You are here

Download A theory of formal deducibility by Haskell Curry PDF

By Haskell Curry

ISBN-10: 0268002746

ISBN-13: 9780268002749

Show description

Read Online or Download A theory of formal deducibility PDF

Similar number theory books

Automorphic forms and Shimura varieties of PGSp (2)

This booklet furthers new and fascinating advancements in experimental designs, multivariate research, biostatistics, version choice and comparable topics. It good points articles contributed by means of many in demand and lively figures of their fields. those articles conceal a wide range of vital concerns in glossy statistical thought, tools and their purposes.

p-adic Numbers: An Introduction

P-adic numbers are of significant theoretical significance in quantity thought, due to the fact that they permit using the language of study to check difficulties concerning toprime numbers and diophantine equations. extra, they provide a realm the place you will do issues which are similar to classical research, yet with effects which are relatively strange.

Extra info for A theory of formal deducibility

Sample text

What about σ? From the preceding, we see that log σ(z + ω) = log σ(z) + η(ω)z + c(ω) for some function c on the lattice. It is convenient to write this as σ(z + ω) = ψ(ω)eη(ω)(z+ω/2) σ(z) thereby defining ψ(ω). Suppose first that ω/2 ∈ / L. Setting z = −ω/2 above and using the fact that σ is odd, we see at once that ψ(ω) = −1. On the other hand, σ(z + 2ω) σ(z + ω) σ(z + 2ω) = σ(z) σ(z + ω) σ(z) Elliptic Functions 45 and so by applying the functional equation twice and using the fact that η(2ω) = 2η(ω), we get ψ(2ω) = ψ(ω)2 .

We will write P ≺ Q if Q dominates P . It is easily verified that if P1 ≺ Q1 and P2 ≺ Q2 , then P1 + P2 ≺ Q1 + Q2 and P1 P2 ≺ Q1 Q2 . Moreover, if Di is the derivative operator with respect to the i-th variable and P ≺ Q, then Di P ≺ Di Q. If the total degree of a polynomial P in n variables is r, then P ≺ size(P )(1 + x1 + · · · + xn )r . We also need some facts about derivations. Recall that a derivation D of a ring R is a map D : R → R such that D(x + y) = D(x) + D(y) and which satisfies D(xy) = D(x)y + xD(y).

We end this chapter by noting the following result for future reference. 4 The numbers ℘(ω1 /2), ℘(ω2 /2) and ℘((ω1 + ω2 )/2) are distinct. Proof. Suppose not. Let L be the lattice spanned by ω1 , ω2 which are linearly independent over R. Let us consider the function f1 (z) = ℘(z) − ℘(ω1 /2). This has a double order zero at z = ω1 /2 since ℘ (ω1 /2) = 0. 4. It follows that any zero must be congruent to ω1 /2 modulo L. If ℘(ω2 /2) = ℘(ω1 /2), then we would have ω1 ≡ ω2 modulo L, contrary to their linear independence over R.

Download PDF sample

Rated 4.05 of 5 – based on 35 votes
Top