Calculating generalised image and discriminant Milnor numbers in
... a number of different meanings to the phrase ‘double point space’ in the literature. The multiple point spaces we shall be interested are those defined by Goryunov in [5, Section 4]; they are also denoted by Dk ð f Þ. In Mond’s earlier definition of multiple point spaces, [15], a curve arising from the ...
... a number of different meanings to the phrase ‘double point space’ in the literature. The multiple point spaces we shall be interested are those defined by Goryunov in [5, Section 4]; they are also denoted by Dk ð f Þ. In Mond’s earlier definition of multiple point spaces, [15], a curve arising from the ...
Chapter4
... factorizations is not efficient because there is no efficient algorithm for finding the prime factorization of a positive integer. ...
... factorizations is not efficient because there is no efficient algorithm for finding the prime factorization of a positive integer. ...
Sums of Digits and the Distribution of Generalized Thue
... s(A(b+1)Y ) ≥ s(Ab(Y +1)) for every Y with the form of k digits. Thus, by Lemma 10 (3) it is easy to verify that the digit sums of (A(b+1)0k )q , . . . , (A(b+1)(X−2))q cover the digit sums of (Ab0k−1 1)q , . . . , (Ab(X − 1))q . Therefore, the digit sums of (AbX)q , . . . , (Ab(q − 1)k )q , (A(b + ...
... s(A(b+1)Y ) ≥ s(Ab(Y +1)) for every Y with the form of k digits. Thus, by Lemma 10 (3) it is easy to verify that the digit sums of (A(b+1)0k )q , . . . , (A(b+1)(X−2))q cover the digit sums of (Ab0k−1 1)q , . . . , (Ab(X − 1))q . Therefore, the digit sums of (AbX)q , . . . , (Ab(q − 1)k )q , (A(b + ...
CMSC 203 / 0202 Fall 2002
... More time in class for students to try to solve problems that we then work through as a class In return: Students agree to review chapter readings before class so we can spend less time on the basics without losing everybody. September1999 October 1999 ...
... More time in class for students to try to solve problems that we then work through as a class In return: Students agree to review chapter readings before class so we can spend less time on the basics without losing everybody. September1999 October 1999 ...
THE FRACTIONAL PARTS OF THE BERNOULLI NUMBERS BY
... It is well known that they are rational numbers and that B n = 0 for odd n > 1 . We have B 2 = 1/6, B 4 = - 1/30, B 6 = 1/42, etc . The fractional parts {B 2k} may be computed easily by the von Staudt-Clausen theorem, which says that B 2k + Y 1/p is an integer, where the sum is taken over all primes ...
... It is well known that they are rational numbers and that B n = 0 for odd n > 1 . We have B 2 = 1/6, B 4 = - 1/30, B 6 = 1/42, etc . The fractional parts {B 2k} may be computed easily by the von Staudt-Clausen theorem, which says that B 2k + Y 1/p is an integer, where the sum is taken over all primes ...
Cantor`s Legacy Outline Let`s review this argument Cantor`s Definition
... The cardinal numbers |N| = ℵ0 < ℵ1 < ℵ2 < … ℵk is the smallest set larger than ℵk-1 Are there any more infinities? Let S = {k | k N } P(S) is provably larger than any of them!! No single infinity is big enough to count the number of infinities! ...
... The cardinal numbers |N| = ℵ0 < ℵ1 < ℵ2 < … ℵk is the smallest set larger than ℵk-1 Are there any more infinities? Let S = {k | k N } P(S) is provably larger than any of them!! No single infinity is big enough to count the number of infinities! ...
1. Expand (a b)n Using Pascal`s Triangle Section 15.4 The Binomial
... 1. Expand (a ⴙ b)n Using Pascal’s Triangle Here are the expansions of (a b) n for several values of n: (a b) 0 1 (a b) 1 a b (a b) 2 a2 2ab b2 (a b) 3 a3 3a2b 3ab2 b3 (a b) 4 a4 4a3b 6a2b2 4ab3 b4 (a b) 5 a5 5a4b 10a3b2 10a2b3 5ab4 b5 Notice th ...
