Chunky and Equal-Spaced Polynomial Multiplication

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many "easy" cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing "sparse" and "dense" methods. We prove that these approaches provide significant improvements in many cases but in the worst case are still comparable to the fastest existing algorithms.Comment: 23 Pages, pdflatex, accepted to Journal of Symbolic Computation (JSC

Multivariate sparse interpolation using randomized Kronecker substitutions

We present new techniques for reducing a multivariate sparse polynomial to a univariate polynomial. The reduction works similarly to the classical and widely-used Kronecker substitution, except that we choose the degrees randomly based on the number of nonzero terms in the multivariate polynomial, that is, its sparsity. The resulting univariate polynomial often has a significantly lower degree than the Kronecker substitution polynomial, at the expense of a small number of term collisions. As an application, we give a new algorithm for multivariate interpolation which uses these new techniques along with any existing univariate interpolation algorithm.Comment: 21 pages, 2 tables, 1 procedure. Accepted to ISSAC 201

POPE: Partial Order Preserving Encoding

Recently there has been much interest in performing search queries over encrypted data to enable functionality while protecting sensitive data. One particularly efficient mechanism for executing such queries is order-preserving encryption/encoding (OPE) which results in ciphertexts that preserve the relative order of the underlying plaintexts thus allowing range and comparison queries to be performed directly on ciphertexts. In this paper, we propose an alternative approach to range queries over encrypted data that is optimized to support insert-heavy workloads as are common in "big data" applications while still maintaining search functionality and achieving stronger security. Specifically, we propose a new primitive called partial order preserving encoding (POPE) that achieves ideal OPE security with frequency hiding and also leaves a sizable fraction of the data pairwise incomparable. Using only O(1) persistent and $O(n^\epsilon)$ non-persistent client storage for $0<\epsilon<1$, our POPE scheme provides extremely fast batch insertion consisting of a single round, and efficient search with O(1) amortized cost for up to $O(n^{1-\epsilon})$ search queries. This improved security and performance makes our scheme better suited for today's insert-heavy databases.Comment: Appears in ACM CCS 2016 Proceeding

Parallel sparse interpolation using small primes

To interpolate a supersparse polynomial with integer coefficients, two alternative approaches are the Prony-based "big prime" technique, which acts over a single large finite field, or the more recently-proposed "small primes" technique, which reduces the unknown sparse polynomial to many low-degree dense polynomials. While the latter technique has not yet reached the same theoretical efficiency as Prony-based methods, it has an obvious potential for parallelization. We present a heuristic "small primes" interpolation algorithm and report on a low-level C implementation using FLINT and MPI.Comment: Accepted to PASCO 201

Computing sparse multiples of polynomials

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=Q and t is constant, we give a polynomial-time algorithm in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650

Response Of Forest Birds To Spotted Wing Drosophila (Drosophila Suzukii Matsumura), A Novel Invasive Fruit Pest, At Allegheny National Forest

