4.2 The Mean Value Theorem (11/9)
... equal to that average velocity. For instance, if a car traveled 140 miles in 2 hours, then the speedometer must have read 70 mph at least once! ...
... equal to that average velocity. For instance, if a car traveled 140 miles in 2 hours, then the speedometer must have read 70 mph at least once! ...
i ≤ n
... So far we only proved that there are ∞ primes. But can we say more than this? We will prove in this lecture a good estimation on the density of primes ...
... So far we only proved that there are ∞ primes. But can we say more than this? We will prove in this lecture a good estimation on the density of primes ...
A counting based proof of the generalized Zeckendorf`s theorem
... Next we show that the sums generated in this way fall between 0 and Gn+1 − 1. This is evident for n ≤ l − 1. For n ≥ l, the largest sum m includes the terms Gn , Gn−1 , . . . , Gn−l+2 and excludes Gn−l+1 , then includes l − 1 consecutive terms if possible, etc. Now simply write Gn+1 as Gn + Gn−1 + · ...
... Next we show that the sums generated in this way fall between 0 and Gn+1 − 1. This is evident for n ≤ l − 1. For n ≥ l, the largest sum m includes the terms Gn , Gn−1 , . . . , Gn−l+2 and excludes Gn−l+1 , then includes l − 1 consecutive terms if possible, etc. Now simply write Gn+1 as Gn + Gn−1 + · ...
Document
... Some time around 1995, after needing to look up several formulas involving the gamma function, Eric Barkan and I began to develop the theory of the gamma function for ourselves using the list of formulas in chapter 6 of the Handbook of Mathematical Functions by Abramowitz and Stegun as a guide. A fe ...
... Some time around 1995, after needing to look up several formulas involving the gamma function, Eric Barkan and I began to develop the theory of the gamma function for ourselves using the list of formulas in chapter 6 of the Handbook of Mathematical Functions by Abramowitz and Stegun as a guide. A fe ...
Modular Diagonal Quotient Surfaces (Survey)
... κ(Z̃N,ε ) = min(2, pg (Z̃N,ε ) − 1), where, as before, pg (Z̃N,ε ) denotes the geometric genus of Z̃N,ε . In particular, Z̃N,ε is a rational surface if and only pg = 0. Similarly, Z̃N,ε is of general type if and only if pg ≥ 3. In fact, the rough classification type of the surface Z̃N,ε is completel ...
... κ(Z̃N,ε ) = min(2, pg (Z̃N,ε ) − 1), where, as before, pg (Z̃N,ε ) denotes the geometric genus of Z̃N,ε . In particular, Z̃N,ε is a rational surface if and only pg = 0. Similarly, Z̃N,ε is of general type if and only if pg ≥ 3. In fact, the rough classification type of the surface Z̃N,ε is completel ...
Full text
... The Stirling numbers of the second kind S(n, k) have been studied extensively. This note was motivated by the enumeration of pairwise disjoint finite sequences of random natural numbers. The two main results presented in this note demonstrate some invariant and minimum properties of the Stirling num ...
... The Stirling numbers of the second kind S(n, k) have been studied extensively. This note was motivated by the enumeration of pairwise disjoint finite sequences of random natural numbers. The two main results presented in this note demonstrate some invariant and minimum properties of the Stirling num ...
Section 2.4 1 Definition of a Limit 2 The Absolute Value Function
... Let us explore what is going on here, we are saying that for any ε > 0, this means ε could be .5, .001, 1, 3000, .0000000001919191878186837634876, whatever, that is no matter how small we make ε then there is also some small δ, such that when the difference of c and x is within δ then the didderence ...
... Let us explore what is going on here, we are saying that for any ε > 0, this means ε could be .5, .001, 1, 3000, .0000000001919191878186837634876, whatever, that is no matter how small we make ε then there is also some small δ, such that when the difference of c and x is within δ then the didderence ...
rendering
... Proposition 2.2. The product of two even numbers is even. Proof. Let n and m be as in (2.1), (2.2), with k, ` ∈ Z. Then n · m = (2k)(2`) = 2(2k`), hence n · m is even (actually, even a multiple of 4). Date: August 26, 2011. ...
... Proposition 2.2. The product of two even numbers is even. Proof. Let n and m be as in (2.1), (2.2), with k, ` ∈ Z. Then n · m = (2k)(2`) = 2(2k`), hence n · m is even (actually, even a multiple of 4). Date: August 26, 2011. ...
Pythagoras Pythagoras A right triangle, such as shown in the figure
... (4) Choose an integer e such that 1 < e < φ(n) and the greatest common factor of e and φ(n) is 1. For example if e = 1 × 2 × 4 and φ(n) = 1 × 3 × 5, the greatest common factor is 1. • The public key consists of n and the public exponent e. Sometimes e = 3 is used. (5) determine d such that d × e = 1 ...
