Set Of Tiles Research Articles

Custom Design of Substratum Topography Increases Biomass Yield in Algal Turf Scrubbers

Abstract Algal turf scrubbers (ATSs) have been used for water remediation in many applications, generating benefits such as water quality improvements and biomass production. Their economical operation is hampered, however, by uncontrolled mixed microbial communities, typical of open cultivation systems that result in lower biomass quality. Design of substratum topography is one approach to control the algal community composition by constraining colonization dynamics. This study seeks to elucidate the role of substratum topography on biomass productivity of benthic algae in an ATS environment. A set of tiles with custom-designed textures of repeated hemispheres across a range of diameters were 3D printed and cast on ceramic clay. An algal turf community dominated by Mougeotia taxa was cultivated on the tiles in a laboratory-scale ATS and harvested for biomass measurements after 21 days. Results show higher biomass on the topographies formed of 500 and 2,000 μm diameter hemispheres, with 74.3% more attache...

Benthic diatoms from Potter Cove, 25 de Mayo (King George) Island, Antarctica: Mucilage and glucan storage as a C-source for limpets

Biofilms were allowed to develop on ceramic tiles placed in closed containers on the shore of Potter Cove, 25 de Mayo (King George) Island. Water pumping from the cove inside the containers extended for 25 days. Diatoms were the dominant microalgae in these biofilms, which were removed from a set of tiles to a) characterize the extracellular mucilage, b) carry out floristic determination and c) perform grazing experiments with the limpet Nacella concinna. Biofilms mucilaginous matrix consisted of proteins and carbohydrates. Room temperature aqueous extraction of the freeze-dried material rendered a fraction enriched in the storage glucan chrysolaminarin, its identity confirmed by methylation structural analyses. Hot water extracted products showed greater heterogeneity in monosaccharide composition, including glucose, mannose, galactose, fucose, xylose and rhamnose. Diatom identification revealed that Pseudogomphonema kamtschaticum was the dominant species followed by several Navicula species, Nitzschia pellucida and Synedra kerguelensis. Photographical survey of colonized tiles placed in glass flasks together with a specimen of Nacella concinna exhibited between 5 and 30% removal of the biofilms coverage after 24 h of exposure to the limpet, suggesting that EPS and chrysolaminarin constitute a C-source for the gastropod.

Co-Scheduling Persistent Periodic and Dynamic Aperiodic Real-Time Tasks on Reconfigurable Platforms

As task preemption/relocation with acceptably low overheads become a reality in today's reconfigurable FPGAs, they are starting to show bright prospects as platforms for executing performance critical task sets while allowing high resource utilization. Many performance sensitive real-time systems including those in automotive and avionics systems, chemical reactors, etc., often execute a set of persistent periodic safety critical control tasks along with dynamic event driven aperiodic tasks. This work presents a co-scheduling framework for the combined execution of such periodic and aperiodic real-time tasks on fully and run-time partially reconfigurable platforms. Specifically, we present an admission control strategy and preemptive scheduling methodology for dynamic aperiodic tasks in the presence of a set of persistent periodic tasks such that aperiodic task rejections may be minimized, thus resulting in high resource utilization. We used the 2D slotted area model where the floor of the FPGA is assumed to be statically equipartitioned into a set of tiles in which any arbitrary task may be feasibly mapped. The experimental results reveal that the proposed scheduling strategies are able to achieve high resource utilization with low task rejection rates over various simulation scenarios.

Revisiting a Tiling Hierarchy (II)

In a recent paper, we revisited Golomb’s hierarchy for tiling capabilities of finite sets of polyominoes. We considered the case when only translations are allowed for the tiles. In this classification, for several levels in Golomb’s hierarchy, more types appear. We showed that there is no general relationship among tiling capabilities for types corresponding to same level. Then we found the relationships from Golomb’s hierarchy that remain valid in this setup and found those that fail. As a consequence we discovered two alternative tiling hierarchies. The goal of this note is to study the validity of all implications in these new tiling hierarchies if one replaces the simply connected regions by deficient ones. We show that almost all of them fail. If one refines the hierarchy for tile sets that tile rectangles and for deficient regions then most of the implications of tiling capabilities can be recovered.

