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### Spherical trig, Research Triangle, and Mathematica

This post will look at the triangle behind North Carolina’s Research

Triangle using Mathematica’s geographic functions.

## Spherical triangles

A spherical triangle is a triangle drawn on the surface of a sphere. It

has three vertices, given by points on the sphere, and three sides. The

sides of the triangle are portions of great circles running between two

vertices. A great circle is a circle of maximum radius, a circle with

the same center as the sphere.

An interesting aspect of spherical geometry is that both the sides and

angles of a spherical triangle are angles. Because the sides of a

spherical triangle are arcs, they have angular measure, the angle formed

by connecting each vertex to the center of the sphere. The arc length of

a side is its angular measure times the radius of the sphere.

## Research triangle

Research Triangle is a (spherical!) triangle with v... mehr anzeigen

### Complex exponentials

Here’s something that comes up occasionally, a case where I have to tell

someone “It doesn’t work that way.” I’ll write it up here so next time I

can just send them a link instead of retyping my explanation.

## Rules for exponents

The rules for manipulating expressions with real numbers carry over to

complex numbers so often that it can be surprising when a rule doesn’t

carry over. For example, the rule

(

*b*^

*x*^)^

*y*^ =

*b*^

*xy*^

holds when

*b*is a positive real number and

*x*and

*y*are real

numbers, but doesn’t necessarily hold when

*x*or

*y*are complex. In

particular, if

*x*is complex,

(

*e*^

*x*^)^

*y*^ =

*e*^

*xy*^

does not hold in general, though it does hold when

*y*is an integer. If

it did hold, and

**this is where people get in**... mehr anzeigen

### Sine sum

Sam Walters posted something interesting on Twitter yesterday I hadn’t

seem before:

The sines of the positive integers have just the right balance ofIf for some reason your browser doesn’t render the embedded tweet, he

pluses and minuses to keep their sum in a fixed interval. (Not hard to

show.)

\#math

pic.twitter.com/RxeoWg6bhn

— Sam Walters

(@SamuelGWalters) November 29,

2018

points out that

{.alignc... mehr anzeigen

### Rényi Differential Privacy

Differential privacy, specifically

**ε-differential privacy**, gives

strong privacy guarantees, but it can be overly cautious by focusing on

worst-case scenarios. The generalization

**(ε, δ)-differential privacy**

was introduced to make ε-differential privacy more flexible.

**Rényi differential privacy**(RDP) is a new generalization

of ε-differential privacy by Ilya Mironov that is comparable to the (ε,

δ) version but has several advantages. For instance, RDP is easier to

interpret than (ε, δ)-DP and composes more simply.

## Rényi divergence

My previous post discussed Rényi

entropy.

**Rényi**

divergenceis to Rényi entropy what Kullback-Leibler divergence is to

divergence

Shannon... mehr anzeigen

### Rényi entropy

{.alignnone

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The most common way of measuring information is Shannon entropy, but

there are others. Rényi entropy, developed by Hungarian mathematician

Alfréd Rényi, generalizes Shannon entropy and includes other entropy

measures as special cases.

## Rényi entropy of order α

If a discrete random variable

*X*has

*n*possible values, where

the

*i*th outcome has probability

*p*~

*i*~, then the Rényi entropy of

order α is defined to be

{.aligncenter

.size-medium width="234" height="54"}

for 0 ≤ α ≤ ∞. In the case α = 1 or ∞ this expression mean... mehr anzeigen

#### The Triple Jeopardy of a Chinese Math Prodigy

To continue, please click the box below to let us know you're not a robot.

HN Discussion: https://news.ycombinator.com/item?id=18499712

Posted by vthallam (karma: 1850)

*Post stats: Points: 123 - Comments: 57 - 2018-11-21T02:16:21Z*

\#HackerNews #chinese #jeopardy #math #prodigy #the #triple

**Article content:**

To continue, please click the box below to let us know youʼre not a robot.

HackerNewsBot debug: Calculated post rank: 101 - Loop: 464 - Rank min: 100 - Author rank: 39

#### Bloomberg - Are you a robot?

^{www.bloomberg.com}

### Prime denominators and nines complement

Let

*p*be a prime. If the repeating decimal for the fraction

*a*/

*p*

has even period, the the second half of the decimals are the 9’s

complement of the first half. This is known as Midy’s theorem.

