### Buscar

## Objetos taggeado con: math

### Passwords and power laws

According to this

paper,

the empirical distribution of real passwords follows a power law. In the

authors’ terms, a Zipf-like distribution. The frequency of the

*r*th

most common password is proportional to something like 1/

*r*. More

precisely,

*f*~

*r*~ =

*C*

*r*^–

*s*^

where

*s*is on the order of 1. The value of

*s*that best fit the data

depended on the set of passwords, but their estimates of

*s*varied from

0.46 to 0.91.

This means that the most common passwords are very common and easy to

guess.

If passwords come from an alphabet of size

*A*and have length

*n*, then

there are

*A*^

*n*^ possibilities. For example, if a password has length

10 and consists of uppercase and lowercase English... ver más

### Riemann hypothesis, the fine structure constant, and the Todd function

This morning Sir Michael Atiyah gave a presentation at the Heidelberg

Laureate Forum with a claimed proof of the Riemann hypothesis. The

Reimann hypothesis (RH) is the most famous open problem in mathematics,

and yet Atiyah claims to have a simple proof.

{.alignnone

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## Simple proofs of famous conjectures

If anyone else claimed a simple proof of RH they’d immediately be

dismissed as a crank. In fact, many people have sent me simple proofs of

RH just in the last few days in response to my blog

post,

and I imagine they’re all cranks. But Atiyah is not a crank. He won the

Fields Medal in 1966 and the Abel prize... ver más

### Three applications of Euler’s theorem

**Fermat’s little theorem**says that if

*p*is a prime and

*a*is not a

multiple of

*p*, then

*a*^

*p*-1^ = 1 (mod

*p*).

**Euler’s generalization**of Fermat’s little theorem says that if

*a*

is relatively prime to

*m*, then

*a*^φ(

*m*)^ = 1 (mod

*p*)

where φ(

*m*) is Euler’s so-called

**totient**function. This function

counts the number of positive integers less than

*m*and relatively

prime to

*m*. For a prime number

*p*, φ(

*p*) =

*p*-1, and to Euler’s

theorem generalizes Fermat’s theorem.

Euler’s totient function is

**multiplicative**, that is, if

*a*and

*b*

are relatively prime, then φ(

*ab*) = φ(

*a*) φ(

*b*). We will n... ver más

https://github.com/monim67/perspective-vue

### News regarding ABC conjecture and Riemann Hypothesis

There have been a couple news stories regarding proofs of major

theorems. First, an update on Shinichi Mochizuki’s proof of the

**, then an announcement that Sir Michael Atiyah claims to**

conjecture

*abc*conjecture

have proven the

**Riemann hypothesis**.

## Shinichi Mochizuki’s proof of the *abc* conjecture

Quanta

Magazine

has a story today saying that two mathematicians have concluded that

Shinichi Mochizuki’s proof of the ABC conjecture is flawed beyond

repair. The story rightly refers to a “clash of Titans” because Shinichi

Mochizuki and his two critics Peter Scholze and Jakob Stix are all three

highly respected.

I first wrote about the

*abc*conjecture when it came out in

... ver más

### Footnote on fifth root trick

Numberphile has a nice video on the fifth root trick: someone raises a

two-digit number to the 5th power, reads the number aloud, and you tell

them immediately what the number was.

Here’s the trick in a nutshell. For any number

*n*,

*n*^5^ ends in the

same last digit as

*n*. You could prove that by brute force or by

Euler’s theorem. So when someone tells you

*n*^5^, you immediately know

the first digit. Now you need to find the first digit, and you can do

that by learning, approximately, the powers (10k)^5^ for

*i*= 1, 2, 3,

…, 9. Then you can determine the first digit by the range.

Here’s where the video is a little vague. It says that you don’t need to

know the powers of 10

*k*very accurately. This is true, but just how

accurately

*do*you need to know the ranges?

