41 research outputs found
Optimal bounds for a colorful Tverberg--Vrecica type problem
We prove the following optimal colorful Tverberg-Vrecica type transversal
theorem: For prime r and for any k+1 colored collections of points C^l of size
|C^l|=(r-1)(d-k+1)+1 in R^d, where each C^l is a union of subsets (color
classes) C_i^l of size smaller than r, l=0,...,k, there are partition of the
collections C^l into colorful sets F_1^l,...,F_r^l such that there is a k-plane
that meets all the convex hulls conv(F_j^l), under the assumption that r(d-k)
is even or k=0.
Along the proof we obtain three results of independent interest: We present
two alternative proofs for the special case k=0 (our optimal colored Tverberg
theorem (2009)), calculate the cohomological index for joins of chessboard
complexes, and establish a new Borsuk-Ulam type theorem for (Z_p)^m-equivariant
bundles that generalizes results of Volovikov (1996) and Zivaljevic (1999).Comment: Substantially revised version: new notation, improved results,
additional references; 12 pages, 2 figure
Joint distribution for the Selmer ranks of the congruent number curves
summary:We determine the distribution over square-free integers of the pair , where is a curve in the congruent number curve family, is the image of isogeny , , and is the isogeny dual to
A result on the size of iterated sumsets in
In this paper we give a different approach to determining the cardinality of
-fold sumsets when has elements. This
enables us to provide more general result with a shorter and simpler proof. We
also obtain an upper bound for the value of when
is a set of elements with simplicial hull.Comment: It was noticed that the proof of the main Theorem could be improved
to give a more general result. Also some motivating examples are provide
Selmer group statistics in the congruent number family of elliptic curves
Prvi deo disertacije se bavi skupovima zbirova hA = {a1 + · · · + ah ∈
Zd : a1, . . . , ah ∈ A}, gde je A konačni skup u Zd. Poznato je da postoji konstanta
h0 ∈ N i polinom pA(X) takav da je pA(h) = |hA| za h ⩾ h0. Međutim, malo se zna
o polinomu, kao i o konstanti h0. Konus CA nad skupom A sadrži informacije
o hA, za svako h ∈ N. Kada skup A ima d + 2 elementa, mogu se eksplicitno
opisati polinom pA i konstanta h0. Kada A ima d + 3 elementa, nalazi se gore
ograniqee za broj elemenata skupa hA.
Drugi deo disertacije se bavi Selmerovim grupama u familiji eliptičkih
krivih pridruenih kongruentnim brojevima. Beskvadratan prirodan broj n je
kongruentan ako i samo ako postoji pravougli trougao sa celobrojnim duinama
stranica qija povrxina je n. Poznato je da je prirodan broj n kongruentan
ako i samo ako je rang eliptiqke krive En : y2 = x3 − n2x kao algebarske
grupe razliqit od nule. Selmerove grupe pridruene izogenijama eliptiqkih
krivih En su zanim ive, jer ihov rang nije mai od ranga krive En, pa kada
je rang Selmerovih grupa nula, tada je i rang krive En jednak nuli. Elementi
Selmerovih grupa se mogu predstaviti kao particije odreenog grafa, pa se na
taj naqin moe na'i distribucija ranga Selmerovih grupa.irst part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Zd :
a1, . . . , ah ∈ A}, where A is a finite set in Zd. It is known that there exists a constant
h0 ∈ N and a polynomial pA(X) such that pA(h) = |hA| for h ⩾ h0. However, little is
known of polynomial pA and constant h0. Cone CA over the set A contains information
about hA, for all h ∈ N. When A has d + 2 elements, polynomial pA and constant
h0 can be explicitly described. When A has d + 3 elements, an upper bound is found
for the number of elements of hA.
Second part of dissertation examines Selmer groups of elliptic curves in the con-
gruent number family. A squarefree natural number is congruent if and only if there
exists a right triangle with area n whose sides all have integer lengths. It is known
that n is a congruent number if and only if elliptic curve En : y2 = x3 − n2x has
nonzero rank as an algebraic group. Selmer groups of isogenies on En are interesting,
because their rank is not smaller than the rank of En, so when the Selmer groups have
rank zero, then the elliptic curve En also has rank zero. Elements of these Selmer
groups can be represented as partitions of a particular graph, from which one may
find the distribution of ranks of Selmer groups