A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.

Volume, Surface, Connectivity and Size Distribution of Soil Pore Space in CT Images: Comparison of Samples at Different Depths from Nearby Natural and Tillage Areas

The study of soil structure, i.e., the pores, is of vital importance in different fields of science and technology. Total pore volume (porosity), pore surface, pore connectivity and pore size distribution are some (probably the most important) of the geometric measurements of pore space. The technology of X-ray computed tomography allows us to obtain 3D images of the inside of a soil sample enabling study of the pores without disturbing the samples. In this work we performed a set of geometrical measures, some of them from mathematical morphology, to assess and quantify any possible difference that tillage may have caused on the soil. We compared samples from tilled soil with samples from a soil with natural vegetation taken in a very close area. Our results show that the main differences between these two groups of samples are total surface area and pore connectivity per unit pore volume.

Fluorescence-intensity distribution analysis and its application in biomolecular detection technology

A methodology, fluorescence-intensity distribution analysis, has been developed for confocal microscopy studies in which the fluorescence intensity of a sample with a heterogeneous brightness profile is monitored. An adjustable formula, modeling the spatial brightness distribution, and the technique of generating functions for calculation of theoretical photon count number distributions serve as the two cornerstones of the methodology. The method permits the simultaneous determination of concentrations and specific brightness values of a number of individual fluorescent species in solution. Accordingly, we present an extremely sensitive tool to monitor the interaction of fluorescently labeled molecules or other microparticles with their respective biological counterparts that should find a wide application in life sciences, medicine, and drug discovery. Its potential is demonstrated by studying the hybridization of 5′-(6-carboxytetramethylrhodamine)-labeled and nonlabeled complementary oligonucleotides and the subsequent cleavage of the DNA hybrids by restriction enzymes.

Columnar distribution of serotonin-dependent plasticity within kitten striate cortex

Recent studies have identified the potential for an important role for serotonin (5-HT) receptors in the developmental plasticity of the kitten visual cortex. 5-HT2C receptors are transiently expressed in a patchy fashion in the visual cortex of kittens between 30–80 days of age complementary to patches demarcated by cytochrome oxidase staining. 5-HT, operating via 5-HT2C receptors, increases cortical synaptic plasticity as assessed both in brain slices and in vivo. Herein, we report that bath application of 5-HT substantially increases the probability of long-term potentiation within 5-HT2C receptor-rich zones of cortex, but this effect is not observed in the 5-HT2C receptor-poor zones. Instead, in these zones, 5-HT application increases the probability of long-term depression. These location-specific effects of 5-HT may promote the formation of compartment-specific cortical responses.

Geometric constraints explain much of the species richness pattern in African birds

The world contains boundaries (e.g., continental edge for terrestrial taxa) that impose geometric constraints on the distribution of species ranges. Thus, contrary to traditional thinking, the expected species richness pattern in absence of ecological or physiographical factors is unlikely to be uniform. Species richness has been shown to peak in the middle of a bounded one-dimensional domain, even in the absence of ecological or physiographical factors. Because species ranges are not linear, an extension of the approach to two dimensions is necessary. Here we present a two-dimensional null model accounting for effects of geometric constraints. We use the model to examine the effects of continental edge on the distribution of terrestrial animals in Africa and compare the predictions with the observed pattern of species richness in birds endemic to the continent. Latitudinal, longitudinal, and two-dimensional patterns of species richness are predicted well from the modeled null effects alone. As expected, null effects are of high significance for wide ranging species only. Our results highlight the conceptual significance of an until recently neglected constraint from continental shape alone and support a more cautious analysis of species richness patterns at this scale.

