A generalized mathematical theory and experimental verification of proportional notches

A theory and generalized synthesis procedure is advocated for the design of weir notches and orifice-notches having a base in any given shape, to a depth a, such that the discharge through it is proportional to any singular monotonically-increasing function of the depth of flow measured above a certain datum. The problem is reduced to finding an exact solution of a Volterra integral equation in Abel form. The maximization of the depth of the datum below the crest of the notch is investigated. Proof is given that for a weir notch made out of one continuous curve, and for a flow proportional to the mth power of the head, it is impossible to bring the datum lower than (2m − 1)a below the crest of the notch. A new concept of an orifice-notch, having discontinuity in the curve and a division of flow into two distinct portions, is presented. The division of flow is shown to have a beneficial effect in reducing the datum below (2m − 1)a from the crest of the weir and still maintaining the proportionality of the flow. Experimental proof with one such orifice-notch is found to have a constant coefficient of discharge of 0.625. The importance of this analysis in the design of grit chambers is emphasized.

Biosynthesis of sulfated mucopolysaccharides in relation to vitamins A and K

1.(1) Incorporation of Na235SO4 into acid mucopolysaccharides of intestine and colon tissue has been studied in normal, vitamin A-deficient and excess vitamin A-fed rats. 2. (2) Vitamin A deficiency resulted in a significant decrease in [35S]sulfate incorporation into mucopolysaccharides isolated from intestines of male rats. There was no significant change in the total mucopolysaccharides per gram of fresh tissue. 3. (3) When rats are fed excessive amounts of retinyl acetate, increased [35S]sulfate incorporation into mucopolysaccharides of rat intestine and colon is observed. 4. (4) Supplementation of vitamin K1 to rats fed excessive amounts of vitamin A restores the incorporation of [35S]sulfate into the acid mucopolysaccharides to the normal level. 5. (5) The implications of these findings with special reference to the role of vitamins A and K in the synthesis of sulfated mucopolysaccharides are discussed.

Thermal stresses in rectangular plates

A method of determining the thermal stresses in a flat rectangular isotropic plate of constant thickness with arbitrary temperature distribution in the plane of the plate and with no variation in temperature through the thickness is presented. The thermal stress have been obtained in terms of Fourier series and integrals that satisfy the differential equation and the boundary conditions. Several examples have been presented to show the application of the method.

Antipyrine complexes of titanium(IV), zirconium(iv), thorium(IV) and uranium(VI) perchlorates

Antipyrine complexes of TiO2+, ZrO2+, Zr4+, Th4+ and UO2+2 perchlorates with molecular formulae TiO(Apy)4(ClO4)2, ZrO(Apy)3(ClO4)2, Zr(Apy)6(ClO4)4, Th(Apy)7(ClO4)4 and UO2(Apy)5(ClO4)2 have been prepared and characterized. The complexes are stable in air at room temperature and decompose exothermally at ~3OO °C. The i.r. study indicates the bonding of the antipyrine to the metal ion through its carbonyl oxygen. The nature of the bonding of the perchlorate and the stereochemistry of the complexes are discussed in the light of infrared spectra, conductivity in solvents of different polarity, and molecular weight measurements. From the UO2+2 group frequencies, the force constant K and rU-o are found to be 6.29 × 105 dynes/ cm-1 and 1.74 Å, respectively.

Stability of systems with power-law non-linearities

It is shown that a sufficient condition for the asymptotic stability-in-the-large of an autonomous system containing a linear part with transfer function G(jω) and a non-linearity belonging to a class of power-law non-linearities with slope restriction [0, K] in cascade in a negative feedback loop is ReZ(jω)[G(jω) + 1 K] ≥ 0 for all ω where the multiplier is given by, Z(jω) = 1 + αjω + Y(jω) - Y(-jω) with a real, y(t) = 0 for t < 0 and ∫ 0 ∞ |y(t)|dt < 1 2c2, c2 being a constant associated with the class of non-linearity. Any allowable multiplier can be converted to the above form and this form leads to lesser restrictions on the parameters in many cases. Criteria for the case of odd monotonic non-linearities and of linear gains are obtained as limiting cases of the criterion developed. A striking feature of the present result is that in the linear case it reduces to the necessary and sufficient conditions corresponding to the Nyquist criterion. An inequality of the type |R(T) - R(- T)| ≤ 2c2R(0) where R(T) is the input-output cross-correlation function of the non-linearity, is used in deriving the results.

