Rearrangeable Networks with Limited Depth

Rearrangeable networks are switching systems capable of establishing simultaneous independent communication paths in accordance with any one-to-one correspondence between their n inputs and n outputs. Classical results show that Ω( n log n ) switches are necessary and that O( n log n ) switches are sufficient for such networks. We are interested in the minimum possible number of switches in rearrangeable networks in which the depth (the length of the longest path from an input to an output) is at most k, where k is fixed as n increases. We show that Ω( n1 + 1/k ) switches are necessary and that O( n1 + 1/k ( log n )1/k ) switches are sufficient for such networks

Pricing A Pig In A Poke: Endogenous Valuations And Storage Unit Auctions

Storage unit auctions have recently received widespread attention from reality shows and “get rich easy” business models. This study examines the pitfalls associated with winning storage unit auctions, commonly observed bidding strategies, participant behavior and auction methods. Storage unit auctions present a classic example of endogenous valuation using incomplete information and imperfect secondary markets. The examination of the contents of a storage unit auction yields a novel use for the facility and sheds light on the process that leads to a lien and sale

Redundancy and error resilience in Boolean Networks

We consider the effect of noise in sparse Boolean Networks with redundant functions. We show that they always exhibit a non-zero error level, and the dynamics undergoes a phase transition from non-ergodicity to ergodicity, as a function of noise, after which the system is no longer capable of preserving a memory if its initial state. We obtain upper-bounds on the critical value of noise for networks of different sparsity.Comment: 4 pages, 5 figure

Clones with finitely many relative R-classes

For each clone C on a set A there is an associated equivalence relation analogous to Green's R-relation, which relates two operations on A iff each one is a substitution instance of the other using operations from C. We study the clones for which there are only finitely many relative R-classes.Comment: 41 pages; proofs improved, examples adde

Making Classical Ground State Spin Computing Fault-Tolerant

We examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation. This model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions. In its most primitive form, systems constructed in this model cannot compute in an error free manner when working at non-zero temperature. However, by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free. We achieve this by using techniques in fault-tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature. We further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class Merlin-Arthur. This provides an interesting connection between the physical behavior of certain many-body spin systems and computational complexity.Comment: 24 pages, 1 figur

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur

Abundance of unknots in various models of polymer loops

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $\sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of $N_0$ for a variety of polymer models. Among models examined, $N_0$ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.Comment: 13 pages, 6 color figure

Tightness of slip-linked polymer chains

We study the interplay between entropy and topological constraints for a polymer chain in which sliding rings (slip-links) enforce pair contacts between monomers. These slip-links divide a closed ring polymer into a number of sub-loops which can exchange length between each other. In the ideal chain limit, we find the joint probability density function for the sizes of segments within such a slip-linked polymer chain (paraknot). A particular segment is tight (small in size) or loose (of the order of the overall size of the paraknot) depending on both the number of slip-links it incorporates and its competition with other segments. When self-avoiding interactions are included, scaling arguments can be used to predict the statistics of segment sizes for certain paraknot configurations.Comment: 10 pages, 6 figures, REVTeX

BioDeepTime : a database of biodiversity time series for modern and fossil assemblages

We thank the Paleosynthesis Project and the Volkswagen Stiftung for funding that supported this project (Az 96 796). M.C.R. acknowledges the German Research Foundation (DFG) for funding through the Cluster of Excellence ‘The Ocean Floor – Earth's Uncharted Interface’ (EXC 2077, grant no. 390741603). E.E.S. acknowledges funding from Leverhulme Trust grant RPG-201170, the Leverhulme Prize and the National Science Research Council grant NE/V011405/1. Q.J.L. and L.N. acknowledge support from the Youth Innovation Promotion Association (2019310) and the Chinese Academy of Sciences (CAS-WX2021SF-0205). A.M.P. acknowledges funding from the Leverhulme Trust through research grant RPG-2019-402. M.D. acknowledges funding from Leverhulme Trust through the Leverhulme Centre for Anthropocene Biodiversity (RC-2018-021) and a research grant (RPG-2019-402), and the European Union (ERC coralINT, 101044975). L. H. L. acknowledges funding from the European Research Council (macroevolution.abc ERC grant no. 724324). K.H.P acknowledges funding from the National Science Foundation Graduate Research Fellowship Program (DGE-2139841). H.H.M.H. acknowledges support from Peter Buck Postdoc Fellowship, Smithsonian Institution. A.T. acknowledges funding from the Slovak Research and Development Agency (APVV 22-0523) and the Slovak Scientific Grant Agency (VEGA 02/0106/23).Motivation We have little understanding of how communities respond to varying magnitudes and rates of environmental perturbations across temporal scales. BioDeepTime harmonizes assemblage time series of presence and abundance data to help facilitate investigations of community dynamics across timescales and the response of communities to natural and anthropogenic stressors. BioDeepTime includes time series of terrestrial and aquatic assemblages of varying spatial and temporal grain and extent from the present-day to millions of years ago. Main Types of Variables Included BioDeepTime currently contains 7,437,847 taxon records from 10,062 assemblage time series, each with a minimum of 10 time steps. Age constraints, sampling method, environment and taxonomic scope are provided for each time series. Spatial Location and Grain The database includes 8752 unique sampling locations from freshwater, marine and terrestrial ecosystems. Spatial grain represented by individual samples varies from quadrats on the order of several cm2 to grid cells of ~100 km2. Time Period and Grain BioDeepTime in aggregate currently spans the last 451?million years, with the 10,062 modern and fossil assemblage time series ranging in extent from years to millions of years. The median extent of modern time series is 18.7?years and for fossil series is 54,872?years. Temporal grain, the time encompassed by individual samples, ranges from days to tens of thousands of years. Major Taxa and Level of Measurement The database contains information on 28,777 unique taxa with 4,769,789 records at the species level and another 271,218 records known to the genus level, including time series of benthic and planktonic foraminifera, coccolithophores, diatoms, ostracods, plants (pollen), radiolarians and other invertebrates and vertebrates. There are to date 7012 modern and 3050 fossil time series in BioDeepTime. Software Format SQLite, Comma-separated values.Publisher PDFPeer reviewe