Tile/line access cache memory based on a multi‐level Z‐order tiling data layout

SummaryIneffective column‐directional cache memory access has become a bottleneck for efficient two‐dimensional (2‐D) data processing utilizing extended single instruction multiple data (SIMD) instructions. To solve this problem, we propose a cache memory with tile (column and row directions) and line (row direction) accessibility for efficient 2‐D data processing. 2‐D data access to the proposed cache memory is enabled via a hardware‐based multi‐mode address translation unit that eliminates the overhead of software‐based address calculation. To reduce the hardware overhead of the proposed cache, we propose a tag memory reduction method that replaces multiple tiles with an aligned tile set (RATS) in the cache. To verify the feasibility of the proposed cache, an LSI layout of a SIMD‐based general purpose‐oriented datapath embedding the proposed cache is designed in a 2.5×5 mm2 area using 0.18‐μm CMOS technology. Under a 3.9‐ns clock period (250 MHz), the read latency is limited to 3 clock cycles, which is the same as that for the conventional cache memory. Using the RATS method, the entire hardware overhead of the proposed cache is reduced to only 7% of that required for a conventional cache. In addition, simulation results for the proposed cache indicate a considerable reduction of L1 and L2 cache confliction misses compared with a conventional cache in power‐of‐two matrix size due to the column‐directional address stride being sufficiently smaller than page size. Therefore, the proposed cache provides efficient column‐directional parallel access as same as row‐directional parallel access so that it enables efficient SIMD operation requiring no transposition in matrix multiplication (MM). For LU decomposition (LUD), the proposed cache can provide almost the same performance to the column‐major–based LUD program as that to the row‐major–based LUD program. These results show that the proposed cache does not restrict our freedom in selecting either row‐ or column‐major order coding.

WANG TILE SIZE IN TERMS OF CIRCULAR PARTICLE DYNAMICS

The main advantage of the Wang tiling concept for material engineering is ability to create large material domains with a relatively small set of tiles. Such idea allows both a reduction of computational demands and preserving heterogeneity of a reconstructed media in comparison with traditional cell concepts. This work is dealing with a random heterogeneous material composed of monodisperse circular hard particles within a matrix. The Wang tile sets are generated via algorithm with molecular dynamics and adaptive boundaries approach. Even though previous works proved usefulness of the Wang tiling for material reconstruction, still plenty of questions remain unanswered. In here we would like to provide simulations with emphasis on the overall particle distribution and the ratio of hard disc number to tile size. The results and discussion should followers help with settings of both tile generations and the tiling algorithms when creating samples of various degree of heterogeneity.

OPTIMIZATION-BASED APPROACH TO TILING OF FINITE AREAS WITH ARBITRARY SETS OF WANG TILES

Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem.

The Simulation Powers and Limitations of Higher Temperature Hierarchical Self-Assembly Systems*

In this paper, we extend existing results about simulation and intrinsic universality in a model of tile-based self-assembly. Namely, we work within the 2-Handed Assembly Model (2HAM), which is a model of self-assembly in which assemblies are formed by square tiles that are allowed to combine, using glues along their edges, individually or as pairs of arbitrarily large assemblies in a hierarchical manner, and we explore the abilities of these systems to simulate each other when the simulating systems have a higher parameter, which is a system wide threshold dictating how many glue bonds must be formed between two assemblies to allow them to combine. It has previously been shown that systems with lower temperatures cannot simulate arbitrary systems with higher temperatures, and also that systems at some higher temperatures can simulate those at particular lower temperatures, creating an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. These previous results relied on two different definitions of simulation, one (strong simulation) seemingly more restrictive than the other (standard simulation), but which have previously not been proven to be distinct. Here we prove distinctions between them by first fully characterizing the set of pairs of temperatures such that the high temperature systems are intrinsically universal for the lower temperature systems (i.e. one tile set at the higher temperature can simulate any at the lower) using strong simulation. This includes the first impossibility result for simulation downward in temperature. We then show that lower temperature systems which cannot be simulated by higher temperature systems using the strong definition, can in fact be simulated using the standard definition, proving the distinction between the types of simulation.

