spraci.info

An

square of an

edge if a rook can legally move from one to the other. That is, two

nodes are connected if they are either in the same row or in the same

column.

A

prime power and congruent to 1 mod 4. Label each node with an element of

the finite group

edge if the difference between their labels is a quadratic residue. That

is, for nodes labeled

We’ll look at a graph which is both a rook graph and a Paley graph: the

3 by 3 Paley... Show more...

Programming languages differ in the names they use for inverse trig

functions, and in some cases differ in their return values for the same

inputs.

If sin(θ) =

Depends on how you define the inverse sine. If -1 ≤

are infinitely many θ’s whose sine is

The conventional choice is for inverse sine to return a value of θ in

the closed interval

[-π/2, π/2].

Inverse tangent uses the same choice. However, the end points aren’t

possible outputs, so inverse tangent typically returns a value in the

open interval

(-π/2, π/2).

For inverse cosine, the usual choice is to return values in the closed

interval

[0, π].

Aside from how to

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I recently had to work with the function

The semicolon between

of

This function is an example of the coming full

circle

theme I’ve written about several times. Here’s how a novice, a

journeyman, and an expert might approach studying our function.

- Novice: arctan(
*k*tan(*x*) ) =*kx.* - Journeyman: You can’t do that!
- Expert: arctan(
*k*tan(*x*) ) ≈*kx*for small*x*.

and so someone might pull the

Suppose you want to compute the natural logarithms of every floating

point number, correctly truncated to a floating point result. Here by

floating point number we mean an IEEE standard 64-bit float, what C

calls a

We’ll get to the hardest logarithm shortly, but we’ll first start with a

warm up problem. Suppose you needed to compute the first digit of a

number

-10^-100^. If ε is positive, you should return 2. If ε is negative, you

should return 1. You know

return the first digit correctly, you need to compute

places.

In this post we’re truncating floating point numbers, i.e. rounding

down, but similar considerations apply when rounding to... Show more...

Three physicists stumbled across an unexpected relationship between some of the most ubiquitous objects in math.

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The previous

post

looked at an example of strange floating point behavior taking from book

End of Error. This post looks at another

example.

This example, by Siegfried Rump, asks us to evaluate

333.75

5.5

at

Here we evaluate Rump’s example in single, double, and quadruple

precision.

`#include <iostream> `

#include <quadmath.h>

using namespace std;

template <typename T> T f(T x, T y) {

T x2 = x\*x;

T y2 = y\*y;

T y4 = y2\*y2;

T y6 = y2\*y4;

T

... Show more...
ivan zlax wrote the following post Tue, 12 Nov 2019 03:16:13 +0300

@ was born 30 years ago in USSR.

At NYU, he studied computer science at The Courant Institute of Mathematical Sciences, where he met the three friends with whom he founded DIASPORA\*, a social networking service, in 2010. The project... Show more...

At NYU, he studied computer science at The Courant Institute of Mathematical Sciences, where he met the three friends with whom he founded DIASPORA\*, a social networking service, in 2010. The project was conceived after the founders had attended a lecture by Columbia Law School professor and free software activist Eben Moglen in February 2010 about the threat to privacy posed by commercial Internet services. According to Moglen, Zhitomirskiy was "immensely talented" and "the most idealistic of the group... He had a choice between graduate school and this project, and he chose to do the project because he wanted to do something with his time that would make freedom"... Show more...

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https://m.phys.org/news/2019-11-longstanding-conjecture-area-negatively-spaces.html

"There are many possible shapes, and nature picks the one using the least amount of energy," Spruck says. It follows that the shape that encloses a given area with the smallest possible perimeter is the circle—or, venturing into three dimensions, the sphere.... Show more...

....But things get trickier when you want to generalize this idea beyond circles and spheres to more complicated situations.

Back in 1926, the conjecture was proved for two dimensions. In 1984, it was proved for four dimensions, and for three in 1992. "Then we did all the other dimensions,"

The challenge, he explains, was that while the conjecture was relatively straightforward—if you're handy with math—in what's known as Euclidean space, things got more complicated in, say, negatively curved space.

Neg

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Seven Bridges - A Stroll & Write Board Game

Come visit the historic European city of Königsberg and cross its famous seven bridges!#russian #lang ru #Königsberg #math #boardgame #crowdfunding #history

Robert Bosch has a new book coming out next

week titled OPT ART: From Mathematical Optimization to Visual Design.

This post will look at just one example from the book: creating images

of Frankenstein’s monster [1]using dominoes.

{.aligncenter

.size-medium width="623" height="913"}

The book includes two images, a low-resolution image made from 3 sets of

dominoes and a high resolution image made from 48 sets. These are double

nine dominoes rather than the more common double six dominoes because

the former allow a more symmetric arrangement of dots. There are 55

dominoes in a double nine set. (Explanation and generalization

here.)

... Show more...

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A couple weeks ago I wrote about how De Bruijn

sequences

can be used to attack locks where there is no “enter” key, i.e. the lock

will open once the right symbols have been entered.

Here’s a variation on this theme: what about locks that let you press

more than one button at a time?

[1]{.aligncenter

.size-medium width="400" height="467"}

You could just treat this as if it were a keypad with more buttons. For

example, suppose you can push either one or two buttons at a time on the

lock pictured above. Then you could treat this as a lock with 15

buttons: the five actual buttons and 10 virtual buttons corresponding to

the ten ways you could choose 2 buttons out of 5 to press... Show more...

Adding up an array of numbers is simple: just loop over the array,

adding each element to an accumulator.

Usually that’s good enough, but sometimes you need more precision,

either because your application need a high-precision answer, or because

your problem is poorly conditioned and you need a moderate-precision

answer [1].

There are many algorithms for summing an array, which may be surprising

if you haven’t thought about this before. This post will look at Kahan’s

algorithm, one of the first non-obvious algorithms for summing a list of

floating point numbers.

We will sum a list of numbers a couple ways using C’s

32-bit integer. Some might object that this is artificial. Does anyone

use

all our precision problems? No and no.

In fact,... Show more...

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Any positive number can be found at the beginning of a factorial. That

is, for every positive positive integer

such that the leading digits of

There’s a tradition in math to use the current year when you need an

arbitrary numbers; you’ll see this in competition problems and

recreational math articles. So let’s demonstrate the opening statement

with

3177! = 2019…000

The factorial of 3177 is a number with 9749 digits, the first of which

are 2019, and the last 793 of which are zeros.

The solution

and there are infinitely more.

Not only does every number appear at the beginning of a factorial, it

appears at the beginning of infinitely many factorials.... Show more...

Are there infinitely many positive integers

David P. Bellamy and Felix Lazebnik [1]asked in 1998 whether there

were infinitely man solutions to |tan(

yes. But at least

as recently as 2014 the problem at the top of the page was still open

[2].

It seems that tan(

five instances for

1\

260515\

37362253\

122925461\

534483448

In fact, there are no more solutions if you search up to over two

billion as the following C code shows.

`#include <math.h> `

#include <stdio.h>

#include <limits.h>

int main() {

... Show more...
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There are variations on Newton’s root finding method that use higher

derivatives and converge faster. Alston Householder developed a sequence

of such methods of the form

{.aligncenter

.size-medium width="243" height="78"}

where the superscripts in parentheses indicate derivatives. When

2, Householder’s method reduces to Newton’s method.

Once Newton’s method is close enough to a root, the error is squared at

each iteration. Said another way, the number of correct decimal places

roughly doubles. With the Householder method of order

correct decimal places roughly increases by a factor of

The Householder method

I wanted to stress test the

π to 10,000 digits a couple different ways.

First I ran

which calculates π as 4 arctan(1). This took 2 minutes and 38 seconds.

I imagine

so smaller arguments should converge faster. So next I used a formula

due to John Machin (1680–1752).

This took 52 seconds.

Both results were correct to 9,998 decimal places.

When you set the

calculations out to

deliver

This quirky little calculator is growing on me. For one thin... Show more...

My attempt at defining randomness:

https://gitlab.com/drummyfish/my_writings/blob/master/randomness.md

e.g. the most random bytes I've found are:

`00100101 `

00101001

01001001

01001010

01010010

01011011

01101011

01101101

10010010

10010100

10100100

10101101

10110101

10110110

11010110

11011010

#randomness #probability #math #python #fun

Last

week

I wrote about a way to

way to

Start with an alphabet of

cycles of that contain every sequence of

possible. Since there are

and every one corresponds to some starting position in the De Bruijn

cycle, an element of

symbols. In fact, the elements of

symbols.

It’s not obvious that

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