User:ArnoldReinhold/sandbox

Data from the Neonatal Research Network's Glutamine Trial showed that the incidence of NEC among extremely low birthweight (ELBW) infants fed with more than 98% human milk was 1.3%, compared with 11.1% among infants fed only preterm formula, and 8.2% among infants fed a mixed diet, suggesting that infant deaths could be reduced by efforts to support production of milk by mothers of ELBW newborns.

Adding the trigonometric identities:
 * $$\cos \left(x-y\right)=\cos x \cos y + \sin x \sin y. \,$$
 * $$\cos \left(x+y\right)=\cos x \cos y - \sin x \sin y, \,$$

we see that
 * $$2\cos x \cos y = \cos \left(x+y\right) + \cos \left(x-y\right)\,$$

Group (mathematics) intro edits
In mathematics, a group is a set together with an operation that combines any two elements of the set to form an element also in the set. The group operation must obey the associative property and be reversible, meaning any operation can be undone by a second operation. A familiar example of a group is the set of integers together with the addition operation.

The requirements to be a group are usually stated as four conditions called the group axioms:
 * 1) the operation must only produce elements within the group (closure)
 * 2) the associative property: (a • b) • c = a • (b • c), where a, b and c are any elements of the group and the symbol • stands for the group operation
 * 3) the group must contain an identity element that leaves other elements unchanged under the group operation
 * 4) every element must have an inverse. A group element when combined with its inverse always produces the identity element.

For the set of integers under addition, zero is the identity element and the negative of a number is its inverse.

A rich theory of groups has developed since the mid-nineteenth century, and they are a powerful tool that mathematicians use to solve a wide variety of theoretical and practical problems. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

Second example: the even numbers
The set of all even integers also forms a group under addition. It is easy to check the group axioms: the sum of two even number is even (closure); addition of integers always obeys the associative rule; the identity element for addition, zero, is an even number; and, finally, the negative of an even number is even, so every even number has an inverse in the set. The group of even integers is a subgroup of the group of integers, and usually written 2Z. Similarly, the subset of the integers that are evenly divisible by 3: {... -12, -9, -6, -3, 0, 3, 6, 9, 12, ...} forms a group. So do the integers that are evenly divisible by 4, or by 5, or by any integer n. The set of all integers that are evenly divisible by n, that is all integers that are the product of n and another integer, form a sugroup of the group of integers Z. Closure is true because na +nb = n(a + b). This subgroup is often called nZ.

We can define a function, f from Z to nZ by f(a) = na. Because n(a + b) = na +nb, the function f has the property that f(a + b) = f(a) + f(b). A function from one group to another that has this property is called a homomorphism. In this case, the function f is one to one and onto, making it also a isomorphism. Because an isomoprhism exists between them, we say Z and nZ are isomorphic. Isomorphic groups are essentially identical for group theory even though their underlying sets may differ.

Third example: Modular addition
The integers are an example of an infinite group, but groups can also be constructed on finite sets. In modular addition, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n-1 forms a group under modular addition. The inverse of any element a is n-a, and 0 is the identity element. These groups are known as the. An everyday example is addition on the face of a clock, where the hour hand is advanced a certain number of hours from an initial position to a new position. If the hand is on 9 and is advanced 4 hours, it ends up on 1. This equivalent to modular addition with 12 as the modulus: 9 + 4 = 13 = 1 mod (12). The group of integers modulo n is sometimes written Zn or Z/nZ.

A finite group can also be defined by a table of operations. Here is the addition table for the integers modulo 3:

Z4 and Klein4

 * {|border="2" cellpadding="5" align="left" style="text-align: center;"

!style="background:#efefef;"|+ !style="background:#efefef;"|0 !style="background:#efefef;"|1 !style="background:#efefef;"|2 !style="background:#efefef;"|3 !style="background:#efefef;"|0 !style="background:#efefef;"|1 !style="background:#efefef;"|2 !style="background:#efefef;"|3
 * 0 || 1 || 2 || 3
 * 1 || 2 || 3 || 0
 * 2 || 3 || 0 || 1
 * 3 || 0 || 1 || 2
 * }

---

The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects.

