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Full-Text Articles in Physical Sciences and Mathematics

The Serendipity Of Discovery: Life Of A Geochemist, Willard S. Moore Jan 2025

The Serendipity Of Discovery: Life Of A Geochemist, Willard S. Moore

Faculty Publications

My strategy for writing this autobiography is to use examples of how working on seemingly different projects can often lead to outcomes more important than originally envisioned. Serendipity is a "happy accident"; specifically, the accident of discovering something useful without directly looking for it. This often occurs when two research projects converge unexpectedly. The main text contains examples of how serendipity has led me to important discoveries including: (a) finding surprisingly high 228Ra activities in the ocean, (b) developing a means of rapidly and quantitatively extracting radium from seawater, (c) devising a rapid, sensitive method of measuring 224Ra …


Development Of A Scituate Reservoir Mitigation Model For Assessing Flood Control During Extreme Scenarios, Supria Paul Jan 2025

Development Of A Scituate Reservoir Mitigation Model For Assessing Flood Control During Extreme Scenarios, Supria Paul

Open Access Dissertations

Effective reservoir management is essential for mitigating downstream flooding, especially with the increasing frequency and intensity of extreme weather events impacting major drinking water reservoirs such as the Scituate Reservoir in Rhode Island. These reservoirs provide water for large populations but can be overwhelmed by intense precipitation, flooding and land cover changes. Climate change is expected to increase precipitation in the study area located in New England, highlighting the need for improved reservoir systems, especially given the aging dams and higher risk of failures. Additionally, land cover changes, like the widespread defoliation caused by the Gypsy Moth infestations, affect water …


Light, Symbiosis, And The Biology Of Corals, Taylor R. Lindsay Jan 2025

Light, Symbiosis, And The Biology Of Corals, Taylor R. Lindsay

Open Access Dissertations

Light is critical for the survival of most shallow water scleractinian corals. Corals are mixotrophs, acquiring energy via both feeding on water column plankton and photosynthesis via a symbiotic relationship with dinoflagellates living within their tissues. Corals rely heavily on photosynthetically derived metabolites through this symbiotic relationship, intrinsically linking their biology to light availability. Corals are known to modulate their biology to maximize light capture. However, there is a limited understanding of the interaction between morphology, physiology, and energetic strategy, as well as how light may affect these relationships. In Chapter 1, we quantified the morphology, physiology and energetic …


Stability And Global Dynamics Of Autonomous And Nonautonomous Discrete Population Models With Stocking, Harvesting, And Cooperation, Sam Habach Jan 2025

Stability And Global Dynamics Of Autonomous And Nonautonomous Discrete Population Models With Stocking, Harvesting, And Cooperation, Sam Habach

Open Access Dissertations

This dissertation investigates the local and global behavior of several classes of discrete population models involving stocking, harvesting, and cooperation. The study is divided into three major manuscripts, each offering new theoretical in sights, bifurcation results, and applications to real-world data.

In Manuscript 1, we study the Beverton-Holt population models with constant and proportional stocking and harvesting, as well as sigmoid Beverton-Holt models under constant stocking and harvesting, both in autonomous and non-autonomous settings. We analyze stability and explore global dynamics using established local and global dynamics theorems, bifurcation theory, and explicit solutions when available. Special attention is given to …


Impacts Of Multiple Coastal Stressors Across Life-History Stages In The Eastern Oyster, Amaelia Zyck Jan 2025

Impacts Of Multiple Coastal Stressors Across Life-History Stages In The Eastern Oyster, Amaelia Zyck

Open Access Dissertations

Urbanized estuaries are characterized as a complex of biotic and abiotic stressors, which currently challenge marine life and are expected to intensify and become increasingly unpredictable under the ongoing impacts of climate change. The persistence of coastal species that inhabit these stressful environments will ultimately depend on their ability to adapt. Many of these species have complex life cycles, featuring distinct morphological and physiological developmental stages that can exhibit unique responses to environmental pressures. However, since all stages share the same genome, selective pressures acting on one stage can have cascading effects throughout the life cycle. The larval stage, being …


Analysis Of Early Interventions To Retain Underrepresented Students In Computer Science, Michael Conti Jan 2025

Analysis Of Early Interventions To Retain Underrepresented Students In Computer Science, Michael Conti

Open Access Dissertations

Computer science, like many STEM disciplines, faces persistent challenges in recruiting and retaining women and individuals from racially and ethnically minoritized backgrounds. This study examines whether targeted interventions can produce sustained improvements in academic performance and sense of belonging among these underrepresented groups. By analyzing longitudinal data, this research aims to evaluate the effectiveness of these interventions in promoting equity and persistence in computer science education.