If the two-digit number is a power of 10, you’ll recogniz... ver más

### An empirical look at the Goldbach conjecture

The Goldbach conjecture says that every even number bigger than 2 is the

sum of two primes. I imagine he tried out his idea on numbers up to a

certain point and guessed that he could keep going. He lived in the 18th

century, so he would have done all his calculation by hand. What might

he have done if he could have written a Python program?

Let’s start with a list of primes, say the first 100 primes. The 100th

prime is

*p*= 541. If an even number less than

*p*is the sum of two

primes, it’s the sum of two primes less than

*p*. So by looking at the

sums of pairs of primes less than

*p*, we’ll know whether the Goldbach

conjecture is true for numbers less than

*p*. And while we’re at it, we

could keep track not just of

*whether*a number is the sum of two

primes, but also

*how many ways*it is a sum of two primes.

`from sympy`

... ver másDoing stuff with data is hard...I love how the data points are exactly the same for each cell 😀 Hat tip to @🛫 Brad Koehn 🛬 and @rss_xkcd@pod.afox.me #math #science #geek #humor https://xkcd.com/2048/

### Group statistics

I just ran across

Mathematica. That would have made some of my earlier

posts

easier to write had I used this instead of writing my own code.

Here’s something I find interesting. For each

*n*, look at the groups of

order at most

*n*and count how many are Abelian versus non-Abelian. At

first there are more Abelian groups, but the non-Abelian groups soon

become more numerous. Also, the number of Abelian groups grows smoothly,

while the number of non-Abelian groups has big jumps, particularly at

powers of 2.

{.aligncenter

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Here’s the Mathematica code:

`fgc = FoldList[Plus, 0, Table[`

... ver más#### Researchers Discover a Pattern to the Seemingly Random Distribution of Prime Numbers

The pattern has a surprising similarity to the one seen in atom distribution in crystals.

^{motherboard.vice.com}

### The permutation symbol

Sometimes simple notation can make a big difference. One example of this

is the

**Kronecker delta**function δ~

*ij*~ which is defined to be 1

if

*i*=

*j*and zero otherwise. Because branching logic is built into

the symbol, it can keep branching logic outside of your calculation.

That is, you don’t have to write “if … else …” in when doing your

calculation. You let the symbol handle it.

The

**permutation symbol**ε~

*ijk*~ is similar. It has some branching

logic built into its definition, which keeps branching out of your

calculation, letting you handle things more uniformly. In other words,

the symbol encapsulates some complexity, keeping it out of your

calculation. This is analogous to how you might reduce the complexity of

a computer program.

[1]

## Definition

The permutation symbol, sometimes called the... ver más

### A strange sort of product rule

Let

*u*be a real-valued function of

*n*variables, and let

*v*be a

vector-valued function of

*n*variables, a function from

*n*variables

to a vector of size

*n*. Then we have the following product rule:

D(

*uv*) =

*v*D

*u*+

*u*D

*v.*

It looks strange that the first term on the right isn’t D

*u*

*v*.

The function

*uv*is a function from

*n*dimensions to

*n*dimensions,

so it’s derivative must be an

*n*by

*n*matrix. So the two terms on the

right must be

*n*by

*n*matrices, and they are. But D

*u*

*v*is a 1 by

1 matrix, so it would not make sense on the right side.

Here’s why the product rule above looks strange: the multiplication by

*u*is a

**scalar**product, not a matrix product. Sometimes you can

think of re... ver más

#### China now the most prolific contributor to physical sciences, engineering, math

Your usage has been flagged as a violation of our terms of service. For inquiries related to this message please contact support. For sales inquiries, please visit…

Article word count: 52

HN Discussion: https://news.ycombinator.com/item?id=17974273

Posted by petethomas (karma: 23316)

*Post stats: Points: 145 - Comments: 103 - 2018-09-12T22:59:19Z*

\#HackerNews #china #contributor #engineering #... ver más

### Duplicates in the classification of finite simple groups

The previous

post

defined the groups PSL(

*n*,

*q*) where

*n*is a positive integer and

*q*

is a prime power. These are finite simple groups for

*n*≥ 2 except for

PSL(2, 2) and PSL(2, 3).

