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Articles 1 - 24 of 24
Full-Text Articles in Mathematics
The Smooth 4-Genus Of (The Rest Of) The Prime Knots Through 12 Crossings, Mark Brittenham, Susan Hermiller
The Smooth 4-Genus Of (The Rest Of) The Prime Knots Through 12 Crossings, Mark Brittenham, Susan Hermiller
Department of Mathematics: Faculty Publications
We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.
Regularity Criteria For The Kuramoto-Sivashinsky Equation In Dimensions Two And Three, Adam Larios, Mohammad Mahabubur Rahman, Kazuo Yamazaki
Regularity Criteria For The Kuramoto-Sivashinsky Equation In Dimensions Two And Three, Adam Larios, Mohammad Mahabubur Rahman, Kazuo Yamazaki
Department of Mathematics: Faculty Publications
We propose and prove several regularity criteria for the 2D and 3D Kuramoto-Sivashinsky equation, in both its scalar and vector forms. In particular, we examine integrability criteria for the regularity of solutions in terms of the scalar solution ∅, the vector solution u ≜ ∇∅, as well as the divergence div(u) = Δ∅, and each component of u and ∇u. We also investigate these criteria computationally in the 2D case, and we include snapshots of solutions for several quantities of interest that arise in energy estimates.
Level And Gorenstein Projective Dimension, Laila Awadalla, Thomas Marley
Level And Gorenstein Projective Dimension, Laila Awadalla, Thomas Marley
Department of Mathematics: Faculty Publications
We investigate the relationship between the level of a bounded complex over a commutative ring with respect to the class of Gorenstein projective modules and other invariants of the complex or ring, such as projective dimension, Gorenstein projective dimension, and Krull dimension. The results build upon work done by J. Christensen [7], H. Altmann et al. [1], and Avramov et al. [4] for levels with respect to the class of finitely generated projective modules.
The concept of level in a triangulated category, first defined by Avramov, Buch- weitz, Iyengar, and Miller [4], is a measure of how many mapping cones …
The Phase Transition Of Discrepancy In Random Hypergraphs, Calum Macrury, Tomáš Masarík, Leilani Pai, Xavier Perez Gimenez
The Phase Transition Of Discrepancy In Random Hypergraphs, Calum Macrury, Tomáš Masarík, Leilani Pai, Xavier Perez Gimenez
Department of Mathematics: Faculty Publications
Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on n vertices and m edges. In the first (edge-independent) model, a random hypergraph H1 is constructed by fixing a parameter p and allowing each of the n vertices to join each of the m edges independently with probability p. In the parameter range in which pn ⟶ ∞ and pm ⟶ ∞, we show that with high probability (w.h.p.) H1 has discrepancy at least Ω(2-n/m √pn) when m = O(n …
Cohomological Blow Ups Of Graded Artinian Gorenstein Algebras Along Surjective Maps, Anthony Iarrobino, Pedro Macias Marques, Chris Mcdaniel, Alexandra Seceleanu, Junzo Watanabe
Cohomological Blow Ups Of Graded Artinian Gorenstein Algebras Along Surjective Maps, Anthony Iarrobino, Pedro Macias Marques, Chris Mcdaniel, Alexandra Seceleanu, Junzo Watanabe
Department of Mathematics: Faculty Publications
We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are …
Bernstein-Sato Polynomials In Commutative Algebra, Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez-Betancourt
Bernstein-Sato Polynomials In Commutative Algebra, Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez-Betancourt
Department of Mathematics: Faculty Publications
This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.
Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo
Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo
Department of Mathematics: Faculty Publications
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.
Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo
Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo
Department of Mathematics: Faculty Publications
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.
A Combinatorial Formula For Kazhdan-Lusztig Polynomials Of Sparse Paving Matroids, George Nasr
A Combinatorial Formula For Kazhdan-Lusztig Polynomials Of Sparse Paving Matroids, George Nasr
Department of Mathematics: Dissertations, Theses, and Student Research
We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as for uniform matroids, our formula has a nice combinatorial interpretation.
