Cauchy's residue theorem for a class of real valued functions

Let $[a,b]$ be an interval in $\mathbb{R}$ and let $F$ be a real valued function defined at the endpoints of $[a,b]$ and with a certain number of discontinuities within $[a,b]$. Having assumed $F$ to be differentiable on a set $[a,b] \backslash E$ to the derivative $f$, where $E$ is a subset of $[a,b]$ at whose points $F$ can take values $\pm \infty$ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to 0 at all points of $E$ and show that $\mathcal{KH-}vt\int_{a}^{b}f=F(b) -F(a)$%, where $\mathcal{KH-}$ $vt$ denotes the total value of the \textit{% Kurzweil-Henstock} integral. The paper ends with a few examples that illustrate the theory.Comment: 6 page

A Study of the Free Oscillations of the Earth

Published observations on the toroidal oscillations of the earth are critically reviewed. A supplementary analysis of the record obtained by the Lamont strain seismometer is presented. Eleven toroidal modes are identified, and it is concluded that the periods are known to within 1 per cent. A perturbation scheme involving the ratio of the angular velocity of the earth to the resonant frequency is used in calculating the effects due to the rotation of the earth on the resonant frequency. The free oscillations are viewed as a superposition of traveling waves. In a nonrotating system two traveling waves combine to produce a stationary standing wave. In a rotating system, the rotation distinguishes between waves that travel in the direction of rotation and those that travel in the opposite direction. Rotation removes a degeneracy and results in a splitting of a spectral peak of order l into 2 times l plus 1 peaks. The fractional displacement in frequency for the lowest-order toroidal oscillations is 1/206 and of the same order as the Q of the peak, so that splitting will probably not be observed in the toroidal oscillations. Viewed locally, rotation causes a particle to precess about a direction parallel to the axis of rotation. This precession will cause a variation of amplitude with time if the motion is recorded by an instrument with an anisotropic response function. Care is therefore needed in studying the time decay of a given spectral peak. Rotation also couples the normal coordinates so that a motion that is initially purely horizontal will develop a vertical component. It is expected that vertical seismometers should record particle motion with the toroidal frequencies. The perturbations of the toroidal oscillations due to core-mantle interaction are treated in detail. An exact expression is obtained for the rate of energy dissipated by a finitely conducting plate oscillating across a magnetic field. The energy dissipated at the core-mantle boundary due to viscous and hydromagnetic coupling is shown to be insignificant as compared with the energy dissipated within the mantle. The toroidal magnetic field leaking into the lower mantle combines with the dipole field, resulting in a stress on the mantle, tending to stiffen the lower boundary. The stress is of sufficient magnitude to produce a displacement toward higher frequency in the lower-order toroidal oscillations. Observations on the (sub 0) T (sub 2) oscillations lead to an estimate of the toroidal magnetic field in the lower mantle. A calculation of elastic energy in the low-order oscillations suggests a value of 10 (sup 18) ergs per cycles per hour for the energy density at low frequencies in the Chilean earthquake. Each mode of oscillation has a characteristic radial distribution of elastic energy associated with it. This distribution determines which parts of the earth contribute most heavily in determining a particular resonant frequency. The distribution of energy for the lower 17 modes for a homogeneous and a Gutenberg model earth is calculated. The resonant frequencies for models of the earth based on the Gutenberg and Lehmann distribution of elastic properties are presented. It is shown that the Gutenberg model earth fits the observations more closely than the Lehmann model and that a slight alteration of the Gutenberg model gives a significantly better fit to the observations. The alteration involves a lower shear-wave velocity in the lower mantle while the Gutenberg velocity distribution is maintained in the upper mantle. Various studies of the earth's oscillations coupled with surface-wave investigations substantiate Gutenberg's hypothesis of a layer of low velocity in the upper mantle. The physical conditions required for the formation of a region of low velocity are examined in detail. The results confirm Birch's earlier statement that a temperature gradient in excess of 6 degrees to 7 degrees per kilometer is needed to produce a decrease in velocity. The low-velocity layer does not require that the temperature approach or exceed the melting temperature. If tile upper mantle is homogeneous, the region of lower velocity should commence at the base of the crust and extend to 150 kilometers under the oceans and about 100 kilometers under continental regions. The distribution of thermal conductivity and radioactivity consistent with the low-velocity layer is also considered