Non-native invasive insect pests can have dramatic impacts on native ecosystems, feeding on plant foliage, wood, or sap. Little is known, however, about how fruit-targeting NNIIPs may affect native ecosystems. Spotted wing Drosophila (Drosophila suzukii Matsumura, SWD) is a recently introduced invasive vinegar fly that parasitizes the fruits of many plant species in the United States. While its activity in agricultural systems is well-documented, little is known about its activity in forest ecosystems, despite growing evidence of its presence and parasitism of fruits there. Parasitism could reduce fruit attractiveness for vertebrate fruit consumers, including migratory birds. As such, this may reduce soft mast food availability for birds in early successional habitat during the post-breeding season and fall migration, interfering with energy accumulation when demands are greatest. This thesis includes 4 chapters focused on understanding factors influencing SWD in invaded forest habitat and whether SWD activity associates with avian abundances, species richness, body condition, and fruit consumption at Allegheny National Forest in northwestern Pennsylvania. In Chapter 1, I provide context and justification for this research. I first describe how NNIIPs are known to influence forest ecosystems. I then review information on how SWD affects fruit crops in agricultural systems and provide a framework detailing how fruit parasitism by SWD may alter forest ecosystems. I lay out the research on fruit parasitism and fruit consumption by wildlife, especially fruit-consuming birds. I relate this to cascading effects on the following: 1) seed dispersal, 2) forest community structure and composition, and 3) food resources for wildlife. Finally, I highlight important research needs to uncover how SWD may affect forest ecosystems. In Chapter 2, I investigate how SWD distribution and abundance are influenced by forest composition, with respect to dominance by black cherry (Prunus serotina), a highly parasitized host, and local fruit resources in a forested ecosystem. From July to October 2019 and 2020 I trapped SWD in 21 regenerating timber harvests and surrounding forest canopies, representing 2 different cover types, and I collected Allegheny blackberry (Rubus allegheniensis) fruit samples in those regenerating timber harvests to observe for emergence of SWD. Relative abundance of SWD in regenerating timber harvests and surrounding forest canopy correlated to host resources at the local (i.e., sampling point) level but not the adjacent cover type. Parasitism of R. allegheniensis fruits positively correlated with local R allegheniensis fruits and basal area of P. serotina in adjacent mature forest. The results of this study provide the first evidence for the importance of local resources over those in an adjacent cover type or surrounding forest for predicting SWD trap captures, and the importance of local fruit resources and host plant density in adjacent cover type for SWD fruit parasitism in invaded forest ecosystems. I suggest these relationships can help predict the timing and abundance of SWD in both natural and semi-natural systems where wild fruiting plants are abundant, providing valuable information for monitoring and management planning. In Chapter 3, I identify factors, primarily those related to SWD abundance and fruit parasitism, influencing frugivorous and non-frugivorous bird relative abundance and species richness in regenerating timber harvests invaded by SWD during the post-breeding season and fall migration. From July to September 2019 and 2020 I sampled bird communities using mist net captures in 21 regenerating timber harvests, and I trapped SWD and sampled R. allegheniensis fruits for SWD parasitism at each mist net. One species, eastern towhee (Pipilo erythrophthalmus), correlated negatively with SWD trap captures. Two species, hooded warbler (Setophaga citrina) and ovenbird (Seiurus aurocapilla), correlated negatively and positively to parasitism of fruits, respectively. Post-breeding and migratory bird relative abundances were otherwise unrelated to SWD variables in regenerating harvests. Frugivorous birds either did not alter fruit consumption in response to SWD, or birds’ abilities to alter diet and foraging behavior masked relationships to SWD. Eastern towhee likely perceived fruits covered with flies as less desirable for consumption or may have been deterred by SWD swarms. Ovenbirds and hooded warblers altered their fruit consumption based on their responses to parasitism in modeling results. Alternatively, SWD activity may have affected arthropod communities, shifting composition and affecting some birds via trophic cascading effects. I highlight the importance of investigating arthropod community responses and the actual consumption of fruit resources in habitats invaded by SWD as critical next steps to understanding the ecological effects of SWD. In Chapter 4, I evaluate how factors, primarily those related to SWD abundance and fruit parasitism, relate to body condition of several bird species, and probability of fruit consumption by frugivorous birds in regenerating timber harvests invaded by SWD during the post-breeding season and fall migration. From July to September 2019 and 2020 I gathered data on body condition indices—subcutaneous fat, feather molt progression, and scaled mass index (SMI)— and collected fecal samples for evidence of fruit consumption for individuals of several bird species captured using mist nets in 21 regenerating timber harvests. I also trapped SWD and sampled R. allegheniensis fruits for SWD parasitism at each mist net. Multiple condition indices for several species correlated with relative abundance of SWD and parasitism of fruits, though the directions of relationships varied and were species-specific. Probability of R. allegheniensis fruit consumption by frugivorous birds correlated positively with parasitism of fruits at both the guild and species level. Frugivorous birds were more likely to consume parasitized fruits, and one species, gray catbird, experienced changes to nutritional intake and subsequent condition as a result. Fruit parasitism and greater abundances of adult SWD may have altered arthropod communities, altering composition and changing an additional food resource for both non-frugivorous and frugivorous birds. Most species improved condition indices as the season progressed, likely due to dietary flexibility. I list several critical avenues for research to better understand the relationships observed in this study: 1) how bird condition changes in relation to foraging microhabitat selection and SWD, 2) selection or rejection of SWD-parasitized fruits by frugivores related to nutritional metrics, and 3) arthropod community responses to SWD

Adapting the Outcome-Based Education Instructional Process to a Fourth Grade Social Studies and Science Curriculum in the Cascade School District

The purpose of this project was to utilize the Outcome-Based Education (OBE) instructional process to develop a curriculum in social studies and science for fourth grade students at Osborn Elementary School, Leavenworth, Washington. To accomplish this purpose, student learning objectives, unit objectives, lesson objectives, and lesson plans were adapted to the OBE instructional model