Microsphere-based super-resolution scanning optical microscope.

High-refractive index dielectric microspheres positioned within the field of view of a microscope objective in a dielectric medium can focus the light into a so-called photonic nanojet. A sample placed in such nanojet can be imaged by the objective with super-resolution, i.e. with a resolution beyond the classical diffraction limit. However, when imaging nanostructures on a substrate, the propagation distance of a light wave in the dielectric medium in between the substrate and the microsphere must be small enough to reveal the sample's nanometric features. Therefore, only the central part of an image obtained through a microsphere shows super-resolution details, which are typically ∼100 nm using white light (peak at λ = 600 nm). We have performed finite element simulations of the role of this critical distance in the super-resolution effect. Super-resolution imaging of a sample placed beneath the microsphere is only possible within a very restricted central area of ∼10 μm2, where the separation distance between the substrate and the microsphere surface is very small (∼1 μm). To generate super-resolution images over larger areas of the sample, we have fixed a microsphere on a frame attached to the microscope objective, which is automatically scanned over the sample in a step-by-step fashion. This generates a set of image tiles, which are subsequently stitched into a single super-resolution image (with resolution of λ/4-λ/5) of a sample area of up to ∼104 μm2. Scanning a standard optical microscope objective with microsphere therefore enables super-resolution microscopy over the complete field-of-view of the objective.

On the Rationality of the Spectrum

Let $$\Omega \subset {\mathbb R}$$ be a compact set with measure 1. If there exists a subset $$\Lambda \subset {\mathbb R}$$ such that the set of exponential functions $$E_{\Lambda }:=\{e_\lambda (x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda \}$$ is an orthonormal basis for $$L^2(\Omega )$$ , then $$\Lambda $$ is called a spectrum for the set $$\Omega $$ . A set $$\Omega $$ is said to tile $${\mathbb R}$$ if there exists a set $$\mathcal T$$ such that $$\Omega + \mathcal T = {\mathbb R}$$ , the set $$\mathcal T$$ is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum.

Adaptive Boundaries of Wang Tiles for Heterogeneous Material Modelling

This contribution deals with algorithms for the generation of modified Wang tiles as a tool for the heterogeneous materials modelling. The proposed approach considers material domains only with 2D hard discs of both equal and different radii distributed within a matrix. Previous works showed potential of the Wang tile principles for reconstruction of heterogeneous materials. The main advantage of the tiling theory for material modelling is to stack large/infinite areas with relative small set of tiles with emphasis on a periodicity reduction in comparison with the traditional Periodic Unit Cell (PUC) concept. The basic units of the Wang Tiling are tiles with codes (colors) on edges. The algorithm for distribution of hard discs is based on the molecular dynamics to avoid particles overlapping. Unfortunately the nature of the Wang tiling together with molecular dynamics algorithms cause periodicity artefacts especially in tile corners of a composed material domain. In this paper a new algorithm with adaptive tile boundaries is presented in order to avoid edge and corner periodicity.

The Tilings of Deficient Squares by Ribbon <i>L</i>-Tetrominoes Are Diagonally Cracked