An integrated circuit for chip-based analysis of enzyme kinetics and metabolite quantification

We have created a novel chip-based diagnostic tools based upon quantification of metabolites using enzymes specific for their chemical conversion. Using this device we show for the first time that a solid-state circuit can be used to measure enzyme kinetics and calculate the Michaelis-Menten constant. Substrate concentration dependency of enzyme reaction rates is central to this aim. Ion-sensitive field effect transistors (ISFET) are excellent transducers for biosensing applications that are reliant upon enzyme assays, especially since they can be fabricated using mainstream microelectronics technology to ensure low unit cost, mass-manufacture, scaling to make many sensors and straightforward miniaturisation for use in point-of-care devices. Here, we describe an integrated ISFET array comprising 216 sensors. The device was fabricated with a complementary metal oxide semiconductor (CMOS) process. Unlike traditional CMOS ISFET sensors that use the Si3N4 passivation of the foundry for ion detection, the device reported here was processed with a layer of Ta2O5 that increased the detection sensitivity to 45 mV/pH unit at the sensor readout. The drift was reduced to 0.8 mV/hour with a linear pH response between pH 2 – 12. A high-speed instrumentation system capable of acquiring nearly 500 fps was developed to stream out the data. The device was then used to measure glucose concentration through the activity of hexokinase in the range of 0.05 mM – 231 mM, encompassing glucose’s physiological range in blood. Localised and temporal enzyme kinetics of hexokinase was studied in detail. These results present a roadmap towards a viable personal metabolome machine

Vicious walkers, friendly walkers and Young tableaux II: With a wall

We derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of Young tableaux, and combinatorial descriptions of symmetric functions. For the problem of $n$-friendly walkers, we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the statement of Theorem 4 and its proof were correcte

Alternatively activated dendritic cells regulate CD4+ T-cell polarization in vitro and in vivo

Interleukin-4 is a cytokine widely known for its role in CD4(+) T cell polarization and its ability to alternatively activate macrophage populations. In contrast, the impact of IL-4 on the activation and function of dendritic cells (DCs) is poorly understood. We report here that DCs respond to IL-4 both in vitro and in vivo by expression of multiple alternative activation markers with a different expression pattern to that of macrophages. We further demonstrate a central role for DC IL-4Rα expression in the optimal induction of IFNγ responses in vivo in both Th1 and Th2 settings, through a feedback loop in which IL-4 promotes DC secretion of IL-12. Finally, we reveal a central role for RELMα during T-cell priming, establishing that its expression by DCs is critical for optimal IL-10 and IL-13 promotion in vitro and in vivo. Together, these data highlight the significant impact that IL-4 and RELMα can have on DC activation and function in the context of either bacterial or helminth pathogens

Comparison of single and combination diuretics on glucose tolerance (PATHWAY-3):protocol for a randomised double-blind trial in patients with essential hypertension

