0
votes
0answers
19 views

Picking a random word in python?

I have a tuple which contains three lists in python and I want the interpreter to pick a random word from each list to make a sentence. How do I do this? Tuple = (['hi','hey','hello] ...
0
votes
0answers
15 views

Integration application, known values

I know how integrals work with functions (academic sense), however I can't seem to get around applying integrals. ie. $$\int_{all \lambda} S(\lambda)P(\lambda)e^{\mu(\lambda)x} d\lambda$$ where I ...
2
votes
2answers
26 views

What makes negative numbers different from positive numbers other than their being (almost) opposite?

To quote from Wikipedia's article on negative numbers Negative numbers represent opposites. If positive represents movement to the right, negative represents movement to the left. If positive ...
0
votes
0answers
17 views

Modulo Arithmetic - Chinese Remainder Theorem

Solve the linear congurence $17x\equiv 3(\mod{2*3*5*7})$ by solving the system: $17x\equiv 3(\mod{2})$ For this one, I simplified to $x\equiv 1(\mod{2})$. Let this $x=5$. $17x\equiv 3(\mod{3})$ ...
3
votes
1answer
35 views

Derivation of Dirac delta function

Is there anyone could give me a hint how to find the distributional derivative of the delta function $\delta$? I don't know how to deal with the infinite point.
2
votes
2answers
28 views

Prove (∀z∈ℂ\{1,-1} : |z|=1)(∃x∈ℝ) where z=(x+i)/(x-i)

I am having trouble proving the next problem: Prove that (∀$z$∈ℂ \ {1,-1} : |$z$|=1)(∃x∈ℝ) where $z=\frac{x+i}{x-i}$ What have I done: I observed complex number $z$ as dots on a circle with radius ...
6
votes
1answer
39 views

$ \int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx$

What is the value of $$ \int_0^\infty \ \frac{(x\cdot\cos x - \sin x)^3}{x^6} \ dx $$ I have no idea how to start with this integral, any hint?
1
vote
2answers
14 views

Commas in propositional logic

I want to know what effect a comma has on a propositional statement. For example: $\{\neg p, p \vee q \} \vDash q$ Does this bit $\{\neg p, p \vee q \}$ mean just $q$? Thanks.
1
vote
1answer
28 views

How to read $\frac{dy}{dx} $ when the term is only given?

When the term $\frac{dy}{dx}$ (not $\frac{d}{dx}y$) is only given, how to read the term between "the derivative $y$ with respect to $x$" and "the quotient of the differential $dy$ by the differential ...
0
votes
0answers
13 views

How do you determine a vector of the form $(x,y,z)$ in 3D?

Knowing the angle between two vectors, the length of each vector being the same for both, and the $(x,y,z)$ form of one of the vectors with both vectors starting from the origin, how do you determine ...
0
votes
0answers
12 views

Show the inverse mapping $\Psi^{-1}:\Psi(\mathcal{O})\to\mathbb{R}^n$ is a smooth change of variables on the open subset of $\mathbb{R}^n$

Suppose that $\Psi:\mathcal{O}\to\mathbb{R}^n$ is a smooth change of variables on an open subset of $\mathbb{R}^n$. Show the inverse mapping $\Psi^{-1}:\Psi(\mathcal{O})\to\mathbb{R}^n$ is a smooth ...
0
votes
3answers
38 views

New very simple Golden Number Ratio PHI construction with Circle and Two Equal Segments of Circle Diameter. Is there prior art? Proofs?

Geogebra gives me the golden number PHI to fifteen decimal places for this simple construction illustrated below wherein the ratio of the blue line i to the red line h is PHI or 1.6180.... The golden ...
0
votes
0answers
29 views

Clarification of meaning of dx in an integral

I would like to have some clarification on the physical meaning of $dx$. I already know the following in the context of the area under the curve: $\lim_{\Delta x \rightarrow 0} \sum f(x) \Delta x ...
0
votes
1answer
25 views

More double integration help.

Im having trouble integrating this function. $$\int_0^{0.5}\int_0^{\pi/4} \frac{r\sin\theta \ln(1-r\cos\theta -r\sin\theta )}{r\cos\theta-\sqrt{(r\cos\theta)^2+(r\sin\theta)^2 }} r \,d\theta \, dr$$ ...
2
votes
2answers
59 views

How to solve $3(a+1)(b+1)=3^a \times 2^b$?

Hi I'm new to logarithms and not sure how to solve equations involving logarithms. I managed to find this equation to answer a problem solving question, however now I do not know how to solve the ...
-1
votes
0answers
9 views

Set language notation

It is given that Universal set { X:X is an integer such that X is equal or bigger than 1 but smaller or equal to K) M = { X:X is a prime number } N = { X: the unit digit of X is 7 } Given that ...
1
vote
0answers
12 views

What is the slowest growing function that is total but not primitive recursive?

