87 research outputs found
Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability
We introduce a new model of algorithmic tile self-assembly called
size-dependent assembly. In previous models, supertiles are stable when the
total strength of the bonds between any two halves exceeds some constant
temperature. In this model, this constant temperature requirement is replaced
by an nondecreasing temperature function that depends on the size of the smaller of the two halves. This
generalization allows supertiles to become unstable and break apart, and
captures the increased forces that large structures may place on the bonds
holding them together.
We demonstrate the power of this model in two ways. First, we give fixed tile
sets that assemble constant-height rectangles and squares of arbitrary input
size given an appropriate temperature function. Second, we prove that deciding
whether a supertile is stable is coNP-complete. Both results contrast with
known results for fixed temperature.Comment: In proceedings of ISAAC 201
Exponential Replication of Patterns in the Signal Tile Assembly Model
Chemical self-replicators are of considerable interest in the field of nanomanufacturing and as a model for evolution. We introduce the problem of self-replication of rectangular two-dimensional patterns in the practically motivated Signal Tile Assembly Model (STAM) [9]. The STAM is based on the Tile Assembly Model (TAM) which is a mathematical model of self-assembly in which DNA tile monomers may attach to other DNA tile monomers in a programmable way. More abstractly, four-sided tiles are assigned glue types to each edge, and self-assembly occurs when singleton tiles bind to a growing assembly, if the glue types match and the glue binding strength exceeds some threshold. The signal tile extension of the TAM allows signals to be propagated across assemblies to activate glues or break apart assemblies. Here, we construct a pattern replicator that replicates a two-dimensional input pattern over some fixed alphabet of size φ with O(φ) tile types, O(φ) unique glues, and a signal complexity of O(1). Furthermore, we show that this replication system displays exponential growth in n, the number of replicates of the initial patterned assembly
Universal Shape Replicators via Self-Assembly with Attractive and Repulsive Forces
We show how to design a universal shape replicator in a self- assembly system with both attractive and repulsive forces. More precisely, we show that there is a universal set of constant-size objects that, when added to any unknown holefree polyomino shape, produces an unbounded number of copies of that shape (plus constant-size garbage objects). The constant-size objects can be easily constructed from a constant number of individual tile types using a constant number of preprocessing self-assembly steps. Our construction uses the well-studied 2-Handed Assembly Model (2HAM) of tile self-assembly, in the simple model where glues interact only with identical glues, allowing glue strengths that are either positive (attractive) or negative (repulsive), and constant temperature (required glue strength for parts to hold together). We also require that the given shape has specified glue types on its surface, and that the feature size (smallest distance between nonincident edges) is bounded below by a constant. Shape replication necessarily requires a self-assembly model where parts can both attach and detach, and this construction is the first to do so using the natural model of negative/repulsive glues (also studied before for other problems such as fuel-efficient computation); previous replication constructions require more powerful global operations such as an “enzyme” that destroys a subset of the tile types.National Science Foundation (U.S.) (Grant EFRI1240383)National Science Foundation (U.S.) (Grant CCF-1138967
Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing
DNA-based self-assembly is an autonomous process whereby a disordered system of DNA sequences forms an organized structure or pattern as a consequence of Watson-Crick complementarity of DNA sequences, without external direction.
Here, we propose self-assembly (SA) hypergraph automata as an automata-theoretic model for patterned self-assembly. We investigate the computational power of SA-hypergraph automata and show that for every recognizable picture language, there exists an SA-hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language.
Moreover, we investigate the computational power of some variants of the Signal-passing Tile Assembly Model (STAM), as well as propose the concept of {\it Smart Tiles}, i.e., tiles with glues that can be activated or deactivated by signals, and which possess a limited amount of local computing capability. We demonstrate the potential of smart tiles to perform some robotic tasks such as replicating complex shapes
Molecular evolution research Annual report
Amino acid syntheses by heating formaldehyde and ammonia mixtures in molecular evolution researc
Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions
In this paper, we derive the time-fractional Cahn-Hilliard equation from
continuum mixture theory with a modification of Fick's law of diffusion. This
model describes the process of phase separation with nonlocal memory effects.
We analyze the existence, uniqueness, and regularity of weak solutions of the
time-fractional Cahn-Hilliard equation. In this regard, we consider
degenerating mobility functions and free energies of Landau, Flory--Huggins and
double-obstacle type. We apply the Faedo-Galerkin method to the system, derive
energy estimates, and use compactness theorems to pass to the limit in the
discrete form. In order to compensate for the missing chain rule of fractional
derivatives, we prove a fractional chain inequality for semiconvex functions.
The work concludes with numerical simulations and a sensitivity analysis
showing the influence of the fractional power. Here, we consider a convolution
quadrature scheme for the time-fractional component, and use a mixed finite
element method for the space discretization
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