Faculty PublicationsCopyright (c) 2021 Stephen F. Austin State University All rights reserved.
https://scholarworks.sfasu.edu/mathandstats_facultypubs
Recent documents in Faculty Publicationsen-usThu, 23 Dec 2021 01:39:44 PST36002-adic Valuations of Quadratic Sequences
https://scholarworks.sfasu.edu/mathandstats_facultypubs/31
https://scholarworks.sfasu.edu/mathandstats_facultypubs/31Tue, 21 Dec 2021 12:24:45 PST
We determine properties of the 2-adic valuation sequences for general quadratic polynomials with integer coefficients directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at terminating nodes. We completely describe the periodic sequences in the bounded case. Throughout, we frame results in terms of trees and sequences.
]]>
Will Boultinghouse et al.A Math Without Words Puzzle
https://scholarworks.sfasu.edu/mathandstats_facultypubs/30
https://scholarworks.sfasu.edu/mathandstats_facultypubs/30Tue, 21 Dec 2021 12:24:42 PST
A visual puzzle by James Tanton forms the basis for a session that has been successfully implemented with various audiences. Designed to be presented with no directions or description, the puzzle requires participants to discover the goals themselves and to generate their own questions for investigation. Solutions, significant facilitation suggestions, and possibilities for deep mathematical extensions are discussed; extensive illustrations are included.
]]>
Jane H. Long et al.The valuation tree for n2+7
https://scholarworks.sfasu.edu/mathandstats_facultypubs/29
https://scholarworks.sfasu.edu/mathandstats_facultypubs/29Tue, 21 Dec 2021 12:24:39 PST
The 2-adic valuation of an integer x is the highest power of 2 which divides x. It is denoted by ν2(x). The goal of the present work is to describe the sequence {ν2(n2+ a)}for 1 6a 67. The first six cases are elementary. The last case considered here, namely a = 7, presents distinct challenges. It is shown here how to represent this family of valuations in the form of an infinite binary tree, with two symmetric infinite branches.
]]>
Olena Kozhushkina et al.The Hodge structure of the coloring complex of a hypergraph
https://scholarworks.sfasu.edu/mathandstats_facultypubs/28
https://scholarworks.sfasu.edu/mathandstats_facultypubs/28Tue, 21 Dec 2021 12:24:36 PST
Let G be a simple graph with n vertices. The coloring complex Δ(G) was defined by Steingrímsson, and the homology of Δ(G) was shown to be nonzero only in dimension n − 3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hn−3(Δ(G)) where the dimension of the jth component in the decomposition, H(j) n−3(Δ(G)), equals the absolute value of the coefficient of λj in the chromatic polynomial of G, χG(λ). Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph,Δ(H), and show that the coefficient of λj inχH(λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of Δ(H). We also examine conditions on a hypergraph, H, for which its Hodge subcomplexes are Cohen–Macaulay, and thus where the absolute value of the coefficient of λj in χH(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Δ(H). We also note that the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of jth term in the associated chromatic polynomial.
]]>
Jane H. Long et al.The Hodge structure of the coloring complex of a hypergraph (extended abstract),
https://scholarworks.sfasu.edu/mathandstats_facultypubs/27
https://scholarworks.sfasu.edu/mathandstats_facultypubs/27Tue, 21 Dec 2021 12:24:33 PST
Let G be a simple graph with n vertices. The coloring complex ∆(G) was defined by Steingr ́ımsson, and the homology of ∆(G) was shown to be nonzero only in dimension n −3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hn−3(∆(G)) where the dimension of the jth component in the decomposition, H(j) n−3(∆(G)), equals the absolute value of the coefficient of λj in the chromatic polynomial of G, χG(λ). Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, ∆(H), and show that the coefficient of λj in χH (λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of ∆(H). We also examine conditions on a hypergraph, H, for which its Hodge subcomplexes are Cohen-Macaulay, and thus where the absolute value of the coefficient of λj in χH (λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of ∆(H).
]]>
Sarah Crown Rundell et al.Delta Function for an Affine Subspace
https://scholarworks.sfasu.edu/mathandstats_facultypubs/26
https://scholarworks.sfasu.edu/mathandstats_facultypubs/26Thu, 27 Feb 2020 12:28:47 PST
The Kubo–Yokoi and Donsker delta functions are well known generalized functions in infinite dimensional distribution theory. In this paper we develop the delta function for an affine subspace and show that it is a generalization of the Kubo–Yokoi and Donsker delta functions. The Wiener– Itˆo expansion of the delta function for an affine subspace is also given.