Distribution and Transport of Cholesterol in Caenorhabditis elegans

Cholesterol transport is an essential process in all multicellular organisms. In this study we applied two recently developed approaches to investigate the distribution and molecular mechanisms of cholesterol transport in Caenorhabditis elegans. The distribution of cholesterol in living worms was studied by imaging its fluorescent analog, dehydroergosterol, which we applied to the animals by feeding. Dehydroergosterol accumulates primarily in the pharynx, nerve ring, excretory gland cell, and gut of L1–L3 larvae. Later, the bulk of dehydroergosterol accumulates in oocytes and spermatozoa. Males display exceptionally strong labeling of spermatids, which suggests a possible role for cholesterol in sperm development. In a complementary approach, we used a photoactivatable cholesterol analog to identify cholesterol-binding proteins in C. elegans. Three major and several minor proteins were found specifically cross-linked to photocholesterol after UV irradiation. The major proteins were identified as vitellogenins. rme-2 mutants, which lack the vitellogenin receptor, fail to accumulate dehydroergosterol in oocytes and embryos and instead accumulate dehydroergosterol in the body cavity along with vitellogenin. Thus, uptake of cholesterol by C. elegans oocytes occurs via an endocytotic pathway involving yolk proteins. The pathway is a likely evolutionary ancestor of mammalian cholesterol transport.

On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

In this paper we provide the proof of a practical point-wise characterization of the set RP defined by the closure set of the real projections of the zeros of an exponential polynomial P(z) = Σn j=1 cjewjz with real frequencies wj linearly independent over the rationals. As a consequence, we give a complete description of the set RP and prove its invariance with respect to the moduli of the c′ js, which allows us to determine exactly the gaps of RP and the extremes of the critical interval of P(z) by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.

Determining Spatial Distribution and Physical Properties of the Vashon Advance Outwash near Mountlake Terrace, Washington

This study is aimed at determining the spatial distribution, physical properties, and groundwater conditions of the Vashon advance outwash (Qva) in the Mountlake Terrace, WA area. The Qva is correlative with the Esperance Sand, as defined at its type section; however, local variations in the Qva are not well-characterized (Mullineaux, 1965). While the Qva is a dense glacial unit with low compressibility and high frictional shear strength (Gurtowski and Boirum, 1989), the strength of this unit can be reduced when it becomes saturated (Tubbs, 1974). This can lead to caving or flowing in excavations, and on a larger scale, can lead to slope failures and mass-wasting when intersected by steep slopes. By studying the Qva, we can better predict how it will behave under certain conditions, which will be beneficial to geologists, hydrogeologists, engineers, and environmental scientists during site assessments and early phases of project planning. In this study, I use data from 27 geotechnical borings from previous field investigations and C-Tech Corporation’s EnterVol software to create three-dimensional models of the subsurface geology in the study area. These models made it possible to visualize the spatial distribution of the Qva in relation to other geologic units. I also conducted a comparative study between data from the borings and generalized published data on the spatial distribution, relative density, soil classification, grain-size distribution, moisture content, groundwater conditions, and aquifer properties of the Qva. I found that the elevation of the top of the Qva ranges from 247 to 477 ft. I found that the Qva is thickest where the modern topography is high, and is thinnest where the topography is low. The thickness of the Qva ranges from absent to 242 ft. Along the northern, east-west trending transect, the Qva thins to the east as it rises above a ridge composed of Pre- Vashon glacial deposits. Along the southern, east-west trending transect, the Qva pinches out against a ridge composed of pre-Vashon interglacial deposits. Two plausible explanations for this ridge are paleotopography and active faulting associated with the Southern Whidbey Fault Zone. Further investigations should be done using geophysical methods and the modeling methods described in this study to determine the nature of this ridge. The relative density of the Qva in the study area ranges from loose to very dense, with the loose end of the spectrum probably relating to heave in saturated sands. I found subtle correlations between density and depth. Volumetric analysis of the soil groups listed in the boring logs indicate that the Qva in the study area is composed of approximately 9.5% gravel, 89.3% sand, and 1.2% silt and clay. The natural moisture content ranges from 3.0 to 35.4% in select samples from the Qva. The moisture content appears to increase with depth and fines content. The water table in the study area ranges in elevation from 231.9 to 458 ft, based on observations and measurements recorded in the boring logs. The results from rising-head and falling-head slug tests done at a single well in the study area indicate that the geometric mean of hydraulic conductivity is 15.93 ft/d (5.62 x 10-03 cm/s), the storativity is 3.28x10-03, and the estimated transmissivity is 738.58 ft2/d in the vicinity of this observation well. At this location, there was 1.73 ft of seasonal variation in groundwater elevation between August 2014 and March 2015.