... 1. Expand (a ⴙ b)n Using Pascal’s Triangle Here are the expansions of (a b) n for several values of n: (a b) 0 1 (a b) 1 a b (a b) 2 a2 2ab b2 (a b) 3 a3 3a2b 3ab2 b3 (a b) 4 a4 4a3b 6a2b2 4ab3 b4 (a b) 5 a5 5a4b 10a3b2 10a2b3 5ab4 b5 Notice th ...
Asymptotic Enumeration of Reversible Maps Regardless of Genus
... Although our primary object is to investigate isomorphism classes of maps on compact orientable connected surfaces without boundary, in order to express the enumeration formulas we need to consider a broader family of maps. Namely, we allow our maps to have semiedges, the underlying surface can be n ...
... Although our primary object is to investigate isomorphism classes of maps on compact orientable connected surfaces without boundary, in order to express the enumeration formulas we need to consider a broader family of maps. Namely, we allow our maps to have semiedges, the underlying surface can be n ...
was the congruence
... Theorem There are infinitely many primes of the form p ≡ −1(mod 4). Proof We argue in a manner similar to Euclid’s proof that there are infinitely many primes. € Suppose there are only finitely many primes of the desired form. List them: q1 = 3,q 2 = 7,… ,qn . Now consider the number N = 4 q2 qn + ...
... Theorem There are infinitely many primes of the form p ≡ −1(mod 4). Proof We argue in a manner similar to Euclid’s proof that there are infinitely many primes. € Suppose there are only finitely many primes of the desired form. List them: q1 = 3,q 2 = 7,… ,qn . Now consider the number N = 4 q2 qn + ...
Infinitesimal Complex Calculus
... of f (z ) , and avoids the contradiction. 2) Infinitesimal Complex Calculus supplies us with a discontinuous complex function that has a derivative. No such result exists in the Calculus of Limits. 3) The Cauchy Integral Formula holds for Hyper-Complex Function analytic in an infinitesimal disk in t ...
... of f (z ) , and avoids the contradiction. 2) Infinitesimal Complex Calculus supplies us with a discontinuous complex function that has a derivative. No such result exists in the Calculus of Limits. 3) The Cauchy Integral Formula holds for Hyper-Complex Function analytic in an infinitesimal disk in t ...
41(3)
... Proof: Part (a) follows from Lemma 2.1 and the fact that Ozl = u. Part (b) follows from the fact that w = T(u) and w' — T(u). D When the conjugates of u are listed as in (2) below3 we observe some interesting phenomena. C o r o l l a r y 3,3 (see [11]): (a) The sequence of words u, Ts(u), T2s(u),... ...
... Proof: Part (a) follows from Lemma 2.1 and the fact that Ozl = u. Part (b) follows from the fact that w = T(u) and w' — T(u). D When the conjugates of u are listed as in (2) below3 we observe some interesting phenomena. C o r o l l a r y 3,3 (see [11]): (a) The sequence of words u, Ts(u), T2s(u),... ...
1 Introduction to Logic
... together to form the product p1p2p3…pn. Add 1 to that product and call the result m (so m = p1p2p3…pn+1). First we have to prove that none of the primes p1, p2, p3, …, pn divides m. Each of p1, p2, p3, …, pn clearly divides the product p1p2p3…pn. If one of the p's also divided the natural number m, ...
... together to form the product p1p2p3…pn. Add 1 to that product and call the result m (so m = p1p2p3…pn+1). First we have to prove that none of the primes p1, p2, p3, …, pn divides m. Each of p1, p2, p3, …, pn clearly divides the product p1p2p3…pn. If one of the p's also divided the natural number m, ...
Note 3
... It looks like we have good news and even better news: The good news is that we have not yet found a counterexample to our claim. The even better news is that there is a surprising pattern emerging — the sum of the first n odd numbers is not just a perfect square, but is equal precisely to n2 ! Motiv ...
... It looks like we have good news and even better news: The good news is that we have not yet found a counterexample to our claim. The even better news is that there is a surprising pattern emerging — the sum of the first n odd numbers is not just a perfect square, but is equal precisely to n2 ! Motiv ...
Full text
... the editor has accepted the referee's recommendation to publish the results in The Fibonacci Quarterly. The author has not yet seen the earlier publication, but understands that the proofs employ the same line of reasoning, although differing in details. If m = r(r +1), then 4m +1 is a square. Our a ...
