I’m glad to announce the release of the full GMac source code as an open source project on GitHub under the GPL v3.0 license. The latest version can be found here: https://github.com/ga-explorer/GMac. The GMac Guides (https://gacomputing.info/about-gmac/) contain detailed information on how to use GMac; in addition to many details about its software architecture. I hope the GA community can benefit from this in the best way possible.

With my best regards

Ahmad Eid

*Source:* Email from Ahmad Eid, ga.computing.eg_AT_gmail.com, 07 Oct. 2017.

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in many situations are considerations and prof dr mircea orasanu and prof horia orasanu and followed for applications and High dimensionless that these can be applied to CONSTRAINTS OPTIMIZATIONS that are used for many problems of optimizations as LAGRANGIAN FORMS . The logical theory of systems and mappings.

The foregoing may suffice for the reader to obtain a grasp of the sense in which sets were at the basis of ideal theory. Ideal theory was genetical or “constructive” in DEDEKIND’s sense: taking the set of algebraic numbers A as given, it employed set-constructions on it, i.e., subsets of A (the ideals), as its basic objects; and this made it possible to define rigorously the operations on ideals, especially multiplication. In this way, it was perfectly coherent with DEDEKIND’s conception of the number system, to which we now have to turn.

4.3.1. DEDEKIND’s program for the foundations of mathematics. In his works on the foundations of the number system, DEDEKIND expressed repeatedly the idea that this system was obtained from the natural numbers through step-by-step definitions or “constructions” (DEDEKIND 1854, 430-431; 1872, 317-318; 1888, 338). In manuscripts that are still preserved, he developed the well-known idea of defining the integers and their operations on the basis of equivalence classes of pairs of natural numbers; and similarly for the rationals on the basis of equivalence classes of pairs of integers. In 1872 he published the much more sophisticated idea of employing so-called ‘DEDEKIND cuts’ on the set of rational numbers for defining the real numbers and the operations on them. In all of these cases, the procedure is genetical in the above sense: taking a set of numbers as given, the next higher set is defined by means of set-constructions on the former, i.e., it is defined as (isomorphic to) the set of some specified subsets of the former. This, and the operations on the ‘lower’ number-set, suffice to define the operations on the new numbers. Since, according to DEDEKIND’s reliable dating, his theory of real numbers was formulated in 1858, it is a mild assumption that the much simpler theories of the integers and rationals should be traced back, at least, to that same year.

1 INTRODUCTION

possible to consider morphisms between those structures.

A reconstruction of DEDEKIND’s thoughts on analysis is necessarily more tentative, because to the best of my knowledge there is no document recording them. But since the real and complex numbers can be defined within DEDEKIND’s framework, and since we have the general notion of mapping at our disposal, real and complex functions are easily obtained. It might well have been along this line that DEDEKIND saw analysis integrated within the general picture of ‘arithmetic’─meaning his conception of classical mathematics as based on numbers, and ultimately on sets and maps.

DEDEKIND’s foundational masterpiece Was sind und was sollen die Zahlen? (1888) was devoted to a detailed presentation of the elements of the whole edifice of his ‘arithmetic’. It developed the “construction” of the natural numbers on the basis of a careful presentation of the theory of sets and mappings. But above all, it has to be read as presenting all the necessary ingredients for a detailed derivation of arithmetic, algebra, and analysis.

4.3.2. Set and mapping as logical notions. In the preface to his 1888 book, DEDEKIND clearly adopted a logicist viewpoint. The work begins with an statement of the author’s epistemological and methodological views:

In science nothing capable of proof ought to be accepted without proof. As evident as this requirement might seem yet I cannot regard it as having been met even in the simplest science, that part of logic which deals with the theory of numbers […] In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the ideas or intuitions of space and time, that I consider it an immediate result from the pure laws of thought. […] It is only through the purely logical construction of the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by relating them to this

also here as p[rof dr mircea orasanu and prof drd horia orasanu approached as followed with prof dr Constantin Udriste and his report in thus with . Area of a Triangle = bh/2

2. Pythagorean Theorem: Euclid’s Windmill Proof

3. Pythagorean Theorem: Chinese Proof (or perhaps the Indian mathematician Bhaskara’s)

4. Pythagorean Theorem: President Garfield’s Trapezoid Proof

5. The Distance Formula: derived from Pythagorean Theorem!

6. Fermat’s Last Theorem: xn + yn = zn has no positive integral solutions for n>2.

Proven recently by Andrew Wiles (omitted here for lack of room in the margin).