INTRODUCTION: Thiazide diuretics are associated with increased risk of diabetes mellitus. This risk may arise from K(+)-depletion. We hypothesised that a K(+)-sparing diuretic will improve glucose tolerance, and that combination of low-dose thiazide with K(+)-sparing diuretic will improve both blood pressure reduction and glucose tolerance, compared to a high-dose thiazide.METHODS AND ANALYSIS: This is a parallel-group, randomised, double-blind, multicentre trial, comparing hydrochlorothiazide 25-50 mg, amiloride 10-20 mg and combination of both diuretics at half these doses. A single-blind placebo run-in of 1 month is followed by 24 weeks of blinded active treatment. There is forced dose-doubling after 3 months. The Primary end point is the blood glucose 2 h after oral ingestion of a 75 g glucose drink (OGTT), following overnight fasting. The primary outcome is the difference between 2 h glucose at weeks 0, 12 and 24. Secondary outcomes include the changes in home systolic blood pressure (BP) and glycated haemoglobin and prediction of response by baseline plasma renin. Eligibility criteria are: age 18-79, systolic BP on permitted background treatment ≥140 mm Hg and home BP ≥130 mm Hg and one component of the metabolic syndrome additional to hypertension. Principal exclusions are diabetes, estimated-glomerular filtration rate &lt;45 mL/min, abnormal plasma K(+), clinic SBP &gt;200 mm Hg or DBP &gt;120 mm Hg (box 2). The sample size calculation indicates that 486 patients will give 80% power at α=0.01 to detect a difference in means of 1 mmol/L (SD=2.2) between 2 h glucose on hydrochlorothiazide and comparators.ETHICS AND DISSEMINATION: PATHWAY-3 was approved by Cambridge South Ethics Committee, number 09/H035/19. The trial results will be published in a peer-reviewed scientific journal.TRIAL REGISTRATION NUMBERS: Eudract number 2009-010068-41 and clinical trials registration number: NCT02351973.</p

Prevention And Treatment of Hypertension With Algorithm-based therapy (PATHWAY) number 2: protocol for a randomised crossover trial to determine optimal treatment for drug-resistant hypertension.

This is the final published version. It first appeared at http://bmjopen.bmj.com/content/5/8/e008951.full.INTRODUCTION: Resistant hypertension is inadequately controlled blood pressure (BP) despite treatment with at least three BP-lowering drugs. A popular hypothesis is that resistant hypertension is due to excessive Na(+)-retention, and that 'further diuretic therapy' will be superior to alternative add-on drugs. METHODS AND ANALYSIS: Placebo-controlled, random crossover study of fourth-line treatment when added to standard (A+C+D) triple drug therapy: ACE inhibitor or Angiotensin receptor blocker (A) +Calcium channel blocker (C)+Diuretic (D). Patients (aged 18-79 years) with clinical systolic BP ≥ 140 mm Hg (135 mm Hg in diabetics) and Home BP Monitoring (HBPM) systolic BP average ≥ 130 mm Hg on treatment for at least 3 months with maximum tolerated doses of A+C+D are randomised to four consecutive randomly allocated 12-week treatment cycles with an α-blocker, β-blocker, spironolactone and placebo. The hierarchical coprimary end point is the difference in HBPM average systolic BP between (in order) spironolactone and placebo, spironolactone and the average of the other two active drugs, spironolactone and each of the other two drugs. A key secondary outcome is to determine whether plasma renin predicts the BP response to the different drugs. A sample size of 346 (allowing 15% dropouts) will confer 90% power to detect a 3 mm Hg HBPM average systolic BP difference between any two drugs. The study can also detect a 6 mm Hg difference in HBPM average systolic BP between each patient's best and second-best drug predicted by tertile of plasma renin. ETHICS AND DISSEMINATION: The study was initiated in May 2009 and results are expected in 2015. These will provide RCT evidence to support future guideline recommendations for optimal drug treatment of resistant hypertension. TRIAL REGISTRATION NUMBER: Clinicaltrials.gov NCT02369081, EUDract number: 2008-007149-30.The study is funded by a special project grant from the British Heart Foundation (number SP/08/002). Further funding is provided by the National Institute for Health Research (NIHR) Comprehensive Local Research Networks. BW is supported by the NIHR UCL/UCL Hospitals Biomedical Research Centre

Electronic transport in two dimensional graphene

We provide a broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature dependent carrier transport in doped or gated graphene structures. A salient feature of our review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g. heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gap- less, massless, chiral Dirac spectrum are highlighted. Experiment and theory as well as quantum and semi-classical transport are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long- range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.Comment: Final version as accepted for publication in Reviews of Modern Physics (in press), 69 pages with 38 figure