We consider tilings of deficient rectangles by the set T4 of ribbon L-tetro-minoes. A tiling exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are paired and each pair tiles a 2×4 rectangle. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The crack divides the square in two parts of equal area. The number of tilings of a (4m+1)×(4m+1) deficient square by T4 is equal to the number of tilings by dominoes of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T4 is twice the number of tilings by dominoes of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. In both cases the counting is realized by an explicit function which is a bijection in the first case and a double cover in the second. If an extra 2×2 tile is added to T4 , we call the new tile set T+4. A tiling of a deficient rectangle by T+4 exists if and only if the rectangle is a square of odd side. The missing cell has to be on the main NW-SE diagonal, in an odd position if the square is (4m+1)×(4m+1) and in an even position if the square is (4m+3)×(4m+3). The majority of the tiles in a tiling follow the rectangular pattern, that is, are either paired tetrominoes and each pair tiles a 2×4 rectangle, or are 2×2 squares. The tiles in an irregular position together with the missing cell form a NW-SE diagonal crack. The crack is located in a thin region symmetric about the diagonal, made out of a sequence of 3×3 squares that overlap over one of the corner cells. The number of tilings of a (4m+1)×(4m+1) deficient square by T+4 is greater than the number of tilings by dominoes and monomers of a 2m×2m square. The number of tilings of a (4m+3)×(4m+3) deficient square by T+4 is greater than twice the number of tilings by dominoes and monomers of a (2m+1)×(2m+1) deficient square, with the missing cell placed on the main diagonal. We also consider tilings by T4 and T+4 of other significant deficient regions. In particular we show that a deficient first quadrant, a deficient half strip, a deficient strip or a deficient bent strip cannot be tiled by T+4. Therefore T4 and T+4 give examples of tile sets that tile deficient rectangles but do not tile any deficient first quadrant, any deficient half strip, any deficient bent strip or any deficient strip.

Aperiodic tilings and entropy

In this paper, we present a construction of Kari–Culik aperiodic tileset. Our construction is self-contained and organized to allow reasoning on properties of the resulting sets of tilings. We prove that this tileset does not have any “unexpected behavior”, i.e., each line of each tiling has an average. Then we prove that this tileset has positive entropy, and that entropy is still positive when one adds some specific restrictions on the tilings. This shows that it is not self-similar, contrarily to all preceding aperiodic tilesets.

Optimal Self-Assembly of Finite Shapes at Temperature 1 in 3D

Tile self-assembly in which tiles may bind in a non-cooperative fashion is often referred to as “temperature 1 self-assembly” or simply “non-cooperative self-assembly”. In this type of self-assembly, a tile may non-cooperatively bind to an assembly via (at least) one of its sides, unlike in cooperative self-assembly, in which some tiles may be required to bind on two or more sides. Cooperative self-assembly leads to highly non-trivial theoretical behavior but two-dimensional non-cooperative self-assembly is conjectured to be only capable of producing highly-regular shapes and patterns, which, in general, cannot be interpreted as complex computation. Remarkably, Cook et al. (Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: Proceedings of the 22nd Annual ACM–SIAM Symposium on Discrete Algorithms 2011) showed that three-dimensional non-cooperative self-assembly is essentially as powerful as cooperative self-assembly, with respect to Turing universality and self-assembly of $$N \times N$$ squares. In this paper, we consider the non-cooperative self-assembly of just-barely 3D shapes. Working in a three-dimensional variant of Winfree’s abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape $$X \subset {\mathbb {Z}}^2$$ , there is a tile set that uniquely self-assembles at temperature 1 into a 3D representation of a scaled-up version of X with optimal program-size complexity, where the “program-size complexity”, also known as “tile complexity”, of a shape is the minimum number of tile types required to uniquely self-assemble it. Moreover, our construction is “just barely” 3D in the sense that it only places tiles in the $$z = 0$$ and $$z = 1$$ planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SIAM J Comput 36(6):1544–1569, 2007).

