A necessary condition for best approximation in monotone and sign-monotone norms

AbstractBest approximation to ƒ ϵ C[a, b] by elements of an n-dimensional Tchebycheff space in monotone norms (norms defined on C[a, b] for which ¦ƒ(x)¦ ⩽ ¦ g(x)¦, a ⩽ x ⩽ b, implies ∥ƒ∥ ⩽ ∥g∥) is studied. It is proved that the error function has at least n zeroes in [a, b], counting twice interior zeroes with no change of sign. This result is best possible for monotone norms in general, and improves the one in [5]. The proof follows from the observation that, for any monotone norm, sgn ƒ(x) = sgn g(x), a ⩽ x ⩽ b, implies ∥ƒ− λg ∥ < ∥ƒ∥ for λ > 0 small enough. This property is shown to characterize a class of norms wider than the class of monotone norms, namely “sign-monotone” norms defined by: ¦ƒ(x)¦ ⩽ ¦g(x)¦, ƒ(x) g(x) ⩾ 0, a ⩽ x ⩽ b, implies ∥ƒ∥ ⩽ ∥g∥. It is noted that various results concerning approximation in monotone norms, are actually valid for approximation in sign-monotone norms

Survival of small populations under demographic stochasticity

We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value

Zonotopal algebra

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This well-known line of study is particularly interesting in case n\eqbd\rank X \ll N. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXi

On the relationships between kinetic schemes and two-state single molecule trajectories

Trajectories of a signal that fluctuates between two states which originate from single molecule activities have become ubiquitous. Common examples are trajectories of ionic flux through individual membrane-channels, and of photon counts collected from diffusion, activity, and conformational changes of biopolymers. By analyzing the trajectory, one wishes to deduce the underlying mechanism, which is usually described by a multi-substate kinetic scheme. In previous works, we divided kinetic schemes that generate two-state trajectories into two types: reducible schemes and irreducible schemes. We showed that all the information in trajectories generated from reducible schemes is contained in the waiting time probability density functions (PDFs) of the two states. It follows that reducible schemes with the same waiting time PDFs are not distinguishable. In this work, we further characterize the topologies of kinetic schemes, now of irreducible schemes, and further study two-state trajectories from the two types of scheme. We suggest various methods for extracting information about the underlying kinetic scheme from the trajectory (e. g., calculate the binned successive waiting times PDF and analyze the ordered waiting times trajectory), and point out the advantages and disadvantages of each. We show that the binned successive waiting times PDF is not only more robust than other functions when analyzing finite trajectories, but contains, in most cases, more information about the underlying kinetic scheme than other functions in the limit of infinitely long trajectories. For some cases however, analyzing the ordered waiting times trajectory may supply unique information about the underlying kinetic scheme

Correctly validating results from single molecule data: the case of stretched exponential decay in the catalytic activity of single lipase B molecules

The question of how to validate and interpret correctly the waiting time probability density functions (WT-PDFs) from single molecule data is addressed. It is shown by simulation that when a stretched exponential WT-PDF, with a stretched exponent alfa and a time scale parameter tau, generates the off periods of a two-state trajectory, a reliable recovery of the input WT-PDF from the trajectory is obtained even when the bin size used to define the trajectory, dt, is much larger than the parameter tau. This holds true as long as the first moment of the WT-PDF is much larger than dt. Our results validate the results in an earlier study of the activity of single Lipase B molecules and disprove recent related critique

Underutilized resources for studying the evolution of invasive species during their introduction, establishment, and lag phases

The early phases of biological invasions are poorly understood. In particular, during the introduction, establishment, and possible lag phases, it is unclear to what extent evolution must take place for an introduced species to transition from established to expanding. In this study, we highlight three disparate data sources that can provide insights into evolutionary processes associated with invasion success: biological control organisms, horticultural introductions, and natural history collections. All three data sources potentially provide introduction dates, information about source populations, and genetic and morphological samples at different time points along the invasion trajectory that can be used to investigate preadaptation and evolution during the invasion process, including immediately after introduction and before invasive expansion. For all three data sources, we explore where the data are held, their quality, and their accessibility. We argue that these sources could find widespread use with a few additional pieces of data, such as voucher specimens collected at certain critical time points during biocontrol agent quarantine, rearing, and release and also for horticultural imports, neither of which are currently done consistently. In addition, public access to collected information must become available on centralized databases to increase its utility in ecological and evolutionary research

What do genetics and ecology tell us about the design of nature reserves?

The SLOSS (single large or several small) debate is no longer an issue in the discussion about the optimal size of nature reserves. The best way to estimate the minimum sizes of reserves may be a three-step process: (1) identify target or keystone species whose disappearance would significantly decrease the value or species diversity of the reserve; (2) determine the minimum number of individuals in a population needed to guarantee a high probability of survival for these species; (3) using known densities, estimate the area needed to sustain the minimum number. The forces that affect population viability and determine MVPs (minimum viable populations) are extremely complex. Thoughtful estimates of MVPs for many animal species are rarely lower than an effective size of a few hundred.Attempts to save only common or smaller species in a community will usually be ill-fated because of the web of ecological relationships between species, including the importance of predation and herbivory in the maintenance of species diversity. Other topics discussed include the complementarity of conservation goals, the problematic function of corridors and the value of buffer zones.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/26318/1/0000405.pd