A ( v , k , λ , μ ) -partial difference set (PDS) is a subset D of a group G such that | G | = v , | D | = k , and every nonidentity element x of G can be written in either λ or μ different ways as a product g h - 1 , depending on whether or not x is in D . Assuming the identity is not in D and D is inverse-closed, the corresponding Cayley graph Cay ( G , D ) will be strongly regular. Partial difference sets have been the subject of significant study, especially in abelian groups, but relatively little is known about PDSs in nonabelian groups. While many techniques useful for abelian groups fail to translate to a nonabelian setting, the purpose of this paper is to show that examples and constructions using abelian groups can be modified to generate several examples in nonabelian groups. In particular, in this paper we use such techniques to construct the first known examples of PDSs in nonabelian groups of order q 2 m , where q is a power of an odd prime p and m ≥ 2 . The groups constructed can have exponent as small as p or as large as p r in a group of order p 2 r . Furthermore, we construct what we believe are the first known Paley-type PDSs in nonabelian groups and what we believe are the first examples of Paley–Hadamard difference sets in nonabelian groups, and, using analogues of product theorems for abelian groups, we obtain several examples of each. We conclude the paper with several possible future research directions.

Denniston constructed partial difference sets (PDSs) with the parameters (23m,(2m+r−2m+2r)(2m−1),2m−2r+(2m+r−2m+2r)(2r−2),(2m+r−2m+2r)(2r−1)) in elementary abelian groups of order 23m for all m≥2,1≤r<m. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters (p3m,(pm+r−pm+pr)(pm−1),pm−pr+(pm+r−pm+pr)(pr−2),(pm+r−pm+pr)(pr−1)) exist in all elementary abelian groups of order p3m for all m≥2,r∈{1,m−1} where p is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.

For nearly a century, mathematicians have been developing techniques for constructing abelian automorphism groups of combinatorial objects, and, conversely, constructing combinatorial objects from abelian groups. While abelian groups are a natural place to start, recent computational evidence strongly indicates that the vast majority of transitive automorphism groups of combinatorial objects are nonabelian. This observation is the guiding motivation for this paper. We propose a new method for constructing nonabelian automorphism groups of combinatorial objects, which could be called the combinatorial transfer method, and we demonstrate its power by finding (1) the first infinite families of nonabelian Denniston partial difference sets (including nonabelian Denniston PDSs of odd order), (2) the first infinite family of Spence difference sets in groups with a Sylow 3-subgroup that is non-normal and not elementary abelian, (3) the first infinite families of McFarland difference sets in groups with a Sylow p-subgroup that is non-normal and is not elementary abelian, (4) new infinite families of partial difference sets in nonabelian p-groups with large exponent, (5) an infinite family of semiregular relative difference sets whose forbidden subgroup is nonabelian, and (6) a converse to Dillon's Dihedral Trick in the PDS setting. We hope this paper will lead to more techniques to explore this largely unexplored topic.

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which-one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil-are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

Spence [9] constructed ( 3^d+1(3^d+1-1)/2, 3^d(3^d+1+1)/2, 3^d(3^d+1)/2) -difference sets in groups K × C_3^d+1 for d any positive integer and K any group of order 3^d+1-1/2 . Smith and Webster [8] have exhaustively studied the d=1 case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in A_4 × C_3 by using (3, 3, 3, 1)-relative difference sets in a non-normal subgroup isomorphic to C_3^2 . Drisko [3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to C_3^d+1 as long as 3^d+1-1/2 is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that 3^d+1-1/2 is a prime power. We conjecture that any group of order 3^d+1(3^d+1-1)/2 with a normal subgroup isomorphic to C_3^d+1 will have a Spence difference set (this is analogous to Dillon’s conjecture in 2-groups, and that result was proved in Drisko’s work). Finally, we present the first known example of a Spence difference set in a group where the Sylow 3-subgroup is nonabelian and has exponent bigger than 3. This new construction, found by computing the full automorphism group Aut(𝒟) of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of Aut(𝒟) , uses (3, 3, 3, 1)-relative difference sets to describe the difference set.