Instability of the self-similar flow into a cavity

In this paper we have investigated the instability of the self-similar flow behind the boundary of a collapsing cavity. The similarity solutions for the flow into a cavity in a fluid obeying a gas law p = Kργ, K = constant and 7 ≥ γ > 1 has been solved by Hunter, who finds that for the same value of γ there are two self-similar flows, one with accelerating cavity boundary and other with constant velocity cavity boundary. We find here that the first of these two flows is unstable. We arrive at this result only by studying the propagation of disturbances in the neighbourhood of the singular point.

The Raman spectrum of cyclohexanol

The Raman spectrum of cyclohexanol has been studied in detail in the liquid state at 30° C. and at about 68° C. and in the solid state at about 13° C. The O-H stretching frequency of cyclohexanol has been found to extend from 3106-3571 cm.-1 in the liquid state at 30° C. and from 3204-3652 cm.-1 at 68° C. The 38 lines recorded in the present investigation have been following frequency shifts: 342, 408, 458, 478, 555, 653, 789, 834, 843, 863, 887, 920, 966, 978, 1024, 1047, 1070, 1139, 1173, 1184, 1210, 1235, 1252, 1301, 1329, 1346, 1362, 1438, 1448, 1464, 2660, 2684, 2710, 2854, 2896, 2925, 2940, 3106 to 3511 (band). Those lines which are italicized are the additional lines observed for the first time. The Raman lines at 966 cm.-1 and 1070 cm.-1 have been assigned to C-OH stretching vibrations of the axial and equatorial isomers. The ratio of the integrated intensity of the 1070 cm.-1 line to the 966 cm.-1 gave the equilibrium constant K as 2·896 at 30° C. and as 2·66 at 68° C. Knowing K, the free energy different Δ F was calculated and it was found to be 0·64 Kcal./mole at 30° C. and 0·66 Kcal./mole at about 68° C. Reasonable assignment has been made for most of the observed Raman lines.

On the theory of the indentation test for the measurement of tensile strength of brittle materials

A theoretical solution has been obtained for the state of stress in a rectangular plate under a pair of symmetrically placed rigid indenters. The stress distributions along the two central axes have been calculated for a square plate assuming the pressure distribution under the indenters as uniform, parabolic and one resulting from 'constant displacement' on a semiinfinite boundary, for different ratios of indenter-width to side of square. The results are compared with those of photoelastic analysis of Berenbaum and Brodie and the validity of the solution is discussed. The solution has been extended to orthotropic materials and numerical results for one type of coal are given.

Importance of carbon dioxide physiological forcing to future climate change

An increase in atmospheric carbon dioxide (CO2) concentration influences climate both directly through its radiative effect (i.e., trapping longwave radiation) and indirectly through its physiological effect (i.e., reducing transpiration of land plants). Here we compare the climate response to radiative and physiological effects of increased CO2 using the National Center for Atmospheric Research (NCAR) coupled Community Land and Community Atmosphere Model. In response to a doubling of CO2, the radiative effect of CO2 causes mean surface air temperature over land to increase by 2.86 ± 0.02 K (± 1 standard error), whereas the physiological effects of CO2 on land plants alone causes air temperature over land to increase by 0.42 ± 0.02 K. Combined, these two effects cause a land surface warming of 3.33 ± 0.03 K. The radiative effect of doubling CO2 increases global runoff by 5.2 ± 0.6%, primarily by increasing precipitation over the continents. The physiological effect increases runoff by 8.4 ± 0.6%, primarily by diminishing evapotranspiration from the continents. Combined, these two effects cause a 14.9 ± 0.7% increase in runoff. Relative humidity remains roughly constant in response to CO2-radiative forcing, whereas relative humidity over land decreases in response to CO2-physiological forcing as a result of reduced plant transpiration. Our study points to an emerging consensus that the physiological effects of increasing atmospheric CO2 on land plants will increase global warming beyond that caused by the radiative effects of CO2.