For a small example, take

1/7 = 0.142857142857…

and notice that 142 + 857 = 999. That is, 8, 5, and 7 are the nine’s

complements of 1, 4, and 2 respectively.

For a larger example, we can use Mathematica to look at the decimal

expansion of 6/47:

```
In: N[6/47, 60]
Out: 0.127659574468085106382978723404255319148936170212765957446809
```

and we can confirm

```
12765957446808510638297 +
87234042553191489361702 =
99999999999999999999999
```

Let’s do another example with 6/43:

```
In: N[6/43, 50]
Out: 0.13953488372093023255813953488372
```

... mehr anzeigen### Big data and privacy

{.size-medium

width="400" height="267"}

How does big data impact privacy? Which is a bigger risk to your

privacy, being part of a little database or a big database?

## Rows vs Columns

People commonly speak of big data in terms of

**volume**—the “four v’s”

of big data being volume, variety, velocity, and veracity—but what we’re

concerned with here might better be called “area.” We’ll think of our

data being in one big table. If there are repeated measures on an

individual, think of them as more columns in a denormalized database

table.

In what sense is the data big: is it wide or long? That is, if we think

of the data as a table with rows for individuals and columns for

different fields of information on individuals, are there a lot of ro... mehr anzeigen

### What is proof-of-work?

The idea of proof of work was first explained in a paper Cynthia Dwork

and Moni Naor [1], though the term “proof of work” came later [2].

It was first proposed as a way to deter spam, but it’s better known

these days through its association with cryptocurrency.

If it cost more to send email, even a fraction of a cent per message,

that could be enough to deter spammers. So suppose you want to charge

anyone \$0.005 to send you an email message. You don’t actually want to

collect they money, you just want proof that they’d be

*willing*to

spend something to email you. You’re not even trying to block robots,

you just want to block

*cheap*robots.

So instead of asking for a micropayment, you could ask the sender to

solve a puzzle, something that would require around \$0.005 worth of

computing resources. If you’re still getting too much spam, you could

increase your... mehr anzeigen

### Graffiti irracional

Lo hallé en las calles de la ciudad de México.

#matemáticas #math #streetart #México #graffiti

#### Making floating point math highly efficient for AI hardware

Radical changes to floating point math make it as much as 16 percent more efficient than int8/32 math, yet still highly accurate for CNNs.

Article word count: 2917

HN Discussion: https://news.ycombinator.com/item?id=18418667

Posted by probdist (karma: 285)

*Post stats: Points: 85 - Comments: 16 - 2018-11-09T22:44:17Z*

\#HackerNews #efficient #floating #for #hardware #highly #... mehr anzeigen

### Logic and applications Twitter account

I stopped posting to the @FormalFact Twitter account last July, but I

didn’t deactivate the account. Now I’m going to restart it.

Unlike my other Twitter

accounts, I don’t plan to

have a regular posting schedule. I may not post often. We’ll see how it

goes.

{.aligncenter

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I’ve changed the account name from @FormalFact to

@LogicPractice. The “formal” part

of the original name referred to formal theorem proving, the initial

focus of the account. The new name reflects a focus on logic more

generally, and practical applications of logic that are less laborious

than formal theorem proving.

{width="1"

height="1"}

http://feedproxy.google.com/~r/TheEndeavour/~3/DLKkwXevg2E/

#johndcook #Math #Logic

#### Mystery Math Whiz and Novelist Advance Permutation Problem

A new proof from the Australian science fiction writer Greg Egan and a 2011 proof anonymously posted online are now being hailed as significant advances on a puzzle mathematicians have been studying…

Article word count: 1633

HN Discussion: https://news.ycombinator.com/item?id=18389080

Posted by beefman (karma: 3205)

*Post stats: Points: 142 - Comments: 37 - 2018-11-06T07:31:56Z*

\#HackerNews #advance #and #math #... mehr anzeigen

### Continued fraction cryptography

Every rational number can be expanded into a continued fraction with

positive integer coefficients. And the process can be reversed: given a

sequence of positive integers, you can make them the coefficients in a

continued fraction and reduce it to a simple fraction.