... the editor has accepted the referee's recommendation to publish the results in The Fibonacci Quarterly. The author has not yet seen the earlier publication, but understands that the proofs employ the same line of reasoning, although differing in details. If m = r(r +1), then 4m +1 is a square. Our a ...
17 Sums of two squares
... ) = −1 as p ≡ 3 mod 4, giving a contradiction. But ( −1 p Questions: 1. What if one considers sums of k squares with k > 2, e.g., 7 = 22 + 12 + ...
... ) = −1 as p ≡ 3 mod 4, giving a contradiction. But ( −1 p Questions: 1. What if one considers sums of k squares with k > 2, e.g., 7 = 22 + 12 + ...
What are the Features of a Good Explanation?
... and a prime p is a factor of rs, then p must be a factor of r, or of s, or of both r and s. Proving irrationality √ follows the same structure as proving that 2 is irrational, which is sometimes used in the high school curriculum as an introduction to proof by contradiction. On the next page are fou ...
... and a prime p is a factor of rs, then p must be a factor of r, or of s, or of both r and s. Proving irrationality √ follows the same structure as proving that 2 is irrational, which is sometimes used in the high school curriculum as an introduction to proof by contradiction. On the next page are fou ...
19Goodarzi copy - Matematiska institutionen
... It is a well-known fact that Hilbert series can be computed by using the graded Betti numbers, so the minimal free resolution is a finer invariant than Hilbert series. On the other hand unlike the case of Hilbert series, the graded Betti numbers depend on the characteristic of the ground field K. Ho ...
... It is a well-known fact that Hilbert series can be computed by using the graded Betti numbers, so the minimal free resolution is a finer invariant than Hilbert series. On the other hand unlike the case of Hilbert series, the graded Betti numbers depend on the characteristic of the ground field K. Ho ...
Modular Arithmetic - Jean Mark Gawron
... And in general the additive inverse of any k mod n is n − k. So additive inverses in modular arithmetic differ from multiplicative inverses. Every number is guaranteed to have an additive inverse mod n. But only numbers relatively prime to have multiplicative inverses. Summarizing the discussion of ...
... And in general the additive inverse of any k mod n is n − k. So additive inverses in modular arithmetic differ from multiplicative inverses. Every number is guaranteed to have an additive inverse mod n. But only numbers relatively prime to have multiplicative inverses. Summarizing the discussion of ...
Chapter 3: Elementary Number Theory And Methods of Proof
... (Unique Factorization Theorem; Fundamental Theorem of Arithmetic) Given any integer n > 1, there exists a positive integer k , distinct prime numbers p1 , p2 ,. . . , pk , and positive integers e1 , e2 , . . . , ek such that n = p1e1 p2e2 p3e3 . . . pkek , and any other way of writing n as a product ...
... (Unique Factorization Theorem; Fundamental Theorem of Arithmetic) Given any integer n > 1, there exists a positive integer k , distinct prime numbers p1 , p2 ,. . . , pk , and positive integers e1 , e2 , . . . , ek such that n = p1e1 p2e2 p3e3 . . . pkek , and any other way of writing n as a product ...
Constructions of the real numbers
... sometimes refer to totally ordered sets only as ordered sets, and to total orders simply as orders. The standard notation for the corresponding weak orders (≤) and converses (>) and (≥) will be used. We are only going to work with commutative rings and semirings with a multiplicative identity and wi ...
... sometimes refer to totally ordered sets only as ordered sets, and to total orders simply as orders. The standard notation for the corresponding weak orders (≤) and converses (>) and (≥) will be used. We are only going to work with commutative rings and semirings with a multiplicative identity and wi ...
A first step towards automated conjecture
... Abstract. We present a framework for encoding information about objects from higher arithmetic geometry. This framework is built around a new kind of data type called a Tannakian symbol. The arithmetic objects we have in mind include modular forms (and more general automorphic representations), elli ...
... Abstract. We present a framework for encoding information about objects from higher arithmetic geometry. This framework is built around a new kind of data type called a Tannakian symbol. The arithmetic objects we have in mind include modular forms (and more general automorphic representations), elli ...