## Duplicates among PSL(*n*, *q*)

There are a couple instances where different values of

*n*and

*q*lead

to isomorphic groups: PSL(2, 4) and PSL(2, 5) are isomorphic, and PSL(2,

7) and PSL(3, 2) are isomorphic. These are the only instances [1].

With the exceptions stated above, distinct values of

*n*and

*q*lead to

distinct groups. Is it possible for different choices of

*n*and

*q*to

lead to groups of the same size, even though the groups are not

isomorphic to each other? Yes, PSL(3, 4) and PSL(4, 2) both... ver más

Der Spruch auf Deutsch:

"Ich hätte lieber Fragen, die nicht beantwortet werden können, als Antworten, die nicht befragt werden können."

Na ja, befragt... wer hat ne bessere Formulierung?

Donna Schwieder wrote the following post Sat, 08 Sep 2018 18:45:52 +0200

### “I would rather have questions that can't be answered than answers that can't be questioned.”~Richard Feynman

\#questions #answers #science #math #FreeSpeech

view full size

### How a Kalman filter works, in pictures

Surprisingly few software engineers and scientists seem to know about it, [even if] it is such a general and powerful tool for combining information in the presence of uncertainty. (...)

You can use a Kalman filter in any place where you have uncertain information about some dynamic system, and you can make an educated guess about what the system is going to do next. Even if messy reality comes along and interferes with the clean motion you guessed about, the Kalman filter will often do a very good job of figuring out what actually happened. And it can take advantage of correlations between crazy phenomena that you maybe wouldn’t have thought to exploit!

Kalman filters are ideal for systems which are continuously changing. They have the advantage that they are light on memory (they don’t need to keep any history other than the previous state), and they are very fast, making them well suited for real time problems and embedded systems. (...)

... ver más

### “I would rather have questions that can't be answered than answers that can't be questioned.”~Richard Feynman

#questions #answers #science #math #FreeSpeech

### Accuracy, precision, and recall

I posted the definitions of accuracy, precision, and recall on

@BasicStatistics this afternoon.

Accuracy = (TP+TN)/(TP+FP+FN+TN)\There seems to be no end of related definitions, and multiple names for

Precision = TP/(TP+FP)\

Recall = TP/(TP+FN)

where

T = true\

F = false\

P = positive\

N = negative

— Basic Statistics (@BasicStatistics) September 6,

2018

the same definitions.

**Precision**is also known as

**positive predictive value**(

**PPV**)

and

**recall**is also known as

**sensitivity**,

**hit rate**,

and

**true positive rate**(

**TPR**).

Not mentioned in the tweet above

are

**specificity**(a.k.a.

**selectivity**... ver más

### Simplest exponential sum

Today‘s exponential

sum curve is simply a triangle.

{.aligncenter

.size-medium width="400"}

But yesterday‘s curve

was more complex

{.aligncenter

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and tomorrow‘s curve

will be more complex as well.

{.aligncenter

.size-medium width="400"}

Why is today’s curve... ver más

### Three ways to sum a divergent series

There’s no direct way to define the sum of an infinite number of terms.

Addition takes two arguments, and you can apply the definition

repeatedly to define the sum of any finite number of terms. But an

infinite sum depends on a theory of convergence. Without a definition of

convergence, you have no way to define the value of an infinite sum. And

with different definitions of convergence, you can get different values.

In this post I’ll review two ways of assigning a meaning to divergent

series that I’ve written about before, then mention a third way.