Beyond The Phonon Gas Model (Pgm): Unraveling The Unique Mechanistic Processes Dictating Thermal Transport In Highly Anharmonic And Disordered Materials, Sandip Thakur Jan 2025

Beyond The Phonon Gas Model (Pgm): Unraveling The Unique Mechanistic Processes Dictating Thermal Transport In Highly Anharmonic And Disordered Materials, Sandip Thakur

Open Access Dissertations

Thermal transport in nonmetallic solids has traditionally been described by the phonon gas model (PGM), in which heat is carried by weakly interacting phonon quasiparticles in an ordered crystalline lattice. However, this model breaks down in materials characterized by strong anharmonicity, dynamic disorder, or structural complexity, features prevalent in many next-generation materials used in energy conversion, optoelectronics, and thermal management. This dissertation investigates thermal transport in such complex materials, including metal halide perovskites (MHPs), covalent organic frameworks (COFs), metal organic frameworks (MOFs), and 2D-3D heterostructures, through large-scale molecular dynamics (MD) simulations and frequency-resolved spectral analyses.

In MHPs, the work reveals …


Bioaccumulation Of Legacy And Novel Pfas In The Environment, Hesham Taher, Rainer Lohmann Jan 2025

Bioaccumulation Of Legacy And Novel Pfas In The Environment, Hesham Taher, Rainer Lohmann

Graduate School of Oceanography Faculty Publications

The bioaccumulation of per- and polyfluoroalkyl substances (PFAS), both legacy and novel, in the environment presents significant ecological and health risks. PFAS are a diverse group of synthetic chemicals known for their persistence and bioaccumulation, which can cause widespread environmental contamination and health risks. Strong carbon-fluorine bonds give these compounds unparalleled stability, preventing them from degrading and enabling them to endure in a range of environmental matrices, including water, soil, and biota. Legacy PFAS, including perfluorooctanoic acid (PFOA) and perfluorooctane sulfonic acid (PFOS), have been extensively studied and regulated, resulting in lower concentrations in some environmental media. However, novel PFAS, …


Estimating The Carbon Footprint Of Sheep Production In Western Australia, Tafesse Estifanos, Christophe D'Abbadie, Johnny Machon, Amir Abadi Abadi Ghadim, Gus Manatsa, Paul Mattingley, Ross Kingwell, Kevin Foster Jan 2025

Estimating The Carbon Footprint Of Sheep Production In Western Australia, Tafesse Estifanos, Christophe D'Abbadie, Johnny Machon, Amir Abadi Abadi Ghadim, Gus Manatsa, Paul Mattingley, Ross Kingwell, Kevin Foster

Farm Systems and Economics published reports

Western Australia’s broadacre mixed farming systems combine cropping and sheep production and are the largest component of the state’s agricultural sector. The south west Western Australia (WA) agricultural area hosts about 98% of the state's sheep flock. The sheep industry contributed $1.5 billion to the state’s economy in 2021-22 and accounts for about a quarter of on-farm greenhouse gas (GHG) emissions. Most sheep industry emissions (over 80%) come from methane produced by the sheep. Despite this environmental impact, only 9.9% of Australian sheep producers have estimated their emissions, which limits the industry's ability to monitor and manage carbon footprints effectively. …


Feature Engineering And Anchor Optimization For Enhancing Faster R-Cnn Detection Of Low-Contrast Steel Surface Defects, Herdianti Darwis, Sitti Nurhalimah, Huzain Azis Jan 2025

Feature Engineering And Anchor Optimization For Enhancing Faster R-Cnn Detection Of Low-Contrast Steel Surface Defects, Herdianti Darwis, Sitti Nurhalimah, Huzain Azis

Knowledge Engineering and Data Science

Detection of defects on low-contrast steel surfaces, especially crazing and rolled-in-scale, remains a major challenge due to their visual similarity to background patterns. Although state-of-the-art methods have achieved high accuracy through complex architectural adjustments, the contribution of preprocessing techniques has not been thoroughly investigated. This study investigates pre-processing-based improvements to Faster R-CNN by combining Bilateral Filtering to reduce noise, CLAHE to enhance local contrast, CIoU Loss for more effective bounding box regression, and customized anchor settings for irregular defect configurations. Evaluated using the NEU-DET dataset, our BF-CIoU Faster R-CNN model achieved a mAP@50 score of 72.32%, with an AP of …


