Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified), for any elements a1, . . . , an of GF (q), there are distinct field elements b1, . . . , bn such that a1b1 + · · ·+ anbn = 0. This implies the classification of hyperplanes lying in the union of the hyperplanes Xi = Xj in a vector space over GF (q), and also the classi...

v det([1− ρ(Frobv)q v ]|W Iv )−1, W Iv = subspace of W fixed by inertia at v. Grothendieck gave a related description of ζ∗ X using continuous p-adic representations GF → GL(H ét,c(XF ,Qp)) =: GL(W ). These are unramified almost everywhere, including at all good places away from p. Here the ith cohomology group W i above vanishes for i > 2 dimX. Grothendieck proved that if we remove the contrib...

We give a proof of a theorem on the common divisibility of polynomials and permuted polynomials (over GF (2)) by a polynomial ( ).

The Galois number Gn(q) is defined to be the number of subspaces of the n-dimensional vector space over the finite field GF (q). When q is prime, we prove that Gn(q) is equal to the number Ln(q) of n-dimensional mod q lattices, which are defined to be lattices (that is, discrete additive subgroups of n-space) contained in the integer lattice Z and having the property that given any point P in t...

We discuss the commutativity of certain rings with unity 1 and one-sided s-unital rings under each of the following conditions: xr[xs, y] = ±[x, yt]xn, xr[xs, y] = ±xn[x, yt], xr[xs, y] = ±[x, yt]ym, and xr[xs, y] = ±ym[x, yt], where r, n, and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s[x, y] = 0 implies [x, y] = 0. Further, we ...

We describe the use of grammars as an approach to automatic function definition in Grammatical Evolution. The automatic generation of functions allows the evolution of both the function and the code belonging to the main body of the program which has the ability to call upon the evolving function. As proof of concept we apply a grammar using automatic function definition to the Santa Fe ant tra...

It is proved that, for a prime number p, showing that an n-element matroid is not representable over GF (p) requires only O(n) rank evaluations.

What do you do to start reading foundations in grammatical evolution for dynamic environments? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored ...

AssumeG is a connected reductive algebraic group defined over an algebraic closure K = Fp of the finite field of prime order p > 0. Furthermore, assume that F : G → G is a Frobenius endomorphism of G. In this article we give a formula for the value of any F -stable character sheaf of G at a unipotent element. This formula is expressed in terms of class functions of GF which are supported on a s...

The existence of a primitive element of GF (q) with certain properties is used to prove that all cycles that could theoretically be embedded in AG(2, q) and PG(2, q) can, in fact, be embedded there (i.e. these planes are ‘pancyclic’). We also study embeddings of wheel and gear graphs in arbitrary projective planes.