ $$\dot{y}=(4+6x+2y)y$$ How would I find the horizontal and vertical isoclines of such a system? I know ...
1
vote
1answer
22 views

Recursion in Integration by Parts

I'm trying to integrate by parts, but I keep getting recursive answers. $I=\int_0^\pi f(x)cos(x)dx$ where $f''(x)=3f(x)$, $f'(0)=-5$, and $f'(\pi)=4$ Thanks.
0
votes
0answers
23 views

Which of the following sets is compact, bounded, closed or open and why? [on hold]

Which of the following sets is compact, bounded, closed or open and why? $M1= [-1,42]$ $M2= (-1,42]$ $M3= (-1,42)$ $M4= (-\infty, +\infty)$ $M5= \{z \in \mathbb C: 0 < \operatorname{Re} z + ...
0
votes
0answers
15 views

Help with constructing a certain ring in GAP

I need to construct the following ring in GAP: $$F_2(u) / \langle u^2=0 \rangle =\{ \; a+bu \; | \; a,b \in F_2 \; \}=\{0,1,u,1+u\}$$. I tried using the commands ...
0
votes
1answer
26 views

Using Newton's Method to solve $f(x)=x^2-2bx+b^2-d^2=0$

What would be the Newton's method in the form $x_{k+1}=g(x_k)$ to solve the equation $$f(x)=x^2-2bx+b^2-d^2=0$$ in which both $b>0,d>0$ are parameters? I also need to show that $|g'(x)|\le 1/2$ ...
0
votes
2answers
15 views

Spectral theorem for unitary operators T o F

Si $T$ es unitaria y $B$ es una base de $V$ formada por vectores propios de $T$ entonces $B$ es un conjunto ortogonal. If $T$ is unitary and $B$ is a basis for $V$ consisting of eigenvectors of $T$ ...
0
votes
1answer
26 views

Predicates about functions in 1st order logic

Given the usual definition of function as a subset of $ D \times C $. What is the correct way to write "All functions $ f $ from $ D $ to $ C $ have property $P(f)$". This is both a question about ...
0
votes
1answer
28 views

Finding vectors in a set.

I am in linear algebra and was given this question as a review: Let $E \subset \mathbb{R}^3$ be the set of all vectors $(x, y, z)$ such that $x + 2y + 3z = 0$. Find two vectors $v, w \in E$ such ...
0
votes
0answers
19 views

condition for having a positive solution to these linear equations.

Consider the following system of linear equations: $\sum_{j=1}^n c_{ij}\cdot x_{ij}=a_i$ for $j=1,\cdots,m$ and $\sum_{i=1}^m c_{ij}\cdot x_{ij}=b_j$ for $j=1,\cdots, n$ where for all $1\leq i\leq m$ ...
0
votes
2answers
18 views

Cosets of submodule

Suppose we have a ring $R$ and an ideal $J \subseteq R$. Let $M$ be an $R$-module and $N$ be a submodule of $M$. Assume for two elements $m$ and $m'$ of $M$ we have $$m+ JM=m'+JM.$$ Since $N ...
11
votes
0answers
54 views

Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them

I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
1
vote
1answer
25 views

Representation of a group and its quotient

Let $G$ be a (finite) group and let $N$ be a normal subgroup of $G$. Suppose that we have a representation $(V,\rho)$ of $N$ and a representation of $(V, \tau)$ of the quotient group $G/N$. Here $V$ ...
0
votes
0answers
13 views

Implicit - simplify last step

Please let me know if this link works. I'm pretty new at posting questions. Maybe there is a better way to post from the derivative-calculator.net site. I'm not sure how they simplify the last step ...
0
votes
0answers
17 views

$\bf{x'}=Ax$ with eigenvalues of multiplicity greater than $1$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, if I solve it by first finding matrix $\bf{P}$ and then ...

15 30 50 per page