A Bayesian Complex-Valued Latent Variable Model Applied To Functional Magnetic Resonance Imaging, Chase J. Sakitis, D. Andrew Brown, Daniel B. Rowe Jan 2025

A Bayesian Complex-Valued Latent Variable Model Applied To Functional Magnetic Resonance Imaging, Chase J. Sakitis, D. Andrew Brown, Daniel B. Rowe

Mathematical and Statistical Science Faculty Research and Publications

In linear regression, the coefficients are simple to estimate using the least squares method with a known design matrix for the observed measurements. However, real-world applications may encounter complications such as an unknown design matrix and complex-valued parameters. The design matrix can be estimated from prior information but can potentially cause an inverse problem when multiplying by the transpose as it is generally ill-conditioned. This can be combat by adding regularizers to the model but does not always mitigate the issues. Here, we propose our Bayesian approach to a complex-valued latent variable linear model with an application to functional magnetic …


High Tech Touts, Sherman J. Clark Jan 2025

High Tech Touts, Sherman J. Clark

Articles

This essay has three interrelated aims. First, it articulates a set of capacities I call virtues of attention—capacities for intuitive discernment, good judgment about what is worth sustained focus, and the ability to engage deeply with worthwhile things. These are eudaimonist virtues in that they help us live well, not merely act rightly. Second, the essay explores what I call poisonous persuasion: the idea that rhetorical appeals, especially those used in marketing, may not only succeed by appealing to certain desires or habits of mind but may also deepen and entrench them. Third, I bring these insights together to examine …


Multimodal Deception Detection Via Audio-Text Fusion With Deep Learning And Asr, Xiangyi Li Jan 2025

Multimodal Deception Detection Via Audio-Text Fusion With Deep Learning And Asr, Xiangyi Li

Master's Projects

Emotion detection plays a crucial role in human-computer interaction, enabling machines to recognize and respond appropriately to human emotional states. This project explores a two-stage approach to emotion detection using multimodal data, first predicting dimensional values (Arousal, Valence, Dominance) from textual and audio inputs, then mapping these representations to discrete emotion categories. We compare this approach with direct categorical classification using transformer-based language models like BERT, RoBERTa, and DeBERTa for text processing, alongside various audio feature extraction methods, including MFCCs and spectrograms. Using the IEMOCAP dataset, we evaluate both approaches across text-only, audio-only, and multimodal configurations. Our findings reveal that …


Evaluating The Emotional Accuracy Of Ai-Generated Facial Expressions In Neurotypical Individuals, Antonio Pagán, Katherine A Loveland, Ronald Acierno Jan 2025

Evaluating The Emotional Accuracy Of Ai-Generated Facial Expressions In Neurotypical Individuals, Antonio Pagán, Katherine A Loveland, Ronald Acierno

Faculty, Staff and Student Publications

This study examines the ability of generative artificial intelligence to produce facial expressions representing basic emotions in a neutral context using black-and-white cartoon imagery. Mentalization, the capacity to recognize and interpret one’s own and others’ mental states, is critical for social interaction and emotional regulation. We explored the emotional validation of artificial intelligence (AI)-generated images by assessing the agreement between human interpretations of emotions and those generated by an AI model. Thirty-four participants evaluated images depicting six basic emotions: sadness, anger, happiness, surprise, fear, and disgust. Our findings revealed significant variability in human agreement, with higher concordance for sadness, anger, …


The Tyrant King Of The Auction House: The Disastrous “Fine Artification” Of The Dinosaur, Amelia C. Kolstad Jan 2025

The Tyrant King Of The Auction House: The Disastrous “Fine Artification” Of The Dinosaur, Amelia C. Kolstad

Anthós

This article examines how fine art auctioneers market dinosaur fossils as art pieces to wealthy collectors, depriving the paleontological field of specimens for scientific research. Drawing from recent examples of fossils sold at fine art auctions, this article notes how rising auction prices for specimens as ‘art’ prevent museums and research institutions from acquiring specimens for research and restoration. Kolstad concludes that the art auction houses’ marketing of fossils as art is detrimental to paleontological research. To protect fossils and their scientific value, Kolstad proposes alternatives, such as selling replicas to collectors and highlighting the importance of original specimens for …


Long Oligos: Direct Chemical Synthesis Of Genes With Up To 1728 Nucleotides, Yipeng Yin, Reed Arneson, Yinan Yuan, Shiyue Fang Jan 2025

