Unique Factorization in the Rings of Integers of Quadratic Fields: A Method of Proof

It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that the FTA does not always hold. For example, in the domain O(√−5), 6 = 2 · 3 and 6 = (1 + √−5)(1 − √−5) are two valid factorizations of 6, with 2, 3, (1 + √−5), (1 − √−5) all irreducible elements in O(√−5). This paper discusses the history and development of the problem of discerning which fields of the form O(√D) are unique factorization domains (UFDs) and concludes by constructing a method of proving unique factorization in some domain using results concerning Euclidean domains and principal ideal domains

Distribution and Habitat Association of the Northern Long-eared Bat

Because of white-nose syndrome, the northern long-eared bat (Myotis septentrionalis) is experiencing rapid population declines across the majority of its range. This decline has prompted increased regulatory protection and thus an increased need for effective habitat management strategies. Insight into the species habitat associations, however, is a prior necessity to ensure management practices are biologically relevant. Understanding these relationships requires a holistic approach that addresses the multiple ways in which the species is interacting with its environment. The objective of my thesis was to address these relationships following a multi-scale approach that assessed the factors associated with roost tree selection, distribution, and habitat use. I conducted a quantitative meta-analysis of all northern long-eared bat roost-selection studies, thus enabling inference across the range of the species. Secondly, I sought to understand the factors associated with the northern long-eared distribution in Nebraska by conducting a multiscale occupancy study with four acoustic survey stations nested within 101 10 km x 10 km grids distributed across Nebraska. Lastly, to assess factors associated with multi-scale occupancy at finer spatial scales, I combined presence/absence results from 5 intensively sampled study sites in Nebraska with on-the-ground habitat measurements. Results from the meta-analysis indicate selected roost trees had a greater amount of bark remaining on the bole, a larger diameter at breast height, a lower decay class, and were taller. Results from the statewide occupancy study provide evidence for relationships between large-scale occupancy (ψ) and forest clumpiness, proximity to potential hibernacula, and summer temperature. Within occupied grids, evidence supported a positive relationship between small-scale occupancy (θ) and forest area within 125 m. Results failed to provide evidence of habitat factors associated with ψ at the 5 study sites likely due to high presence of the species. Evidence, however, did support a positive relationship between canopy closure and θ. Reported occupancy estimates between the two studies results serve as a pre-white-nose syndrome baseline, as I collected all data prior to the detection of white-nose syndrome in Nebraska. Advisor: Craig R. Alle

Effect of Corn Silage Inclusion Level and Steroidal Implant Type on Growth Performance, Dietary Net Energy Utilization, Beef Production Per Acre and Apparent Total Tract Digestibility in Feedlot Finishing Steers

Objective The experimental objective was to determine the influence corn silage inclusion level and terminal implant type have on animal performance, carcass traits, and beef production per acre of cropland in finishing steers harvested at a common fatness endpoint. Study Description Maine-Anjou × Angus beef steers (n = 156 steers; initial BW 807 ± 81.9 lbs) were used in a 2 × 2 factorial arrangement finishing experiment at the Ruminant Nutrition Center in Brookings, SD. Steers were weighed on 2 consecutive days and assigned into 5 weight blocks. Dietary treatments consisted (DM basis) of 1) 15% (CS15) or 2) 30% corn silage (CS30). Implant treatments consisted of 1) Coated implant, 200 mg trenbolone acetate (TBA), 28 mg estradiol benzoate (EB) (Synovex ONE Feedlot, ONE) or 2) Non-coated implant, 200 mg TBA, 28 mg EB (Synovex PLUS; PLUS)

Effects of Harvest Maturity and/or Kernel Processing on Corn Silage Processing Score and Particle Size of Corn Silage

Objective A single corn hybrid was used to evaluate harvest maturity (Mat) and/or kernel processing (KP) effects on corn silage processing score (CSPS) and particle size (PS). Study Description Treatments were arranged in a 2 × 2 factorial of: 1) Mat (early and late) and 2) KP (no or yes). A single corn field was planted on April 27, 2020. There were 12 loads (experimental unit) per simple effect treatment mean. Data were analyzed as a completely randomized design. Early harvest (3/4 milk line) occurred on August 28, 2020 [yield (as is) = 17.40 tons/acre; DM = 43.1%; CP, NDF, and starch = 6.5, 46.0, and 32.9%, respectively (DM basis)]. Late harvest (black layer) occurred on September 9, 2020 [yield = 16.83 ton/acre (as is); DM = 49.2%; CP, NDF, and starch = 6.6, 49.8, and 37.5%, respectively (DM basis)]. The same equipment was used for both Mat with KP achieved by narrowing processing rollers. The CSPS was determined as the proportion of starch retained below a 0.19 in sieve. Grain content (DM basis) of the corn silage was calculated from: starch content/0.72. Particle size was assessed using the Penn State Particle Separator. A Mat × KP interaction (P = 0.05) was detected for CSPS. Early/no and late/no had decreased (P ≤ 0.05) CSPS compared to early/yes and late/yes had the greatest CSPS (P ≤ 0.05) compared to others. Grain content was 13.9% greater in late compared to early (P = 0.01). A Mat × KP interaction (P = 0.03) was detected for PS. Early/no had the greatest (P ≤ 0.05) PS, early/yes and late/no were intermediate, and late/yes had decreased PS compared to others (P ≤ 0.05). Tak