## Asymptotic series

A few months ago I wrote about an asymptotic series

solution

to the differential equation

You end u... ver más

### The right level of abstraction

Mark Dominus wrote a blog post yesterday entitled Why I never finish my

Haskell programs (part 1 of

∞). In a nutshell,

there’s always another layer of abstraction. “Instead of just adding

lists of numbers, I can do addition-like operations on list-like

containers of number-like things!”

Is this a waste of time? It depends entirely on context.

I can think of two reasons to pursue high levels of abstraction. One is

**reuse**. You have multiple instances of things that you want to handle

simultaneously. The other reason is

**clarity**. Sometimes abstraction

makes things simpler, even if you only have one instance of your

abstraction. Dijkstra had the latter in mind when he said

The purpose of abstraction is not to be vague, but to create a new... ver más

semantic level in which one

### Pi primes

I was reading a recent blog

post

by Evelyn Lamb where she mentioned in passing that 314159 is a prime

number and that made me curious how many such primes there are.

Lets look at numbers formed from the digits of π to see which ones are

prime.

Obviously 3 and 31 are prime. 314 is even. 3141 is divisible by 9

because its digits sum to 9, and 31415 is clearly divisible by 5. And

now we know that 314159 is prime. What’s the next prime in the sequence?

Here’s a little Python code to find out.

`from sympy import pi, isprime `

M = 1000

digits = "3" + str(pi.evalf(M+1))[2:]

for i in range(1, M+1):

n = int(digits[:i])

if isprime(n):

print(n)

This looks at numbers formed from the first digit up to the thousandth

digit in th... ver más

### Distribution of prime powers

The prime number theorem says that π(

*x*), the number of primes less

than or equal to

*x*, is asymptotically

*x*/ log

*x*. So it’s easy to

estimate the number of primes below some number

*N*. But what if we want

to estimate the number of prime

*powers*less than

*N*? This is a

question that comes up in finite

fields, for example,

since there is a finite field with

*n*elements if and only if

*n*is a

prime power. It’s also important in finite simple

groups

because these groups are often indexed by prime powers.

**Riemann’s prime-power counting function**Π(

*x*) counts the number of

prime powers less than or equal to

*x*. Clearly... ver más

### Orders of finite simple groups

**Simple groups**are to groups as prime numbers are to numbers.A simple

group has no non-trivial normal subgroups, just as a prime number has no

non-trivial factors.

[]{#more-34567}

## Classification

Finite simple groups have been

classified

into five broad categories:

- Cyclic groups of prime order
- Alternating groups
- Classical groups
- Exceptional groups of Lie type
- Sporadic groups.

The

**cyclic groups**of prime order are simply the integers mod

*p*

where

*p*is prime. These are the only Abelian finite simple groups.

The

**alternating gr**... ver más

### How fast can you multiply matrices?

Suppose you want to multiply two 2 × 2 matrices together. How many

multiplication operations does it take? Apparently 8, and yet in 1969

Volker Strassen discovered that he could do it with 7 multiplications.

## Upper and lower bounds

The obvious way to multiply two

*n*×

*n*matrices takes

*n*³

operations: each entry in the product is the inner product of a row from

the first matrix and a column from the second matrix. That amounts to

*n*² inner products, each requiring

*n*multiplications.

You can multiply two square matrices with O(

*n*³) operations with the

method described above, and it must take at least O(

*n*²) operations

because the product depends on all of the 2

*n*² entries of the two

matrices. Strassen’s result suggests that the optimal algorithm for

multiplying matrices takes O(

*n*^

*k*... ver más

### Drawing Spirograph curves in Python

I was looking back over an old blog

post and

noticed some code in the comments that I had overlooked. Tom Pollard

gives the following code for drawing

Spirograph-like curves.

{.aligncenter .size-medium

width="614" height="461"}

`import matplotlib.pyplot as plt `

from numpy import pi, exp, real, imag, linspace

def spiro(t, r1, r2, r3):

"""

Create Spirograph curves made by one circle of radius r2 rolling

around the inside (or outside) of another of radius r1. The pen

is a distance r3 from the center of the first circle.