Improved k-t BLAST for fast fMR imaging

A popular dynamic imaging technique, k-t BLAST (ktB) is studied here for BAR imaging. ktB utilizes correlations in k-space and time, to reconstruct the image time series with only a fraction of the data. The algorithm works by unwrapping the aliased Fourier conjugate space of k-t (y-f-space). The unwrapping process utilizes the estimate of the true y-f-space, by acquiring densely sampled low k-space data. The drawbacks of this method include separate training scan, blurred training estimates and aliased phase maps. The proposed changes are incorporation of phase information from the training map and using generalized-series-extrapolated training map. The proposed technique is compared with ktB on real fMRI data. The proposed changes allow for ktB to operate at an acceleration factor of 6. Performance is evaluated by comparing activation maps obtained using reconstructed images. An improvement of up to 10 dB is observed in thePSNR of activation maps. Besides, a 10% reduction in RMSE is obtained over the entire time series of fMRI images. Peak improvement of the proposed method over ktB is 35%, averaged over five data sets. (C)2010 Elsevier Inc. All rights reserved.

Distribution of products around a burning droplet: an analytical approach

A knowledge of the concentration distribution around a burning droplet is essential if accurate estimates are to be made of the transport coefficients in that region which influence the burning rate. There are two aspects of this paper; (1) determination of the concentration profiles, using the simple assumption of constant binary diffusion coefficients for all species, and comparison with experiments; and (2) postulation of a new relation for the therinal conductivity, which takes into account the variations of both temperature and concentrations of various species. First, the theoretical concentration profiles are evaluated and compared with experimental results reported elsewhere [5]. It is found that the agreement between the theory and experiment is fairly satisfactory. Then, by the use of these profiles and the relations proposed in the literature for the thermal conductivity of a mixture of nonpolar gases, a new relation for thermal conductivity: K = (A1 + B1 T) + (A2 + B2 T) xr (21). is suggested for analytical solutions of droplet combustion problems. Equations are presented to evaluate A1, A2, B1, and B2, and values of these terms for a few hydrocarbons are tabulated.

Polarographic and redox potential studies on copper(I) and copper(II) and their complexes. Part I. Copper(I and II)-ammonia and methylamine complexes

The equilibrium between cuprous ion, cupric ion and metallic copper has been studied using polarographic and redox potential measurements, by reducing cupric ion with copper gauze until equilibrium. Using the well-defined anodic diffusion current plateau, an amperometric method for estimating cuprous copper based on the titration of cuprous ion with dichromate or permanganate has been developed. The diffusion current constant and the disproportionation constant of cuprous ion and the standard potential for the reduction reaction of Cu2+ → Cu+ have been determined. Polarograms have been taken after reducing cupric complexes of ammonia and methylamine with copper until equilibrium. In the case of the copper-ammonia system, reduction to the cuprous state is practically complete while in the case of the cupric-methylamine system, the first cathodic wave occurs to some extent. A new method, called the polarographic-redox potential method, for determining the stability constants of cuprous and cupric complexes has been developed. The method depends upon the determination of the concentration of complexes by polarographic wave heights, and free cupric anc cuprous ions by redox potentials. The stability constants of the following complexes have been obtained: Cu(NH3)2+4, Cu(NH3)+2, Cu(CH3NH2)2(OH)2, Cu(CH3NH2)+2. The stability constants determined by the new method and the half-wave potential shift method agree and the value for the cupric-ammonia complex is in good agreement with Bjerrum method, indicating the reliability of this method.

Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,