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How to Explicitly Solve a Thue-Mahler Equation
, M.Sc. Thesis Sharif University of Technology ; Ebrahimi Borojeni, Javad (Supervisor) ; Ghadermarzi, Amir (Supervisor)
Abstract
David Hilbert posed a list of mathematical problems in 1900. Hilbert's tenth problem was "Is there an algorithm to determine whether a given Diophantine equation, has a solution with all unknowns taking integer values." Although Matiyasevich Showed that the answer to this problem is negative in the general case; the answer of this problem for a specific diophantine equation $f(x,y) $ that has rational coefficients, is unknown. Thue-Mahler equation is a diophantine equation of the form: $$F(x,y)=c\cdot p_1^{z_1}\cdots p_v^{z_v}$$ where $F$ is homogeneous with integer coefficients, degree $n\geq3$ and $p_1,\ldots p_v$ are distinct rational primes ($v\geq1$). All the unknowns are...
On the size of the minimum critical set of a Latin square
, Article Discrete Mathematics ; Volume 293, Issue 1-3 , 2005 , Pages 121-127 ; 0012365X (ISSN) ; Hatami, H ; Mahmoodian, E. S ; Sharif University of Technology
2005
Abstract
A critical set in an n×n array is a set C of given entries, such that there exists a unique extension of C to an n×n Latin square and no proper subset of C has this property. For a Latin square L, scs(L) denotes the size of the smallest critical set of L, and scs(n) is the minimum of scs(L) over all Latin squares L of order n. We find an upper bound for the number of partial Latin squares of size k and prove thatn2-(e+o(1))n10/6≤maxscs(L)≤n2- π2n9/6.This improves on a result of Cavenagh (Ph.D. Thesis, The University of Queensland, 2003) and disproves one of his conjectures. Also it improves the previously known lower bound for the size of the largest critical set of any Latin square of order...
On the diameters of commuting graphs
, Article Linear Algebra and Its Applications ; Volume 418, Issue 1 , 2006 , Pages 161-176 ; 00243795 (ISSN) ; Mohammadian, A ; Radjavi, H ; Raja, P ; Sharif University of Technology
2006
Abstract
The commuting graph of a ring R, denoted by Γ (R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ≥ 3. In this paper we investigate the diameters of Γ(Mn(D)) and determine the diameters of some induced subgraphs of Γ(Mn(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in Mn(D). For every field F, it is shown that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) ≤ 6. We conjecture that if Γ(Mn(F)) is a connected graph, then diam Γ(Mn(F)) ≤ 5. We show that if F is an algebraically closed field or n...