Long Oligos: Direct Chemical Synthesis Of Genes With Up To 1728 Nucleotides, Yipeng Yin, Reed Arneson, Yinan Yuan, Shiyue Fang

Michigan Tech Publications

The longest oligos that can be chemically synthesized are considered to be 200-mers. Here, we report direct synthesis of an 800-mer green fluorescent protein gene and a 1728-mer Φ29 DNA polymerase gene on an automated synthesizer. Key innovations that enabled this breakthrough include conducting the synthesis on a smooth surface rather than within the pores of traditional supports, and the use of the powerful catching-by-polymerization (CBP) method for isolating the full-length oligos from a complex mixture. Conducting synthesis on a smooth surface not only eliminated the steric hindrance that would otherwise prevent long oligo assembly, but also, surprisingly, drastically reduced …


A Critical Juncture: The Convergence Of Natural Resource Management And Tribal Sovereignty, Anna Aleta Butterfield Jan 2025

A Critical Juncture: The Convergence Of Natural Resource Management And Tribal Sovereignty, Anna Aleta Butterfield

Graduate Student Portfolios, Professional Papers, and Capstone Projects

This portfolio reflects the work I’ve done in my four years as a dual degree student, pursuing a Master of Science in Environmental Studies and a juris doctorate degree, and focuses on policies and laws surrounding natural resource management. It touches on how those policies relate to and expand or constrict the exercise of sovereignty by indigenous populations.

This piece first considers the effects of plastic on salmon populations and those indigenous groups that rely on salmon for subsistence and cultural purposes. The remaining three pieces focus on various aspects of the Confederated Salish and Kootenai Tribes - Montana Water …


Analysis Of Cytokine Data In A Case-Control Study Of Me/Cfs, Navya Amaratunga Jan 2025

Analysis Of Cytokine Data In A Case-Control Study Of Me/Cfs, Navya Amaratunga

Electronic Theses & Dissertations (2024 - present)

This study investigated the role of cytokines in Myalgic Encephalomyelitis/Chronic Fatigue Syndrome (ME/CFS), a debilitating multi-system disease with unknown etiology. Using data from a case-control study (60 cases, 61 controls) previously collected by Dr. Roxana Moslehi’s lab, serum levels of 48 cytokines were analyzed. Statistical analyses, including T-tests, Wilcoxon Rank Sum tests, and logistic regression (unadjusted and adjusted for medication use), were performed to identify differences between cases and controls.

Results revealed a statistically significant difference in mean Fractalkine levels between cases and controls (p=0.031). Borderline significant differences were also observed for IL-6 (p=0.046) and MIP-1α (p=0.046). Unadjusted logistic regression …


Using Convolutional Neural Networks To Identify Rare Weather Events: Application To Kona Low Classifications With Large-Scale Wind Pattern, Anadhi Sharma Jan 2025

Using Convolutional Neural Networks To Identify Rare Weather Events: Application To Kona Low Classifications With Large-Scale Wind Pattern, Anadhi Sharma

Electronic Theses & Dissertations (2024 - present)

Extreme events pose challenges in prediction on account of their low frequency and erratic nature. Among these events, Kona lows are rare subtropical cyclones that significantly impact the weather pattern of Hawaii, sometimes causing heavy rainfall, strong winds, and coastal flooding. In the past, studies have used traditional statistical methods such as synoptic climatology, regression analysis, and manual classification that rely on expert judgment to identify Kona lows; however, they often struggle to capture the complex spatial patterns of the Kona storms. Convolutional Neural Networks (CNNs) exhibit a property called translation invariance, which allows them to learn and recognize Kona …


Towards Human Explainable Digital Forensics: Generating Human Interpretable Evidence For Semantic Understanding In Manipulated Images And Text, Yuwei Chen Jan 2025

Towards Human Explainable Digital Forensics: Generating Human Interpretable Evidence For Semantic Understanding In Manipulated Images And Text, Yuwei Chen

Electronic Theses & Dissertations (2024 - present)

Detecting and characterizing manipulations in digital media continues to pose a significant challenge within the field of digital forensics. Despite notable advancements, the discipline often remains in a reactive stance against emerging threats. Current state-of-the-art methods, typically evaluated within academic settings, fails to mirror the complexities of real-world disinformation scenarios. These methods generally prioritize high performance based on quantitative metrics, yet they demonstrate a considerable dependency on training data and lack adaptability to new novel attack signatures. With the rapid evolution of attack methodologies, the dependency on highly accurate models that do not generalize or adapt well to unseen threats …


