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You searched for +publisher:"University of Notre Dame" +contributor:("Yu Xie, Committee Member"). Showing records 1 – 3 of 3 total matches.

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University of Notre Dame

1. Bernadette M. Boyle. On the Unimodality of Pure O-Sequences</h1>.

Degree: Mathematics, 2012, University of Notre Dame

In this dissertation, we will discuss some properties of pure O-sequences, which, due to Macaulay’s Inverse Systems, are in bijective correspondence with the Hilbert functions of Artinian level monomial algebras. In particular, we will focus on their unimodality. A sequence of numbers is unimodal if it does not increase after a strict decrease. There has been previous progress in this area. Specifically, it is known that all algebras in two variables are unimodal, due to Macaulay’s Maximal Growth Theorem. Furthermore, it is known that all pure O-sequences of type two in three variables are unimodal and the Hilbert function of monomial complete intersections are unimodal; these results are due to the Weak Lefschetz Property. In addition to these families of pure O-sequences which are unimodal, there are known families of pure O-sequences which fail to be unimodal. In particular, for any r > 2, there exists a monomial Artinian level algebra in r variables whose Hilbert function fails unimodality with an arbitrary number of peaks. We will focus on the question of whether particular socle types or socle degrees in a fixed number of variables guarantee unimodality. Since the Weak Lefschetz Property is only guaranteed to hold in the cases mentioned above, we will use different techniques in our approach. We will show that all pure O-sequences of type three in three variables and all pure O-sequence of type two in four variables are strictly unimodal. We will also show that all pure O-sequences with socle degree less than or equal to nine in three variables are unimodal and all pure O-sequences (except for possibly two cases) with socle degree less than or equal to four in four variables are unimodal. Finally we will show that for r greater than or equal to 4 and e greater than or equal to 7, there exists a non-unimodal pure O-sequence in r variables with socle degree e. Advisors/Committee Members: Dr. Claudia Polini, Committee Member, Dr. Juan Migliore, Committee Chair, Dr. Nero Budur, Committee Member, Dr. Yu Xie, Committee Member.

Subjects/Keywords: pure O-sequence; Hilbert function; unimodal; level monomial Artinian algebra

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Boyle, B. M. (2012). On the Unimodality of Pure O-Sequences</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/n296ww74n34

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Boyle, Bernadette M.. “On the Unimodality of Pure O-Sequences</h1>.” 2012. Thesis, University of Notre Dame. Accessed July 07, 2020. https://curate.nd.edu/show/n296ww74n34.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Boyle, Bernadette M.. “On the Unimodality of Pure O-Sequences</h1>.” 2012. Web. 07 Jul 2020.

Vancouver:

Boyle BM. On the Unimodality of Pure O-Sequences</h1>. [Internet] [Thesis]. University of Notre Dame; 2012. [cited 2020 Jul 07]. Available from: https://curate.nd.edu/show/n296ww74n34.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Boyle BM. On the Unimodality of Pure O-Sequences</h1>. [Thesis]. University of Notre Dame; 2012. Available from: https://curate.nd.edu/show/n296ww74n34

