I was saddened to learn this week of the passing of Solomon Golomb. Solomon Golomb. Can you imagine the world without Tetris? What about the world without GPS or cell phones? If you think about it, an ordinary inch ruler is kind of inefficient.

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A: MATH. GEN , " We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile densities. In this context, and under rather mild assumptions, we pr In this context, and under rather mild assumptions, we prove a generalization of the first random tiling hypothesis which connects the maximum of the entropy with the symmetry of the ensemble.

Explicit examples are obtained through the re-interpretation of several exactly solvable models. This also leads to a counterexample to the analogue of the second random tiling hypothesis about the form of the entropy function near its maximum. Show Context Citation Context The following section is devoted to the introduction of our concept and the thermodynamic formalism used.

In the section on symmetry versus entropy, we present an argument on how to derive the first ra Voids exist in proteins as packing defects and are often associated with protein functions. We study the statistical geometry of voids in two-dimensional lattice chain polymers. We define voids as topological features and develop a simple algorithm for their detection.

For short chains, void geometr For short chains, void geometry is examined by enumerating all conformations. For long chains, the space of void geometry is explored using sequential Monte Carlo importance sampling and resampling techniques. We characterize the relationship of geometric properties of voids with chain length, including probability of void formation, expected number of voids, void size, and wall size of voids.

We formalize the concept of packing density for lattice polymers, and further study the relationship between packing density and compactness, two parameters frequently used to describe protein packing.

We find that both fully extended and maximally compact polymers have the highest packing density, but polymers with intermediate compactness have low packing density. To study the conformational entropic effects of void formation, we characterize the conformation reduction factor of void formation and found that there are strong end-effect.

Voids are more likely to form at the chain end. The critical exponent of end-effect is twice as large as that of self-contacting loop formation when existence of voids is not required. We also briefly discuss the sequential Monte Carlo sampling and resampling techniques used in this study. It is well-known that the question of whether a given finite region can be tiled with a given set of tiles is NP-complete.

We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process, we show that Monotone 1-in-3 Sati In the process, we show that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs. In higher dimensions, we show NP-completeness for the domino and straight tromino for general regions on the cubic lattice, and for simply-connected regions on the four-dimensional hypercubic lattice.

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## Polyominoes : puzzles, patterns, problems, and packings

Zulutilar This is the version described by Redelmeier. Brent Dalrymple Riccardo Giacconi Beyond rectangles, Golomb gave his hierarchy for single polyominoes: Golomb, University Professor at the University of Southern California, teaches in the Departments of Mathematics and Electrical Engineering, invents mathematical puzzles, and publishes in many areas of science and technology. Beginning with an initial square, number the adjacent squares, clockwise from the top, 1, 2, 3, pplyominoes 4. Martin David Kruskal Generalizations of polyominoes to other base shapes other that squares are known as polyformswith the best-known examples being the polyiamonds and polyhexes. Robert Huebner Ernst Mayr. Huffman Solomon W.

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## Polyominoes

In other words, An grows exponentially. Define the upper-right square to be the rightmost square in the uppermost row of the polyomino. Define the bottom-left square similarly. Refinements of this procedure combined with data for An produce the lower bound given above. The upper bound is attained by generalizing the inductive method of enumerating polyominoes. Instead of adding one square at a time, one adds a cluster of squares at a time.