Limitations Of The High-Resolution Rapid Refresh Gust Algorithm, Alex M. Glasser Jan 2025

Limitations Of The High-Resolution Rapid Refresh Gust Algorithm, Alex M. Glasser

Electronic Theses & Dissertations (2024 - present)

ABSTRACT:

Accurate wind and gust forecasts are crucial in operational meteorology, yet there is a distinct gap in the literature regarding the verification of operational gust forecasts. In the present work, we assess the performance of the High-Resolution Rapid Refresh's (HRRR) gust potential, verifying it against the Automated Surface Observing System's (ASOS) observed gusts across the Continental United States (CONUS). The HRRR gust's diurnal cycle reveals a significant underprediction of the gust during daylight hours and a slight overprediction of the gust during hours of darkness. The daytime underprediction undermines the assumption that forecasts represent a potential that tends to …


Large Scale Machine Learning Over Knowledge Graphs, Bedirhan Gergin Jan 2025

Large Scale Machine Learning Over Knowledge Graphs, Bedirhan Gergin

Electronic Theses & Dissertations (2024 - present)

Knowledge graphs (KGs) have become popular across various fields, providing convenient access to web-based knowledge while storing and formalizing domain-specific information. By analyzing KGs, patterns, connections, and dependencies can be identified across different data sources, enabling the inference of new knowledge from given facts. As the use of KGs expands, the size of modern KGs has grown significantly, making them impossible to process within the main memory of a single computer. Distributed computing offers a viable solution to this challenge by leveraging the combined capabilities of multiple servers within a cluster. This thesis explores how distributed computing can be effectively …


Further Results On Learning Quantum Measurement Classes: Quantum Pac Model For Povm Hypothesis Classes, Arka Prabha Das Jan 2025

Further Results On Learning Quantum Measurement Classes: Quantum Pac Model For Povm Hypothesis Classes, Arka Prabha Das

Electronic Theses & Dissertations (2024 - present)

This thesis investigates the problem of learning from quantum systems, where each example consists of a quantum state paired with a classical outcome. The task centers on choosing an effective measurement rule from a fixed set to enable accurate prediction of the classical outcome from the quantum state. A central focus lies in understanding whether joint measurement strategies that cannot be separated into local operations offer a real benefit in terms of the number of examples needed for successful learning. We examine conditions under which a non-separable measurement within a given hypothesis class achieves strictly better sample complexity bounds compared …


Using The High Resolution Rapid Refresh To Validate A Novel Forecast Methodology, Sylvia M. Garmong Jan 2025

Using The High Resolution Rapid Refresh To Validate A Novel Forecast Methodology, Sylvia M. Garmong

Electronic Theses & Dissertations (2024 - present)

NOAA’s High-Resolution Rapid Refresh (HRRR) is frequently used by forecasters for short term predictions and although the model performs well in many metrics, previous study (Fovell and Gallagher 2020) identified temperature and windspeed biases within the boundary layer using radiosonde and surface observations. The current work uses a forecast drift metric to study how model forecasts evolve with time and distance to radiosonde sites. The metric is calculated by subtracting model analysis from forecasts at the same valid time, creating a proxy for bias. HRRR data on native model levels are obtained from the Google Cloud and Amazon Web Services …


Enhancing Crispr-Cas12a Activity Through Guide Rna Methylation, Hilal Dagci Jan 2025

Enhancing Crispr-Cas12a Activity Through Guide Rna Methylation, Hilal Dagci

Electronic Theses & Dissertations (2024 - present)

CRISPR-Cas12a is a rapidly evolving technology for improved gene editing in cells, as well as in vivo. The system, consisting of Cas12a nuclease and approximately 40-nt long guide RNA (gRNA), can efficiently recognize and cut genomic double-stranded DNA (dsDNA). Chemical modifications to the gRNA can potentially improve its dsDNA binding and enhance Cas12a-mediated nuclease activity. In this work, 5-methyl cytidine (m5C) and N6-methyl adenosine (m6A) were explored to improve the binding between gRNA and target DNA. Phosphoramidites of m5C and m6A were used to construct a library of gRNAs using solid phase RNA synthesis. Tm studies were carried out …


Two Computational Problems On String Rewriting Systems, Wei Du Jan 2025

Two Computational Problems On String Rewriting Systems, Wei Du

Electronic Theses & Dissertations (2024 - present)

String rewriting systems are widely used computational models in theoretical computer science research such as artificial intelligence, software and hardware verification, and symbolic cryptographic protocol analysis. In this dissertation, we investigate two interesting problems concerning these systems, namely the common left multiplier problem and the SYMBOL-ORDER problem.