Video processing for panoramic streaming using HEVC and its scalable extensions

Panoramic streaming is a particular way of video streaming where an arbitrary Region-of-Interest (RoI) is transmitted from a high-spatial resolution video, i.e. a video covering a very “wide-angle” (much larger than the human field-of-view – e.g. 360°). Some transport schemes for panoramic video delivery have been proposed and demonstrated within the past decade, which allow users to navigate interactively within the high-resolution videos. With the recent advances of head mounted displays, consumers may soon have immersive and sufficiently convenient end devices at reach, which could lead to an increasing demand for panoramic video experiences. The solution proposed within this paper is built upon tile-based panoramic streaming, where users receive a set of tiles that match their RoI, and consists in a low-complexity compressed domain video processing technique for using H.265/HEVC and its scalable extensions (H.265/SHVC and H.265/MV-HEVC). The proposed technique generates a single video bitstream out of the selected tiles so that a single hardware decoder can be used. It overcomes the scalability issue of previous solutions not using tiles and the battery consumption issue inherent of tile-based panorama streaming, where multiple parallel software decoders are used. In addition, the described technique is capable of reducing peak streaming bitrate during changes of the RoI, which is crucial for allowing a truly immersive and low latency video experience. Besides, it makes it possible to use Open GOP structures without incurring any playback interruption at switching events, which provides a better compression efficiency compared to closed GOP structures.

Tiling sets and spectral sets over finite fields

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6]). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Zp5 for all odd primes p and Zp4 for all odd primes p such that p≡3 mod 4. Although counterexamples in low dimensional groups over cyclic rings Zn were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis.

(Invited) Electron Channeling Contrast Imaging: Potential for Future Metrology in Semiconductor Industry

For future scaling of CMOS technology beyond the 10 nm node it is necessary to integrate high mobility semiconductors including (Si)Ge and III/V compounds on silicon substrates. The lattice mismatch between these materials and the Silicon platform can lead to the formation of defects which will cause a degradation of the final device performance. Therefore, a metrology technique is needed to monitor the presence and density of defects in blanket layer as well as in patterned structures such as fins. There are several techniques for defect metrology including optical methods, X-ray diffraction or transmission electron microscopy, however, expected defect densities (<105 cm-2) make them ineffective for this task. Electron channeling contrast imaging (ECCI) is scanning electron microscopy (SEM) technique used for the visualization and characterization of crystalline defects including dislocations or stacking faults. Distortions of the crystal lattice of a material due to the presence of such defects cause the variation of the backscattered electron intensity allowing for the visualization of the defects. Application of ECCI proved to be a consistent method for assessing the density and Burgers vector of different defect types in various materials with reliability comparable to that of transmission electron microscopy (TEM). Moreover, TEM limitations related to the destructive sample preparation necessity, limited viewed area accessible for each sample and possible difference in the behavior of thin samples compared to bulk ones brings SEM based ECCI as excellent candidate for non-destructive investigation of large sample areas with low defect densities. For such an analysis, areas exceeding 105 μm2 in size need to be examined to ensure proper statistics. Therefore, automated acquisition of a set of tiles that can be stitched into a single image with a resolution allowing for the detection of single threading dislocations is required. Important parameter for high contrast ECCI of crystalline defects is the sample orientation with respect to the electron beam. The sample should be oriented close to the Bragg condition. This is facilitated by the acquisition of Electron Channeling Patterns and simultaneous sample tilting and/or rotation. For the majority of samples based on silicon wafers only require a couple of degrees tilt from the zone axis. Hence, the technique can be applied to 300 mm wafers easily.The presented technique allows the visualizing and analyzing the density and distribution of threading dislocations in blanket SiGe layers of different defect densities, strain relaxed buffers as well as on (Si)Ge and III/V fin structures. Therefore we are able to demonstrate that automated image acquisition of ECCI has big potential as possible metrology tool for future semiconductor devices process.

