73 research outputs found
Path ideals of rooted trees and their graded Betti numbers
Let be a rooted tree and let be a positive integer. We study
algebraic invariants and properties of the path ideal generated by monomial
corresponding to paths of length in . In particular, we give a
recursive formula to compute the graded Betti numbers, a general bound for the
regularity, an explicit computation of the linear strand, and we characterize
when this path ideal has a linear resolution.Comment: 18 page
ALGEBRAIC PROPERTIES OF EDGE IDEALS
Given a simple graph G, the corresponding edge ideal IG is the ideal generated by the edges of G. In 2007, Ha and Van Tuyl demonstrated an inductive procedure to construct the minimal free resolution of certain classes of edge ideals. We will provide a simplified proof of this inductive method for the class of trees. Furthermore, we will provide a comprehensive description of the finely graded Betti numbers occurring in the minimal free resolution of the edge ideal of a tree. For specific subclasses of trees, we will generate more precise information including explicit formulas for the projective dimensions of the quotient rings of the edge ideals. In the second half of this thesis, we will consider the class of simple bipartite graphs known as Ferrers graphs. In particular, we will study a class of monomial ideals that arise as initial ideals of the defining ideals of the toric rings associated to Ferrers graphs. The toric rings were studied by Corso and Nagel in 2007, and by studying the initial ideals of the defining ideals of the toric rings we are able to show that in certain cases the toric rings of Ferrers graphs are level
Fibonacci numbers and resolutions of domino ideals
This paper considers a class of monomial ideals, called domino ideals, whose generating sets correspond to the sets of domino tilings of a tableau. The multi-graded Betti numbers are shown to be in one-to-one correspondence with equivalence classes of sets of tilings. It is well-known that the number of domino tilings of a tableau is given by a Fibonacci number. Using the bijection, this relationship is further expanded to show the relationship between the Fibonacci numbers and the graded Betti numbers of the corresponding domino ideal
Finite Sum Representations of Elements in R and R2
In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a masterâs student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2
Finite Sum Representations of Elements in R and R2
In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a masterâs student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2
Quasi-ordered photonic structures colour the bluespotted ribbontail ray
Due to the scarcity of blue colour exhibited by natural organisms, highlighting the underlying this colour mechanisms is always very impactful for the understanding of the natural world. In this research, the colour of the blue rounded spots occurring in the skin of Taeniura lymma stingray was unveiled by a combination of experimental and numerical techniques. Our results demonstrated that this blue colour arises from coherent scattering in quasi-ordered photonic structures occurring in the skin of this stingray.</p
Quasi-ordered photonic structures colour the bluespotted ribbontail ray
Due to the scarcity of blue colour exhibited by natural organisms, highlighting the underlying this colour mechanisms is always very impactful for the understanding of the natural world. In this research, the colour of the blue rounded spots occurring in the skin of Taeniura lymma stingray was unveiled by a combination of experimental and numerical techniques. Our results demonstrated that this blue colour arises from coherent scattering in quasi-ordered photonic structures occurring in the skin of this stingray.</p
Structure of 12Be: intruder d-wave strength at N=8
The breaking of the N=8 shell-model magic number in the 12Be ground state has
been determined to include significant occupancy of the intruder d-wave
orbital. This is in marked contrast with all other N=8 isotones, both more and
less exotic than 12Be. The occupancies of the 0 hbar omega neutron p1/2-orbital
and the 1 hbar omega, neutron d5/2 intruder orbital were deduced from a
measurement of neutron removal from a high-energy 12Be beam leading to bound
and unbound states in 11Be.Comment: 5 pages, 2 figure
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
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