A geometric approach to quantum circuit lower bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2(n)). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

Information geometric measurements of generalisation

Neural networks can be regarded as statistical models, and can be analysed in a Bayesian framework. Generalisation is measured by the performance on independent test data drawn from the same distribution as the training data. Such performance can be quantified by the posterior average of the information divergence between the true and the model distributions. Averaging over the Bayesian posterior guarantees internal coherence; Using information divergence guarantees invariance with respect to representation. The theory generalises the least mean squares theory for linear Gaussian models to general problems of statistical estimation. The main results are: (1)~the ideal optimal estimate is always given by average over the posterior; (2)~the optimal estimate within a computational model is given by the projection of the ideal estimate to the model. This incidentally shows some currently popular methods dealing with hyperpriors are in general unnecessary and misleading. The extension of information divergence to positive normalisable measures reveals a remarkable relation between the dlt dual affine geometry of statistical manifolds and the geometry of the dual pair of Banach spaces Ld and Ldd. It therefore offers conceptual simplification to information geometry. The general conclusion on the issue of evaluating neural network learning rules and other statistical inference methods is that such evaluations are only meaningful under three assumptions: The prior P(p), describing the environment of all the problems; the divergence Dd, specifying the requirement of the task; and the model Q, specifying available computing resources.

The topographic distribution of the magnetic P100M to full- and half-field stimulation

Visual evoked magnetic responses were recorded to full-field and left and right half-field stimulation with three check sizes (70′, 34′ and 22′) in five normal subjects. Recordings were made sequentially on a 20-position grid (4 × 5) based on the inion, by means of a single-channel direct current-Superconducting Quantum Interference Device second-order gradiometer. The topographic maps were consistent on the same subjects recorded 2 months apart. The half-field responses produced the strongest signals in the contralateral hemisphere and were consistent with the cruciform model of the calcarine fissure. Right half fields produced upper-left-quadrant outgoing fields and lower-left-quadrant ingoing fields, while the left half field produced the opposite response. The topographic maps also varied with check size, with the larger checks producing positive or negative maximum position more anteriorly than small checks. In addition, with large checks the full-field responses could be explained as the summation of the two half fields, whereas full-field responses to smaller checks were more unpredictable and may be due to sources located at the occipital pole or lateral surface. In addition, dipole sources were located as appropriate with the use of inverse problem solutions. Topographic data will be vital to the clinical use of the visual evoked field but, in addition, provides complementary information to visual evoked potentials, allowing detailed studies of the visual cortex. © 1992 Kluwer Academic Publishers.

Computational mechanics of molecular systems:quantifying high-dimensional dynamics by distribution of Poincaré recurrence times

A framework that connects computational mechanics and molecular dynamics has been developed and described. As the key parts of the framework, the problem of symbolising molecular trajectory and the associated interrelation between microscopic phase space variables and macroscopic observables of the molecular system are considered. Following Shalizi and Moore, it is shown that causal states, the constituent parts of the main construct of computational mechanics, the e-machine, define areas of the phase space that are optimal in the sense of transferring information from the micro-variables to the macro-observables. We have demonstrated that, based on the decay of their Poincare´ return times, these areas can be divided into two classes that characterise the separation of the phase space into resonant and chaotic areas. The first class is characterised by predominantly short time returns, typical to quasi-periodic or periodic trajectories. This class includes a countable number of areas corresponding to resonances. The second class includes trajectories with chaotic behaviour characterised by the exponential decay of return times in accordance with the Poincare´ theorem.