Correction of second‐order‐diffraction errors in spectrophotometry

AbstractConventional diffraction gratings are designed so that most of the light striking the grating is diffracted toward the “first‐order beam” direction, but in practice, some of the light may be diffracted towards the “second‐order beam” direction, so that the light striking a particular sensor pixel may be a combination of both first‐ and second‐order components. This causes errors in the output of diffraction‐grating based UV–visible spectrometers. A new method has been developed to correct such errors by calibrating the optical system with a UV‐excluded standard (such as an orange tile) and a UV‐included standard (such as a white tile). These two calibrations can predict the relationship between first‐order‐only signals and first‐and‐second‐order combined signals at sensor pixels (nominal wavelengths) where there is second‐order light, and thus help correct the second‐order diffraction error. This compensation method applied on the reflectance measurement of a set of color ceramic tiles showed significant improvement on the accuracy of the result. © 2016 Wiley Periodicals, Inc. Col Res Appl, 42, 189–192, 2017

A Globally Optimum Parallelizable Whole Slide Image Registration Algorithm

Introduction/ Background Advancements in imaging, communication and computing technologies lead to commercially available digital slide scanners for pathology. These devices help to increase the quality of consultation, diagnosis, research, education and archiving processes as well as contribute to cancer research. At this point, high resolution imaging of the whole pathological specimen is one of the most important parts of the digitalization process. The imaged tiles must be perfectly combined to create the While Slide Image (WSI). In the literature, there are publicly available registration schemes to create the WSI [1] [2]. These methods mainly suffer from the high computational times and high computational resource usage (mainly RAM). To overcome these weaknesses, we have developed a novel, fast, globally optimum WSI registration algorithm. Aims In this study, we plan to develop a new globally optimum WSI registration scheme for digital pathology. Time consumption, RAM usage, starting tile invariance and compliance to parallel processing are selected as the main design concerns of the registration algorithm. Methods The proposed registration scheme has 2 main steps: In the first step, Phase Correlation Method (PCM) given in [3] is utilized to compute the translational offsets between an image and its 4-neighbors. Since PCM requires computing Fourier and inverse Fourier transforms, only overlapping parts of neighbours plus an error region is used to reduce the amount of calculations. Then, all the translational offsets are stored in memory to be used in the global optimization step. In the second step iterations are run to globally optimize the registration according to the neighbouring tile relations. First, the priory knowledge from the motorized table is used to assign global coordinates to the tiles. Then, tiles best position is calculated by maximizing the translational offsets of the neighbors. Here, the tile coordinates are not updated, but the shift values both in X and Y coordinates are saved. After finishing the shift calculation process for all tiles, the shift map is updated, which concludes the first iteration. The iterations continue until zero shift is calculated for all images. Results The performance of the proposed scheme is evaluated using 2 basic scenarios. In the first scenario, 40 different pathology slides are registered using the proposed scheme and then stitched using a bilinear blender. The resulting WSI’s are reviewed by 3 pathologists for stitching errors, and no faulty registration is recognized. In the second scenario, registration time and maximum RAM usage are employed to evaluate the performance of the proposed scheme under different number of tiles. In the evaluations the proposed scheme is compared with Grid Stitching algorithm [1] implemented in ImageJ using the same tile sets and the same PC. The results given in below table show that the proposed scheme has significant RAM and processing time advantages which are very important for real-world applications.

Nondeterministic self-assembly of two tile types on a lattice

Self-assembly is ubiquitous in nature, particularly in biology, where it underlies the formation of protein quaternary structure and protein aggregation. Quaternary structure assembles deterministically and performs a wide range of important functions in the cell, whereas protein aggregation is the hallmark of a number of diseases and represents a nondeterministic self-assembly process. Here we build on previous work on a lattice model of deterministic self-assembly to investigate nondeterministic self-assembly of single lattice tiles and mixtures of two tiles at varying relative concentrations. Despite limiting the simplicity of the model to two interface types, which results in 13 topologically distinct single tiles and 106 topologically distinct sets of two tiles, we observe a wide variety of concentration-dependent behaviors. Several two-tile sets display critical behaviors in the form of a sharp transition from bound to unbound structures as the relative concentration of one tile to another increases. Other sets exhibit gradual monotonic changes in structural density, or nonmonotonic changes, while again others show no concentration dependence at all. We catalog this extensive range of behaviors and present a model that provides a reasonably good estimate of the critical concentrations for a subset of the critical transitions. In addition, we show that the structures resulting from these tile sets are fractal, with one of two different fractal dimensions.