"""

return r3*exp(1j*t*(r1+r2)/r2) + (r1+r2)*exp(1j*t)

def circle(t, r):

return r

... ver más### Epi and mono

My first math classes used the terms

**one to one**and

**onto**to

describe functions. These Germanic names have largely been replaced with

their French equivalents

**injective**and

**surjective**.

**Monic**and

**epic**are the category theory analogs of injective and

surjective respectively. This post will define these terms and show how

they play out in several contexts. In some categories these new terms

correspond exactly to their traditional counterparts, but in others they

do not.[]{#more-34324}

## Sets and functions

A function

*f*from a set

*A*to a set

*B*is injective (one-to-one) if

different points on

*A*go to different points in

*B*. That is,

*f*(

*x*)

=

*f*(

*y*) only if

*x*=

*y*... ver más

### A tale of two elliptic curves

A few days ago I blogged about the elliptic

curve secp256k1

and its use in Bitcoin. This curve has a sibling,

**secp256r1**. Note

the “r” in the penultimate position rather than a “k”. Both are defined

in SEC 2: Recommended Elliptic Curve

Domain Parameters. Both are elliptic curves over a field

*z*~

*p*~ where

*p*is a 256-bit prime (though different primes for each curve).

The “k” in sepc256k1 stands for

**Koblitz**and the “r” in sepc256r1

stands for

**random**. A Koblitz elliptic curve has some special

properties that make it possible to implement the group operation more

efficiently. It is believed that there is a small security trade-off,

that more... ver más

### Distribution of zeros of the Riemann zeta

A recent video by Quanta Magazine says that the eigenvalues of random

matrices and the zeros of the Riemann zeta function have the same

distribution.

I assume by “random matrices” the video is referring specifically to

Gaussian orthogonal

ensembles.

By zeros of the Riemann zeta function, they mean the imaginary parts of

the zeros in the critical strip. You can download the first 100,000

zeros of the Riemann zeta function

here. So, for example,

the first zero is 14.134725142, which actually means 0.5 + 14.134725142

*i*.

Here’s the histogram of random matrix eigenvalues from my previous post:

{.aligncenter

.size-medium width="401" height="293"}

And here’s a histogram of the spacing between the first two thousand

zeros of the Riemann zeta function:

{.aligncenter

.size-medium width="401" height="293"}

{width="1"

height="1"}

http://feedproxy.google.com/~r/TheEndeavour/~3/DDR6qsozdws/

#johndcook #Math #Numbertheory #ProbabilityandStatistics

### RSA numbers and factoring

It’s much easier to multiply numbers together than to factor them apart.

That’s the basis of RSA encryption.

In particular, the RSA encryption scheme rests on the assumption that

given to large primes

*p*and

*q*, one can quickly find the product

*pq*

but it is much harder to recover the factors

*p*and

*q*. For the size

numbers you’ll see in math homework, say two or three digits, factoring

is a little harder than multiplication. But when you get into numbers

that are hundreds of digits long, factoring is orders of magnitude more

difficult. Or so it seems. There’s no proof that factoring has to be as

difficult as it appears to be. And sometimes products have special

structure that makes factoring much easier, hence the need for so-called

safe

primes... ver más

### Projecting the globe onto regular solids

I was playing around with some geographic features of Mathematica this

morning and ran across an interesting example in the documentation for

the

given here.

Here’s what you get when you project a map of the earth onto each of the

five regular (Platonic) solids.

{.aligncenter

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[]{#more-34220}\

{.aligncenter

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... ver más

Was ist denn bitte eine Nullachse? xD

https://youtu.be/TiPVEsnvGnc?t=5m17s

#berlintagundnacht #berlin #mathe #math

### From elementary addition to Markov chains

I wrote a new blog post this morning, but for some reason it posted with

an earlier date and so it doesn’t show up at the top of the blog. Here

it is: Addition carries and Markov

chains

{width="1"

height="1"}

http://feedproxy.google.com/~r/TheEndeavour/~3/lV5W2nsIoI4/

#johndcook #Math

### Distribution of carries

Suppose you add two long numbers together. As you work your way toward

the result, adding digits from right to left, sometimes you carry a 1

and sometimes you don’t. How often do you carry 1?

