Cayley's hyperdeterminant: a combinatorial approach via representation theory

Cayley's hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a 2 x 2 x 2 array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra sl_2(C) to reduce the problem of finding the invariant polynomials for a 2 x 2 x 2 array to a combinatorial problem on the enumeration of 2 x 2 x 2 arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley's hyperdeterminant generates all the invariants. In the last section we show how this approach can be applied to general multidimensional arrays.Comment: 20 page

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Is there a role for surgery in the management of reflux oesophagitis

Mineralisation of 2,4-dichlorophenoxyacetic acid by acoustic or hydrodynamic cavitation in conjunction with the advanced Fenton process

The mineralisation of 2,4-dichlorophenoxyacetic acid (2,4-D) in the presence of zero-valent iron and hydrogen peroxide (the Advanced Fenton process – AFP) whilst being subjected to acoustic or hydrodynamic cavitation is reported. If the reaction is merely stirred then there is 57% removal of TOC whilst on irradiation the figure is 64% although the latter reaction is more rapid. Use of ultrasound alone results in only 11% TOC removal in 60 min of treatment time. Addition of iron powder marginally enhances the extent of degradation but an appreciable increase is observed in the presence of hydrogen peroxide which acts as a source for hydroxyl radicals by Fenton chemistry as well as by dissociation in the presence of ultrasound. The use of hydrodynamic cavitation in conjunction with the advanced Fenton process has also been found to be a useful tool for continuous remediation of water contaminated with 2,4-D. After 20 minutes of treatment the residual TOC is reduced to 30% and this probably represents the remaining highly recalcitrant small organic molecules

Intensification of hydroxyl radical production in sonochemical reactors

The efficacy of sonochemical reactors in chemical processing applications has been well established in the laboratory scale of operation though at a given set of operating parameters and no efforts have been directed in terms of maximizing the free radical production. In the present work, the effect of different operating parameters viz. pH, power dissipation into the system, effect of additives such as air, haloalkanes, titanium dioxide, iron and oxygen on the extent of hydroxyl radical formation in a sonochemical reactor have been investigated using salicylic acid dosimetry. Possible mechanisms for oxidation of salicylic acid in the presence of different additives have also been established. It has been observed that acidic conditions under optimized power dissipation in the presence of iron powder and oxygen result in maximum liberation of hydroxyl radicals as quantified by the kinetic rate constant for production of 2,5- and 2,3-dihydroxybenzoic acid. The study has enabled the optimization of the conditions for maximum efficacy of sonochemical reactors where free radical attack is the controlling mechanism for the chemical processing applications

Industrial wastewater treatment using hydrodynamic cavitation and heterogeneous advanced Fenton processing

A combination of hydrodynamic cavitation and heterogeneous advanced Fenton process (AFP) based on the use of zero valent iron as the catalyst has been investigated for the treatment of real industrial wastewater. The effect of various operating parameters such as inlet pressure, temperature, and the presence of copper windings on the extent of mineralization as measured by total organic carbon (TOC) content have been studied with the aim of maximizing the extent of degradation. It has been observed that increased pressures, higher operating temperature and the absence of copper windings are more favourable for a rapid TOC mineralization. A new approach of latent remediation has also been investigated where hydrodynamic cavitation is only used as a pre-treatment with an aim of reducing the overall cost of pollutant degradation. It has been observed that approach of latent remediation works quite well with about 50–60% removal of TOC using only minimal initial treatment by hydrodynamic cavitation

On the Maximum Expected Electric Field in Electrically Small, Undermoded Enclosures

This paper reports the experimental validation of an improved statistical model for the prediction of maximum expected electric field in electrically small and under-moded enclosures. The aerospace community is interested in application of Hills statistical models to design of avionics boxes for shielding effectiveness and for tailoring EMC test requirements for critical applications. However, it is observed that the probability distribution for mean-squared electric field (|E(sub x)|(exp 2)) in an electrically small enclosure differs from the exponential distribution which is widely used in reverberation chamber testing. It is postulated here that the difference is attributable to the under-moded character of the small enclosure. We will define under-moded as the condition where a single excitation frequency does not excite enough closely spaced resonant modes to achieve Hills assumption of an isotropic (or fully diffuse) plane wave field in the enclosure