]]>
Jeremy BecnelRecovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform
https://scholarworks.sfasu.edu/mathandstats_facultypubs/25
https://scholarworks.sfasu.edu/mathandstats_facultypubs/25Thu, 27 Feb 2020 11:56:53 PST
The Radon transform maps a function on n-dimensional Euclidean space onto its integral over a hyperplane. The fields of modern computerized tomography and medical imaging are fundamentally based on the Radon transform and the computer implementation of the inversion, or reconstruction, techniques of the Radon transform. In this work we use the Radon transform with a Gaussian measure to recover random variables from their conditional expectations. We derive reconstruction algorithms for random variables of unbounded support from samples of conditional expectations and discuss the error inherent in each algorithm.
]]>
Jeremy Becnel et al.Probability Inequalities for the Sum of Random Variables When Sampling Without Replacement
https://scholarworks.sfasu.edu/mathandstats_facultypubs/24
https://scholarworks.sfasu.edu/mathandstats_facultypubs/24Thu, 27 Feb 2020 11:56:48 PST
Exponential-type upper bounds are formulated for the probability that the maximum of the partial sample sums of discrete random variables having finite equispaced support exceeds or differs from the population mean by a specified positive constant. The new inequalities extend the work of Serfling (1974). An example of the results are given to demonstrate their efficacy.
]]>
Jeremy Becnel et al.Global existence and finite time blow-up in a class of stochastic nonlinear wave equations
https://scholarworks.sfasu.edu/mathandstats_facultypubs/23
https://scholarworks.sfasu.edu/mathandstats_facultypubs/23Thu, 27 Feb 2020 11:56:43 PST
We consider a stochastic extension of a class of wave equations with nonlinear viscoelastic damping and nonlinear forcing. We show the global existence of the solution of the stochastic equation and, additionally, when the source term dominates the damping term and when the initial data are large enough, we show that the expected value of the L p norm of the solution, blows up in finite time. In the presence of noise, we extend the previously known range of initial data corresponding to blow-up. Furthermore we use a spectral stochastic Galerkin method to perform numerical simulations that verify certain special cases of our theoretical results.
]]>
Rana D. Parshad et al.A Limiting Process to Invert the Gauss-Radon Transform
https://scholarworks.sfasu.edu/mathandstats_facultypubs/22
https://scholarworks.sfasu.edu/mathandstats_facultypubs/22Thu, 27 Feb 2020 11:56:37 PST
In this work we extend the finite dimensional Radon transform [23] to the Gaussian measure. We develop an inversion formula for this GaussRadon transform by way of Fourier inversion formula. We then proceed to extend these results to the infinite dimensional setting.
]]>
Jeremy BecnelUse of Trees by the Texas Ratsnake (Elaphe obsoleta) in Eastern Texas
https://scholarworks.sfasu.edu/mathandstats_facultypubs/21
https://scholarworks.sfasu.edu/mathandstats_facultypubs/21Fri, 15 Nov 2019 09:48:59 PST
We present information on the use of trees by Elaphe obsoleta (Texas Ratsnake) in a mesic pine-hardwood forest in eastern Texas. Using radiotelemetry, seven snakes (3 females, 4 males) were relocated a total of 363 times from April 2004 to May 2005, resulting in 201 unique locations. Snakes selected trees containing cavities and used hardwoods and snags for a combined 95% of arboreal locations. Texas Ratsnake arboreal activity peaked during July and August, well after the peak of avian breeding activity, suggesting arboreal activity involves factors other than avian predation.
]]>
Josh B. Pierce et al.Habitat Selection by Anolis carolinensis (Green Anole) in Open Pine Forests in Eastern Texas
https://scholarworks.sfasu.edu/mathandstats_facultypubs/20
https://scholarworks.sfasu.edu/mathandstats_facultypubs/20Fri, 15 Nov 2019 09:48:54 PST
We initiated a mark-recapture study to determine the effects of shrub density on Anolis carolinensis (Green Anole) populations. Green Anole perch site, shrub species, and shrub volume preferences were also examined. We established two study plots of different shrub densities in open pine forests on the Angelina National Forest in eastern Texas. In late spring, the Green Anole population at the higher shrub-density plot was estimated to be 16 times greater than the population at the lower shrub-density plot. Green Anoles most commonly perched on live shrubs, but exhibited very little preference or avoidance of any particular species of live shrub or shrub-level vine. However, shrubs used by Green Anoles were 4–6 times greater in volume than plot averages.