Advisers: Kyungyong Lee and Jamie Radclie
Bootstrap Percolation On Random Geometric Graphs, Alyssa Whittemore
Bootstrap Percolation On Random Geometric Graphs, Alyssa Whittemore
Department of Mathematics: Dissertations, Theses, and Student Research
Bootstrap Percolation is a discrete-time process that models the spread of information or disease across the vertex set of a graph. We consider the following version of this process:
Initially, each vertex of the graph is set active with probability p or inactive otherwise. Then, at each time step, every inactive vertex with at least k active neighbors becomes active. Active vertices will always remain active. The process ends when it reaches a stationary state. If all the vertices eventually become active, then we say we achieve percolation.
This process has been widely studied on many families of graphs, deterministic …
Free Complexes Over The Exterior Algebra With Small Homology, Erica Hopkins
Free Complexes Over The Exterior Algebra With Small Homology, Erica Hopkins
Department of Mathematics: Dissertations, Theses, and Student Research
Let M be a graded module over a standard graded polynomial ring S. The Total Rank Conjecture by Avramov-Buchweitz predicts the total Betti number of M should be at least the total Betti number of the residue field. Walker proved this is indeed true in a large number of cases. One could then try to push this result further by generalizing this conjecture to finite free complexes which is known as the Generalized Total Rank Conjecture. However, Iyengar and Walker constructed examples to show this generalized conjecture is not always true.
In this thesis, we investigate other counterexamples of …
Results On Nonorientable Surfaces For Knots And 2-Knots, Vincent Longo
Results On Nonorientable Surfaces For Knots And 2-Knots, Vincent Longo
Department of Mathematics: Dissertations, Theses, and Student Research
A classical knot is a smooth embedding of the circle into the 3-sphere. We can also consider embeddings of arbitrary surfaces (possibly nonorientable) into a 4-manifold, called knotted surfaces. In this thesis, we give an introduction to some of the basics of the studies of classical knots and knotted surfaces, then present some results about nonorientable surfaces bounded by classical knots and embeddings of nonorientable knotted surfaces. First, we generalize a result of Satoh about connected sums of projective planes and twist spun knots. Specifically, we will show that for any odd natural n, the connected sum of the n-twist …
Near-Optimal Learning Of Tree-Structured Distributions By Chow-Liu, Arnab Bhattacharyya, Sutanu Gayen, Eric Price, N. V. Vinodchandran
Near-Optimal Learning Of Tree-Structured Distributions By Chow-Liu, Arnab Bhattacharyya, Sutanu Gayen, Eric Price, N. V. Vinodchandran
Department of Mathematics: Faculty Publications
We provide finite sample guarantees for the classical Chow-Liu algorithm (IEEE Trans. Inform. Theory, 1968) to learn a tree-structured graphical model of a distribution. For a distribution P on Σn and a tree T on n nodes, we say T is an ε-approximate tree for P if there is a T-structured distribution Q such that D(P || Q) is at most ε more than the best possible tree-structured distribution for P. We show that if P itself is tree-structured, then the Chow-Liu algorithm with the plug-in estimator for mutual information with eO (|Σ| …
Modernization Of Scienttific Mathematics Formula In Technology, Iwasan D. Kejawa Ed.D, Prof. Iwasan D. Kejawa Ed.D
Modernization Of Scienttific Mathematics Formula In Technology, Iwasan D. Kejawa Ed.D, Prof. Iwasan D. Kejawa Ed.D
Department of Mathematics: Faculty Publications
Abstract
Is it true that we solve problem using techniques in form of formula? Mathematical formulas can be derived through thinking of a problem or situation. Research has shown that we can create formulas by applying theoretical, technical, and applied knowledge. The knowledge derives from brainstorming and actual experience can be represented by formulas. It is intended that this research article is geared by an audience of average knowledge level of solving mathematics and scientific intricacies. This work details an introductory level of simple, at times complex problems in a mathematical epidermis and computability and solvability in a Computer Science. …
Chudnovsky's Conjecture And The Stable Harbourne-Huneke Containment, Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thái Thành Nguyên
Chudnovsky's Conjecture And The Stable Harbourne-Huneke Containment, Sankhaneel Bisui, Eloísa Grifo, Huy Tài Hà, Thái Thành Nguyên
Department of Mathematics: Faculty Publications
We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.