Denniston constructed partial difference sets (PDSs) with the parameters $(2^{3m}, (2^{m+r} - 2^m + 2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2), (2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ for all $m \geq 2, 1 \leq r < m$. In this paper, we show that PDSs with the Denniston parameters $(p^{3m}, (p^{m+r} - p^m + p^r)(p^m-1), p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ exist in all elementary abelian groups of order $p^{3m}$ for all $m \geq 2, r \in \{1, m-1\}$ where $p$ is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes.

The traditionally held assumption that photo-dependent processes are the predominant source of H2O2 in natural waters has been recently questioned by an increrasing body of evidence showing the ubiquitiousness of H2O2 in dark water bodies and in groundwater. In this study, we conducted field measurement of H2O2 in an intra-meander hyporheic zone and in surface water at East River, CO. On-site detection using a sensitive chemiluminescence method suggests H2O2 concentrations in groundwater ranging from 6 nM (at the most reduced region) to ~ 80 nM (in a locally oxygen-rich area) along the intra-meander transect with a maxima of 186 nM detected in the surface water in an early afternoon, lagging the maximum solar irradiance by ∼ 1.5 h. Our results suggest that the dark profile of H2O2 in the hyporheic zone is closely correlated to local redox gradients, indicating that interactions between various redox sensitive elements could play an essential role. Due to its transient nature, the widespread presence of H2O2 in the hyporheic zone indicates the existence of a sustained balance between H2O2 production and consumption, which potentially involves a relatively rapid succession of various biogeochemically important processes (such as organic matter turnover, metal cycling and contaminant mobilization). More importantly, this study confirmed the occurrence of reactive oxygen species at a subsurface redox transition zone and further support our understanding of redox boundaries on reactive oxygen species generation and as key locations of biogeochemical activity.

A $$(v,k,\lambda )$$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group $$C_8 \times C_4 \times C_2$$ as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families.

This chapter reviews the surface chemical behavior of aluminum oxides and hydroxides, with an emphasis on adsorption studies and the electrochemical behavior of the alumina–water interface. The coordination behavior of aluminum cation toward hydroxide ions explains the marked dependence of aluminum hydroxide solubility on pH. Thus, the initial surface structure can be related to the crystal structure and the way in which the surface was formed. Furthermore, the surface chemical behavior of aluminum oxides and hydroxides can be explained in part by surface structure expected for particular crystals. All strongly dried aluminum oxides and hydroxides chemisorb at least a monolayer of water when exposed to moisture at room temperature. Defects and elemental substitutions in the gibbsite crystal structure may lead to a net charge imbalance. The surface acidity (ability to release protons from surface sites) of aluminum oxides is of long-standing interest, for both practical and theoretical reasons.

Incentives designed to spur the adoption of agroforestry practices by farmers across the United States have largely been ineffective. Unfortunately, most Conservation Reserve Program (CRP) contracts are composed of temporary grassland conversions, not permanent forest land conversions. Using data collected from forestry and livestock producers, we estimate the minimum payment necessary for producers to adopt the agroforestry practice of silvopasture. Results indicate that CRP contracts based on silvopasture or other integrated systems may be a more efficient alternative than contracts based on complete conversion to forest land. However, successful implementation of such a policy requires a well-informed producer base whose behavior can be nudged towards integrated systems.

This chapter presents preliminary results of small-scale, natural-gradient tracer tests conducted in oxic and suboxic zones of a shallow, unconfined sand and gravel aquifer near Falmouth, Massachusetts. A more fundamental, mechanistic approach that uses the surface ionization and coordination concept in conjunction with aqueous speciation models is now widely accepted for modeling the interactions of metal ions with mineral surfaces. The physical processes that lead to macrodispersion and fluid mixing may affect aqueous speciation of metals via dilution of complexing ligands or the addition of new solutes from dissolving mineral phases, each of which may have direct or indirect effects on the sorption of metal ions. However, the application of these models to metal transport in groundwaters is still limited by inadequate consideration of adsorption-desorption reactions within the models and the availability of relevant kinetic and thermodynamic data to describe metal sorption processes in the groundwater environment.