]]>
Richard R. Schaefer et al.Diel activity patterns of the Louisiana pine snakes (Pituophis ruthveni) in eastern Texas
https://scholarworks.sfasu.edu/mathandstats_facultypubs/19
https://scholarworks.sfasu.edu/mathandstats_facultypubs/19Fri, 15 Nov 2019 09:21:08 PST
This study examined the diel activity patterns of six Louisiana pine snakes in eastern Texas using radio-telemetry. snakes were monitored for 44 days on two study areas from May to October 1996. Louisana pine snakes were primarily diurnal with moderate crepuscular activity, spending the night within pocket gopher burrows or inactive on the surface. During daylight hours, snakes spent approximately 59% of their time underground within gopher burrows, burned out/rotten stumps, or nine-branded armadillo (Dasypus novemcinctus) burrows. Remaining time was spent on the surface either close to subteranean refuge, or in long distance movements that generally terminet at another pocket gopher burrow system. Long distance movements occurred on 45% of the days snakes were monitored and averaged 163 m/movement. When snakes were active, movements related to ambientair temperature; 82% of these movements occurred between 1000 and 1800 hours. These resutls confirm that Louisiana pine snakes are diurnal and closely associated with Baird's pocket gophers and their burrow systems, and have provided new insight on the ecology of this rare snake.
]]>
Marc J. Ealy et al.Spatial Ecology of the Coachwhip, Masticophis flagellum (Squamata: Colubridae), in Eastern Texas
https://scholarworks.sfasu.edu/mathandstats_facultypubs/18
https://scholarworks.sfasu.edu/mathandstats_facultypubs/18Fri, 15 Nov 2019 09:21:02 PST
We radio-tracked nine Masticophis flagellum (Coachwhips) to determine home range, habitat use, and movements in eastern Texas from April to October 2000. Home ranges of Coachwhips contained more oak savanna macrohabitat than early-successional pine plantation or forested seep, based on the availability of these three macrohabitats in the study area. Likewise, within their individual home ranges, Coachwhips used oak savanna more than the other two macrohabitats, based on availability. An analysis of microhabitat use revealed that, relative to random sites within their home range, Coachwhips were found at sites with fewer pine trees and more herbaceous vegetation taller than 30 cm. Results of the two analyses, macrohabitat and microhabitat, were consistent: oak savannas contained relatively few pine trees but much herbaceous vegetation taller than 30 cm. Coachwhips made frequent long-distance moves, which resulted in large home ranges. Core activity areas, however, were small. These core activity areas were always within the oak savanna macrohabitat. Long movements, large home ranges, and small core activity areas likely were a result of the preferred oak savanna macrohabitat being patchily distributed in the landscape.
]]>
Richard W. Johnson et al.Cooper’s Hawk Nest Site Characteristics in the Pineywoods Region
https://scholarworks.sfasu.edu/mathandstats_facultypubs/17
https://scholarworks.sfasu.edu/mathandstats_facultypubs/17Fri, 15 Nov 2019 09:20:55 PST
Early accounts describe the Cooper’s Hawk (Accipiter cooperi) as a species in decline in much of North America during the early twentieth century (Bent 1937), particularly when in close proximity to humans (Eaton 1914). This decreasing population trend continued to be recognized later in the century in both Texas (Oberholser 1974) and Louisiana (Lowery 1974). Shooting and trapping during the first half of the 1900s, and pesticide use (especially DDT) after World War II are suggested as primary causes of the decline (Henny and Wight 1972, Bednarz et al. 1990). The Migratory Bird Treaty Act of 1972 and the ban on DDT during that same year, along with changes in human behaviors and attitudes have guided Cooper’s Hawk populations toward recovery in areas negatively impacted (Bednarz et al. 1990, Johnsgard 1990). The overall North American population has increased substantially since the 1990s (Curtis et al. 2006), and the species is increasing as a breeder in parts of Texas, particularly in urban areas (Lockwood and Freeman 2004).