Complementary methods to investigate the development of clogging within a horizontal sub-surface flow tertiary treatment wetland

A combination of experimental methods was applied at a clogged, horizontal subsurface flow (HSSF) municipal wastewater tertiary treatment wetland (TW) in the UK, to quantify the extent of surface and subsurface clogging which had resulted in undesirable surface flow. The three dimensional hydraulic conductivity profile was determined, using a purpose made device which recreates the constant head permeameter test in-situ. The hydrodynamic pathways were investigated by performing dye tracing tests with Rhodamine WT and a novel multi-channel, data-logging, flow through Fluorimeter which allows synchronous measurements to be taken from a matrix of sampling points. Hydraulic conductivity varied in all planes, with the lowest measurement of 0.1 md1 corresponding to the surface layer at the inlet, and the maximum measurement of 1550 md1 located at a 0.4m depth at the outlet. According to dye tracing results, the region where the overland flow ceased received five times the average flow, which then vertically short-circuited below the rhizosphere. The tracer break-through curve obtained from the outlet showed that this preferential flow-path accounted for approximately 80% of the flow overall and arrived 8 h before a distinctly separate secondary flow-path. The overall volumetric efficiencyof the clogged system was 71% and the hydrology was simulated using a dual-path, dead-zone storage model. It is concluded that uneven inlet distribution, continuous surface loading and high rhizosphere resistance is responsible for the clog formation observed in this system. The average inlet hydraulic conductivity was 2 md1, suggesting that current European design guidelines, which predict that the system will reach an equilibrium hydraulic conductivity of 86 md1, do not adequately describe the hydrology of mature systems.

What determines the size frequency distribution of β-amyloid (Aβ) deposits in Alzheimer's disease patients?

The factors determining the size of individual β-amyloid (A,8) deposits and their size frequency distribution in tissue from Alzheimer's disease (AD) patients have not been established. In 23/25 cortical tissues from 10 AD patients, the frequency of Aβ deposits declined exponentially with increasing size. In a random sample of 400 Aβ deposits, 88% were closely associated with one or more neuronal cell bodies. The frequency distribution of (Aβ) deposits which were associated with 0,1,2,...,n neuronal cell bodies deviated significantly from a Poisson distribution, suggesting a degree of clustering of the neuronal cell bodies. In addition, the frequency of Aβ deposits declined exponentially as the number of associated neuronal cell bodies increased. Aβ deposit area was positively correlated with the frequency of associated neuronal cell bodies, the degree of correlation being greater for pyramidal cells than smaller neurons. These data suggested: (1) the number of closely adjacent neuronal cell bodies which simultaneously secrete Aβ was an important factor determining the size of an Aβ deposit and (2) the exponential decline in larger Aβ deposits reflects the low probability that larger numbers of adjacent neurons will secrete Aβ simultaneously to form a deposit. © 1995.

An examination of the calcium and strontium site distribution in bioactive glasses through isomorphic neutron diffraction, X-ray diffraction, EXAFS and multinuclear solid state NMR.

Strontium has been substituted for calcium in the glass series (SiO2)49.46(Na2O)26.38(P2O5)1.07(CaO)23.08x(SrO)x (where x = 0, 11.54, 23.08) to elucidate their underlying atomic-scale structural characteristics as a basis for understanding features related to the bioactivity. These bioactive glasses have been investigated using isomorphic neutron and X-ray diffraction, Sr K-edge EXAFS and solid state 17O, 23Na, 29Si, 31P and 43Ca magic-angle-spinning (MAS) NMR. An effective isomorphic substitution first-order difference function has been applied to the neutron diffraction data, confirming that Ca and Sr behave in a similar manner within the glass network, with residual differences attributed to solely the variation in ionic radius between the two species. The diffraction data provides the first direct experimental evidence of split Ca–O nearest-neighbour correlations in these melt quench bioactive glasses, together with an analogous splitting of the Sr–O correlations; the correlations are attributed to the metal ions correlated either to bridging or to non-bridging oxygen atoms. Triple quantum (3Q) 43Ca MAS NMR corroborates the split Ca–O correlations. Successful simplification of the 2 < r (A) < 3 region via the difference method has also revealed two distinct Na environments. These environments are attributed to sodium correlated either to bridging or to nonbridging oxygen atoms. Complementary multinuclear MAS NMR, Sr K-edge EXAFS and X-ray diffraction data supports the structural model presented. The structural sites present will be intimately related to their release properties in physiological fluids such as plasma and saliva, and hence the bioactivity of the material. Detailed structural knowledge is therefore a prerequisite for optimising material design.