Demailly's Conjecture And The Containment Problem, Sankhaneel Bisui, Eloisa Grifo, Huy Tài Hà, Thái Thành Nguyên
Demailly's Conjecture And The Containment Problem, Sankhaneel Bisui, Eloisa Grifo, Huy Tài Hà, Thái Thành Nguyên
Department of Mathematics: Faculty Publications
We investigate Demailly’s Conjecture for a general set of sufficiently many points. Demailly’s Conjecture generalizes Chudnovsky’s Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly’s bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.
N-Fold Matrix Factorizations, Eric Hopkins
N-Fold Matrix Factorizations, Eric Hopkins
Department of Mathematics: Dissertations, Theses, and Student Research
The study of matrix factorizations began when they were introduced by Eisenbud; they have since been an important topic in commutative algebra. Results by Eisenbud, Buchweitz, and Yoshino relate matrix factorizations to maximal Cohen-Macaulay modules over hypersurface rings. There are many important properties of the category of matrix factorizations, as well as tensor product and hom constructions. More recently, Backelin, Herzog, Sanders, and Ulrich used a generalization of matrix factorizations -- so called N-fold matrix factorizations -- to construct Ulrich modules over arbitrary hypersurface rings. In this dissertation we build up the theory of N-fold matrix factorizations, proving analogues of …
Constructing Non-Proxy Small Test Modules For The Complete Intersection Property, Benjamin Briggs, Eloísa Grifo, Josh Pollitz
Constructing Non-Proxy Small Test Modules For The Complete Intersection Property, Benjamin Briggs, Eloísa Grifo, Josh Pollitz
Department of Mathematics: Faculty Publications
A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category Df(R), which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in Df(R) is proxy small. In this paper, we study a …
Frobenius And Homological Dimensions Of Complexes, Taran Funk
Frobenius And Homological Dimensions Of Complexes, Taran Funk
Department of Mathematics: Dissertations, Theses, and Student Research
Much work has been done showing how one can use a commutative Noetherian local ring R of prime characteristic, viewed as algebra over itself via the Frobenius endomorphism, as a test for flatness or projectivity of a finitely generated module M over R. Work on this dates back to the famous results of Peskine and Szpiro and also that of Kunz. Here I discuss what work has been done to push this theory into modules which are not necessarily finitely generated, and display my work done to weaken the assumptions needed to obtain these results.
Adviser: Tom Marley
Gauge-Invariant Uniqueness And Reductions Of Ordered Groups, Robert Huben
Gauge-Invariant Uniqueness And Reductions Of Ordered Groups, Robert Huben
Department of Mathematics: Dissertations, Theses, and Student Research
A reduction φ of an ordered group (G,P) to another ordered group is an order homomorphism which maps each interval [1, p] bijectively onto [1, φ(p)]. We show that if (G,P) is weakly quasi-lattice ordered and reduces to an amenable ordered group, then there is a gauge-invariant uniqueness theorem for P -graph algebras. We also consider the class of ordered groups which reduce to an amenable ordered group, and show this class contains all amenable ordered groups and is closed under direct products, free products, and hereditary subgroups.
Adviser: Mark Brittenham and David Pitts
Free Semigroupoid Algebras From Categories Of Paths, Juliana Bukoski
Free Semigroupoid Algebras From Categories Of Paths, Juliana Bukoski
Department of Mathematics: Dissertations, Theses, and Student Research
Given a directed graph G, we can define a Hilbert space HG with basis indexed by the path space of the graph, then represent the vertices of the graph as projections on HG and the edges of the graph as partial isometries on HG. The weak operator topology closed algebra generated by these projections and partial isometries is called the free semigroupoid algebra for G. Kribs and Power showed that these algebras are reflexive, and that they are semisimple if and only if each path in the graph lies on a cycle. We extend …
On The Bounded Negativity Conjecture And Singular Plane Curves, Alexandru Dimca, Brian Harbourne, Gabriel Sticlaru
On The Bounded Negativity Conjecture And Singular Plane Curves, Alexandru Dimca, Brian Harbourne, Gabriel Sticlaru
Department of Mathematics: Faculty Publications
There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the H-constant of a rational curve C with at most 9 singular points satisfies H(C) > -2 regardless of the characteristic.
Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu
Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu
Department of Mathematics: Faculty Publications
We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.
Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu
Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu
Department of Mathematics: Faculty Publications
We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.