Now suppose you add three numbers at a time. Now your carry might be 0,

1, or 2. How often does each appear? How do things change if you’re not

working in base 10?

John Holte goes into this problem in detail in [1]. We’ll only look at

his first result here.

Suppose we’re adding

*m*numbers together in base

*b*, with each digit

being uniformly distributed. Then at each step of the addition process,

the probability distribution of the amount we carry

*out*only depends

on the amount we carried

*in*from the previous step. In other words,

the carries form a

**Markov chain**!

This means that we can do more than describe the distribution of the

amounts car... ver más

### Ada Lovelace and Bernoulli numbers

Many consider

**Ada Lovelace**to be the first programmer. An article

from a couple days ago asks what did her program

do?

In a sense, nothing. It was written for a computer that did not exist,

so it was never executed. But it was written to compute the 8th

**Bernoulli number**.

The article’s author created a

translation

of Lovelace’s program into C.

**More posts on Bernoulli numbers**:{width="1"

height="1"}

http://feedproxy.google.com/~r/TheEndeavour/~3/JTYsDAIGi0M/

#johndcook #Math

### Carbon curvature

It’s been known for some time that carbon can form structures with

**positive**curvature (fullerenes) and structures with

**zero**

curvature (graphene). Recently researches discovered a form of carbon

with

**negative**curvature (schwartzites). News story

here.

{.aligncenter

.size-medium width="400" height="300"}

**More curvature posts**:{width="1"

height="1"}

http://feedproxy.google.com/~r/TheEndeavour/~3/I1Xg5zsYIdg/

#johndcook #Math #Science

### Bitcoin signatures and elliptic curves over finite fields

Bitcoin uses the Elliptic Curve Digital Signature Algorithm (

**ECDSA**)

based on elliptic curve cryptography. The particular elliptic curve is

known as

**secp256k1**, which is the curve

*y*² =

*x*³ + 7

over a finite field to be described shortly.

{.size-medium

.aligncenter width="400" height="300"}

Addition on elliptic curves in the plane is defined geometrically in

terms of where lines intercept the curve. We won’t go into the geometry

here, except to say that it boils down to a set of equations involving

real numbers. But we’re not working over real numbers; we’re working

over a finite field.

## Finite field modulus

The idea is to take the equations motivated by the geometry in the... ver más

### Equal Earth map projection

There’s no perfectly satisfying way to map the globe on to a flat

surface. Every projection has its advantages and disadvantages. The

Mercator projection, for example, is much maligned for the way it

distorts area, but it has the property that lines of constant bearing

correspond to straight lines on the map. Obviously this is convenient if

you’re sailing without GPS. But for contemporary use, say in a

classroom, minimizing area distortion is often a higher priority than

keeping bearing lines straight.

Bojan Šavrič, Tom Patterson, and Bernhard Jenny have developed a new map

projection called

**Equal Earth**that nicely balances several competing

criteria, including aesthetics.

{.aligncenter

.size-medium width="500" height="239"}

The Equal Earth projection sa... ver más

### The other butterfly effect

{.alignnone

.size-medium width="400" height="286"}

## The original butterfly effect

The

**butterfly effect**is the semi-serious claim that a butterfly

flapping its wings can cause a tornado half way around the world. It’s a

poetic way of saying that some systems show

**sensitive dependence on**

initial conditions, that the slightest change now can make an enormous

initial conditions

difference later. Often this comes up in the context of

**nonlinear**,

**chaotic**systems but it isn’t limited to that. I give an example

here

of a linear differential equation whose solutions start out the

essentially the same but eventually diverge completely.