First, we consider the common left multiplier problem for forward-closed convergent string rewriting systems. The task is to discover, given two distinct strings α and β, a target string W such that W α and W β will be equivalent with respect to the provided forward-closed convergent string rewriting system. We describe an …


Misoscale Vortices In The Eyewall Boundary Layer Of A Tropical Cyclone In A Large Eddy Simulation, Emily A. Lucy Jan 2025

Misoscale Vortices In The Eyewall Boundary Layer Of A Tropical Cyclone In A Large Eddy Simulation, Emily A. Lucy

Electronic Theses & Dissertations (2024 - present)

Since the late 1980s, there have been numerous observations of extreme updrafts, gusts, and small-scale rotation in the eyewall of tropical cyclones (TCs). Research aircraft, dropsondes, and, more recently, mobile radars have collected these observations, most often from within the boundary layer. Previous studies have documented dropsondes associated with extreme gusts and updrafts, while others have discussed specific incidents of misoscale features in tropical cyclones. Several high-resolution numerical simulations of TCs have documented similar features. This study uses one such simulation from Worsnop et al. (2017). A small but intense tropical cyclone in a quiescent environment is simulated.

This simulation …


"Assessment Of The Likelihood To Stay Below 2 Degree Celsius Warming: Combining Modeling And Scenario Uncertainty.", Peter M. Booker Jan 2025

"Assessment Of The Likelihood To Stay Below 2 Degree Celsius Warming: Combining Modeling And Scenario Uncertainty.", Peter M. Booker

Electronic Theses & Dissertations (2024 - present)

ABSTRACT

Uncertainty in how much the Earth will warm in response to rising atmospheric CO₂ concentrations remains a major challenge for climate science and policy. In particular, estimates of equilibrium climate sensitivity (ECS) and transient climate response (TCR) directly affect projections of global mean surface temperature (GMST) and the allowable carbon budget for limiting warming to 2°C by 2100.

This study assesses the performance of a two-layer energy balance model (EBM) in reproducing TCR and ECS values under CMIP6 forcing scenarios, with a focus on SSP2-4.5. EBM outputs are compared against General Circulation Models (GCMs), and agreement is evaluated using …


Food Rescue, Food Waste, And Policy In New York: A Mixed-Methods Study Of Food System Impacts, Mariana Torres Arroyo Jan 2025

Food Rescue, Food Waste, And Policy In New York: A Mixed-Methods Study Of Food System Impacts, Mariana Torres Arroyo

Electronic Theses & Dissertations (2024 - present)

This dissertation investigates how food donation policies in New York State influence the recovery, redistribution, and waste of surplus fresh produce, with a focus on fruits and vegetables. Using an interdisciplinary, systems-based approach that integrates system dynamics modeling with qualitative, community-engaged research, the study analyzes the effectiveness and unintended consequences of three major policies: Nourish New York, the Farm to Food Bank tax credit, and the Food Donation and Food Scraps Recycling Law.

Chapter 2 uses simulation modeling to examine how the tax credit and waste ban affect produce recovery and redistribution. Results show that while both policies can boost …


Betti Numbers Of Generic Ideals, Jason R. Howell Jan 2025

Betti Numbers Of Generic Ideals, Jason R. Howell

Electronic Theses & Dissertations (2024 - present)

We present results related to Betti numbers of so called generic ideals over a polynomial ring $T = \Bbbk[x_1, \ldots, x_n]$. For each $m$ define $T(m) = T/(x_{m+1}, \ldots, x_n)$ and for a homogeneous ideal $J$ we use the notation $J(m) = JT(m)$. Also we set $Q(m) = T(m)/J(m)$, and $L(m) = ann_{Q(m)}(x_m)$. The first main result is Theorem \ref{Long Exact Sequence} where we produce the following long exact sequence \begin{align*} \cdots \rightarrow &Tor_{k+1,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-1,j+1}^{T(m-1)}(L(m),\Bbbk)_{j-1} \rightarrow Tor_{k,j}^{T(m)}(Q(m),\Bbbk) \rightarrow \\ &Tor_{k,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-2,j-1}^{T(m-1)}(L(m),\Bbbk) \rightarrow \cdots. \end{align*} The second main result is Theorem \ref{Theorem F(j,m) equiv k(j,m)} where we explicitly describe …