');
}
var Pages = new Array (
new Array('Home','./'),
new Array('CV','cv.pdf'),
new Array('Research','research.html'),
new Array('Teaching','teaching.html')
// new Array('Origami','origami.html'),
// new Array('Links','links.html')
);
function makeNavbar()
{
document.write('
');
document.write('
');
for(var i = 0; i < Pages.length; i++)
{
var page = Pages[i];
if (currentPage == page[0]) {
document.write('
');
}
function toggleDiv(divid)
{
if (document.getElementById(divid).style.display == 'none') {
document.getElementById(divid).style.display = 'block';
} else {
document.getElementById(divid).style.display = 'none';
}
}
var Papers = new Array (
new Array(
"An elementary (number theory) proof of Touchard's congruence",
'G.Hurst and A.Schultz',
'Under review',
"We use well known recurrence relations for Bell and Stirling numbers to develop a new recurrence relation on Bell numbers. This allows us to give a new proof of Touchard's congruence which uses some techniques approachable by undergraduates.",
'http://front.math.ucdavis.edu/0906.0696'
),
new Array(
'Galois module structure of Galois cohomology for embeddable cyclic extensions of degree pn',
'N.Lemire, J.Mináč, A.Schultz, and J.Swallow',
'Accepted pending revisions by Journal of the London Mathematical Society',
'In the case that E/F is a cyclic, degree pn extension which embeds in a cyclic, degree pn+1 extension L/F, we give the Galois module structure of the reduced Milnor k-groups kmE.',
'http://front.math.ucdavis.edu/0904.3719'
),
new Array(
'Cyclic algebras, Schur indices, norms, and Galois modules',
'J.Mináč, A.Schultz, and J.Swallow',
'Under review',
"Let p be a prime and suppose that K/F is a cyclic extension of degree pn with group G. Let J be the FpG-module K*/K*p of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be pi+1 for some 0≤ i < n. We show that for all primes p and all 0 ≤ i< n, there exists a field extension K/F with a summand of dimension pi+1.",
'http://arxiv.org/pdf/math.NT/0410536'
),
new Array(
'Hilbert 90 for Galois cohomology',
'N.Lemire, J.Mináč, A.Schultz, and J.Swallow',
'Accepted pending revisions by Communications in Algebra',
"Assuming the Bloch-Kato Conjecture, we determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n.",
'http://arxiv.org/pdf/math.NT/0601663'
),
new Array(
'Galois module structure of Milnor K-theory mod ps in characteristic p',
'J.Mináč, A.Schultz, and J.Swallow',
'Appeared in New York Journal of Mathematics 14: 225--233 (2008).',
"Let E be a cyclic extension of pth-power degree of a field F of characteristic p. For all m, s in N, we determine K_mE/p^sK_mE as a (Z/p^sZ)[Gal(E/F)]-module. We also provide examples of extensions for which all of the possible nonzero summands in the decomposition are indeed nonzero.",
'http://arxiv.org/pdf/math.NT/0602546'
),
new Array(
'Automatic realizations of Galois groups with cyclic quotient of order pn',
'J.Mináč, A.Schultz, and J.Swallow',
'Appeared in Journal de Théorie des Nombres de Bordeaux 20: 419--430 (2008).',
"We establish automatic realizations of Galois groups among groups M semidirect G, where G is a cyclic group of order p^n for a prime p and M is a quotient of the group ring Fp[G].",
'http://arxiv.org/pdf/math.NT/0603594'
),
new Array(
'Hilbert 90 for biquadratic extensions',
'R.Dwilewicz, J.Mináč, A.Schultz, and J.Swallow',
'Appeared in American Mathematical Monthly 114:7 (August-September 2007), 577--587',
"Hilbert's Theorem 90 is a classical result in the theory of cyclic extensions. The quadratic case of Hilbert 90, however, generalizes in noncyclic directions as well. Informed by a poem of Richard Wilbur, the article explores several generalizations, discerning connections among multiplicative groups of fields, values of binary quadratic forms, a bit of module theory over group rings, and even Galois cohomology.",
'http://arxiv.org/pdf/math.NT/0510154'
),
new Array(
'Galois module structure of Galois cohomology (thesis)',
'A.Schultz',
'Final Draft',
"My thesis, in which I explore the structure of certain Galois cohomology groups in the vein of Mináč and Swallow. This includes generalizations to p-adic extensions, the first result on pro-p groups in this family of projects.",
'http://www.math.uiuc.edu/~acs/thesis.pdf'
),
new Array(
'Galois module structure of pth-power classes of cyclic extensions of degree pn',
'J.Mináč, A.Schultz, and J.Swallow',
'Appeared in Proc. London Math. Soc.92 (2006), 307--341',
"In the mid-1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F.",
'http://arxiv.org/pdf/math.NT/0409532'
),
new Array(
'A five color zero-sum generalization',
'D.Grynkiewicz and A. Schultz',
'Appeared in Graphs and Combinatorics22 (2006), 351 -- 360',
"Several theorems of Ramsey-type have been generalized by considering Z/mZ-colorings and zero-sum configurations rather than 2-colorings and monochromatic configurations; such generalizations are called EGZ. There is also a notion of EGZ generalizations of r-colorings for r>2. Recently in a sequence of three papers the first author showed that a certain Ramsey-type theorem with 4-colors admitted an EGZ generalization. In this paper we show that a Ramsey-type problem with 5-colors considered in a paper of the second author admits an EGZ generalization. This is the only EGZ generalization in 5 colors which known to the authors at the time of this paper.",
'GS.pdf'
),
new Array(
'On a Modification of a Problem of Bialostocki, Erdös, and Lefmann',
'A.Schultz',
'Appeared in Discrete Math.306 (2006), 244--253',
"For positive integers m and r, one can easily show there exist integers N such that for every map D:{1,2,...,N} -> {1,2,...,r} there exist 2m integers x_1 < ... < x_m < y_1 < ... < y_m which satisfy:
D(x_1) = ... = D(x_m),
D(y_1) = ... = D(y_m), and
2(x_m-x_1) \leq y_m-x_1.
In this paper we investigate the minimal such integer, which we call g(m,r). We compute g(m,2) for m \geq 2; g(m,3) for m \geq 4; and g(m,4) for m \geq 3. Furthermore, we consider g(m,r) for general r. Along with results that bound g(m,r), we compute g(m,r) exactly for the following infinite families of r: {f_{2n+3}}, {2f_{2n+3}}, {18f_{2n}-7f_{2n-2}}, and {23f_{2n}-9f_{2n-2}}, where here f_i is the ith Fibonacci number defined by f_0 = 0 and f_1=1.",
'http://arxiv.org/pdf/math.CO/0506545'
)
);
function makePapers()
{
document.write('
');
for(var i = 0; i < Papers.length; i++)
{
var paper = Papers[i];
document.write('