... (4) Choose an integer e such that 1 < e < φ(n) and the greatest common factor of e and φ(n) is 1. For example if e = 1 × 2 × 4 and φ(n) = 1 × 3 × 5, the greatest common factor is 1. • The public key consists of n and the public exponent e. Sometimes e = 3 is used. (5) determine d such that d × e = 1 ...
Proofs - faculty.cs.tamu.edu
... essentially use the same method. However, formal proofs are not very appealing to humans (the intended readership of our proofs), so we should try to formulate our proofs in plain English! ...
... essentially use the same method. However, formal proofs are not very appealing to humans (the intended readership of our proofs), so we should try to formulate our proofs in plain English! ...
CONGRUENCE PROPERTIES OF VALUES OF L
... Let L(ED , s) be the Hasse-Weil L-function for ED . For modular E, Kolyvagin [Ko] proved that if L(ED , 1) 6= 0, then ED has rank zero. Theorem 1 together with Kolyvagin’s theorem implies: Corollary 1. If E/Q is a modular elliptic curve, then the number of |D| ≤ X for which ED has rank zero is ÀE X/ ...
... Let L(ED , s) be the Hasse-Weil L-function for ED . For modular E, Kolyvagin [Ko] proved that if L(ED , 1) 6= 0, then ED has rank zero. Theorem 1 together with Kolyvagin’s theorem implies: Corollary 1. If E/Q is a modular elliptic curve, then the number of |D| ≤ X for which ED has rank zero is ÀE X/ ...
Fermat`s Last Theorem - Math @ McMaster University
... all of the positive integer solutions to the equation. We then produce new positive integers r , s, and t that are also a solution to the equation, but with t < w . This will contradict the minimality of w , and so we conclude that the equation cannot have any positive integer solutions. ...
... all of the positive integer solutions to the equation. We then produce new positive integers r , s, and t that are also a solution to the equation, but with t < w . This will contradict the minimality of w , and so we conclude that the equation cannot have any positive integer solutions. ...
In Class Slides
... 5. Keep your reader informed about the status of each statement in your proof. – Your reader should never be in doubt about whether something in your proof has been assumed, established, or is still to be deduced. • If it is assumed use words like “Suppose” or “Assume” • If it is still to be shown u ...
... 5. Keep your reader informed about the status of each statement in your proof. – Your reader should never be in doubt about whether something in your proof has been assumed, established, or is still to be deduced. • If it is assumed use words like “Suppose” or “Assume” • If it is still to be shown u ...
1 Introduction 2 What is Discrete Mathematics?
... relations, counting, probability and graph theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. Some of these applications are listed below. • The analysis of algorithms requires the ability to count the number of operations in an alg ...
... relations, counting, probability and graph theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. Some of these applications are listed below. • The analysis of algorithms requires the ability to count the number of operations in an alg ...
a simple derivation of jacobi`s four-square formula
... important that he devoted an entire chapter to its discussion. Following Dickson we briefly here record that the theorem was conjectured by Bachet in 1621, was claimed to have been proved by Fermât, but was not actually proved until Lagrange did so in 1770. It should also be mentioned that Lagrange ...
... important that he devoted an entire chapter to its discussion. Following Dickson we briefly here record that the theorem was conjectured by Bachet in 1621, was claimed to have been proved by Fermât, but was not actually proved until Lagrange did so in 1770. It should also be mentioned that Lagrange ...
Axioms and Theorems
... There is a number 0, which has the property that for any number n n+0=n There is a number 1 which has the property that for any number n nx1=n For every number n, there is a number k such that n+k=0 For any numbers m, n and k if k ≠ 0 and kn = km, then n = m ...
... There is a number 0, which has the property that for any number n n+0=n There is a number 1 which has the property that for any number n nx1=n For every number n, there is a number k such that n+k=0 For any numbers m, n and k if k ≠ 0 and kn = km, then n = m ...
Examples of mathematical writing
... Prime numbers are essential to crytography, Euclid’s famous theorem has held generations of mathematicians spellbound in it’s inescapable beauty. Theorem. (Euclid, 400) There are infinitely many prime numbers, where a prime is a number only divisible by itself and 1. (Throughout this project, number ...
... Prime numbers are essential to crytography, Euclid’s famous theorem has held generations of mathematicians spellbound in it’s inescapable beauty. Theorem. (Euclid, 400) There are infinitely many prime numbers, where a prime is a number only divisible by itself and 1. (Throughout this project, number ...
Full text
... where N^j is the number of incongruent solutions of f^(x) E 0 (mod p . ) , see [8, Theorem 1]. This totient function is multiplicative and it is very general. As special cases, we obtain Jordan1s well-known totient J^(n) [3, p. 147] for f\(x) ...
... where N^j is the number of incongruent solutions of f^(x) E 0 (mod p . ) , see [8, Theorem 1]. This totient function is multiplicative and it is very general. As special cases, we obtain Jordan1s well-known totient J^(n) [3, p. 147] for f\(x) ...