Evolutionary dynamics of group cooperation with asymmetrical environmental feedback

In recent years, there has been growing interest in studying evolutionary games with environmental feedback. Previous studies exclusively focus on two-player games. However, extension to multi-player game is needed to study problems such as microbial cooperation and crowdsourcing collaborations. Here, we study coevolutionary public goods games where strategies coevolve with the multiplication factors of group cooperation. Asymmetry can arise in such environmental feedback, where games organized by focal cooperators may have a different efficiency than the ones by defectors. Our analysis shows that co-evolutionary dynamics with asymmetrical environmental feedback can yield oscillatory convergence to persistent cooperation, if the relative changing speed of cooperators' multiplication factor is above a certain threshold. Our work provides useful insights into sustaining group cooperation in a changing world

New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification

We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in ${\mathbb C}$ for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar graphs. The classification has an explicit criterion. The new P-time cases in (2) provably cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local connection to #CSP, defined in terms of a "loop space". This is the first substantive advance toward a planar Holant classification with not necessarily symmetric constraints. We introduce M\"obius transformation on ${\mathbb C}$ as a powerful new tool in hardness proofs for counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202

Inter-Grade and Inter-Batch Variability of Pharmaceutical-Grade Sodium Alginate

Polymeric excipients are generally the least well-characterized components of pharmaceutical formulations. The aim of this dissertation work is to facilitate the quality-by-design (QbD) approach to pharmaceutical formulation and manufacturing by evaluating the inter-grade and inter-batch variability of pharmaceutical-grade polymeric excipients. Sodium alginate, a widely used polymeric excipient, was selected for evaluation using appropriate analytical methods and test conditions, especially rheological methods. The materials used were six different grades of sodium alginate and an additional ten batches of one of the grades. To compare the six grades, steady shear measurements were conducted on solutions at 1, 2, and 3% w/w, consistent with their use as thickening or binding agents. Small amplitude oscillation (SAO) measurements were conducted on sodium alginate solutions at higher concentrations (4-13% w/w) corresponding to their use in controlled release matrices. In order to compare the ten batches of one grade, steady shear and SAO measurements were performed on their solutions at 2% w/w and 8% w/w, respectively. Results show that rheological properties of sodium alginate solutions are influenced by both molecular weight and chemical composition of sodium alginate. ¡§One-point¡¨ apparent viscosity data obtained at one low concentration and one shear rate is not representative of the complex rheological behavior of various grades of sodium alginate solutions at higher concentrations or other shear rates. The potential interchangeability of these different grades used as thickening or binding agents could be established by comparing the apparent viscosities of their solutions as a function of both alginate concentration and shear conditions. For sodium alginate used in controlled release formulations, both steady shear (at one low concentration, e.g., 2% w/w) and SAO measurements (at one high concentration indicative of polymer gel state, e.g., 8% w/w) are recommended to be performed on sodium alginate solutions to ensure interchangeability. Furthermore, among batches of the same grade, significant differences in rheological properties were observed, especially at the high solution concentration (i.e., 8% w/w). In summary, inter-grade and inter-batch variability of sodium alginate can be determined using steady shear and SAO methods. The influence of inter-grade and inter-batch variability of sodium alginate on the functionality of sodium alginate used in matrix tablets was investigated with a focus on compression properties, swelling, erosion behavior of alginate matrix tablets, and drug release from matrix tablets. The compression behavior of four grades and three batches of sodium alginate were studied by compaction energetics, out-of-die Gurnham, and out-of-die Heckel analysis. It was found that sodium alginates deform less plastically than microcrystalline cellulose (MCC PH102) but similar to lactose anhydrous. Sodium alginates also demonstrate more elastic deformations during compression than both MCC PH102 and lactose anhydrous. Compacts prepared from multiple batches of the same grade varied in porosity. The same tensile strength of compacts can be achieved by compressing the multiple batches to the same porosity. Sodium alginate tablets undergo both swelling and erosion in water. Grades with substantially higher apparent viscosities at low solution concentration exhibit a higher percentage of water uptake and a low percentage of erosion. Those batches not significantly different in their apparent viscosities at low solution concentration but significantly different in viscoelasticity at high solution concentrations do demonstrate significant differences in their swelling and erosion behavior. Acetaminophen release from sodium alginate matrix tablets prepared from the four grades and three batches can be well described by a zero-order equation. Significant differences in release profile were observed among various grades and batches. In conclusion, the inter-grade and inter-batch variability of sodium alginate has a significant influence on the swelling, erosion, and drug release behavior of sodium alginate matrix tablets. Apparent viscosities of sodium alginate solution at low concentration alone are not sufficient to predict the functionality of sodium alginate used in matrix tablets. Viscoelastic properties of sodium alginate solutions at high concentrations indicative of polymer gel state are appropriate to be characterized. Further study was conducted to determine whether sodium alginate solutions\u27 rheological parameters are relevant to sodium alginate\u27s use in the formulation of calcium alginate gels. Among the grades with similar guluronic acid percentage (%G), there is a significant correlation between gel fracture force and apparent viscosity. However, the results for the partial correlation analysis for all six grades of sodium alginate show that gel fracture force is significantly correlated with %G, but not with the rheological properties of the sodium alginate solutions. Studies of the ten batches of one grade of sodium alginate show that apparent viscosities of their solutions do not correlate with gel fracture force while tan δ values are significantly, but minimally, correlated to gel fracture force. Inter-batch differences in the rheological behavior for one specific grade of sodium alginate are insufficient to predict the corresponding calcium alginate gel\u27s mechanical properties. In summary, rheological methods, including steady shear and small amplitude oscillation, are able to identify the inter-grade and inter-batch variability of sodium alginate. Inter-grade and inter-batch variability of sodium alginate could lead to substantial differences in the functionality of sodium alginate in matrix tablets and in calcium alginate gels. Rheological properties of sodium alginate in solution are suggestive of its functionality as thickeners, or as controlled release agent. However, rheological properties of sodium alginate in solution do not seem to be sufficient to predict the mechanical properties of the corresponding calcium alginate gels