Onc... ver más

### Hom functors and a glimpse of Yoneda

Given two objects

*A*and

*B*, Hom(

*A*,

*B*) is simply the set of

functions between

*A*and

*B*. From this humble start, things get more

interesting quickly.

## Hom sets

To make the above definition precise, we need to say what kinds of

objects and what kinds of functions we’re talking about. That is, we

specify a category

*C*that the object belong to, and the functions are

the morphisms of that category [1]. For example, in the context of

groups, Hom(

*A*,

*B*) would be the set of group homomorphisms

[2]between

*A*and

*B*, but in the context of continuous groups (Lie

groups), we would restrict Hom(

*A*,

*B*) to be

*continuous*group

homomorphisms.

To emphasize that Hom refers to a set of morphisms in a particular

category, sometimes you’ll see the name of the category as a subscript,

as... ver más

### Deconstructing a parametric plot

Today’s exponential

sum looks like three

octagons, slightly rotated with respect to each other.

{.aligncenter

.size-medium width="613" height="460"}

I thought that the graph was tracing out one octagon, then another, then

another. But that’s not what it’s doing at all. You can see for yourself

by going to the exponential sum page and clicking the animate link.

The curve is being traced out in back-and-forth strokes. It’s drawing

thin wedges, not octagons. You can get a better picture of this by

looking at the real and imaginary parts (

*x*and

*y*coordinates)

separately.

{.aligncenter

.size-medium width="613" height="443"}

It would be hard to look at the first plot an imagine the second, or to

look at the second plot and imagine the first.

{width="1"

height="1"}

http://feedproxy.google.com/~r/TheEndeavour/~3/wH2OQ2Uyd6Y/

#johndcook #Math

### Sine of five degrees

Today’s the first day of a new month, which means the exponential sum

of the day will be simpler than

usual. The exponential sum of the day plots the partial sums of

{.aligncenter .size-medium

width="245"}

where

*m*,

*d*, and

*y*are the month, day, and (two-digit) year.

The

*n*/

*d*term is simply

*n*, and integer, when

*d*= 1 and so it has

no effect because exp(2π

*n*) = 1. Here’s today’s

sum, the plot formed by

the partial sums above.

{.aligncenter

.size-medium width="613"... ver más

### Distribution of eigenvalues for symmetric Gaussian matrix

## Symmetric Gaussian matrices

Theprevious

post looked at

the distribution of eigenvalues for very general random matrices. In

this post we will look at the eigenvalues of matrices with more

structure. Fill an

*n*by

*n*matrix

*A*with values drawn from a

standard normal distribution and let

*M*be the average of

*A*and its

transpose, i.e.

*M*= ½(

*A*+

*A*^T^). The eigenvalues will all be real

because

*M*is symmetric.

This is called a “Gaussian Orthogonal Ensemble” or GOE. The term is

standard but a little misleading because such matrices may not be

orthogonal.

## Eigenvalue distribution

The joint probability distribution for the eigenvalues of

*M*has three

terms: a constant term that we will ign... ver más

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### Circular law for random matrices

Suppose you create a large matrix

*M*by filling its components with

random values. If

*M*has size

*n*by

*n*, then we require the

probability distribution for each entry to have mean 0 and variance

1/

*n*. Then the Girko-Ginibri circular law says that the eigenvalues

of

*M*are approximately uniformly distributed in the unit disk in the

complex plane. As the size

*n*increases, the distribution converges to

a uniform distribution on the unit disk.

The probability distribution need not be normal. It can be any

distribution, shifted to have mean 0 and scaled to have variance 1/

*n*,

provided the tail of the distribution isn’t so thick that the variance

doesn’t exist. If you don’t scale the variance to 1/

*n*you still get a

circle, just not a

*unit*circle.

We’ll illustrate the circular law with a uniform distribution.... ver más