1916 SIGNAL BOOK
q.v. File:Seneca code instructions.agr.jpg

Finite St. Petersburg game Draft 250321
The classical St. Petersburg game assumes that the casino or banker has infinite resources. This assumption has long been challenged as unrealistic. Alexis Fontaine des Bertins pointed out in 1754 that the resources of any potential backer of the game are finite. More importantly, the expected value of the game only grows logarithmically with the resources of the casino. As a result, the expected value of the game, even when played against a casino with the largest resources realistically conceivable, is quite modest. In 1777, Georges-Louis Leclerc, Comte de Buffon calculated that after 29 rounds of play there would not be enough money in the Kingdom of France to cover the bet.

If the casino has finite resources, the game must end when there are no longer enough funds to cover the next play. Suppose the total resources (or maximum jackpot) of the casino are W dollars (more generally, W is measured in units of half the game's initial stake). Then the maximum number of times the casino can play before it no longer can fully cover the next bet is L = floor(log2(W)). Assuming the game ends when the casino can no longer cover the bet, the expected value E of the lottery then becomes:
 * $$\begin{align}

E &= \sum_{k=1}^{L} \frac{1}{2^k} \cdot 2^k = L\,. \end{align}$$ The following table shows the expected value E of the game with various potential bankers and their bankroll W:

Note: Under rules that say if the player wins more than the bankroll they will be paid all the bank has, the additional expected value is less than it would be if the bank had enough funds to cover one more round, i.e. less than $1.

The premise of infinite resources produces a variety of apparent paradoxes in economics. In the martingale betting system, a gambler betting on a tossed coin doubles his bet after every loss so that an eventual win would cover all losses; this system fails with any finite bankroll. The gambler's ruin concept shows a persistent gambler will go broke, even if the game provides a positive expected value, and no betting system can avoid this inevitability.

Categories with Short Descriptions mockups

 * User:ArnoldReinhold/Category-IEEE standards SD test
 * User:ArnoldReinhold/Category-Theorems in combinatorics
 * User:ArnoldReinhold/Category-Theorems in Number Theory SD test

Misc

 * . <--annotated link


 * User:ArnoldReinhold/sandbox/Mirifici Logarithmorum Canonis

Vacuum tube notes
While the history of mechanical aids to computation goes back centuries, if not millennia, the history of vacuum tube computers is confined to the middle of the 20th century. Lee De Forest invented the triode in 1906. The first example of using vacuum tubes for computation, the Atanasoff–Berry computer, was demonstrated in 1939. By the early 1960s vacuum tibe computers were obsolete, replaced by similar designs using transistors.

However, much of what we now consider part of digital computers evolved during the vacuum tube era. Initially, vacuum tube computers performed the same operations as earlier mechanical computers, only at much higher speeds. Gears and mechanical relays operate in milliseconds, whereas vacuum tubes can switch in microseconds. The first departure from what was possible prior to vacuum tubes was the incorporation of large memories that could store and randomly access, at high speeds, thousands of bits of data —there was no way to do this with mechanical designs. That, in turn, allowed the storage of machine instruction in the same memory as data —the stored program concept, a breakthrough which today is a hallmark of a digital computer. Other inventions included the use of magnetic tape to store large volumes of data in compact form, UNIVAC I and the Invention of random access secondary storage IBM RAMAC 305, the direct ancestor of all the hard disk drives we use today. Even computer graphics began during the vacuum tube era with the IBM 740 CRT Data Recorder.

Many programming originated in the vacuum tube era including we still used today such as Fortran, lisp, Algol Z22 and COBOL.

Daisy Bell computer speech synthesis by the IBM 704 in 1961, Core memory which was the mainstay of second generation computing was introduced in this era

Networking whirlwind fsq7 light pen interactive computer graphics

GM-NAA I/O

Antikythera Mechanism

[Text incorporated into Vacuum-tube computer Sept. 18, 2023]