Not strictly academic geogebra app, more of visualisation onto how #bias and #variance in fact works.

Great visualisation for beginners, in my humble opinion.

Use sliders to change bias and variance of prediction of random dataset.

Geogebra applet with 4 datapoints, fitting line with best square distance (from point to line)

square here, as well as in variance concept – is because distance can well be negative, but we need to be able to sum it neatly somehow, therefore square is rational with MMSE and variance

Have met a concept of Bayes theorem, and would pretty much like to visualize it here. As per purpose of this blog, its studying diary, so pretty much – visualizations, and making of those – would give me myself broader understanding.

Not to mention, that keeping notes publicly does make wanders to being neat in thoughtful way, so – lets begin.

Suppose we hear the description of a man: “He is strong and with a loud strong but assuring voice”.

Who is this man, do we want to guess – is he a librarian or a military drill sergeant?

Добрый день. Раз пошла такая пьянка, нужен наверное сайт, куда удобно скидывать более постоянные артикли, которые можно редактировать.

Следуюший, вполне логичный шаг, после создание видео-вопроса.

См. следующую статью

Suppose we have two countably infinite sets, which means we can find injection (one-to-**one**) relation to natural numbers.

In other words, we could literally assign natural number indexes to only one of each member of the set.

So, we have with cardinality , and with cardinality

Then cartersian product (set of all possible pairs from A and from B) – is also infinite, but countable. I mean, for , we can assign single natural number for each member of P.

Lets prove it. Suppose we injections *f* and *g, *so that and

Lets define function this way:

Then for and , because of unique division into primes and *f,g* injectivity **(a,b)=(c,d)**.

Therefore there is injection from *p(a,b)* to natural numbers, and cartesian product is infinitely countable.

### About number construction

helmi 13

– Let us construct **natural** numbers (0,1,2,3..)

– Then from naturals, lets construct **integer** numbers (…,-2,-1,0,1,2…)

– Then from integers, lets construct **rationals **(…,-0.3,…,0,…,1/2,..,0.75,..)

– And finally from rationals – **real** numbers (…,-0.3,…,0,…,1/π,…,1/2,..,0.75,..,e,…,π,…)

Read the rest of this entry »

Lets consider linear cellular automata with Wolfram rule 90

Elementary cellular automata is essentially string of two symbols, lets call them 0 and 1, equipped with rules (and starting configuration, of course)

Rule takes symbol in the position, as well as left and right neighbors. Ie – when calculating symbol of the cell, we should consider 3 cells, cell to the left, cell itself and cell to the right.

Ie – suppose we have set of rules(*):*111 - 0**110 - 1**101 - 0**100 - 1**011 - 1**010 - 0**001 - 1**000 - 0*

Or graphically:

So if we have **10011** for starting configuration, calculation goes like this(from top to bottom):

1 |
0 |
0 |
1 |
1 |

(0) |
1 |
1 |
1 |
(1) |

(1) |
1 |
0 |
0 |
(1) |

*(italics)* means that we dont know which symbol is to the left and right to the starting configuration, for simplicity I consider here absent symbols to be *0*s (ie …*00000***10011***0000*…)

Now, what is 90, which is called *Wolfram rule*.

It is obvious, that 3 cells define state, and since we have two symbols, all possible combinations are , and since we have two possible symbols for each combination – rule of CA can be expressed as (0,2⁸=256) number.

90 in binary would be (*), which is enough to describe elementary ca behavior.

There were question in exam, which I understood very close to correct, thus absolutely wrong.

Roughly translated: “*Define real axis ℝ Lebesque outermeasure on ℝ² surface*“

Lets get into it. Bit intuitively first.. Suppose we have segment of a line* (a,b)*

segment (*s*) itself can be expressed as

Actually, lets forget about *s*, lets consider without segments, ie

Then outer measure can be expressed as union of open sets:

for all

Where intervals are:

Thus length of intervals are from one integer to another – ,.. all integers

Height of intervals are

Ie , where biggest height is , which is quater of epsilon.

Therefore

or shorter: and for all

It is obvious from here

This article was inspired by **Margarita**.

### Opiskelu.org – reborn

helmi 7

I think I’d better come back to good old opiskelu.org, instead of spamming facebook page.

Oh yea, it all seems to work now.. kinda enjoy new editor for WordPress. So – here are plans for the future:

- Lebesque integral and measure, few notes regarding course and exam (english/finnish)
- Cellular Automata (english)
- Bitcoin shop with tor in mind (english)
- Немножко политики/russian politics (russian/english)

There. Lebesque integral notes&measure is coming today.

CA – on this weekend, based on university lecture.

Bitshop – next week.

Russian politics – when feel like it.