elkebir

Law of El'Kebir

(Generalized law of first digit distributions)

Law of El'Kebir states that;

1-Data sets; generated by mathematical functions, have a unique pattern of first digit distribution.

2-Under some specific circumstances this distribution pattern may tend to approach Benford's law (There are countless many functions which fulfill this condition). Functions are changed randomly between certain prespecified values of their variables.

3-If two functions are identical or one is a serial representation of the other, then their first digit distribution patterns are similar**

4-From some aspects, this law resembles "fingerprints of human beings" or "DNA profiles of living organisms".

** You may check this situation by downloading similaris1.zip or similaris2.zip

You may download the following excel files from our site for illustrated examples .

function	Filesize	variables	constants	notes	definition	download winzip (excel) file
f(a,x)=k.x^q.[sinhⁿ(a)]	3.628KB	x , a	k , q , n	approach to benf. dist. under some specific circumstances	invented by our team	wrkbnch7
f(a,x)=k.x^q.[sinⁿ(a)]	3.576KB	x , a	k , q , n	approach to benf. dist. under some specific circumstances	invented by our team	wrkbnch6
f(a,x)=k.x^q.[coshⁿ(a)]	3.651KB	x , a	k , q , n	approach to benf. dist. under some specific circumstances	invented by our team	wrkbnch5
f(a,x)=k.x^q.[arcsinⁿ(a)]	3.577KB	x , a	k , q , n	approach to benf. dist. under some specific circumstances	invented by our team	wrkbnch4
f(x,y)=k . x^p. y^q	2.970KB	x , y	k , p , q	( p >4 and q >4 ) ; then the first digits approach Benf. dist	invented by our team	wrkbnch3C
f(x,y)=k . x^p. asinh(y)	2.991KB	x , y	k , p	approach to benf. dist. under some specific circumstances	invented by our team	wrkbnch2C
	2.392KB	x	r , m	approach to benf. dist. under some specific circumstances	Famous Gaussian function	wrkbnch1

Information and Help on usage of the provided excel worksheets

After downloading (any or all) of the workbench (wrkbnch) files, unzip them with winzip program. The excel worksheets are made with (excel 2000) and contain 55000 lines of data (quite a lot!). Every file is equipped with macro buttons, when you click on this macro button, the variables are changed randomly between the specified limits stated on the worksheet, the constants such as (k , p , m , r , q ) may also be changed on purpose. When you change the (limits of variables) or (constants) on the woksheet, the resulting first digit distribution is changed immediately on the workbench.

The final resulting column is multiplied by 10¹² and then taken the integer part; this is to isolate the first digits of functions (because we are always coping with the initial digits). Hope you understand this !

Since there is a lot of data entry on the spreadsheets; you need to have at least 128 MB of RAM and Pentium II processor, if you have less than these, then the time necessary for the execution of macros will be exceedingly long and you'll be bored.

How to invent your own functions ?

Try to make minor changes and make exercises on the provided sheets; then invent your own functions by changing the corresponding final column. Seeing is believing, you'll understand the main philosophy of this law by scrutinizing your own datasets.

A proposition for "generalized law of first digit distributions"

It is sometimes possible to assign a function f(x) to a single column of data, such that; between prespecified limits of its variables (x₁ and x₂), the function represents the random data faithfully.

Example;

Suppose we have a column of random numerical data received by Arecibo radio telescope at Puerto Rico, which consists of about 20.000 entries. There are no other data available except this single column, that is; we cannot plot (x,y) graph since there are no values for x-axis. Under these circumstances we can perform the following operations to assign a representative function f(x) to the random numerical data received.

For a more precise analysis, we take the first three digital distributions (over first, second, third digits). We accept at fourth and higher digits, the distributions are uniform over (0.......9). By an ingenious software or by sheer luck or by trial and error, lets assume; we have found an f(x), which quite closely fits the first three digital distributions of the original data between the limits (x₁ and x₂) on the randomly generated numerical data between the said limits. Then we may state that, this function f(x) represents our radio dataset quite closely between the limits (x₁ and x₂).

This proposition arises from the rule of the law which states that; "If two functions are identical or one is a serial representation of the other, then their first digit distributions are similar" (see the definition of the law at the top of page). Converse may not be always true!!. Additional and special precautions should be taken in order to hold this proposition true.

You may download a zipped excel worksheet file (arecibo1C.zip) from our site, in order to visualize the situation as a step by step illustrated example.