For what I have in mind is the Ackermann-Buck function. If there isn't a slowest growing function do you have examples of other function slower growing than Ackermann-Buck's function?
0
votes
1answer
21 views

Parametric Problem

i have a question on parametric.. The question states A vector equation $(x,y) = (2,-1) + t(3,2)$. Write as a parametric equation. Show a table with x,y values. Sketch a picture of vector ...
3
votes
0answers
18 views

Eigenvectors of Generalized Sylvester Equation $AX+XB^\text{T}=\lambda CXD^\text{T}$

Ok here's what I mean with the Sylvester equation eigenvectors. The simplest case, where $C = D = I$, has already been solved in the literature (Matrix Calculus by W.H. Steeb). $$A X + X ...
0
votes
2answers
37 views

Prove that $n$ has at least four distinct prime factors

If $n$ is composite and $\phi{(n)} | (n - 1)$ then prove that $n$ has at least four distinct prime factors. Attempt: Since $n$ is not a prime, let's first take the case that $n$ is squarefree. ...
2
votes
1answer
18 views

Stuck trying to solve a PDE by method of characteristics

I've been trying to solve the inhomogeneous PDE IVP $u_t+c u_x = e^{2x}$; $u(x,0)=f(x)$, but got stuck. Would appreciate some help. Here's what I did by trying to use the method of characteristics: ...
1
vote
0answers
9 views

Moving a polynomial coefficient so it's irreducible

Given a polynomial over $Z$, $P(X)$, I'd like to show that for any $i$ in $N$, there is a whole number $r$ so that $P(x)+r*x^i$ is irreducible. I'd preferably would love a elementary argument so ...
0
votes
0answers
14 views

Math jobs in visualization?

What kinds of jobs can mathematics graduates (with an undergrad or grad degree) acquire in fields such as modelling, graphics, etc.? Links would be very helpful too. Thanks!
1
vote
1answer
27 views

Is there another name for a vector?

I am writing a program uses contains both vectors (direction and magnitude) and vectors (a matrix with one row/column) and my head is spinning. I could replace the latter kind of vector with ...
0
votes
0answers
15 views

Power series representation with Gamma function

This is taken from Stein and Shakarchi's Complex Analysis (Chapter 6, Exercise 4): Prove that if we take $$f(z) = \frac{1}{(1-z)^\alpha}$$ for $|z|<1$ (defined in terms of the principal branch ...
0
votes
0answers
14 views

Surface of Class $C^{\infty,1}$.

Please help me for two questions Is the boundary of a ball of class $C^{\infty, 1}$ ? For any Lipschitz surface S , is there smooth surfaces $S_n$ such that $S=\cup S_n$ ($S$ is a union of ...
0
votes
0answers
7 views

Derivative under supremum norm

I have the following property written in a book but I can't understand why this implication is true. I would be glad if anyone could help me let $A \in \mathbb{R}^N$. $$\frac{d}{dt} \nabla A(x,t) = ...
0
votes
1answer
37 views

Find the derivative of $\left(\frac{4x+2}{x-2}\right)^5$

Hey helpful people I have one more question I am stuck on! $$f(x) = \left(\frac{4x+2}{x-2}\right)^5$$ I know the answer is $$\frac{-50(4x+2)^4}{(x-2)^2(x-2)^4}$$ But I really can't figure out how ...
0
votes
0answers
12 views

A inequality concerns with the Legendre polynomial of n-th degree of $\cos\theta$

I am reading a paper, where the author concluded that $${\left| {{P_n}(\cos \theta )} \right|^2} \leqslant \frac{2} {{n\pi \sin \theta }},\,\,\forall \theta \in \left( {0,\pi } \right).$$ Here ...
0
votes
2answers
23 views

Graphing log with number in front of “log”

When I have something like $y = log_2(x)$ I know that I have to turn it into exponential form and get: $2^y = x$. Next, I make a table for $X,Y$ and choose about 5 values for $y$, typically $-1, 0, 1, ...
0
votes
1answer
10 views

If every vertex in a graph G has degree >=d, then show that G must contain a circuit of length at least d+1. (Applied Combinatorics, 1.5.8)

There are two questions that essentially asked the same thing. One is categorized as a repetition but I just don't feel the other one's answers are valid. Let $G$ be a graph of minimum degree ...
2
votes
1answer
9 views

Trouble understanding definition of an attracting set

From Wiggins' book, "Let $\cal{M}$ be a trapping region. Then $A=\cap_{t>0}\phi(t,\cal{M})$ is called an attracting set". Then he gives an example: $\dot{x}=x-x^3$ $\dot{y}=-y$, and claims that ...
1
vote
3answers
20 views

Determine the function to be injective when $f: A\to N$, where $A=\{1,4,3\}$ and $f(x)=x^{2}$.

