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Beiträge die mit math getaggt sind

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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

I just accidentally did a perfect orange peel.

See this arXiv paper for a mathematical explanation:

#geometry #math #fruit #science #food


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

— Sam Walters
(@SamuelGWalters) November 29,
If for some reason your browser doesn’t render the embedded tweet, he
points out that

Bild/Foto{.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
. Rényi
is to Rényi entropy what Kullback-Leibler divergence is to
Shannon... mehr anzeigen

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Rényi entropy

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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 ith outcome has probability p~i~, then the Rényi entropy of
order α is defined to be

.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:
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


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 = 

Let’s do another example with 6/43:
In:  N[6/43, 50] 
    Out: 0.13953488372093023255813953488372
... mehr anzeigen


Big data and privacy

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

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:
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
, I don’t plan to
have a regular posting schedule. I may not post often. We’ll see how it

.size-medium width="200" height="200"}

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.

#johndcook #Math #Logic
Logic and applications Twitter account

John D. Cook: Logic and applications Twitter account


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:
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

.size-medium width="440" height="220"}

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.


The idea of k-anonymity is that every database record appears at
least 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


The red line has length 2 and is perpendicular to the base.

What is the yellow area?
#maths #geometry #puzzle #math


The area of the square is 4.

What is the area of the rectangle?
#maths #puzzle #geometry #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

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

using long division!

.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

#johndcook #Math #Integration
Integration by long division

John D. Cook: Integration by long division


Modal and temporal logic for computer security

.size-medium width="440" height="220"}

In the previous
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

Bild/Foto{.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
... 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

.....said the "professor". yikes. #math #fail #funny #lol "ed-u-muh-cay-shun" #education

.....said the "professor". yikes. #math #fail #funny #lol "ed-u-muh-cay-shun" #education


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 

Related posts

... mehr anzeigen

Hi all, I'm #newhere. Love #math, #science, #logic, #philosophy, and healthy conversation about #politics.


Comparing trig functions and Jacobi functions

My previous

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:

.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

Developing low pass filter for the Geonkick synthesizer. #synthesizer #music #math


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

Bild/Foto{.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

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

Hey everyone, I’m #newhere. I’m interested in #math and #scala.


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