Collections of difference sets called linking systems have been used to construct new families of linked systems of symmetric designs. In this paper, we define relative and almost linking systems, collections of difference sets and almost difference sets with very similar linking properties to linking systems. These linking systems have connections to bent sets and vectorial bent functions. We construct examples of relative and almost linking systems using a technical lemma.

A non-electrostatic generalized composite surface complexation model (SCM) was developed for U(VI) sorption on contaminated F-Area sediments from the U.S. Department of Energy Savannah River Site, South Carolina. The objective of this study was to test if a simpler, semi-empirical, non-electrostatic U(VI) sorption model (NEM) could achieve the same predictive performance as a SCM with electrostatic correction terms in describing U(VI) plume evolution and long-term mobility. One-dimensional reactive transport simulations considering key hydrodynamic processes, Al and Fe minerals, as well as H+ and U surface complexation, with and without electrostatic correction terms, were conducted. The NEM was first calibrated with laboratory batch H+ and U(VI) sorption data on F-Area sediments, and then the surface area of the NEM was adjusted to match field observations of dissolved U(VI). Modeling results indicate that the calibrated NEM was able to perform as well as the previously developed electrostatic model in predicting the long-term evolution of H+ and U(VI) at the site, given the variability of field-site data. The electrostatic and NEM models yield somewhat different results for the time period when basin discharge was active; however, it is not clear which modeling approach may be better to model this early time period because groundwater quality data during this period were not available. A key finding of this study is that the applicability of NEM (and thus robustness of its predictions) to the field system evolves with time and is strongly dependent on the pH range that was used to develop the model.

This study focuses on the effects of fulvic acid (FA) on uranium(VI) sorption kinetics to a silica sand. Using a tritium-labeled FA in batch experiments made it possible to investigate sorption rates over a wide range of environmentally relevant FA concentrations (0.37-23 mg L-1 TOC). Equilibrium speciation calculations were coupled with an evaluation of U(VI) and FA sorption rates based on characteristic times. This allowed us to suggest plausible sorption mechanisms as a function of solution conditions (e.g., pH, U(VI)/FA/surface site ratios). Our results indicate that U(VI) sorption onto silica sand can be either slower or faster in the presence of FA compared to a ligand-free system. This suggests a shift in the underlying mechanisms of FA effects on U(VI) sorption, from competitive sorption to influences of U(VI)-FA complexes, in the same system. Changes in metal sorption rates depend on the relative concentrations of metals, organic ligands, and mineral surface sites. Hence, these results elucidate the sometimes conflicting information in the literature about the influence of organic matter on metal sorption rates. Furthermore, they provide guidance for the selection of appropriate sorption equilibration times for experiments that are designed to determine metal distribution coefficients (K-d values) under equilibrium conditions.

The prediction of U(VI) adsorption onto montmorillonite clay is confounded by the complexities of: (1) the montmorillonite structure in terms of adsorption sites on basal and edge surfaces, and the complex interactions between the electrical double layers at these surfaces, and (2) U(VI) solution speciation, which can include cationic, anionic and neutral species. Previous U(VI)-montmorillonite adsorption and modeling studies have typically expanded classical surface complexation modeling approaches, initially developed for simple oxides, to include both cation exchange and surface complexation reactions. However, previous models have not taken into account the unique characteristics of electrostatic surface potentials that occur at montmorillonite edge sites, where the electrostatic surface potential of basal plane cation exchange sites influences the surface potential of neighboring edge sites (‘spillover’ effect).A series of U(VI) – Na-montmorillonite batch adsorption experiments was conducted as a function of pH, with variable U(VI), Ca, and dissolved carbonate concentrations. Based on the experimental data, a new type of surface complexation model (SCM) was developed for montmorillonite, that specifically accounts for the spillover effect using the edge surface speciation model by Tournassat et al. (2016a). The SCM allows for a prediction of U(VI) adsorption under varying chemical conditions with a minimum number of fitting parameters, not only for our own experimental results, but also for a number of published data sets. The model agreed well with many of these datasets without introducing a second site type or including the formation of ternary U(VI)-carbonato surface complexes. The model predictions were greatly impacted by utilizing analytical measurements of dissolved inorganic carbon (DIC) concentrations in individual sample solutions rather than assuming solution equilibration with a specific partial pressure of CO2, even when the gas phase was laboratory air. Because of strong aqueous U(VI)-carbonate solution complexes, the measurement of DIC concentrations was even important for systems set up in the ‘absence’ of CO2, due to low levels of CO2 contamination during the experiment.