I claimed the function is injective since the elements in domain maps uniquely to elements in the codomain when plug into $f(x)=x^{2}$ . which gives $\{1,16,9\}$ where $\{1,16,9\}$ is found in the ...
0
votes
1answer
38 views

What computer science skills would be beneficial for a maths major?

I'm currently entering my third year of my general science/maths undergrad degree (somewhat like a combined honours in life sciences and maths), and I'm a little wary of what I'm going to be doing ...
0
votes
1answer
11 views

how do i do this similarity question

olivia wants to find the the height of a building. She stands so that the top of her shadow hits the same spot as the top of the buildings shadow. Olivia is $1.6\mathrm m$ tall and her shadow is ...
0
votes
0answers
19 views

The bases for the set of all functions f:[0,1]→[0,1]

Let $X = [0, 1]^{[0,1]}$, the set of all functions $f : [0, 1] \rightarrow [0, 1]$. Given a subset $A \subseteq [0, 1]$, let $U_A = \{ f \in X : f(x) = 0 \forall x \in A \}$ . Show that $B := \{U_A : ...
0
votes
0answers
12 views

Function crossings

Determine the number of positive real roots of $a^x = x^a$, ($a>0$). For $a=2$ there are $2$. For other positive integers other than $a$ there's $1$. I suspect there's range of $a$ for which ...
3
votes
2answers
32 views

Probability of reaching a path

Let's say we have a coordinate plane with only the top right quadrant (i.e. $x \geq 0$ and $y \geq 0)$. If we want to reach an arbitrary point $(m,n)$, then there are $\binom{m+n}{n} = \binom{m+n}{m}$ ...
3
votes
1answer
15 views

Determinant of hankel matrix of hyperbolic functions, $a_n=\frac{n}{\sinh(\pi n)}$

I am trying to learn about the properties of Hankel matrices, and they appear to have nice closed forms for quite a large class of sequences. The class I am interested is when the elements $a_n$ are ...
1
vote
0answers
31 views

Have I found ALL the solutions to this diff eq & boundary conditions?

If we find a solution to a differential equation and its boundary conditions, how can we know if we have found ALL the solutions? For example, let g(x) be a smooth continuous function of x: (Eq 1) ...
-1
votes
0answers
13 views

PTM using Hastings-metropolis

[Compute the 4 × 4 PTM (pij ) under the T = 2 dynamics of Hastings–Metropolis][1]
0
votes
1answer
13 views

Decayment of Fourier coefficients of infinitely differentiable function

For a $C^n[-\pi,\pi]$ function $f$ we have that $|\hat{f}(k)|\in O(1/k^n)$. This implies that if $f$ is $C^\infty[-\pi,\pi]$ then its $k-th$ Fourier coefficient decays faster than any $1/k^n$, ...
0
votes
1answer
19 views

Ito formula when g(t,x) is an integral

Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ ...
0
votes
2answers
28 views

What will the value of an account be after 12 years if the account earns 4.91% a year and if someone invests $20,000?

Second National Bank offers an account that earns 4.91% per year, compounded continuously. If a person invests $20,000 in this account, what will be the value of the account at the end of 12 years? ...
2
votes
2answers
142 views

I don't understand this definition of the integers.

Definition: The set of integers is $\mathbb{Z}:=\frac{\mathbb{N} \times \mathbb{N}}{R}=\{[(m,n)]:(m,n)\in \mathbb{N}\times \mathbb{N}\}$. I understand this is the set of all the equivalence classes ...
0
votes
0answers
26 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
0
votes
0answers
10 views

Composition of symplectomorphisms not a symplectomorphism?

Given $R^{2n}=C^{n}$ with its standard symplectic form $\Omega$, and v an arbitrary vector, the individual transvections $\tau(p)=p+\Omega(v,p)v$ and $\sigma(p)=p+\Omega(iv,p)iv$ preserve the ...
1
vote
3answers
46 views

Logical limitations of Proofs by Contradiction

In general proofs by contradiction go as follows: Given an arbitrary hypothesis, $\ p \implies q$, we assume $\left(p\implies q\right) = T$, and then we show that by assuming the hypothesis to be ...
0
votes
0answers
25 views

Impossible Covering Properties of the Reals

I've been reading more about selection principles (covering properties) recently. Below is terminology. Let $X$ be the set of reals ($X = \mathbb{R}$) or a space homeomorphic to sets of reals, and ...
4
votes
2answers
42 views

For which values of real $\alpha, \beta$ does $\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$ converge?

I was wondering how does the series $$\sum_{n,m \ge 1} \frac{1}{n^{\alpha}+ m^{\beta}}$$ behave for real $\alpha, \beta > 0$. My approach: firstly I considered the case $\alpha = \beta > 2$. ...

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