WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

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Methods of Numerical Integration

Soil microbial communities are essential for ecosystem function, but linking community composition to biogeochemical processes is challenging because of high microbial diversity and large spatial variability of most soil characteristics. We investigated soil bacterial community structure in a switchgrass stand planted on soil with a history of grassland vegetation at high spatial resolution to determine whether biogeographic trends occurred at the centimeter scale. Moreover, we tested whether such heterogeneity, if present, influenced community structure within or among ecosystems. Pronounced heterogeneity was observed at centimeter scales, with abrupt changes in relative abundance of phyla from sample to sample. At the ecosystem scale (> 10 m), however, bacterial community composition and structure were subtly, but significantly, altered by fertilization, with higher alpha diversity in fertilized plots. Moreover, by comparing these data with data from 1772 soils from the Earth Microbiome Project, it was found that 20% of bacterial taxa were shared between their site and diverse globally sourced soil samples, while grassland soils shared approximately 40% of their operational taxonomic units with the current study. By spanning several orders of magnitude, the analysis suggested that extreme patchiness characterized community structure at smaller scales but that coherent patterns emerged at larger length scales.

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Spatial scale drives patterns in soil bacterial diversity

An algorithm is presented for the calculation of the Kramers-Kronig transform of a spectrum via a piecewise Laurent polynomial method. This algorithm is demonstrated to be highly accurate, while also being computationally efficient. The algorithm places no requirements on data point spacing and is capable of integrating across the full spectrum (i.e. from zero to infinity). Further, we present a computer application designed to aid in calculating the Kramers-Kronig transform on near-edge experimental X-ray absorption spectra (extended with atomic scattering factor data) in order to produce the dispersive part of the X-ray refractive index, including near-edge features.

/pdf/calculation-of-the-kramers-kronig-transform-of-x-ray-spectra-3mabsyl1te.pdf

Calculation of the Kramers-Kronig transform of X-ray spectra by a piecewise Laurent polynomial method.

Mir-132 is a neuronal activity-regulated microRNA that controls the morphology of dendritic spines and neuronal transmission. Similar activities have recently been attributed to matrix metalloproteinase-9 (MMP-9), an extrasynaptic protease. In the present study, we provide evidence that miR-132 directly regulates MMP-9 mRNA in neurons to modulate synaptic plasticity. With the use of luciferase reporter system, we show that miR-132 binds to the 3'UTR of MMP-9 mRNA to regulate its expression in neurons. The overexpression of miR-132 in neurons reduces the level of endogenous MMP-9 protein secretion. In synaptoneurosomes, metabotropic glutamate receptor (mGluR)-induced signaling stimulates the dissociation of miR-132 from polyribosomal fractions and shifts it towards the messenger ribonucleoprotein (mRNP)-containing fraction. Furthermore, we demonstrate that the overexpression of miR-132 in the cultured hippocampal neurons from Fmr1 KO mice that have increased synaptic MMP-9 level provokes enlargement of the dendritic spine heads, a process previously implicated in enhanced synaptic plasticity. We propose that activity-dependent miR-132 regulates structural plasticity of dendritic spines through matrix metalloproteinase 9.

/pdf/mir-132-regulates-dendritic-spine-structure-by-direct-1n1bruu12p.pdf

miR-132 Regulates Dendritic Spine Structure by Direct Targeting of Matrix Metalloproteinase 9 mRNA

The authors give a simple, efficient, and easy-to-implement prescription for numerically evaluating finite Hilbert transforms.

On the Evaluation of Finite Hilbert Transforms

A new exact representation of the error function of real arguments justifies an accurate and simple analytical approximation. Two of the most widely used functions in physical sciences are the error function erf(x) and its related complimentary error function erfc(x). These functions occur extensively in problems relating to diffusion, heat conduction, and probability. When the argument x is real, rational approximations for these functions provide a high accuracy. In addition, the Faddeeva function, which is a variant of the error function for the complex argument z = x + iy, is used extensively in nuclear physics and spectroscopy.

A Note on the Error Function

Accurate and compact computing recipe for evaluating correction terms for the Gaussian integration of analytic functions offers economy and high accuracy for many integrals of interest in physical sciences. The requirement of numerical integration in physical sciences is quite pervasive because many practical functions aren't integrable in closed form. A simple example is the error function Erβx) defined by1 Erf(x) 2 /<; dt.

Mohankumar Nandagopal

Papers

On the Evaluation of Finite Hilbert Transforms

A Note on the Error Function

On the Evaluation of Correction Terms of Gaussian Integration