Partial geometric difference sets (PGDSs) were defined in Olmez (J Combin Des 22(6):252–269, 2014). They are used to construct partial geometric designs. We use the framework of extended building sets to find infinite families of PGDSs in abelian groups. Included in our new families of PGDSs are generalizations of the Hadamard, McFarland, Spence, Davis-Jedwab, and Chen difference sets.

Organic carbon (OC) concentrations in subsurface sediments are typically 10 to 200 times lower than in surface soils, posing a distinct challenge for characterization. In this study, a range of chemical extractions were evaluated for extraction of natural organic matter (NOM) from two low-carbon (< 0.2%) alluvial sediments. The OC extraction efficiency followed the order pyrophosphate (PP) > NaOH > HCl, hydroxylamine hydrochloride > dithionite, water. A NOM extraction and purification scheme was developed using sequential extraction with water (MQ) and sodium pyrophosphate at pH 10 (PP), combined with purification by dialysis and solid phase extraction in order to isolate different fractions of sedimentassociated NOM. Characterization of these pools of NOM for metal content and by Fourier transform infrared spectroscopy (FITR) showed that the water soluble fraction (MQ-SPE) had a higher fraction of aliphatic and carboxylic groups, while the PP-extractable NOM (PP-SPE and PP > 1kD) had higher fractions of C=C groups and higher residual metals. This trend from aliphatic to more aromatic is also supported by the specific UV absorbance at 254 nm (SUVA(254)) (3.5 vs 5.4 for MQ-SPE and PP-SPE, respectively) and electrospray ionization Fourier transform ion cyclotron resonance spectrometry (ESI-FTICR-MS) data which showed a greater abundance of peaks in the low O/C and high H/C region (0-0.4 O/C, 0.8-2.0 H/ C) for the MQ-SPE fraction of NOM. Radiocarbon measurements yielded standard radiocarbon ages of 1020, 3095, and 9360 years BP for PP-SPE, PP > 1kD, and residual (non-extractable) OC fractions, indicating an increase in NOM stability correlated with greater metal complexation, apparent molecular weight, and aromaticity. Published by Elsevier Ltd.

The commonly held assumption that photodependent processes dominate H2O2 production in natural waters has been recently questioned. Here, we present evidence for the unrecognized and light-independent generation of H2O2 in groundwater of an alluvial aquifer adjacent to the Colorado River near Rifle, CO. In situ detection using a sensitive chemiluminescent method suggests H2O2 concentrations ranging from lower than the detection limit (<1 nM) to 54 nM along the vertical profiles obtained at various locations across the aquifer. Our results also suggest dark formation of H2O2 is more likely to occur in transitional redox environments where reduced elements (e.g., reduced metals and NOM) meet oxygen, such as oxic-anoxic interfaces. A simplified kinetic model involving interactions among iron, reduced NOM, and oxygen was able to reproduce roughly many, but not all, of the features in our detected H2O2 profiles, and therefore there are other minor biological and/or chemical controls on H2O2 steady-state concentrations in such aquifer. Because of its transient nature, the widespread presence of H2O2 in groundwater suggests the existence of a balance between H2O2 sources and sinks, which potentially involves a cascade of various biogeochemically important processes that could have significant impacts on metal/nutrient cycling in groundwater-dependent ecosystems, such as wetlands and springs. More importantly, our results demonstrate that reactive oxygen species are not only widespread in oceanic and atmospheric systems but also in the subsurface domain, possibly the least understood component of biogeochemical cycles.