Physical Sciences and Mathematics Commons^™

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

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

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

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

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

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

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

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.

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

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

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

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.

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 …

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

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

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.