7. Hypotenuses in a “square root spiral” are of length sqrt 2, sqrt 3, sqrt 4, sqrt 5,…

(Inductive proof)

8. The square root of 2 is irrational.

a. (p/q) 2 = 2 p2 = 2q2, then apply Fundamental Theorem of Arithmetic lhs has even # of prime factors, rhs has odd #, QED.

b. (p/q) 2 = 2 p2 = 2q2 p is even … q is even, QED.

9. “Nearly all” real numbers are irrational!

a. The integers are countable (as are evens, primes, powers of 10, …)

b. Integer pairs – Z2 – are countable (dovetailing!)

c. Integer triplets, etc – Z3 , Z4,… – are countable.

d. Rationals are countable.

e. Algebraics are countable.

f. Reals are not countable (diagonalization!)

g. Thus, “nearly all” reals are irrational (even non-algebraic, hence transcendental!)

7. The Circle Problem and Pascal’s Triangle

a. How many intersections of chords connecting N vertices?

b. How does this relate to Pascal’s Triangle?

8. Patterns in Pascal’s Triangle (see http://www.kosbie.net/lessonPlans/pascalsTriangle/)

a. Simple Patterns

i. Natural Numbers (1,2,3,4…)

ii. Triangular Numbers (1,3,6,10,…)

iii. Binomial Coefficients (nCk) Pascal’s Binomial Theorem

iv. Tetrahedral Numbers (1,4,10,20,…)

v. Pentatope Numbers (1,5,15,35,70…)

b. More Challenging Patterns

i. Powers of 2 (2,4,8,16,…)

ii. Hexagonal Numbers (1,6,15,28,…)

iii. Fibonacci Numbers (1,1,2,3,5,8,…) Prove This!

iv. Sierpinski’s Triangle

v. Catalan Numbers (1,2,5,14,42,…) Prove This!

vi. Powers of 11 (11, 121, 1331, 14641,…)

9. Applications of the Binomial Theorem

a. Find the coefficient of x3 in (x + 5) 3

b. Prove: nC0 + nC1 + … + nCn= 2n (Hint: 2 = 1+1, so what does 2n = ?)

Summer Math Series: Week 4

10. π = C/D (By observation! Since Babylonian times, where π =~ 3.125)

11. Area of a Polygon = ½ hQ (1/2 * apothem * perimeter)

12. Archimedes’ Proof that A = πr2

a. Approximate circle with inscribed (2n)-gons

b. Rephrasing of argument on page 93:

i. Apolygon = ½ hQ, but:

1. As sides infinity, h r (apothem radius)

2. As sides infinity, Q C (perimeter circumference)

ii. So:

As sides infinity, Apolygon ½ r C Acircle

(area of polygon area of circle)

c. Last step (p. 96): combine:

i. A = ½ r C

1 . INTRODUCTION

A high-quality mathematics program is essential for all students and provides every student with the opportunity to choose among the full range of future career paths. Mathematics, when taught well, is a subject of beauty and elegance, exciting in its logic and coherence. It trains the mind to be analytic – providing the foundation for intelligent and precise thinking.

To compete successfully in the worldwide economy, today’s students must have a high degree of comprehension in mathematics. For too long schools have suffered from the notion that success in mathematics is the province of a talented few. Instead, a new expectation is needed: all students will attain California’s mathematics academic content standards, and many will be inspired to achieve far beyond the minimum standards.

These content standards establish what every student in California can and needs to learn in mathematics. They are comparable to the standards of the most academically demanding nations, including Japan and Singapore – two high-performing countries in the Third International Mathematics and Science Study (TIMSS). Mathematics is critical for all students, not only those who will have careers that demand advanced mathematical preparation but all citizens who will be living in the twenty-first century. These standards are based on the premise that all students are capable of learning rigorous mathematics and learning it well, and all are capable of learning far more than is currently expected. Proficiency in most of mathematics is not an innate characteristic; it is achieved through persistence, effort, and practice on the part of students and rigorous and effective instruction on the part of teachers. Parents and teachers must provide support and encouragement.

The standards focus on essential content for all students and prepare students for the study of advanced mathematics, science and technical careers, and postsecondary study in all content areas. All students are required to grapple with solving problems; develop abstract, analytic thinking skills; learn to deal effectively and comfortably with variables and equations; and use mathematical notation effectively to model situations. The goal in mathematics education is for students to:

Develop fluency in basic computational skills.