Evaluation of Dietary Roughage Inclusion in a Single or Two-Diet System for Backgrounding and Finishing

Study Description: Pre-conditioned crossbred beef steers (n = 46; initial shrunk [4%] BW = 621 ± 89.1 lbs) were used in a 210-d grow-finish experiment at the Ruminant Nutrition Center (RNC) in Brookings, SD. Steers were fed once daily, and bunks were managed according to a slick bunk management system. Cattle were fed in 25 × 25 ft concrete surface pens (n = 10 pens; 5 pens/treatment) with 25 linear ft of bunk space and heated, concrete, continuous flow waterers. Treatments included: 1) A single-diet program (targeted a 59 Mcal/cwt NEg diet fed for 210-d; 1D) or 2) two-diet program (targeted a 55 Mcal/cwt NEg diet fed for 98-d, a 59 Mcal/cwt NEg diet fed for 14-d, and a 63 Mcal/cwt NEg diet fed for 98-d; 2D). All steers were implanted initially (d 1) with a 100 mg trenbolone acetate (TBA) and 14 mg estradiol benzoate (EB) implant (Synovex Choice) and re-implanted with a 200 mg TBA and 28 mg EB implant on d 112

Quantitative characterization of viscoelastic behavior in tissue-mimicking phantoms and ex vivo animal tissues.

Viscoelasticity of soft tissue is often related to pathology, and therefore, has become an important diagnostic indicator in the clinical assessment of suspect tissue. Surgeons, particularly within head and neck subsites, typically use palpation techniques for intra-operative tumor detection. This detection method, however, is highly subjective and often fails to detect small or deep abnormalities. Vibroacoustography (VA) and similar methods have previously been used to distinguish tissue with high-contrast, but a firm understanding of the main contrast mechanism has yet to be verified. The contributions of tissue mechanical properties in VA images have been difficult to verify given the limited literature on viscoelastic properties of various normal and diseased tissue. This paper aims to investigate viscoelasticity theory and present a detailed description of viscoelastic experimental results obtained in tissue-mimicking phantoms (TMPs) and ex vivo tissues to verify the main contrast mechanism in VA and similar imaging modalities. A spherical-tip micro-indentation technique was employed with the Hertzian model to acquire absolute, quantitative, point measurements of the elastic modulus (E), long term shear modulus (η), and time constant (τ) in homogeneous TMPs and ex vivo tissue in rat liver and porcine liver and gallbladder. Viscoelastic differences observed between porcine liver and gallbladder tissue suggest that imaging modalities which utilize the mechanical properties of tissue as a primary contrast mechanism can potentially be used to quantitatively differentiate between proximate organs in a clinical setting. These results may facilitate more accurate tissue modeling and add information not currently available to the field of systems characterization and biomedical research

Impact of Corn Silage Moisture and/or Kernel Processing at Harvest on Finishing Steer Growth Performance, Efficiency of Dietary Net Energy Utilization and Carcass Traits

Study Description: Red Angus steers (n = 192; initial shrunk BW = 983 ± 62.3 lbs) were used in the 112 d finishing experiment at the Southeast Research Farm (SERF) of the South Dakota Agricultural Experiment Station in Beresford. Steers were from a single source and obtained from a local SD auction facility. Steers were received 2 weeks prior to trial initiation. Steers were offered a common diet containing 60% concentrate upon arrival. Steers were transitioned to a 90% concentrate diet over the course of 14 d. Steers were consuming the finishing diet (Table 1) at the initiation of the experiment. Fresh feed was manufactured once daily for each treatment in a single batch using a stationary mixer and bunks were managed for ad libitum access to feed. Actual diet formulation and composition is based upon weekly DM analyses, tabular nutrient values, and corresponding feed batching records. Diets presented in Table 1 are actual DM diet composition, tabular nutrient concentrations, and tabular energy values (Preston, 2016)