A patient preference study that evaluated fluticasone furoate and mometasone furoate nasal sprays for allergic rhinitis

Background: Corticosteroid nasal sprays are the mainstay of treatment for allergic rhinitis. These sprays have sensory attributes such as scent and/or odor, taste and aftertaste, and run down the throat and/or the nose, which, when unpleasant, can affect patient preference for, and compliance with, treatment. Objective: This study examined patient preference for fluticasone furoate nasal spray (FFNS) or mometasone furoate nasal spray (MFNS) based on their sensory attributes after administration in patients with allergic rhinitis. Methods: This was a multicenter, randomized, double-blind, cross-over study. Patient preferences were determined by using three questionnaires (Overall Preference, Immediate Attributes, and Delayed Attributes). Results: Overall, 56% of patients stated a preference for FFNS versus 32% for MFNS (p _ 0.001); the remaining 12% stated no preference. More patients stated a preference for FFNS versus MFNS for the attributes of “less drip down the throat” (p _ 0.001), “less run out of the nose” (p _ 0.05), “more soothing” (p _ 0.05), and “less irritating” (p _ 0.001). More patients responded in favor of FFNS versus MFNS for the immediate attributes, “run down the throat” (p _ 0.001), and “run out of the nose” (p _ 0.001), and, in the delayed attributes, “run down the throat” (p _ 0.001), “run out of the nose” (p _ 0.01), “presence of aftertaste” (p _ 0.01), and “no nasal irritation” (p _ 0.001). Conclusion: Patients with allergic rhinitis preferred FFNS versus MFNS overall and based on a number of individual attributes, including “less drip down the throat,” “less run out of the nose,” and “less irritating.” Greater preference may improve patient adherence and thereby improve symptom management of the patient’s allergic rhinitis

Intensification of oxidation capacity using chloroalkanes as additives in hydrodynamic and acoustic cavitation reactors

The effect of the presence and absence of the chloroalkanes, dichloromethane (CH2Cl2), chloroform (CHCl3) and carbon tetrachloride (CCl4) on the extent of oxidation of aqueous I- to I3- has been investigated in (a) a liquid whistle reactor (LWR) generating hydrodynamic cavitation and (b) an ultrasonic probe, which produces acoustic cavitation. The aim has been to examine the intensification achieved in the extent of oxidation due to the generation of additional free radicals/oxidants in the reactor as a result of the presence of chloroalkanes. It has been observed that the extent of increase in the oxidation reaction is strongly dependent on the applied pressure in the case of the LWR. Also, higher volumes of the chloroalkanes favour the intensification and the order of effectiveness is CCl4> CHCl3 > CH2Cl2. However, the results with the ultrasonic probe suggest that an optimum concentration of CH2Cl2 or CHCl3 exists beyond which there is little increase in the extent of observed intensification. For CCl4, however, no such optimum concentration was observed and the extent of increase in the rates of oxidation reaction rose with the amount of CCl4 added. Stage wise addition of the chloroalkanes was found to give marginally better results in the case of the ultrasonic probe as compared to bulk addition at the start of the run. Although CCl4 is the most effective, its toxicity and carcinogenicity may mean that CH2Cl2 and CHCl3 offer a safer viable alternative and the present work should be useful in establishing the amount of chloroalkanes required for obtaining a suitable degree of intensification

The partially alternating ternary sum in an associative dialgebra

The alternating ternary sum in an associative algebra, $abc - acb - bac + bca + cab - cba$, gives rise to the partially alternating ternary sum in an associative dialgebra with products $\dashv$ and $\vdash$ by making the argument $a$ the center of each term: $a \dashv b \dashv c - a \dashv c \dashv b - b \vdash a \dashv c + c \vdash a \dashv b + b \vdash c \vdash a - c \vdash b \vdash a$. We use computer algebra to determine the polynomial identities in degree $\le 9$ satisfied by this new trilinear operation. In degrees 3 and 5 we obtain $[a,b,c] + [a,c,b] \equiv 0$ and $[a,[b,c,d],e] + [a,[c,b,d],e] \equiv 0$; these identities define a new variety of partially alternating ternary algebras. We show that there is a 49-dimensional space of multilinear identities in degree 7, and we find equivalent nonlinear identities. We use the representation theory of the symmetric group to show that there are no new identities in degree 9.Comment: 14 page