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In 1992 Barry Cipra posed an interesting combinatorial counting problem. In essence, it asks for the number Sk,σ of configurations possible if a circular arrangement of k cups, each having σ stones, is modified by applying a particular transition rule that changes the distribution of stones. Ettestad and Carbonara (2010 and 2011) noted that this system is a finite Cellular Automaton, showed two interesting non-recursive formulas for Sk,1 and showed that the shape of the non-zero terms in the reduced matrix for the Cups and Stones Counting Problem (CSCP) with exactly 2n + 1 cups is equivalent to Sierpinski's gasket. We call this the Sierpinski property. In this paper we extend the problem by numbering the stones, thereby revealing several new and interesting properties of the game. In particular we extend a slightly modified version of the Sierpinski property to the CSCP with any number of cups by defining a "home cup" and referencing all the other cups to the home cup. [ABSTRACT FROM AUTHOR]

In 1992 Barry Cipra posed an interesting combinatorial counting problem. In essence, it asks for the number Sk, σ of configurations possible if a circular arrangement of k cups, each having σ stones, is modified by applying a particular transition rule that changes the distribution of stones. Carbonara and Green (1998) studied the integer sequence Sk, 1 and presented a recursive formula for it: Sk = Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. + 2r-j Sd+1 + d2r - 2r+1 where k = 2r + 1 + d >2, r ≥ 0 and 0 < d ≤ 2r. Ettestad and Carbonara (2010) noted that this system is a finite Cellular Automaton, showed two interesting non-recursive formulas for Sk, and claimed that the shape of the non-zero terms in the reduced matrix for the Cups and Stones Counting Problem (CSCP) with 2n + 1 cups is equivalent to the Sierpinski Gasket. We are proving that claim in this paper. In doing this, we extend the classic definition of the Sierpinski Gasket to discrete geometric figures. [ABSTRACT FROM AUTHOR]

In 1992 Barry Cipra posed an interesting combinatorial counting problem. In essence, it asks for the number Sk, σ of configurations possible if a circular arrangement of k cups, each having σ stones, is modified by, applying a particular transition rule that changes the distribution of stones. Carbonara and Green (1998) studied the integer sequence Sk, 1 (a.k.a. Sk) and presented a recursive formula for it: Sk = 2S2r+1 + 2r-JSd+1 + d2r- 2r+l wherek = 2r + 1 + d > 2, r ≥ 0 and 0 < d ≤ 2r. We note that this system is a finite Cellular Automaton. Taking advantage of previous work by Ettestad and Carbonara where properties of this CA are studied, we present here two non-recursive formulas for Sk, 1, one of which involves the number of odd entries in the first nk rows of Pascal's triangle. [ABSTRACT FROM AUTHOR]

In this investigation, supervised machine learning (ML) was utilized to accurately predict the optimum bromine doping concentration in single-junction MASnI 3- x Br x devices. Data-driven optimizations were carried out on 42 000 unique devices built utilizing a solar cell capacitance simulator (SCAPS). The devices were investigated through variations of bromine doping %, bandgap, electron affinity, series resistance, back-contact metal, and acceptor concentration─parameters that were specifically chosen because of their tunable nature and ability to be modified through facile experimental fabrication techniques of the device. Five different algorithms were utilized to explore feature engineering. The first step before bromine doping within the device included validation studies of a pure tin-based system, MASnI 3 : a power conversion efficiency (PCE) of 6.71% was achieved, having close congruence with experimental data. ML analyses for optimal bromine doping resulted in the discovery of two devices with bromine concentrations of 22.43% (Br22) and 25.63% (Br25), with the latter being a more fine-tuned value obtained through extra rigorous analysis. To understand the total and relative impact of each feature on power conversion efficiency (PCE), Br22 and Br25 were analyzed with a state-of-the-art algorithm, namely, the SHapley Additive exPlanations (SHAP) algorithm. Focusing on the two discovered devices, further device optimizations were carried out utilizing SCAPS. Modulations of absorber thickness, bulk and interfacial defect density, and choice of electron transport layer (ETL) and hole transport layer (HTL) materials were tried. Device stability was analyzed through carrier lifetime studies. Following these optimization steps, Br22 and Br25 demonstrated final high PCE values of 20.72 and 17.37%, respectively. The ML-assisted quantitative analysis of the current work provides significant confidence for optimal bromine-doped tin-based devices to be considered as viable and competitive nontoxic alternatives to traditional technologies.