In 1954, Arthur Porges published a one-page article pointing out that

continued fractions could be used to create a cipher. To encrypt your

cleartext, convert it to a list of integers, use them as continued

fraction coefficients, and report the resulting fraction. To decrypt,

expand the fraction into a continued fraction and convert the

coefficients back to text.

We can implement this in Mathematica as follows:

```
encode[text_] := FromContinuedFraction[ ToCharacterCode[ text ]]
decode[frac_] := FromCharacterCode[ ContinuedFraction[ frac ]]
```

So, for example, suppose we want to encrypt “adobe.” If... mehr anzeigen

### Earth mover distance and t-closeness

{.alignnone

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There’s an old saying that if you want to hide a tree, put it in a

forest. An analogous principle in privacy is that a record preserves

privacy if it’s like a lot of other records.

*k*-anonymity

The idea of

**is that every database record appears at**

*k*-anonymityleast

*k*times. If you have a lot of records and few fields, your value

of

*k*could be high. But as you get more fields, it becomes more likely

that a combination of fields is unique. If

*k*= 1, then

*k*-anonymity

offers no anonymity.

Another problem with

*k*-anonymity is that it doesn’t offer group

privacy. A database could be

*k*-anonymous but reveal information about

a group... mehr anzeigen

### Add or remove one blue dot to make this statement true.

*(I'm sure some of you have seen this one before so please hold back for half an hour or so to make the newbies squirm a bit.)*

#maths #puzzle #arithmetic #math

### Integration by long division

Since integration is the inverse of differentiation, you can think of

integration as “dividing” by

*d*.

J. P. Ballantine [1]shows that you can formally divide by

*d*and get

the correct integral. For example, he arrives at

{.aligncenter

.size-medium width="336" height="40"}

using long division!

{.aligncenter

.size-medium width="476" height="198"}

[1]J. P. Ballantine. Integration by Long Division. The American

Mathematical Monthly, Vol. 58, No. 2 (Feb., 1951), pp. 104-105

{width="1"

height="1"}

http://feedproxy.google.com/~r/TheEndeavour/~3/0jHwb1W1Ab8/

#johndcook #Math #Integration

### Modal and temporal logic for computer security

{.alignnone

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In the previous

post,

I mentioned that modal logic has a lot of interpretations and a lot of

axiom systems. It can also have a lot of operators. This post will look

at

**Security Logic**, a modal logic for security applications based on

a seminal paper by Glasgow

*et al*[1]. It illustrates how modal and

temporal logic can be applied to computer security, and it illustrates

how a logic system can have a large number of operators and axioms.

## Knowledge axioms

Security Logic starts with operators

*K*~

*i*~ that extend the box

operator □. For a proposition

*p*,

*K*~

*i*~

*p*... mehr anzeigen

### Typesetting modal logic

**Modal logic**extends propositional logic with two new operators, □

(“box”) and ◇ (“diamond”). There are many interpretations of these two

symbols, the most common being necessity and possibility respectively.

That is, □

*p*means the proposition

*p*is necessary, and ◇

*p*means

that

*p*is possible. Another interpretation is using the symbols to

represent things a person knows to be true and things that may be true

as far as that person knows.

There are also many axiom systems for inference concerning these

operators. For example, some axiom systems include the rule

{.aligncenter .size-medium

width="87" height="16"}

and some do not. If you interpret □ as necessity, this axiom says

whatever is necessary is necessari... mehr anzeigen

### Fermat’s factoring trick and cryptography

Many encryption algorithms rely on the difficulty of factoring a large

number

*n*. If you want to make

*n*hard to factor, you want it to have

only two factors. Otherwise, the more factors

*n*has, the smaller the

smallest factor must be.

So if you want

*n*to be the product of two large primes,

*p*and

*q*,

you want to pick these primes to be roughly the same size so that the

smaller factor is as large as possible. If you’re limited on the size

of

*n*, then you want

*p*and

*q*to be roughly of size √

*n*. But not

too close to √

*n*. You may see in a description of a cryptographic

algorithm, such as RSA, “Pick two large primes

*p*and

*q*, but not too

close together, …” Why is that?

