any integer that divides 1. Then there exists an integer n such that 1 = mn. By Theorem‚ either both m and n are positive or both m and n are negative. If both m and n are positive‚ then m is a positive integer divisor of 1. By Theorem‚ m ≤ 1‚ and‚ since the only positive integer that is less than or equal to 1 is 1 itself‚ it follows that m = 1. On the other hand‚ if both m and n are negative‚ then‚ by Theorem‚ (−m)(−n) = mn = 1. In this case −m is a positive integer divisor of 1‚
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companies are much less than that on western developed firms. In order to have a whole picture of the capital structure in Chinese listed companies‚ this essay will first briefly introduce some relative theories. For instance‚ the Modigliani-Miller theorem without and with taxation‚ trade-off theory and pecking order theory. Also some previous empirical evidences will be given. Secondly‚ this essay will move to discuss the situation in Chinese listed companies. In this part‚ it will first list the special
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JTG- Ch.2 Euclid’s Proof of the Pythagorean Theorem Century and a half between Hippocrates and Euclid. Plato esteemed geometry to be the entrance to his Academy. Let no man ignorant of geometry enter here. “Logical scandal” Theorems were believed to be correct as stated but they lacked the material to prove them. Euclid’s Elements was said to become the staple of mathematics or the standard. 13 books‚ 465 propositions (not all Euclid but rather a collection of great mathematicians
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The Ancient Greeks contributed a lot to modern society‚ but the biggest contribution of them was their contributions to the field of science and mathematics. To start off‚ the Pythagorean theorem contributed a lot to the field of mathematics. What the Pythagorean theorem does is help us to calculate the lengths of the sides of right triangles. Secondly‚ Archimedes contributed a ton to both fields. One of the most famous things that Archimedes did was find out if a crown that the king had ordered
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medium‚ provided the original work is properly cited. 1. Introduction In game theory‚ the existence of equilibrium was uniformly obtained by the application of a fixed point theorem. In fact‚ Nash [3‚ 4] shows the existence of equilibria for noncooperative static games as a direct consequence of Brouwer [1] or Kakutani [2] theorems. More precisely‚ under some regularity conditions‚ given a game‚ there always exists a correspondence whose fixed points coincide with the equilibrium points of the game. However
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"The Elements" which included the composition of many other famous mathematicians together. He began exploring math because he felt that he needed to compile certain things and fix certain postulates and theorems. His book included‚ many of Eudoxus’ theorems‚ he perfected many of Theaetetus’s theorems also. Much of Euclid’s background is very vague and unknown. It is unreliable to say whether some things about him are true‚ there are two types of extra information stated that scientists do not know
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rows 0-16 of the triangle. Around the same time‚ the traingle was discussed by mathematicians in Persia‚ Al-Karaji‚and mathematician Omar Khayyám‚ reffered to the traingle as the "Khayyam triangle" in Iran. Many theorems related to the triangle were known‚ including the binomial theorem. In fact Khayyam used a method of finding nth roots based on the binomial expansion‚ and therefore on the binomial coefficients. Chinese mathematician Chia Hsien showed that he was using the triangle to extract square
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We understand that the theorem is called the proof but it is also ironic that Catherine can provide no proof whatsoever as to her writing it. Catherine also demands proof of Hal’s love for her when he does not believe she wrote the theorem. Auburn writes‚ “Hal: Your dad might have written it and explained it to you later…I’m just saying there’s no proof that you wrote this” (1218)
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Eigen values and eigen vectors. Calculus: Functions of single variable‚ Limit‚ continuity and differentiability‚ Mean value theorems‚ Evaluation of definite and improper integrals‚ Partial derivatives‚ Total derivative‚ Maxima and minima‚ Gradient‚ Divergence and Curl‚ Vector identities‚ Directional derivatives‚ Line‚ Surface and Volume integrals‚ Stokes‚ Gauss and Green’s theorems. Differential equations: First order equations (linear and nonlinear)‚ Higher order linear differential equations with
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Aryabhata (476–550 CE) was the first in the line of greatmathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499 CE‚ when he was 23 years old) and the Arya-siddhanta. Name While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix‚ his name is properly spelled Aryabhata: every astronomical text spells his name thus‚[1] including Brahmagupta’s references to
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