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation


University of Notre Dame

2. Bonnie Bradberry Smith. Cores of Monomial Ideals</h1>.

Degree: Mathematics, 2010, University of Notre Dame

In this dissertation, we describe the cores of several classes of monomial ideals. We also find bounds on the reduction numbers of these ideals. The first class of ideals which we consider is one coming from graph theory, the strongly stable ideals of degree two. We prove that a strongly stable ideal I of degree two in k[x1,…,xd] has the Gd property if and only if x{g-1}xd is in I, where g is the height of I. We also prove that any strongly stable ideal of degree two in k[x1,…,xd] which has the Gd property satisfies the Artin-Nagata property AN{d-1}. We then show that the core of such an ideal I is given by core(I)=I mg-1, where g is the height of I, and m is the homogeneous maximal ideal (x1,…,xd). We also show that the reduction number r(I) is at most g-1. Specifically, we prove that an ideal J=JI whose elements correspond to diagonals in the tableau associated to I is a minimal reduction of I, and that rJ(I) < g. The other classes of ideals which we consider are zero-dimensional monomial ideals in the polynomial ring R=k[x1,…,xd]. We show that, if such an ideal I is an almost complete intersection, then the core of I satisfies a (d+1)-fold symmetry property coming from the generators of I, and hence that the shape of core(I) is closely related to the shape of I. Using this symmetry property, we completely describe the shape of core(I) in the case where I has a monomial minimal reduction. Then, in the two-dimensional case, we give an algorithm for computing the core of I which allows us to prove that the minimal number of generators of core(I) is 2r+2, where r is the reduction number of I. We prove that this result holds even for ideals which do not have a minimal reduction which is monomial. We also describe the core of an ideal I in k[x,y] having more generators, in the case where I has a monomial minimal reduction. Finally, we consider a strongly stable ideal I in k[x,y] having a monomial minimal reduction J. We prove that rJ(I) leq mu(I)-2, and we give an algorithm for obtaining the core of I via its first coefficient ideal. Advisors/Committee Members: Claudia Polini, Committee Chair, Yu Xie, Committee Member, Nero Budur, Committee Member, Juan Migliore, Committee Member.

Subjects/Keywords: strongly stable ideals; almost complete intersections; cores of ideals; reductions of ideals

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Smith, B. B. (2010). Cores of Monomial Ideals</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/mg74qj74z7p

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Smith, Bonnie Bradberry. “Cores of Monomial Ideals</h1>.” 2010. Thesis, University of Notre Dame. Accessed July 07, 2020. https://curate.nd.edu/show/mg74qj74z7p.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Smith, Bonnie Bradberry. “Cores of Monomial Ideals</h1>.” 2010. Web. 07 Jul 2020.

Vancouver:

Smith BB. Cores of Monomial Ideals</h1>. [Internet] [Thesis]. University of Notre Dame; 2010. [cited 2020 Jul 07]. Available from: https://curate.nd.edu/show/mg74qj74z7p.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Smith BB. Cores of Monomial Ideals</h1>. [Thesis]. University of Notre Dame; 2010. Available from: https://curate.nd.edu/show/mg74qj74z7p

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation


University of Notre Dame

3. Angela Kohlhaas. The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>.

Degree: Mathematics, 2010, University of Notre Dame

Given an ideal I in a Noetherian ring R, the core of I is the intersection of all ideals contained in I with the same integral closure as I. The core naturally arises in the context of the Brianc{c}on-Skoda theorem as an ideal which contains the adjoint of a certain power of I. As the arbitrary-characteristic analog of the multiplier ideal, the adjoint is an important tool in the study of resolutions of singularities. The question of when the core and the adjoint of a power of I are equal has been tied to a celebrated conjecture of Kawamata about the non-vanishing of sections of line bundles. We show for certain classes of monomial ideals in the polynomial ring k[x1,ldots,xd] over a field of characteristic zero , core(I)=adj(Id) if and only if core(I) is integrally closed. In order to prove our main result, we further develop the theory of coefficient ideals in regular local rings of dimension two and study the combinatorial properties of the core of a monomial ideal via the symmetry of its exponent set. Advisors/Committee Members: Claudia Polini, Committee Member, Nero Budur, Committee Member, J. Chris Howk, Committee Chair, Juan Migliore, Committee Member, Yu Xie, Committee Member.

Subjects/Keywords: exponent set; commutative algebra; birational geometry

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Kohlhaas, A. (2010). The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/bv73bz62h3d

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Kohlhaas, Angela. “The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>.” 2010. Thesis, University of Notre Dame. Accessed July 07, 2020. https://curate.nd.edu/show/bv73bz62h3d.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Kohlhaas, Angela. “The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>.” 2010. Web. 07 Jul 2020.

Vancouver:

Kohlhaas A. The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>. [Internet] [Thesis]. University of Notre Dame; 2010. [cited 2020 Jul 07]. Available from: https://curate.nd.edu/show/bv73bz62h3d.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Kohlhaas A. The core of an ideal and its relationship to the adjoint and coefficient ideals</h1>. [Thesis]. University of Notre Dame; 2010. Available from: https://curate.nd.edu/show/bv73bz62h3d

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.