The answer goes back to Fermat (1607–1665). His factoring trick is to

start with... mehr anzeigen

### Excessive precision

“There is no point in being precise when you don’t know what you’re

talking about.” — John Tukey

It’s a familiar trope in science fiction that the smartest character

will answer questions with excess precision. On Star Trek, Scottie might

give a number to one significant figure and Spock will correct him

giving the same result to four significant figures.

The trope works on two levels. The innumerate viewer will think “Wow,

the smart guy is really smart! He knows a lot more than the other guy.”

The mathematically savvy viewer will see it as a kind of joke,

intentional or unintentional. In the Star Trek series, I assume the

writers are winking at the audience when precision is excessive. If

Scottie says the ship will blow up in 20 seconds, there’s no point in

Spock replying 19.81 seconds, because it would take more than 0.19

seconds for him to state his correction.... mehr anzeigen

### Integer odds and prime numbers

For every integer

*m*> 1, it’s possible to choose

*N*so that the

proportion of primes in the sequence 1, 2, 3, …

*N*is 1/

*m*. To put it

another way, you can make the odds against one of the first

*N*natural

numbers being prime any integer value you’d like [1].

For example, suppose you wanted to find

*N*so that 1/7 of the first

*N*

positive integers are prime. Then the following Python code shows you

could pick

*N*= 3059.

```
from sympy import primepi
m = 7
N = 2\*m
while N / primepi(N) != m:
N += m
print(N)
```

## Related posts

... mehr anzeigen### Comparing trig functions and Jacobi functions

My previous

post

looked at Jacobi functions from a reference perspective: given a Jacobi

function defined one way, how do I relate that to the same function

defined another way, and how would you compute it?

This post explores the analogy between trigonometric functions and

Jacobi elliptic functions.

## Related basic Jacobi functions to trig functions

In the previous post we mentioned a connection between the argument

*u*

of a Jacobi function and the amplitude φ:

{.aligncenter

.size-medium width="173" height="45"}

We can use this to define the functions

**sn**and

**cn**. Leaving the

dependence on

*m*implicit, we have

... mehr anzeigen

### Clearing up the confusion around Jacobi functions

The

**Jacobi elliptic functions**sn and cn are analogous to the

trigonometric functions sine and cosine. The come up in applications

such as nonlinear oscillations and conformal mapping. Unfortunately

there are multiple conventions for defining these functions. The purpose

of this post is to clear up the confusion around these different

conventions.

{.aligncenter .size-medium

width="450" height="234"}

The image above is a plot of the function sn [1].

## Modulus, parameter, and modular angle

Jacobi functions take two inputs. We typically think of a Jacobi

function as being a function of its first input, the second input being

fixed. This second input is a “dial” you can turn that changes their

behavior.

There are several ways to specify this di... mehr anzeigen

### This dizzying labyrinth will host next year’s party for math’s ‘Nobel’ prize

#dizzying #host #labyrinth #math #next #nobel #party #prize #will #year

#### This dizzying labyrinth will host next year’s party for math’s ‘Nobel’ prize

New maze mimics mathematical patterns in nature and the ancient world

^{www.sciencemag.org}

### Prime interruption

Suppose you have a number that you believe to be prime. You start

reading your number aloud, and someone interrupts “Stop right there! No

prime starts with the digits you’ve read so far.”

It turns out the person interrupting you shouldn’t be so sure. There are

no restrictions on the digits a prime number can begin with. (Ending

digits are another matter. No prime ends in 0, for example.) Said

another way, given any sequence of digits, it’s possible to add more

digits to the end and make a prime.

[1]For example, take today’s date in ISO format: 20181008. Obviously not a

prime. Can we find digits to add to make it into a prime? Yes, we can

add 03 on to the end because 2018100803 is prime.

What about my work phone number: 83242286846? Yes, just add a 9 on the

end because 832422868469 is prime.

This works in any base you’d like. For example, the hexadecimal number

CAFEB... mehr anzeigen