]]>
Richard R. Schaefer et al.Winter Movements of Louisiana Pine Snakes (Pituophis ruthveni) in Texas and Louisiana
https://scholarworks.sfasu.edu/mathandstats_facultypubs/16
https://scholarworks.sfasu.edu/mathandstats_facultypubs/16Fri, 15 Nov 2019 09:20:49 PST
Despite concerns that the Louisiana Pine Snake (Pituophis ruthveni) has been extirpated from large portions of its historic range, only a limited number of studies on their movement patterns have been published. Winter movement patterns are of particular interest since it has been hypothesized that impacts of management practices would be reduced during the winter. Using radiotelemetry, we determined winter movement patterns of Louisiana Pine Snakes (11 males, 8 females) in 5 study areas (2 in Louisiana and 3 in Texas). Movements during winter (November–February) were greatly curtailed compared to the remainder of the year; however, snakes occasionally undertook substantial movements. Relocations were typically within the snake’s previous active-season home range, and movements were more frequent in the early portion of winter. All hibernation sites were within Baird’s Pocket Gopher (Geomys breviceps) burrow systems at depths ranging from 13–25 cm. Louisiana Pine Snakes did not use communal hibernacula, nor did individual snakes return to previously used sites in successive years.
]]>
Josh B. Pierce et al.Extending the Support Theorem to Infinite Dimensions
https://scholarworks.sfasu.edu/mathandstats_facultypubs/15
https://scholarworks.sfasu.edu/mathandstats_facultypubs/15Tue, 05 Nov 2019 12:55:35 PST
The Radon transform is one of the most useful and applicable tools in functional analysis. First constructed by John Radon in 1917 [9] it has now been adapted to several settings. One of the principle theorems involving the Radon transform is the Support Theorem. In this paper, we discuss how the Radon transform can be constructed in the white noise setting. We also develop a Support Theorem in this setting.
]]>
Jeremy J. BecnelBiological control via "ecological" damping: An approach that attenuates non-target effects
https://scholarworks.sfasu.edu/mathandstats_facultypubs/14
https://scholarworks.sfasu.edu/mathandstats_facultypubs/14Tue, 05 Nov 2019 12:33:23 PST
In this work we develop and analyze a mathematical model of biological control to prevent or attenuate the explosive increase of an invasive species population in a three-species food chain. We allow for finite time blowup in the model as a mathematical construct to mimic the explosive increase in population, enabling the species to reach “disastrous” levels, in a finite time. We next propose various controls to drive down the invasive population growth and, in certain cases, eliminate blow-up. The controls avoid chemical treatments and/or natural enemy introduction, thus eliminating various non-target effects associated with such classical methods. We refer to these new controls as “ecological damping”, as their inclusion dampens the invasive species population growth. Further, we improve prior results on the regularity and Turing instability of the three-species model that were derived in [43]. Lastly, we confirm the existence of spatio-temporal chaos.
]]>
Rana D. Parshad et al.A Nonlinear Splitting Algorithm for Systems of Partial Differential Equations with self-Diffusion
https://scholarworks.sfasu.edu/mathandstats_facultypubs/13
https://scholarworks.sfasu.edu/mathandstats_facultypubs/13Tue, 05 Nov 2019 12:27:17 PST
Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attempts to construct efficient and accurate numerical approximations of the underlying systems of equations. In this paper, a new nonlinear splitting algorithm is designed for a partial differential equation that incorporates self diffusion. We present a general model that incorporates self-diffusion and develop a numerical approximation. The numerical analysis of the approximation provides criteria for stability and convergence. Numerical examples are used to illustrate the theoretical results.
]]>
Matthew Beauregard et al.CHARACTERIZING GORENSTEIN RINGS USING CONTRACTING ENDOMORPHISMS
https://scholarworks.sfasu.edu/mathandstats_facultypubs/12
https://scholarworks.sfasu.edu/mathandstats_facultypubs/12Tue, 05 Nov 2019 12:16:33 PST
We prove several characterizations of Gorenstein rings in terms of vanishings of derived functors of certain modules or complexes whose scalars are restricted via contracting endomorphisms. These results can be viewed as analogues of results of Kunz (in the case of the Frobenius) and Avramov-Hochster-Iyengar-Yao (in the case of general contracting endomorphisms).
]]>
Brittney Falahola et al.