Dimer Covering and Percolation Frustration

Covering a graph or a lattice with non-overlapping dimers is a problem that has received considerable interest in areas such as discrete mathematics, statistical physics, chemistry and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by non-overlapping dimers has not evidently been considered. Here, we propose a novel procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattice, on the remaining bonds connecting the dimers. We obtain $p_c=0.367713(2)$ and $p_c=0.235340(1)$ for random coverings of the square and the triangular lattice, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice, and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this work, has an unusually low percolation threshold, and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this work, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly-coordinated networks.Comment: 11 pages, 7 figure

Percolation in Networks with Voids and Bottlenecks

A general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks, in the limit that the sub-net mesh size goes to zero. The validity of this method is tested for bond percolation on filled checkerboard and "stack-of-triangle" lattices. Thresholds for the checkerboard lattices of different mesh sizes are estimated using the gradient percolation method, while for the triangular system they are found exactly using the triangle-triangle transformation. The values of the thresholds approach the asymptotic values of 0.64222 and 0.53993 respectively as the mesh is made finer, consistent with a direct determination based upon the predicted critical corner-connection probability.Comment: to appear, Physical Review E. Small changes from first versio

Computational investigation of structure, dynamics and nucleation kinetics of a family of modified Stillinger-Weber model fluids in bulk and free-standing thin films

In recent decades, computer simulations have found increasingly widespread use as powerful tools of studying phase transitions in wide variety of systems. In the particular and very important case of aqueous systems, the commonly used force-fields tend to offer quite different predictions with respect to a wide range of thermodynamic and kinetic properties, including the ease of ice nucleation, the propensity to freeze at a vapor-liquid interface, and the existence of a liquid-liquid phase transition. It is thus of fundamental and practical interest to understand how different features of a given water model affect its thermodynamic and kinetic properties. In this work, we use the forward-flux sampling technique to study the crystallization kinetics of a family of modified Stillinger-Weber (SW) potentials with energy ($\epsilon$) and length ($\sigma$) scales taken from the monoatomic water (mW) model, but with different tetrahedrality parameters ($\lambda$). By increasing $\lambda$ from 21 to 24, we observe the nucleation rate increases by 48 orders of magnitude at a supercooling of ${\zeta}=T/T_m=0.845$. Using classical nucleation theory, we are able to demonstrate that this change can largely be accounted for by the increase in $|\Delta\mu|$, the thermodynamic driving force. We also perform rate calculations in freestanding thin films of the supercooled liquid, and observe a crossover from a surface-enhanced crystallization at $\lambda = 21$ to a bulk-dominated crystallization for $\lambda\ge22$.Comment: 10 pages, 9 figures, five table

Universal condition for critical percolation thresholds of kagome-like lattices

Lattices that can be represented in a kagome-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P_3, the probability that all three vertices in the triangle connect, and P_0, the probability that none connect. A linear approximation for P_3(P_0) is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for P_3(P_0) gives thresholds for the kagome, site-bond honeycomb, (3-12^2), and "stack-of-triangle" lattices that compare favorably with numerical results.Comment: Several new figures and small change

Towards a historical ecology of intertidal foraging in the Mafia Archipelago: archaeomalacology and implications for marine resource management

Understanding the timing and nature of human influence on coastal and island ecosystems is becoming a central concern in archaeological research, particularly when investigated within a historical ecology framework. Unfortunately, the coast and islands of eastern Africa have not figured significantly within this growing body of literature, but are important given their historically contingent environmental, social, and political contexts, as well as the considerable threats now posed to marine ecosystems. Here, we begin developing a longer-term understanding of past marine resource use in the Mafia Archipelago (eastern Africa), an area of high ecological importance containing the Mafia Island Marine Park. Focusing on the comparatively less researched marine invertebrates provides a means for initiating discussion on potential past marine ecosystem structure, human foraging and environmental shifts, and the implications for contemporary marine resource management. The available evidence suggests that human-environment interactions over the last 2000 years were complex and dynamic; however, these data raise more questions than answers regarding the specific drivers of changes observed in the archaeomalacological record. This is encouraging as a baseline investigation and emphasizes the need for further engagement with historical ecology by a range of cognate disciplines to enhance our understanding of these complex issues

Peningkatan Kemampuan Komunikasi Matematik Melalui Pembelajaran Matematika Realistik

Tujuan penelitian ini adalah untuk mengetahui pencapaian dan peningkatan kemampuan komunikasi matematik siswa Sekolah Menengah Pertama Kota Bengkulu melalui Pembelajaran Matematika Realistik. Metode penelitian yang digunakan adalah kuasi eksperimen dengan non-equivalent control group design. Hasil penelitian sebagai berikut. Pembelajaran Matematika Realistik lebih efektif dalam pencapaian dan peningkatan kemampuan komunikasi matematik siswa dibandingkan pembelajaran konvensional. Besarnya pencapaian kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik adalah 63,96 dan pembelajaran konvensional adalah 47,46. Sementara itu, besarnya peningkatan kemampuan komunikasi matematik siswa yang diajar melalui pembelajaran matematika realistik adalah 0,51 dan pembelajaran konvensional 0,24. The purpose of this study is to determine the achievement and improvement of students' mathematical communication ability in Bengkulu City Junior High School through realistic mathematics learning. The method used is a quasi-experimental non-equivalent control group design. The results of this study as follows. The realistic mathematics learning is more effective in the achievement and improvement of students' mathematical communication ability than conventional learning. The magnitude of the achievement of students' mathematical communication ability in realistic mathematics learning is 63,96 and conventional learning is 47,46. Meanwhile, the magnitude of the improvement of students' mathematical communication ability is realistic mathematics learning is 0,51 and conventional learning is 0,2

Membangun Kemandirian Belajar Siswa Melalui Pembelajaran Matematika Realistik

The Purpose of this study is to determine the achievement and improvement of self regulated learning student through realistics mathematics learning. This study use a quasi-experimental design by the non-equivalent control group. The results are realistics mathematics learning more effectively in the achievement and improvement of self regulated learning student is compared conventional learning. Differences achievement and improvement in both groups is significant. The magnitude of the achievement of self regulated learning student is taught through realistics mathematics learning is 176.85, while self regulated learning student is taught through conventional learning is 172.96. While, the magnitude of the increase in self regulated learning student is taught realistic mathematics learning is 0.1, if it is compared with conventional learning

A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger

We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings